MA 343 / FI 645

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Strike or delivery price

(K) the agreed-upon delivery price

The spot price of the underlying asset at maturity.

(ST)

Maturity

(T) The time at which payment is due

Call Intrinsic Value

= MAX of ((ST) stock price - less (K) strike price OR zero) - You are buying the stock. - You want to buy as low as possible. - Therefore, if strike is lower, option has greater intrinsic value.

backwardation

When a futures price is below the spot price; Caused by hedgers to insure against price declines in the future; Some markets are described as having normal backwardation.

credit derivatives

derivatives used to reduce a lender's exposure to credit risk

Tranches

different classes of securities that comprise a single bond issuance

standardized instruments

on exchanges are administered the same way each time and scored the same way each time exchange-traded derivatives are standardized, highly regulated, transparent transactions, gauranteed

3.4 Cross Hedging

when asset underlying futures contract is different than asset whose price is being hedged.

equivalent annual interest rate

when m=1

Hedging Example

•A US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract •An investor owns 1,000 Microsoft shares currently worth $28 per share. A two-month put with a strike price of $27.50 costs $1. The investor decides to hedge by buying 10 contracts

Long Hedge

•A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price -hedgers with long positions usually avoid any possibility of having to take delivery by closing out their positions before the delivery period

Short Hedge

•A short futures hedge is appropriate when you know you will sell an asset in the future and want to lock in the price -A short hedge is appropriate when the hedger already owns or will own an asset and expects to sell it at some time in the future.

Arbitrage Example

•A stock price is quoted as £100 in London and $150 in New York • The current exchange rate is 1.5300 • What is the arbitrage opportunity?

Speculation Example

•An investor with $2,000 to invest feels that a stock price will increase over the next 2 months. The current stock price is $20 and the price of a 2-month call option with a strike of 22.50 is $1 • What are the alternative strategies?

Basis Risk

•Basis is usually defined as the spot price minus the futures price •Basis risk arises because of the uncertainty about the basis when the hedge is closed out -If the asset to be hedged and the asset underlying the futures contract are the same, the basis should be zero at the expiration -prior to basis may be positive or negative -increase: strengthening basis -imporives short positions -decrease: weakening basis - improves long positions Basis = Spot price of asset to be hedged - Futures price of contract used

Arguments in Favor of Hedging

•Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables

Call Option

Right to buy an asset by a certain date (T) for a certain price (K = strike price)

4.4 An investor receives $1,100 in one year in return for an investment of $1,000 now. Calculate the percentage return per annum with: (a) Return with Annual compounding: (b) Return with Semiannual compounding per annum (c) Monthly compounding: per annum (d) Continuous compounding:

(a) Return with Annual compounding: 1100/1000 -1 = .10 = 10% per annum (b) Return with Semiannual compounding 1000(1+R/2)^2=1100 =1+R/2 = sqrt(1.1) = 1.0488 = R = .0976 = 9.76% per annum (c) Monthly compounding: 1000(1+R/12)^12=1100 =(1+R/12)=12sqrt(1.1) = 1.00797 R=.0957 = 9.57% per annum (d) Continuous compounding: 1000exp(r) = 1100 =exp(r)=1.1 r = ln1.1 = .0953 = 9.53% per annum

Current Time

(t)

4.7 Forward Rate Agreements

- A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period Measured w compounding frequency: RK: The rate of interest agreed to in the FRA RF : The forward LIBOR interest rate for the period between times T1 and T2, calculated today using annual compounding (ie .5 = semi annual) RM : The actual LIBOR interest rate observed in the market at time T1 for the period between times T1 and T2 L: The principal underlying the contract. - An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate - An FRA can be valued by assuming that the forward interest rate is certain to be realized

CBOE(Chicago Board Options Exchange)

- Chicago Board Options Exchange - Largest options (derivatives) exchange in the U.S.

Normal Backwardation and Contango

- This is where we start involved the EXPECTED Spot price. S0=current spot price E=expected value Et(ST)= Expected future spot price F< E(S0) - Normal Backwardation F > E(S0) - Normal Contango What causes this is the balance of hedges, the supply & demand Long underlying -> Short Futures Contract (Short hedger) Short Underlying -> Long Futures Contract (long hedgers)

investment-grade corporate bond fund

- moderate income & risk - taxable at all levels

2.4 Margins

-A cash or marketable securities deposited by an investor with his or her broker -The balance in the margin account is adjusted to reflect daily settlement -Margins minimize the possibility of a loss through a default on a contract -settled at close of day -maintenance margin - If balance in the margin account falls below the maintenance margin, the investor receives a margin call and is expected to top up the margin account to the initial margin level by the end of the next day. -usually about 75% of the initial margin. -if variation margin is not deposited, the broker closes out position -T-bills (90%) and shares are sometimes accepted in lieu of cash (50% value)

5.7 Valuing a Forward Contract

-A forward contract is worth zero -Later it may have a positive or negative value -Suppose that K is the delivery price -F0 is the forward price for a contract that would be negotiated today At beginning of contract F0=delivery price K=delivery price=forward price =stays same as time passes f=value of the contract=0 f=(F0-K)exp(-rT) By considering the difference between a contract with delivery price K and a contract with delivery price F0 we can deduce that: the value of a long forward contract, ƒ, is: (F0 - K )exp(-rT) the value of a short forward contract is (K - F0 )exp(-rT)

10.7 The Impact of Dividends on Lower Bounds to Option Prices

-A: one European call plus cash equal to D+ Ke^(-rT) -B: one share c≥S0-D-Xe^(-rT) -C: one European put plus one share -D: cash equal to D+ Ke-rT p≥D+Xe^(-rT)-S0

2.2 Futures Contracts

-Available on a wide range of assets -Exchange traded -Specifications need to be defined: -(The Asset) What can be delivered (speicfy grades acceptable) -(Contract Size) How much can be delivered -(Delivery Arrangements) Where it can be delivered -(Delivery Months) When it can be delivered Key Take Awaya -Settled daily -Closing out a futures position involves entering into an offsetting trade -Most contracts are closed out before maturity

Forward Contracts vs Futures Contracts

-Both contracts are agreements to buy or sell an asset for a certain price at a certain future time - A forward contract is private contract, traded in the OTC market, one delivery date, held to end of its life and settled at end, delivery usually takes place, some credit risk - Futures contract is traded on exchange range of delivery dates, settled daily and closed out before maturity, no credit risk

Bermudan Options (B)

-Exercise only at time {T1, T2, ... Tn, T} at price K. -Dates are specified in contract -A sequence of European options

4.10 Theories of the Term Structure of Interest Rates

-Expectations Theory: forward rates equal expected future zero rates - Market Segmentation: short, medium and long rates determined independently of each other - Liquidity Preference Theory: forward rates higher than expected future zero rates

4.6.c Upward vs Downward Sloping Yield Curve

-For an upward sloping yield curve: Fwd Rate > Zero Rate > Par Yield - For a downward sloping yield curve Par Yield > Zero Rate > Fwd Rate

Foreign Exchange Quotes

-Futures exchange rates are quoted as the number of USD per unit of the foreign currency -Forward exchange rates are quoted in the same way as spot exchange rates. -This means that GBP, EUR, AUD, and NZD are quoted as USD per unit of foreign currency. Other currencies (e.g., CAD and JPY) are quoted as units of the foreign currency per USD.

2.9 Regulation of Futures

-Futures markets in the United States are currently regulated federally by the Commodity Futures Trading Commission (CFTC; www.cftc.gov), which was established in 1974. -Regulators try to protect the public interest and prevent questionable trading practices

2.10 Accounting & Tax

-Hedge Accounting -Ideally hedging profits (losses) should be recognized at the same time as the losses (profits) on the item being hedged -Ideally profits and losses from speculation should be recognized on a mark-to-market basis Roughly speaking, this is what the accounting and tax treatment of futures in the U.S. and many other countries attempt to achieve If contract does not qualify for hedge: Ex: Mar 2012 futures contract purchase 5,000 bushels for 250 cents, Closes out Feb 2012 End of 2011 270 cents 5,000 bushels* (2.7-2.5)=$1,000 in 2011 5000 bushels * (2.8-2.7)=$500 If company hedging 5000bushels of corn in Feb 2012, the entire gain is realized in 2012. -For a noncorporate taxpayer, short-term capital gains are taxed at the same rate as ordinary income, but long-term capital gains are subject to a maximum capital gains tax rate of 15%. -(Long-term capital gains are gains from the sale of a capital asset held for longer than one year; short-term capital gains are the gains from the sale of a capital asset held one year or less.) For a noncorporate taxpayer, capital losses are deductible to the extent of capital gains plus ordinary income up to $3,000 and can be carried forward indefinitely.

4.2.a Continuous Compounding

-In the limit as we compound more and more frequently we obtain continuously compounded interest rates - $100 grows to $100exp(RT) when invested at a continuously compounded rate R for time T - $100 received at time T discounts to $100exp(-RT) at time zero when the continuously compounded discount rate is R A= amount invested n= # of years R = interest rate per annum m = # of times interest is compounded per year Formula is: A(1+(r/m))^(mn) Initial investment * (1 + Annual interest rate / Compounding periods per year) ^ (Years * Compounding periods per year)

QUESTION 7 A trader takes the long position and a hedge fund takes a short position on ten 1-month S&P 500 futures contracts at 1300. A single S&P 500 futures contract equals ($250) x (Index Value). The initial margin is $325,000 and the maintenance margin is $245,000 for both accounts. Ten trading days later, the futures price of the index drops to 1,260 triggering a margin call for the trader. What is the change margin call for the trader. $100,000 no margin call $80,000 $20,000 $325,000

-Initial margin for 10 S&P 500 futures contracts is $325,000 -Required maintenance margin is $245,000 Futures price drops to 2460 from 2500 Gain/Loss per contract for long position in S&P 500 futures: (current price - buying price) = (2460 -2500)*250 = (- 40 * 250) per contract. Total Gain/loss for 10 long positions for trader = - 40 * 250 *10 = -$100,000. Hence, the net balance in his account is now = Initial margin deposited by trader - Loss in his positions = 325,000 - 100,000 = $225,000 < required maintenance margin of $245,000 $100,000 needs to be deposited in the account in order to bring the account value from $225,000 back to $325,000 (initial margin). Amount of margin call to trader = $325,000 - $225,000 = $100,000. Similarly, Gain/Loss per contract for short position in S&P 500 futures is calculated as (selling price - current price) = (2500-2460)*250 = (40 * 250) per contract. Total Gain/loss for 10 short positions for trader = 40 * 250 *10 = $100,000. So, the hedge fund who is short in S&P 500 futures will have a gain of $100,000.

QUESTION 6 A trader takes the long position and a hedge fund takes a short position on ten 1-month S&P 500 futures contracts at 1300. A single S&P 500 futures contract equals ($250) x (Index Value). The initial margin is $325,000 and the maintenance margin is $245,000 for both accounts. Ten trading days later, the futures price of the index drops to 1,260 triggering a margin call for the trader. What is the change in the margin account balance (indicate gain or loss) for the hedge fund. $100,000 loss $100,000 gain $80,000 gain $80,000 loss $245,000

5.2 Short Selling

-Short selling involves selling securities you do not own -Your broker borrows the securities from another client and sells them in the market in the usual way -The investor takes a profit if the stock price has declined and a loss if it has risen. -At some stage you must buy the securities so they can be replaced in the account of the client You must pay dividends and other benefits the owner of the securities receives -There may be a small fee for borrowing the securities

4.6.a Formula for Forward Rates

-Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded. - The forward rate (RF) for the period between times T1 and T2 is: RF= (R2*T2-R1*T1)/(T2-T1)

4.2 Measuring interest rates

-The compounding frequency used for an interest rate is the unit of measurement -The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers

4.6.b Instantaneous Forward Rate

-The instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T. RF=Forward Interest rate -P(0,T) = price of zero coupon bond maturing at time T P(0,T)=exp(-RT) RF=(@/@T)lnP(0,T) RF= R+T(@R/@T) where R is the T year rate

4.4 Bond Pricing

-To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate -In our example, the theoretical price of a two year bond providing a 6% coupon semiannually is 3exp(-.05*.5)+3exp(-.058*1)+3exp(-.064*1.5)+103exp(-.068*2)=98.39

4.7a Forward Rate Agreement Valuation Formulas

-Value of FRA where a fixed rate RK will be received on a principal L between times T1 and T2 is L(RK-RF)(T2-T1)(exp(-R2T2) - Value of FRA where a fixed rate is paid is L(RF-RK)(T2-T1)(EXP-R2T2) - RF is the forward rate for the period and R2 is the zero rate for maturity T2 - RK, RM, and RF are expressed with a compounding frequency corresponding to T2-T1 whereas R2 is expressed with continuous compounding

5.2 Short Selling Example

-You short 500 shares in April when the price per share is $120 and close out the short position three months later in July when the price is $100. -During the three months a dividend of $1 per share is paid in May -What is your profit? 1. The investor receives 500 shares x $120 per share = $60,000 gross profit in April when short position is initiated. 2. The dividend leads to payment by the investor of 500 share x $1 dividend = $500 in May. 3. The investor also pays 500 shares x $100 per share = $50,000 for shares when the position is closed out in July. 4. Net Profit = $60,000 gross profit - $500 Dividend - $50,000 cost to buy shares = $9,500 profit less any commision fees for borrowing the shares 5. margin account may be required

collateralization

-agreement applying to the transaction where the transaction is valued each day -significantly reduces the credit risk in OTC contracts.

Clearing House

-becomes the counterparty to both A and B, two sides of a futures trade.) -takes on the credit risk of both A and B. -Collateral automatically has to be posted -manages this risk by requiring an initial margin and daily variation margins from them. -reduces risk in the financial system

5.4 Forward Price For an Investment Asset

-easiest forward contract to value is one written on an investment asset that provides the holder with no income. -Ex: Non-dividend-paying stocks and zero-coupon bonds

Example 3.2 Date: June 8 -company needs to purchase 20,000 barrels of crude oil at some time in October or November. -Oil futures contracts traded for delivery every month -contract size: 1,000 barrels -co. takes a long position in 20 December contracts -futures price on June 8 is $68.00 per barrel. -co closes out futures contract on November 10 -The spot price and futures price on November 10 are $70.00 per barrel and $69.10 per barrel.

-gain on futures contract is 69.10 - 68.00 = $1.10 per barrel. -basis when the contract is closed out is 70.00 - 69.10 = $0.90 per barrel. -Effective price paid (in dollars per barrel) is the final spot price less the gain on the futures, or $70.00 - 1.10 = 68.90 -initial futures price plus the final basis: $68.00 + 0.90 = 68.90 -Total price paid: 68.90 X 20,000 = 1,378,000

Liquidity Issues

-in any hedging situation there is a danger that losses will be realized on the hedge while gains on the underlying exposure are unrealized -this can create liquidity problems

hedge-and-forget strategies

-no attempt is made to adjust the hedge once it has been put in place. -hedger simply takes a futures position at the beginning of the life of the hedge and closes out the position at the end of the life of the hedge

4.4.a Bond Yield

-the return an investor would receive on a bond if it were purchased and held to maturity -The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond -Suppose that the market price of the bond in our example equals its theoretical price of 98.39 - The bond yield (y)(continuously compounded) is given by solving 3exp(-y*.5)+3exp(-y*1.0)+3exp(-y*1.5)+3exp(-yX2.0) = 98.39 y= 6.76%

2.7 Delivery

-very few futures contracts that are entered into lead to delivery of the underlying asset. -Most are closed out early. -The party taking delivery is then responsible for all warehousing costs. -In the case of livestock futures, there may be costs associated with feeding and looking after the animals -To avoid the risk of having to take delivery, an investor with a long position should close out his or her contracts prior to the first notice day. -Cashe Settlement - for futures contracts for S&P 500 -final settlement price = spot price as open or close

Suppose that: -The spot price of gold is US$1,200 -The 1-year forward price of gold is US$1,200 -The 1-year US$ interest rate is 5% per annum Is there an arbitrage opportunity?

1. Compute the present value of future price of the gold in the following manner: Present value: Future value / (1+R)^#Years $1,200/ (1.05)^1 = 1,238.09 Since 1,142.85 < 1200 = -57.14 There is not an arbitrage opportunity

Suppose that: The spot price of gold is US$1,200 The 1-year forward price of gold is US$1,300 The 1-year US$ interest rate is 5% per annum Is there an arbitrage opportunity?

1. Compute the present value of future price of the gold in the following manner: Present value: Future value / (1+R)^#Years $1,300/ (1.05)^1 = 1,238.09 Since 1,238.09 > 1200 = 38.09 There is an arbitrage opportunity

Suppose Microsoft (MSFT) is currently trading at $45 per share. Assume that a 6-month $50 strike European call option on MSFT stock sells for $7, and the stock price at maturity is $60 (S0=$45, ST=$60, c=7, T =.5) Is the option exercised?

1. Each options contract costs = 100 x 7 = 700 2. Strike price = K = 50 =>exercise call if ST>50 cost to exercise = 100 x 50 = 5000 3. Suppose ST=60: Gross Profit = (60-50)*100=1000 Net Profit = 1000-700 =300 4. Suppose ST =48 => Option is NOT exercised Loss = 700

5.4 The Forward Price

1. F0 > S0erT , arbitrageurs can buy the asset and short forward contracts on the asset 2. If F0 < S0erT , they can short the asset and enter into long forward contracts on it If the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then F0 = S0exp(rT) where r is the T-year risk-free rate of interest. In our examples, S0 =40, T=0.25, and r=0.05 so that F0 = 40e0.05×0.25 = 40.50

Ex 3.3 An airline expects to purchase 2 million gallons of jet fuel in 1 month and decides to use heating oil futures for hedging. We suppose that Table 3.2 gives, for 15 successive months, data on the change, Δ S, in the jet fuel price per gallon and the corresponding change, Δ F, in the futures price for the contract on heating oil that would be used for hedging price changes during the month. In this case, the usual formulas for calculating standard deviations and correlations give σF=0.0313, σS = 0.0263, and Þ =0.928. 1. Find min variance hedge ratio and 2. Optimal number of contracts

1. From equation (3.1), the minimum variance hedge ratio, h*, is therefore 0.928 x (0.0263/0.0313) = 0.7777 =.78 2. Each heating oil contract traded on NYMEX is on 42,000 gallons of heating oil. From equation (3.2), the optimal number of contracts is (0.78 x 2,000,000 )/42,000 = 37.03 or, rounding to the nearest whole number, 37.

1.6 types of traders

1. Hedgers 2. Speculators 3. Arbitrageurs

QUESTION 1 You sell one December futures contracts when the futures price is $1,010 per unit. Each contract is on 100 units and the initial margin per contract that you provide is $2,000. The maintenance margin per contract is $1,500. During the next day the futures price rises to $1,012 per unit. What is the balance of your margin account at the end of the day? $1,800 $3,300 $2,200 $3,700

1. Initial margin = 2,000 Since you short sold the futures contract, you profit if the price goes down. Profit = 1,010 - 1,008 = 2 per unit Profit per contract = 2 * 100 = 200 Margin account balance at the end of the day = Initial margin + profit Margin account balance at the end of the day = 2,000 + 200 = $2,200

5.14 Futures Prices & Expected Future Spot Prices Suppose that a speculator puts the present value of the futures price into a risk-free investment while simultaneously taking a long futures position. The proceeds of the risk-free investment are used to buy the asset on the delivery date. The asset is then immediately sold for its market price. 1. What are the cash flows to the speculator? 2. How do we value this investment?

1. The cash flows to the speculator are as follows: Today:-F0e(-rT) End of futures contract: +ST F0=futures prices today ST=spot price at T r= rfr k=investors required return(disc rate) 2. The present value of this investment is: -F0e(-rT)+E(ST)e(-kT)=0 or F0=E(ST)e((r-k)T) E=expected value All investments in securities have a net zero present value Et(ST)= Expected future spot price No Systematic Risk k = r F0 = E(ST) disc rate/inv req ret = rfr Positive Systematic Risk -expect futures price to understate the expected future spot price k > r disc rate/inv req ret > rfr F0 < E(ST) Negative Systematic Risk -expect futures price to overstate the expected future spot price k < r disc rate/inv req ret < rfr F0 > E(ST)

Chapter 10. Properties of stock options

10.1 Factors affecting option prices 10.2 Assumptions and notation 10.3 Upper and lower bounds for option prices 10.4 Put-call parity 10.5 Calls on a non-dividend-paying stock 10.6 Puts on a non-dividend-paying stock . 10.7 Effect of dividends

Chapter 11. Trading strategies involving options

11.1 Principal-protected notes 11.2 Trading an option and the underlying asset . 11.3 Spreads 11.4 Combinations 11.5 Other payoffs Summary

Chapter 12. Binomial trees

12.1 A one-step binomial model and a no-arbitrage argument 12.2 Risk-neutral valuation 12.3 Two-step binomial trees 12.4 A put example 12.5 American options 12.6 Delta 12.7 Matching volatility with u and d 12.8 The binomial tree formulas 12.9 Increasing the number of steps 12.10 Using DerivaGem 12.11 Options on other assets Summary Appendix: Derivation of the Black-Scholes-Merton option-pricing formula from a binomial tree

Basis Risk: Choice of Contract

2 components to basis risk: 1. The choice of the asset underlying the futures contract (choose one most highly correlated) 2. The choice of the delivery month (choose a delivery month that is as close as possible to but later than the expieration of the hedge) Ex: Delivery months Mar, Jun, Sep, Dec for futures contract. Hedge exp in Dec, jan, Feb = Mar contract chosen

Chapter 5 Determination of Forward and Futures Prices

5.1 Investment assets vs. consumption assets 5.2 Short selling 5.3 Assumptions and notation 5.4 Forward price for an investment asset 5.5 Known income 5.6 Known yield 5.7 Valuing forward contracts 5.8 Are forward prices and futures prices equal? 5.9 Futures prices of stock indices 5.10 Forward and futures contracts on currencies 5.11 Futures on commodities 5.12 The cost of carry 5.13 Delivery options 5.14 Futures prices and the expected future spot price how forward prices and futures prices are related to the spot price of the underlying asset Forward contracts - easier, only 1 payment at maturity, no daily settlement -relationship between forward price and spot price - results obtained for forwards are usually also true for futures.

Short Put Profit

=-MAX(K-ST,0)-P where ST = Stock price at maturity ie $97 K=strike price ie $100 P=call premium ie $3 -(MAX ($100 - 97, 0) = -(3, 0) = -3 - 3 = $-600 https://stock-screener.org/options-profit-calculator.aspx

Short Call Profit

=-MAX(ST-K,0) +C where ST = Stock price at maturity ie $97 K=strike price ie $100 C=call premium ie $3 -(MAX ($97 - $100, 0) = (-3, 0) = -(0-3) = -(3) = 3*100= $300

Long Put Profit

=MAX(K-ST,0)-P where ST = Stock price at maturity ie $97 K=strike price ie $100 P=call premium ie $3 (MAX ($100 - 97, 0) = (3, 0) = 3 - 3 = $0 https://stock-screener.org/options-profit-calculator.aspx

Long Put (seller-bearish) Payoff

=MAX(K-St,0) The maximum of whichever is more: the strike price - Stock Price at maturity OR 0. Does not take into account cost of the premium (p) For example, at a stock price of $97, has a strike price, K, of 100: (MAX ($100 - $97, 0) = 3

Long Call Payoff (buyer- bullish)

=MAX(ST-K,0) The maximum of which ever is more: the Stock price (ST) at maturity less the strike price (K) OR zero. Does not take into account cost of the premium (c) For example, at a stock price of $97: (MAX ($97 - $100, 0) = -3, 0 = 0 At a stock price of $106, (MAX ($106 - $100, 0) = 6

Long Call Profit

=MAX(ST-K,0)-c where ST = Stock price at maturity ie $97 K=strike price ie $100 C=call premium ie $3 (MAX ($97 - $100, 0) = (-3, 0) = 0-3 = -3*100 = -300 https://stock-screener.org/options-profit-calculator.aspx

QUESTION 2 The price of a stock on February 1 is $124. A trader sells 200 put options on the stock with a strike price of $120 when the option price is $5. The options are exercised when the stock price is $110. The trader's net profit or loss is A) Gain of $1,000 B) Loss of $2,000 C) Loss of $2,800 D) Loss of $1,000

@The payoff that must be made on the options is 200×(120−110) or $2000. The amount received for the options is 5×200 or $1000. The net loss is therefore 2000−1000 or $1000.

5.10 Futures and Forwards on Currencies

A foreign currency is analogous to a security providing a yield F0=S0exp((r-Rf)T) S0=current spot price in USD of one unit of foreign currency F0=forward/futures price in USD of one unit of currency Rf=value of foreign rfr r=USD rfr The yield is the foreign risk-free interest rate It follows that if rf is the foreign risk-free interest rate

Mutual Funds

A pool of money used by a company to purchase a variety of stocks, bonds or money market instruments. Provides diversification and professional management for investors. •must: -disclose investment policies, -make shares redeemable at any time, -limit use of leverage

Long Call

A call buyer owns the RIGHT TO BUY 100 shares of a specific stock at the strike price before the expiration if he chooses to exercise.

Short Call

A call writer (seller) has the OBLIGATION TO SELL 100 shares of a specific stock at the strike price if the buyer exercises the contract.

4.5.a Zero curve

A chart showing the zero rate as a function of maturity (yrs) for the bootstrap method

Repurchase Agreements

A contract where an investment dealer who owns securities agrees to sell them to another company now and buy them back later at a slightly higher price.

1.2 OVER-THE-COUNTER MARKETS

A decentralized market in which traders working for banks, fund managers and corporate treasurers contact each other directly. Trade stocks, commodities, currencies, or other instruments directly between two parties and without a central exchange or broker. Became regulated in 2008, reduce risk and increase transparency Liquidity in the OTC market may come with a premium fee attached.

Asset Backed Securities

A derivative contract in which a portfolio of debt instruments is assembled and claims are issued on the portfolio in the form of tranches, which have different priorities of claims on the payments made by the debt securities ie prepayments or credit losses are allocated to most junior-tranches first and most-senior tranches last.

derivative

A financial instrument that derives its performance from the performances of an underlying asset

Min Function

A function that determines the minimum value in a set of values. Note: Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign. MIN(a ,0) = { a if a ≤ 0 0 if a > 0 MIN(a ,0) = -max(-a,0) eg: min(-5,0) =-5 ; min(5,0) = 0

Max Function

A function that returns the largest value in a set of values. Note: Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign. MAX(a ,0) = { a if a ≥ 0 0 if a < 0 eg: max (10,0) = 10 ; max(-10,0) = 0

Speculating using futures

A futures contract allows a trader to speculate on the direction of movement of a commodity's price. Provide access to leverage with small margin up front. Futures has both a large potential loss as well as potential gain

QUESTION 5 Which of the following derivatives provide payoffs that are non-linearly related to the payoffs of the underlying? A) Options B) Forwards C) Interest rate swaps

A is correct. Options are classified as a contingent claim which provides payoffs that are non-linearly related to the performance of the underlying. B is incorrect because forwards are classified as a forward commitment, which provides payoffs that are linearly related to the performance of the underlying. C is incorrect because interest-rate swaps are classified as a forward commitment, which provides payoffs that are linearly related to the performance of the underlying.

Covariance

A measure of linear association between two variables. Positive values indicate a positive relationship; negative values indicate a negative relationship

Double-barreled municipal bond

A municipal bond in which the interest and principal payments are pledged by two distinct entities—the revenue from a defined project and the issuer and its taxing power.

Hedge fund

A private investment organization that employs complex risky strategies that often made huge profits for investors not subject to the same rules as mutual funds (disclose inv policies, redeemable, limit use of leverage) and cannot offer their securities publicly.

Long Put

A put buyer owns the right to sell 100 shares of a specific stock at the strike price before the expiration if he chooses to exercise

Short Put

A put writer (seller) has the obligation to buy 100 shares of a specific stock at the strike price if the buyer exercises the contract

1.7. Suppose that you write a put contract with a strike price of $40 and an expiration date in 3 months. The current stock price is $41 and the contract is on 100 shares. What have you committed yourself to? How much could you gain or lose?

A put writer (seller) has the obligation to buy 100 shares of a stock as the strike price of $40 per share if the buyer of the contract exercises the contract when it is below $40. Stock Price S0 41 Strike K 40 Premium P0=S0-K 1 Maximum Loss (100*number of contracts*strike price)-premium received $ 3,900 Max Gain Premium per share $ 100

QUESTION 3 If the basis is defined as the amount by which the spot price exceeds the futures price, which of the following is true? A long hedger's position improves when the basis strengthens unexpectedly and worsens when the basis weakens unexpectedly. A short hedger's position improves when the basis strengthens unexpectedly and worsens when the basis weakens unexpectedly. A short hedger's position worsens when the basis strengthens unexpectedly and improves when the basis weakens unexpectedly. A long hedger's position is insensitive to unexpected changes in the basis.

A short hedger's position improves when the basis strengthens unexpectedly and worsens when the basis weakens unexpectedly. -increase: strengthening basis -improves short positions -worsens long positions -decrease: weakening basis - improves long positions worsens short positions Basis = Spot price of asset to be hedged - Futures price of contract used

preferred stock

A special type of stock whose owners, though not generally having a say in running the company, have a claim to profits before other stockholders do.

Premium

A specific sum of money paid by the insured to the insurance company in exchange for financial protection against loss.

Mid-cap stock

A stock whose total market value falls somewhere between $2 billion and $10 billion.

interest

A sum paid or charged for the use of money or for borrowing money

4.5 Determining Treasury Zero Rates from Tbond using the Bootstrap method

A technique based on the price-yield equation using different rates for each of the 6-month terms, as determined by market prices: ie: amount earned in 3 months = 100-97.5=$2.5 -3 month quarterly compounded rate: 4*2.5/97.5 = 10.256% -3-month continuously compounded rate = 100=97.5exp(r*.25) = 10.127% -6-month & 1 yr rates continuously compounding = 10.468% and 10.536% -To calculate 1.5 year rate we solve 4exp(-.10469*.5)+4exp(-.10536*1)+104exp(-R*1.5)=96 R=.10681 = 10.681% -Similarly 2-yr rate is 10.808% https://www.symbolab.com/

Speculative stock

A very high risk stock from a company with potential for substantial earnings in the future.

4.3 Zero Rates (Spot Rates)

A zero rate (or spot rate), for maturity T, is the rate of interest earned on an investment that provides a payoff only at time T P x e^(r*T)

An investor can avoid reinvestment rate risk by purchasing A) A zero- coupon bond B) A REIT C) A blue-chip stock fund D) Long-term bonds

A) A zero- coupon bond

What are the two components of total return in common stock? A) Dividend income and price change B) Price change and tax loss C) Bid-ask spread and tax loss D) Dividend income and interest income

A) Dividend income and price change

A closed-end fund investor that is most concerned with issuer risk in an income portfolio should feel most comfortable with which of the following funds? A) Investment-grade corporate bond fund B) A high yield bond fund C) An emerging market debt fund D) A global income fund

A) Investment-grade corporate bond fund

Before any recommendations can be made in communications, FINRA members must conduct a due diligence investigation on the product recommended. This process is called A) Reasonable basis suitability B) Supervisory investigation C) Customer-specific suitability D)Blue-sky investigation

A) Reasonable basis suitability

The stock of a company with no earnings history, but high potential for appreciation is most likely classified as a(n) A) Speculative stock B) Income stock C) Mid-cap stock D) Cyclical stock

A) Speculative stock

QUESTION 1 Which of the following is NOT true A) When a CBOE call option on IBM is exercised, IBM issues more stock B) An American option can be exercised at any time during its life C) A call option will always be exercised at maturity if the underlying asset price is greater than the strike price D) A put option will always be exercised at maturity if the strike price is greater than the underlying asset price.

A) When a CBOE call option on IBM is exercised, IBM issues more stock

The risk that stock cannot be sold easily or promptly is known as A) marketability risk B) financial risk C) reinvestment rate risk D) systemic risk

A) marketability risk

QUESTION 7 The expected return on the S&P 500 is 12% and the risk-free rate is 5%. What is the expected return on the investment with a beta of 0.2? 6.4% 7.0% 12.0% 5.0%

According to the CAPM, what is the market risk premium given an expected return on a security of Er%, a stock beta of .2, and a risk-free interest rate of 5%? Er = 5 + .2 × (7); MRP = 7% = 6.4% Expected return on portfolio - Risk-free interest rate = 1.5 x (Return on index - Risk-free interest rate)

One Contract

An agreement to buy or sell 100 shares

1.3 Forward Contract

An agreement to buy or sell an asset at a certain future time for a certain price. A future obligation between 2 parties. It is traded in the over-the-counter market—usually between two financial institutions or between a financial institution and one of its clients. Purpose: to remove uncertainty in future prices / hedging currency risk

Example 5.6.2 Suppose next that the 2-year forward rate is 0.6600 (greater than the 0.6453 value given by equation (5.9)). What can an arbitrager do?

An arbitrageur can: 1. Borrow 1,000 USD at 7% per annum for 2 years, convert to 1,000/0.6200 = 1,612.90 AUD, and invest the AUD at 5%. 2. Enter into a forward contract to sell 1,782.53 AUD for 1,782.53 x 0.66 = 1,176.47 USD. The 1,612.90 AUD that is invested at 5% grow to 1,612.90e(0:05x2) = 1,782.53 AUD in 2 years. The forward contract has the effect of converting this to 1,176.47 USD. The amount needed to payoff the USD borrowings is 1,000e(0.07x2) = 1,150.27 USD. The strategy therefore gives rise to a riskless profit of 1,176.47 - 1,150.27 = 26.20 USD.

Example 5.6.1 Suppose that the 2-year interest rates in Australia and the United States are 5% and 7%, respectively, and the spot exchange rate between the Australian dollar (AUD) and the US dollar (USD) is 0.6200 USD per AUD. From equation (5.9), the 2-year forward exchange rate should be F0=S0exp((r-Rf)T) .62e((.07-.05)x2)=.6453 Suppose that the 2-year forward exchange rate is less than this, say .63 What can an arbitrager do?

An arbitrageur can: 1. Borrow 1,000 AUD at 5% per annum for 2 years, convert to 620 USD and invest the USD at 7% (both rates are continuously compounded). 2. Enter into a forward contract to buy 1,105.17 AUD for 1,105.17 x 0.63 = 696.26 USD The 620 USD that is invested at 7% grow to 620e0.07x2 = 713.17 USD in 2 years. Of this, 696.26 USD is used to purchase 1,105.17 AUD under the terms of the forward contract. This is exactly enough to repay principal and interest on the 1,000 AUD that is borrowed (1,000e(0.05x2) = 1,105.17). The strategy therefore gives rise to a riskless profit of 713.17 - 696.26 = 16.91 USD.

intrinsic value

An estimate of a stock's "true" value based on accurate risk and return data. The intrinsic value can be estimated but not measured precisely.

Example of a Futures Trade (page 27-29)

An investor takes a long position in 2 December gold futures contracts on June 5 contract size is 100 oz. futures price is US$1250 initial margin requirement is US$6,000/contract*2= US$12,000 in total maintenance margin is US$4,500/contract *2=US$9,000 in total

What is the Payoff of 2 options with a strike price of 40? Solve: MAX(2ST - 40,0)?

Answer: MAX(a ,0) = { a if a ≥ 0 0 if a < 0 a = (2ST -40) 1. replace a with variable = { (2ST -40) if (2ST -40) ≥ 0 0 if (2ST -40) < 0 2. solve by isolating ST = { (2ST -40) if T ≥ 20 0 if T < 20

QUESTION 3 In July 2012, a small chocolate factory receives a large order for chocolate bars to be delivered in November. The spot price for Cocoa is $2,400 per metric ton. It will need 10 metric tons of Cocoa in September to fill this order. Because of limited storage capacity and volatility in the world cocoa prices, the company decides the best strategy is to buy 10 call options for $53 each with strike price of $2,400 (equal to the current price) with a maturity date of September 2012. When the options expire in September, how much will the company pay (including the cost of the options) for cocoa if the spot price in September proves to be $2,300. $23,000 $22,470 $24,530 $23,530 $23,470

Answer: Market Price at Expiration: $2,300 Trade OTM $2,400 = not exercised Premium -$53/ea * 10 = -$530 purchase 10@ market price $2300 = -$23,000 =$23,530

Which of the following options position provides the most protection for an investor's short stock position?

Answer: Long call Explanation: An investor that is short stock must eventually buy the stock to deliver it. A long call provides the best protection for a short stock position because the investor has locked in the price at which the stock can be purchased. Textbook Reference: Please see textbook section 6.2.4

A bank's forwards quotes for bonds Suppose an investor enters into a long 6-month forward contract to buy L250,000. What is the profit/loss to this investor if the spot price in 6 months is: a) 1.5 b) 1.3

Answer: The 6-month forward bid/ask spread is: 1.4416 - 1.4422 K = 1.4422 ( to purchase GBP) Profit = 250,000(ST-11.4422) a) ST= 1.5000 profit=ST-K profit = 250,000(1.5000-1.4422) =$14,450 -> Gain b) ST = 1.3000 profit = 250,000(1.3000-1.4422) =-$35,550 ->Loss

Suppose an investor is long 3-month forward contracts to purchase L1,000,000 GBP. What is the profit/loss to the investor if the spot price in 3 months is: a) 1.5100 b) 1.3500 c) 1.4415

Answer: 3-month forward price of GBP is K= 1.4415 Profit = 1,000,000(ST-1.4415) a) ST 1.5100 Profit = 1,000,000(1.5100-1.4415) = $68,500 Gain b) ST = 1.3500 Profit = 1,000,000(1.3500-1.4415) = $-91,500 Loss c)ST=1.4415 Profit = 1,000,000(1.4415-1.4415)= 0

A bank's forwards quotes for bonds Suppose an investor wants to sell L250,000. worth of GBP in 6-months. What is the profit/loss to this investor if the spot price in 6 months is: a) 1.5 b) 1.3

Answer: Delivery price = K = 1.4416 Profit = 250,000(1.4416-ST) a)250,000(1.4416-ST) ST=1.5000 = Loss of $14,600 b) 250,000(1.4416-ST) ST = 1.3000 = Gain $35,400

XYZ stock is currently trading at $20/share. A $22-strike 2 month European call option on XYZ costs $2. An investor has $20,000 to buy either options only or shares only a) What are the payoff and profit in each scenario? Show using a diagram b) What should the price of the stock be at maturity for the options to be more valuable than the stock only strategy?

Answer: S0= $20 Investment = $20,000 # contracts purchasable = 20,000/20 = 1,000 shares for all stock/100 = 10 call contracts ST = Stock price at maturity ie $? K=strike price ie $22 c=call premium ie $2 a.1) All stock Stock Payoff = the profit OR loss from the sale of an item or service after the costs of selling it 1000(ST-S0) Stock Payoff=1000(ST) Stock Profit = 1000(ST - 20) a.2) Options Strategy 1. calculate number of options contracts amount to invest/call premium = # shares /100 shares per contract 20,000/2 =10,000 shares/100 spc = 100 contracts 2. calculate payoff per option = MAX(ST-K,0) = MAX(ST-22, 0) 3. Calculate call profit: a = (ST -22), 0 PROFIT=MAX(ST - k) - c =MAX(ST-22,0)-2 =MAX(ST-22-2,0-2) PROFIT = ST-24, -2 4. Sub a for if statement a ≥ 0 ST -22 ≥ 0 ST ≥ 22 5. Combine profit w if statements ={ST - 24, if ST ≥ 22 -2, if ST < 22 6. MULTIPLY BY number of shares to find profit for 100 contracts Profit = { 10,000(ST-24, if ST ≥ 22 -20,000 if ST < 22 you would lose all $20k if price does not go above 22 B. Option strategy > stock strategy in terms of profit: 1. Set profit on put option to greater than that of stock strategy: 10,000[MAX(ST-22,0)-2] > 1000(ST - 20) 2. Simplify 10[MAX(ST-22,0)-2] > (ST - 20) 3. Solve for ST 10[MAX(ST-22,0)-2] > ST - 20 10MAX(ST-22,0)-20 > ST - 20 10MAX(ST-22,0) > ST 4. Solve for ST > 22 ONLY DO NOT CONSIDER BASE CASE FOR ZERO 10(ST-22)>ST 10ST-220>ST 9ST>220 ANSWER: THE OPTIONS STRATEGY IS MORE PROFITABLE WHEN ST > 24.44

QUESTION 3 An investor sells a futures contract of an asset when the futures price is $1,500. Each contract is on 100 units of the asset. The contract is closed out when the futures price is $1,540. Which of the following is true A) The investor has made a gain of $4,000 B) The investor has made a loss of $4,000 C) The investor has made a gain of $2,000 D) The investor has made a loss of $2,000

Answer: B An investor who buys (has a long position) has a gain when a futures price increases. An investor who sells (has a short position) has a loss when a futures price increases.

2.3 CONVERGENCE OF FUTURES PRICE TO SPOT PRICE

As the delivery period for a futures contract is approached, the futures price converges to the spot price of the underlying asset. When the delivery period is reached, the futures price equals—or is very close to—the spot price. If the future price is above the spot price, traders will exploit this arbitrage opportunity, by 1. Selling short futures contract, 2. Buying the asset 3. Making the delivery; the futures price will fall and vice versa.

10.2 Assumptions and notation

Assumptions 1.There are no transaction costs. 2.All trading proﬁts (net of trading losses) are subject to the same tax rate. 3.Borrowing and lending are possible at the risk-free interest rate. Notation c : European call option price p : European put option price S0 : Stock price today K : Strike price T : Life of option s: Volatility of stock price C : American Call option price P : American Put option price ST :Stock price at option maturity D : Present value of dividends during option's life r : Risk-free rate for maturity T with cont. comp.

5.3 Assumptions and Notation for Forwards and Futures Contracts

Assumptions: 1. Subject to no transaction costs when they trade. 2. Subject to the same tax rate on all net trading profits. 3. Can borrow money at the same risk-free rate of interest as they can lend money. 4. Take advantage of arbitrage opportunities as they occur. Notation: S0: Spot price today F0: Futures or forward price today T: Time until the delivery date r: Risk-free interest rate for maturity T (usually LIBOR or Treasury Rates

The overall objective of asset allocation is to ensure that investors have A) An equally balanced portfolio of stocks, bonds, and cash. B) A blended portfolio of assets that will react differently under different market conditions. C) Sufficient cash available to make investments that are recommended by prominent investment managers. D) A portfolio of securities tailored to their specific goals. Previous

B) A Blended portfolio of assets that will react differently under different market conditions

One important component of an investor profile is called risk tolerance. What is the best definition of this term? A) A customer's history of investing in risky investments B) A customer's willingness to risk losing part or all of an investment in exchange for higher potential return C) The amount of investment loss that would cause the investor severe hardship or emotional distress D) The likelihood that a customer will lose money, given his current investment mix and asset allocation

B) A customer's willingness to risk losing part or all of an investment in exchange for higher potential return

According to the customer-specific suitability standards, a broker must determine that a recommendation is A) Sometimes suitable for that customer B) Always suitable for that customer C) Suitable based on that customer's prior investment history D) Occasionally suitable for that customer

B) Always suitable for that customer

Which of the following investment products would not be appropriate for an individual who says her current investment objective is growth? A) Small-cap stock B) High-grade municipal bond C) Exchange-traded fund D) Equity mutual fund

B) High-grade municipal bond

Which of the following is a significant risk of a raw land limited partnership program that must be disclosed by a registered representative? A) Limited appreciation potential B) Lack of liquidity C) All of these are risks of raw land partnerships that must be disclosed D) Significant tax losses that are only useful to investors that have other passive income

B) Lack of liquidity

An investor with a relatively high-risk tolerance would like to add a growth fund to his portfolio. Of the following, which type of fund may be most appropriate? A) Junk bond fund B) Small cap stock fund C) Hybrid fund D) Commodity fund

B) Small cap stock fund

To ensure that suitable recommendations are made to customers, broker-dealers are required to make reasonable efforts to obtain all of the following types of customer information EXCEPT: A) investment objectives B) educational background C) financial status D) tax status

B) educational background

Investors, who indicate they are seeking current income from an investment would be least likely to consider which of the following products? A) High-grade corporate bonds B) Zero-coupon bonds C) REITs D) Equity mutual fund

B) zero coupon bonds

Put-call inequality

Based on the payoffs of two portfolio combinations, a fiduciary call and protective put Call with Strike X + Present Value of X = Stock Price + Put with Strike X -if given call price, Ca, we can compute no-arbitrage price range for puts and vice versa. Dividend paying: Ca+K*exp(-rt)-S0 <= Pa<= Ca = K-S0 (Nondividend paying stock) Ca + K*exp(-rt)-S0<= Pa<=Ca+K+)-S0 (Dividend paying stock) Price range for calls: given Pa, the no-ard price of Ca is : Pa+S0-D-K<=Ca<

Example 5.1 Consider a 4-month forward contract to buy a zero-coupon bond that will mature 1 year from today. (This means that the bond will have 8 months to go when the forward contract matures.) The current price of the bond is $930. We assume that the 4-month risk-free rate of interest (continuously compounded) is 6% per annum.

Because zero-coupon bonds provide no income, we can use equation (5.1) with: T =4/12, r = 0.06, and S0 = 930. The forward price, F0, is given by F0 = 930exp(0.06x4/12) = $948.79 This would be the delivery price in a contract negotiated today.

Lehman Bankruptcy

Biggest bankruptcy in history Sept 15 2008 Active participant in the OTC derivatives market took high risks and unable to roll over its short term funding hundreds of thousands of transactions outstanding with 8,000 counterparties

Long-term bonds

Bonds issued by companies to raise debt finance, often with a fixed rate of interest (*debentures*)

Treasury Bonds

Bonds issued by the federal government, sometimes referred to as government bonds.

10.5 Calls on a NDPS

Bound for European Options:No Dividends -Call C≥c≥S0-Ke^(-rT) -Put p≥Ke^(-rT)-S0 P≥K-S0 -Put-Call Parity c+Ke^(-rT) = p + S0 -Early Exercise -Usually there is some chance that an American option will be exercised early -An exception is an American call on a non-dividend paying stock -This should never be exercised early -Consider an American call on a non-dividend-paying stock with 3 months to expiration . S0 = 40. K=70. -What should you do if: -You want to hold the stock for the next 3 months? -No income is sacrificed -You delay paying the strike price -Holding the call provides insurance against stock price falling below strike price -You do not feel that the stock is worth holding for the next 3 months? -Exercise the option and sell the stock immediately? -In this case, the investor is better off selling the option than exercising it. -Payoff from exercising: S0 - K -Payoff from selling the option: C -C>=S0 -Ke-rT C>S0 -K when T>0 -It's better to sell the option rather than exercising it! -Reasons For Not Exercising a Call Early -No income is sacrificed -You delay paying the strike price -Holding the call provides insurance against stock price falling below strike price

10.6 Puts on a NDPS

Bounds for a put option -European put: Max(Ke-rT - S0 )<=p<=Ke-rT -American put: Max(K - S0 )<=p<=K n

call option

the option to buy shares of stock at a specified time in the future trade on both markets

QUESTION 4 In July 2012, a small chocolate factory receives a large order for chocolate bars to be delivered in November. The spot price for Cocoa is $2,400 per metric ton. It will need 10 metric tons of Cocoa in September to fill this order. Because of limited storage capacity and volatility in the world cocoa prices, the company decides the best strategy is to buy 10 call options for $53 each with strike price of $2,400 (equal to the current price) with a maturity date of September 2012. When the options expire in September, how much will the company pay (including the cost of the options) for cocoa if the spot price in September proves to be $2,600 $23,530 $22,470 $23,470 $24,530 $23,000

Buy 10 call options for $53 each with strike price of $2,400 per ton Trade - ITM $2,600>$2,400 Market Price in Sept = $2,600 * 10 = $26,000 Purchase Price = $2,400 * 10 = $24,000 Premium pd up front= $53 *10 = $530 Total Cost = $24,530 Profit = $26,000 - 24,530 = 1470

long

Buyers of calls & Sellers of puts have _____ positions

QUESTION 4 Which of the following is NOT true A) A call option gives the holder the right to buy an asset by a certain date for a certain price B) A put option gives the holder the right to sell an asset by a certain date for a certain price C) The holder of a call or put option must exercise the right to sell or buy an asset D) The holder of a forward contract is obligated to buy or sell an asset

C The holder of a call or put option must exercise the right to sell or buy an asset

An 80-year-old individual would be least likely to purchase a A) Bank CD B) Treasury bond C) Hedge fund D) Money market fund

C) Hedge fund An individual who is retired, or in the later stages of life, would not be likely to make an investment that could result in the complete loss of their capital, or one that would require a long-term investment horizon. Textbook Reference: Please see textbook section 7.3.1

Broker-dealers and their representatives are held to suitability standards that apply to virtually all securities A) Purchases B) Transactions C) Recommendations D) Accounts

C) Recommendations

An investor in a high tax bracket is subject to federal, state and local income taxes. If the investor is seeking current income with minimum tax liability, which of the following choices is most appropriate? A) Private activity municipal bonds B) Double barreled municipal bonds C) US. Virgin Island Utility bonds D) Zero Coupon Treasury Strips

C) US. Virgin Island Utility bonds

Options Exchanges

CBOE - Chicago Board Options Exchange NYSE - New York Stock Exchange AMEX - American Stock Exchange

Popular Futures Exchanges

CBOT - Chicago Board of Trade CME - Chicago Mercantile Exchange NYME - New York Mercantile Exchange LIFFE - London International Financial Futures Exchange InterContinental Exchange BM&F (Sao Paulo, Brazil) TIFFE (Tokyo)

put option

the option to sell shares of stock at a specified time in the future trade on both markets

QUESTION 5 Suppose that there are no storage costs for crude oil and the interest rate for borrowing or lending is 4% per annum. Could you make money risk-free if the June and December futures contracts for a particular year trade at $50 and $56? True False

Calculate Profit: 1. Take June futures contract as a long position as $50 and Dec Futures contract as a short position as $56. 2. Then borrow $50 and get delivery of oil. 3. Sell the oil at $56 and pay the interest expense and principal = =($50x(.04/2))+$50=$51 Profit = $56-$51 =$5

Option Types

Call; right to buy. Put: Right to sell

Derivatives

Can be used for: 1. Hedging risks 2. Speculation (position on direction of market) 3. Arbitrage (to lock in an arbitrage profit) A derivative is an instrument whose value depends on (or derives from) the values of a more basic variable(s), called the underlying assets facilitate the transfer of risks in economy, enable the creation of strategies and payoffs, lower transaction costs, reduce amount of capital required (examples: futures, forwards, swaps, options, interest rate, stock price, commodities, debt instruments, electricity prices, insurance payouts, weather, etc)

5.9 Stock Index

Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore F0 = S0 e((r-q )T) where q is the average dividend yield on the portfolio represented by the index during life of contract For the formula to be true it is important that the index represent an investment asset In other words, changes in the index must correspond to changes in the value of a tradable portfolio The Nikkei index viewed as a dollar number does not represent an investment asset (See Business Snapshot 5.3, page 113)

Delta (Options)

Change in Option Price / Change in Underlying

Central Counterparty (CCP)

Clearhousing for standard OTC transactions

2.1 Closing out Futures Positions

Closing out a position means entering into the opposite trade to the original one. For example, the New York investor who bought a July corn futures contract on March 5 can close out the position by selling (i.e., shorting) one July corn futures contract on, say, April 20.

An investor writes 2 Feb 45 calls for 2.25. The stock price at which the investor breaks even is

Correct Answer: 47.25 Explanation: The breakeven for the call option writer is the strike price plus the premium. 45 + 2.25 = 47.25. The call writer does not begin to lose money until after the price of underlying stock exceeds this price. Textbook Reference: Please see textbook section 6.2.2

When growth is the primary investment objective, an investor's portfolio would likely be focused on A) Municipal bonds B) Preferred stock C) Treasury bonds D) Common stock

Correct Answer: Common stock Explanation: Where growth is the primary investment objective, there would be a focus on equity securities, particularly common stock, which over the course of many years will tend to increase in value. Textbook Reference: Please see textbook section 7.3.4

An investor that holds 100 shares of XYZ stock expects little short-term movement in the stock's price but desires to generate additional income from the shares. Which strategy meets the objectives of this investor?

Correct Answer: Covered call writing Explanation: Covered call writing is a strategy used by investors with long stock positions to generate additional income. By writing calls, premiums are received, and if the market price of the stock does not surpass the exercise price, the call will not be exercised. Textbook Reference: Please see textbook section 6.2.2.1

Which two of the following positions permit an investor to exercise the contract prior to expiration? I. Long call II. Short call III. Long put IV. Short put

Correct Answer: I and III Explanation: I. Long call (buyer-right) II. Short call (seller-obligation) III. Long put (put buyer owns the right to sell ) IV. Short put Buyers of options have rights, sellers have obligations. A call buyer holds the right to buy stock at the exercise price and may exercise that right any time prior to the expiration of the contract. A put buyer holds the right to sell stock at the strike price, and may also exercise that right any time prior to the expiration. Textbook Reference: Please see textbook section 6.1

Which two of the following positions permit an investor to exercise the contract prior to expiration? I. Long call II. Short call III. Long put IV. Short put

Correct Answer: I and III Explanation: Buyers of calls (long call) & Sellers of puts (long put) have _____ positions have rights to buy/sell Sellers of calls (short call) & buyers of puts (short put) have a ______ position have obligations Buyers of options have rights, sellers have obligations. A long call buyer holds the right to buy stock at the exercise price and may exercise that right any time prior to the expiration of the contract. A put buyer holds the right to sell stock at the strike price, and may also exercise that right any time prior to the expiration. Textbook Reference: Please see textbook section 6.1

Which two of the following options strategies are bearish? I. Long call (call buyer) II. Short call (call writers) III. Long put (put buyer) IV. Short put (put writer)

Correct Answer: II and III Explanation: A bearish investor profits when market prices fall. Call writers and put buyers are bearish. Put buyers have the right to sell at the exercise price in a declining market, while call sellers are not assigned when market prices fall. They may keep the premium received without having to sell stock. Textbook Reference: Please see textbook section 6.3.1

When the XXXX Index is 2125, which of the following positions would be exercised? A) Long 2025 Call B) Long 2100 Put C) Short 2150 Call D) Short 2075 Put

Correct Answer: Long 2025 Call Explanation: A call option would be exercised when the market price of the security is higher than the strike price. A put option would be exercised when the market price of the security is lower than the strike price. Textbook Reference: Please see textbook section 6.2.1.1 A) Long 2025 Call - bullish call option (buyer is long) C) Short 2150 Call - flat or bearish (seller is short) D) Short 2075 Put - flat or bullish (buyer is short) B) Long 2100 Put - bearish put option (seller is long)

An investor that owns 100 shares of ABC stock wishes to protect his position as much as possible from a potential market correction. Which of the following positions provides the best protection?

Correct Answer: Long an ABC put Explanation: An investor that owns stock can best protect it from downside risk by purchasing a put. The put gives the investor the right to sell at the exercise price. A short call also provides some protection but only in the amount of the premium received. As you can see in this example, although the profits are reduced when the stock goes up in value, the protective put limits the risk to the unrealized gains during a decline. Textbook Reference: Please see textbook section 6.3.4

Which of the following organizations guarantees the performance of standardized options contracts?

Correct Answer: OCC Explanation: The Options Clearing Corporation (OCC) is the world's largest equity derivatives clearing house. As a clearinghouse, the OCC also acts as guarantor, ensuring that the obligations of the contracts it clears are fulfilled. Textbook Reference: Please see textbook section 6.6.1

Political risk may be mitigated when an investor focuses attention

Correct Answer: On US domestic securities Explanation: Political risk can be reduced by investing in US domestic securities in lieu of overseas investments. Textbook Reference: Please see textbook section 7.2.9

To partially protect a long stock position and increase income to her portfolio, an investor could do which of the following?

Correct Answer: Sell calls Explanation: Additional portfolio income can be generated by writing calls against a stock position that is currently held. The premium from the calls delivers the additional income. Textbook Reference: Please see textbook section 6.3.4

In which of the following positions does the investor face unlimited risk?

Correct Answer: Short call Explanation: When an investor is short a call with no other positions, the investor is obligated to buy stock if assigned. There is no limit on the price to which the stock can rise, so the investor's risk is theoretically unlimited. Textbook Reference: Please see textbook section 6.2.2

What is the settlement date that applies to the exercise of index options?

Correct Answer: The business day following the date of exercise Explanation: Cash settlement for index options must take place on the business day following the date of exercise. Textbook Reference: Please see textbook section 6.4

A call option is out of the money when

Correct Answer: The strike price is above the market price of the underlying security Explanation: A call option is out of the money when the market price of the underlying security is below the strike price of the option. Textbook Reference: Please see textbook section 6.2

Which income-producing options strategy requires an investor to purchase shares at a predetermined price if exercised?

Correct Answer: Write a put Explanation: The sale of a put requires an investor to buy shares when assigned. The income is earned through the receipt of the premium. Textbook Reference: Please see textbook section 6.1

An investment denominated in a foreign currency may lose value or depreciate as the US dollar strengthens. This is an example of

Correct Answer: currency risk Explanation: This is the essence of currency risk. As one currency weakens, another currency will strengthen. Textbook Reference: Please see textbook section 7.2.10

When recommending a specific municipal bond to a client, which of the following factors is least relevant in making a suitability determination? A) Investment time horizon B) Other investments held C) Investment experience D) Ages of their children

D) Ages of their children

Which of the following best describes central clearing parties A) Help market participants to value derivative transactions B) Must be used for all OTC derivative transactions C) Are used for futures transactions D) Perform a similar function to exchange clearing houses

D) Perform a similar function to exchange clearing houses

The risk that a technology or product will become obsolete, causing the company to discontinue operations is A) systemic risk B) market risk C) economic risk D) business risk

D) business risk

4.5 Conversion Formulas

Define Rc : continuously compounded rate Rm: same rate with compounding m times per year Rc=mln(1+Rm/m) Rm=m(exp(rc/m)-1)

Discrete Random Variables

Discrete - whole numbers, things that are separate Random - multiple outcomes with out bias all options equally likely Variable -number can vary/change

4.8 Duration

Duration of a bond that provides cash flow ci at time ti is (see picture) where B is its price and y is its yield (continuously compounded) - This leads to (see diagram) When the yield y is expressed with compounding m times per year - The expression is referred to as the "modified duration"

Bounds on Options Prices

E=european Ce=Call european Pe= put european Rule (0): cE<=Ca and PE<=Pa (american options are more valuable than euro) Rule(1): Upper bounds a. c<=S0 and P<=K b. P <=K*exp(-(r)(t)) -> euro only } prepaid strike (what you pay when purchasing option) Rule 3. Lower bounds (LB) Premium => LB D=) for NDPS European: call (c) option (LB^E): max(0,S0-D-Kexp(-rT) put(p) max(0,K*exp((-(r)(T)+D-S0) American (can exercise at any time): call (c): max(0,S0-K,S0-D-K*exp(-rT) put(p): max(0,K-S0,K*exp((-(r)(T)+D-S0) -cE>=LB^E -pE>=LB^E -CA>=LBA -PA>=LBA Rule 4. Lower Bounds with dividend yield -premium>= LB European: Call(c) max (0,S0*exp(-(q)(T))-K*exp(-(r)(T)) Put(p): max(0,K*exp(-(rT)-S0*exp(-(q)(T))

Electronic Markets

Electronic markets (or electronic marketplaces) are information systems (IS) that allow buyers and suppliers to meet and trade with each other.

Example Find number of contracts that should be shorted to hedge the portfolio and the gain from the short futures position Value of S&P 500 index = 1,000 S&P 500 futures price is 1,010 VF= 250 x 1,010 = $252,500 VA= Value of Portfolio is $5.05 million Beta of portfolio is 1.5 Risk-free interest rate = 4% per annum Dividend yield on index = 1% per annum What position in futures contracts on the S&P 500is necessary to hedge the portfolio? One futures contract is for delivery of $250 times the index. As before, VF = 250 x 1,010 = 252,500. From equation (3.5), the number of futures contracts that should be shorted to hedge the portfolio is: B x VA/VF 1.5 x 5,050,000/ 252,500 = 30 Suppose the index turns out to be 900 in 3 months and the futures price is 902. The gain from the short futures position is then: 30 x (1010 - 902) x250=$810,000 The loss on the index is 10%. The index pays a dividend of 1% per annum or 0.25% per 3 months. When dividends are taken into account, an investor in the index would therefore earn -9.75% over the 3-month period. Because the portfolio has a Beta of 1.5, the capital asset pricing model gives: Expected return on portfolio - Risk-free interest rate = 1.5 x (Return on index - Risk-free interest rate) The risk-free interest rate is approximately 1% per 3 months. It follows that the expected return (%) on the portfolio during the 3 months when the 3-month return on the index is -9.75% is 1.0 [1.5 x (9.75 - 1:0)] =-15.125 Expected value: $5,050,000 x (1-.15125) = $4,286,187 Gain on hedge is: $4,286,187 + $810,000 = $5,096,187

1.1 EXCHANGE-TRADED MARKETS

Exchange is an organized and regulated market, wherein trading of stocks takes place between buyers and sellers in a safe, transparent and systematic manner. ie CME Group = CME + CBOT + NYME

European Options (E)

Exercised only at time T at price K

4.9 What rate of interest with continuous compounding is equivalent to 15% per annum with monthly compounding?

Exp(r) = (1+.15/12)^12 R = 12ln(1+.15/12) = .1491 = 14.91%

Forward Price Calculation

F=S×e^(r+q)×t where: F=the contract's forward price (S-D) x e^(rxt) D = sum of ea div's pres value = d(1)×e^−(r×t(1))+d(2)×e^−(r×t(2))+⋯+ d(x)×e^−(r×t(x)) S=the underlying asset's current spot price e=the mathematical irrational constant approximated by 2.7183 r= quoted interest rate that applies to the life of the forward contract t=the delivery date in years q= carrying costs

5.4 If Short Sales Are Not Possible..

Formula still works for an investment asset because investors who hold the asset will sell it and buy forward contracts when the forward price is too low -Short sales are not possible for all investment assets -sometimes a fee is charged for borrowing assets

5.4 An Arbitrage Opportunity? Suppose that: The spot price (S0) of a non-dividend-paying stock is $40 The 3-month forward price (F0) is $43 The 3-month US$ interest rate is 5% per annum Is there an arbitrage opportunity?

Forward Price Formula: S0 =40, T=0.25, and r=0.05 so that F0 = 40e0.05×0.25 = 40.50 1. An arbitrageur can borrow $40 at the risk-free interest rate of 5% per annum, buy one share, and short a forward contract to sell one share in 3 months. 2. At the end of the 3 months, the arbitrageur delivers the share and receives $43. The sum of money required to pay off the loan is: 40exp(0.05x3/12) = $40.50 3. By following this strategy, the arbitrageur locks in a profit of: $43.00 -$40:50 = $2:50 at end of the 3-month period.

5.4 Another Arbitrage Opportunity? Suppose that: The spot price (S0) of nondividend-paying stock is $40 The 3-month forward price (F0) is US$39 The 1-year US$ interest rate is 5% (R) per annum Is there an arbitrage opportunity?

Forward Price Formula: S0 =40, T=0.25, and r=0.05 so that F0 = 40e0.05×0.25 = 40.50 1. An arbitrageur can short one share, invest the proceeds of the short sale at 5% per annum for 3 months, 2. Take a long position in a 3-month forward contract. 3. The proceeds of the short sale grow to: 40exp(0.05x3/12) = $40.50 in 3 months. 4. At the end of the 3 months, the arbitrageur pays $39, takes delivery of the share under the terms of the forward contract, and uses it to close out the short position. 5. A net gain of: $40.50 x $39.00 = $1.50 is therefore made at the end of the 3 months.

Payoffs from forward contracts

Forward payoff: If you long a forward on an asset with a delivery price K, and the underlying spot price of the asset at expiry (time T), then the payoff you have from this investment is (ST −K). If you short this forward contract, your payoff is (K −ST ).

5.11 Futures on Commodities Consumption Assets: Storage is Negative Income

Forward price of a commodity that is an investment asset is given by: F0=S0e(rT) If U is the present value of all storage costs: F0=(S0+U)e(rT) F0 ≤ S0 exp((r+u )T) where u is the storage cost per unit time as a percent of the asset value. Alternatively, F0 ≤ (S0+U )e(rT) where U is the present value of the storage costs.

2.6 Market Quotes

Futures quotes are available from exchanges and from several online sources: Ex: www.futures.tradingcharts.com/marketquotes Prices - first 3 numbers in each row -opening price - -highest price achieved -lowest price achieved -settlement price - closing price -change -from previous day -trading volume -open interest- number of contractts open

Trading for time value

Generally, the more time that remains until the option expires, the greater the time value of the option.

QUESTION 1 Suppose the one-year gold lease rate is 1.5% and the one-year risk-free rate is 5.0%. Both rates are compounded annually. Use the discussion in Business Snapshot 3.1 to calculate the maximum one-year forward price Goldman Sachs should quote for gold when the spot price is $1,200.

Goldman Sachs can borrow 1 ounce of gold and sell it for $1200. It invests the $1,200 at 5% so that it becomes $1,260 at the end of the year. It must pay the lease rate of 1.5% on $1,200. This is $18 and leaves it with $1,242. It follows that if it agrees to buy the gold for less than$1,242 in one year it will make a profit

1.2. Explain carefully the difference between hedging, speculation, and arbitrage.

Hedgers use derivatives to reduce the risk from variation of a market variable in the future. There is no gaurantee that the outcome of hedgin will necessarily be better than not hedging. Of course, one must think of these scenarios in terms of ensembles. Hedging can be done using forward contracts and options. The former (forward contracts) is designed to reduce risk by fixing the price that the hedger will pay or recieve for the underlying asset. The latter (options) provide insurance by offering a way for the investors to protect themselves against adverse price movements in the future while allowing themselves to benefit from favorable price movements. Speculators use derivatives to bet on the future direction of a market variable. The same two financial instruments (forward and option contracts) can be used to speculate. Speculators wish to take a position in the market and are betting that either the price of the asset will go up or will go down. When a speculator uses futures, the potential loss as well as potential gain is large. When options are used, no matter how bad things get the speculator's loss is limited to the amount paid for the options. Arbitrageurs take ofsetting position in two or more instruments to lock in a profit. They participate in futures, forward and options markets. Arbitrage involves locking in a riskless profit by simultaneously entering into transactions in two or more markets. Example: Suppose a stoc price is $140 in New York and $100 in London when the exchange rate is $1.4300 per pound. Then an arbitrageur can buy 100 shares of stock in New York and sell them in London to obtain a risk free profit of 100×[$1.43×100−$140]100×[$1.43×100−$140] or $300. These opportunities are very temporary as supply and demand would cause the dollar price to rise and the sterling price to drop. Existence of profit hungry arbitageurs makes it unlikely that a major disparity between the sterling and dollar prices exist.

5.5 When an Investment Asset Provides a Known Income

If F0 > S0-I)exp(rT), buy asset and short a forward contract on asset. If F0 < (S0-I)e(rT), Ex: Stocks that pay dividends and coupon-bearing bonds F0 = (S0 - I )exp(rT) short asset take long position in forward contract where I is the present value of the income during the life of forward contract Ex: long forward contract to purchase a coupon-bearing bond whose S0=current price = $900. T=9/12=.75 Time to maturity = 9 I = coupon payment = $40 after 4 months. T=4/9=.75 4 month and 9 month risk free interest rates (continuously compounded) = 3% and r= 4% per annum Ex 1: Forward price (F0) = $910 An arbitrauger can borrow the $900 to buy the bond and short a forward contract. The coupon payment has a present value of I=40exp(-.03x4/12)=$39.6 F0 = (900 - 39.6 )exp(.004*.75)= $886.6 Of the 900 39.60 is borrowed at 3% per annum for 4 months so that it can be repaid with the coupon payment. the remaining $860.40 is borrowed at 4% per annum for 9 months The amount owing at the end of the 9 month period is $860.40(exp(.04*.75)=$886.6 A sum of $910 is received for bond under terms of forward contract. Net profit - 910-886.6=$23.4 Ex2: Forward price is $870 An investor can short the bond and enter into a long forward contract. Of the $900 realized from shorting the bond, $39.60 is invested for 4 months at 3% per annum so that it grows into an amount sufficient to pay the coupon of the bond. The remaining $860.40 is invested for 9 months at 4% per annum and grows to $886.6. Under the terms of the forward contract, $870 is paid to buy the bond and the short position is closed out. Net profit = 886.6 - 870 =$16.6

10.3 Upper and lower bounds for Option prices

If an option price is above the upper bound or below the lower bound, then there are profitable opportunities for arbitrageurs American vs European Options American option (C) is worth atleast as much as ≥ a European option (c) American put option (P) is at least as much as ≥ a European Put (p) Upper Bounds Call options: A call option can never be worth more than the stock price: •European Call; c ≤ S0 •American Call; C ≤ S0 Put Options: A put option can never be worth more than the strike price, K: •European Put (p): p≤〖Ke〗^(-rt) •American Put (P): P≤K Lower bounds (NDPS): Call option: worst case call expires worthless, therefore: c ≥ 0 c ≥ max(S0 - Ke^(-rT), 0) Put Option: worst case is that it expires worthless, so it cannot be negative: p ≥ max(Ke^(-rT) - S0, 0)

The Forward Price of Gold (ignores the gold lease rate)

If the spot price of gold is S and the forward price for a contract deliverable in T years is F, then F = S (1+r )T where r is the 1-year (domestic currency) risk-free rate of interest. In our examples, S = 1200, T = 1, and r =0.05 so that F= 1200(1+0.05) = 1,260

QUESTION 9 A company has derivatives transactions with Banks A, B, and C which are worth +$20 million, −$15 million, and −$25 million, respectively to the company. How much margin or collateral does the company have to provide? The transactions are cleared bilaterally and are subject to one-way collateral agreements where the company posts variation margin, but no initial margin. The transactions are cleared centrally through the same CCP and the CCP requires a total initial margin of $10 million. $20 million $10 million $60 million $40 million $30 million

If the transactions are cleared centrally they are netted against each other and the company's total variation margin (in millions of dollars) is -20 + 15 + 25 = $20 million in total. The total margin required (including the initial margin of $10 million) is therefore $30 million.

Explain the difference between bilateral and central clearing for OTC derivatives.

In bilateral clearing, two market participants enter into an agreement with each other covering all outstanding derivative transactions between the two parties. Typically the agreement covers collateral arrangements, events of default, the circumstances under which one side can terminate the transactions, etc. In central clearing, a CCP (central clearing party) stands between the two sides of an OTC derivative transaction in much the same way that the exchange clearing house does for exchange-traded contracts. The CCP and its members absorb the credit risk, but initial as well as variation margin is required from each side.

Changing Beta Being insensitive to the market return implies something more such as being "beta neutral" or "zero beta." Simply, being beta neutral means that a portfolio's return will be insensitive to whether the market is up, down or flat. •What position is necessary to reduce the beta of the portfolio to 0.75? •What position is necessary to increase the beta of the portfolio to 2.0?

In the example in Table 3.4, the beta of the hedger's portfolio is reduced to zero so that the hedger's expected return is almost independent of the performance of the index. Sometimes futures contracts are used to change the beta of a portfolio to some value other than zero. Continuing with our earlier example: S&P 500 index = 1,000 S&P 500 futures price = 1,010 Value of portfolio = $5,050,000 Beta of portfolio = 1.5 As before, VF = 250 x 1,010 = 252,500 and a complete hedge requires 1.5 x 5,050,000/ 252,500 = 30 contracts to be shorted. To reduce the beta of the portfolio from 1.5 to 0.75, the number of contracts shorted should be 15 rather than 30. To increase the beta of the portfolio to 2.0, a long position in 10 contracts should be taken; and so on. In general, to change the beta of the portfolio from B to B*, where B > B*, a short position in (B-B*)VA/VF contracts is required. When B < B*, a long position in (B*-B)VA/VF contracts is required.

Example 5.5 Consider a 3-month futures contract on an index. Suppose that the stocks underlying the index provide a dividend yield of 1% per annum, that the current value of the index is 1,300, and that the continuously compounded risk-free interest rate is 5% per annum.

In this case, r=.05, S0=1,300, T=0.25, q=0.01 Hence, the futures price, F0=1,300exp((.05-.01)x.25)=1313.07

Example 5.4 A long-forward contract on a non-dividend-paying stock was entered into some time ago. It currently has 6 months to maturity. The risk-free rate of interest (with continuous compounding) is 10% per annum, the stock price is $25, and the delivery price is $24.

In this case, S0 = 25, r = 0.10, T = 0.5, and K = 24. The 6-month forward price, F0, is given by F0 F0=S0exp(rT) F0=25exp(.1*.5)=26.28 The present value of the forward contract is (eqn 5.4) f=(F0-K)exp(-rT) f=(26.28-24)exp(-.1*.5)=2.17 Value of forward contract on an investment asset -no income (eqn 5.5): f=S0-I-Kexp(-rT) -income (5.6): f=S0-I-Kexp(-rT) -known yield (5.7): f=S0exp(-qT)-Kexp(-rT)

5.1 Investment Assets vs. Consumption Assets

Investment asset - an asset that is held for investment purposes by significant numbers of investors. -Ex: Stocks and bonds, Gold and silver -use arbitrage argument to determine forward and futures prices from its spot price Consumption assets - are assets held primarily for consumption, not held for investment -Ex: copper, oil

small-cap stock funds

Invests in both growth- and value-oriented securities to temper the generally higher risk of small-company stocks. Reliance on intensive in-house research to help uncover opportunities. Small companies tend to be riskier than large companies.

QUESTION 2 In the corn futures contract, the following delivery months are available: March, May, July, September, and December. Which of the following statements is the least accurate? July is the contract that should be used for hedging when the expiration of the hedge is in June July is the contract that should be used for hedging when the expiration of the hedge is in July March is the contract that should be used for hedging when the expiration of the hedge is in January September is the contract that should be used for hedging when the expiration of the hedge is in July

July is the contract that should be used for hedging when the expiration of the hedge is in July

junk bond funds

Junk bonds are bonds that carry a higher risk of default than most bonds issued by corporations and governments. ... Junk bonds represent bonds issued by companies that are financially struggling and have a high risk of defaulting or not paying their interest payments or repaying the principal to investors.

Short position payoff (forward contract)

K - ST If you short a forward on an asset with a delivery price K, and the underlying spot price of the asset at expiry (time T )

Margin Call Formula

Loan / (1 - Maintenance Margin)

A company enters into a short futures contract to sell 5,000 bushels of wheat for 750 cents per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price change would lead to a margin call? Price up by 30 cents Price down by 20 cents Price up by 20 cents Price down by 25 cents

Loan = 750 * 5000 = 3750000 cents or $ 37,500 initial margin = $3000 maintenance margin = $2000 margin call formula = Loan / (1 - Maintenance Margin) 3,750,000 cents / (1-200000) = -199,999 37,500 / (1-2000) = -18.75 Answer = sell short = long, Price down by 20 cents

Options Payoffs

Long Call Payoff = max(ST - K, 0) where ST = stock price at maturity and K = Strike Price

Types of Hedge Funds

Long/Short Equities, Convertible Arbitrage, Distressed Securities, Emerging Markets, Global Macro, Merger Arbitrage

Solve the following Max function: MAX(ST - 50,0) =

MAX(a ,0) = { a if a ≥ 0 0 if a < 0 a = (ST-50) Answer: 1. ={ST-50 if ST-50 ≥ 0 0 if ST-50 < 0 2. Solve by isolating ST: ST-50 if ST ≥ 50 0 if ST < 50

Margin Cash Flows

Main Task is to keep track of all the transactions that take place during a day, so that it can calculate the net positions of each of its members -clearing margin - brokers required to maintain margin accounts with a clearing house member and them with the house

Why Hedge Equity Returns

May want to be out of the market for a while. Hedging avoids the costs of selling and repurchasing the portfolio Suppose stocks in your portfolio have an average beta of 1.0, but you feel they have been chosen well and will outperform the market in both good and bad times. Hedging ensures that the return you earn is the risk- free return plus the excess return of your portfolio over the market.

Supervisory Investigation

Means an investigation of a civilian complaint conducted by a subject officer's supervisor or higher level official within the officer.

forward commitments

Obligates the parties to engage in a transaction at a future date on terms agreed upon in advance. K = delivery price in the contract (fixed) =same as Forward price measured from contract date F0=Forward/futures price today (@t=0) ST= spot price of underlying at maturity

Offer Price

Offer price is the Price at which the security is sold to the investors quotes per contract (100 shares). ie Strike price of $520, offer price $32 of one share December 2010 Exp Date, Investor must pay $3,200 (32 * 100 Google shares) to be remitted to the exchange through the broker to purchase the contract. If strike price is not $520 at date of maturity, the investor loses the $3,200 If stock price at time of redemption is over $520, then the investor is allowed to purchase and sell shares for a profit.

1.9. You would like to speculate on a rise in the price of a certain stock. The current stock price is $29 and a 3-month call with a strike price of $30 costs $2.90. You have $5,800 to invest. Identify two alternative investment strategies, one in the stock and the other in an option on the stock. What are the potential gains and losses from each?

One strategy would be to buy 200 shares ($5,800/29). Another would be to buy 2,000 options (5,800/2.9). If for example, the share price goes up to $40 you gain (40-29)*200= 2,200 from the first strategy. With the options strategy you would earn (40-30-2.9)=7.1*2000=14,200 However, if the share price goes down to $25, the first strategy leads to a loss of 4*200=-$800 whereas the second strategy leads to a loss of the whole $5,800 investment. So although the options strategy would give you additional leverage, you would also risk losing a lot more than if you owned the stocks out right.

Assume a security is currently trading at $100 per unit that pays a 50-cent dividend every 3 months. An investor wants to enter into a forward contract that expires in one year. The current annual risk-free interest rate is 6%. Calculate the present value of each dividend and the forward price.

PV(d(1))=$0.5×e^−(0.06×3/12)=$0.493 PV(d(2)) = $0.5 ×e−(0.06×126)=$0.485 PV(d(3))=$0.5×e−(0.06×129)=$0.478 PV(d(4))=$0.5×e−(0.06×1212)=$0.471 The sum of these is $1.927. This amount is then plugged into the dividend-adjusted forward price formula: Forward Price F=($100−$1.927)×e(0.06×1)=$104.14

QUESTION 12 On a particular day, there were 2,000 trades in a particular futures contract. This means that there were 2000 buyers (going long) and 2000 sellers (going short). Of the 2,000 buyers, 1,400 were closing out positions and 600 were entering into new positions. Of the 2,000 sellers, 1,200 were closing out positions and 800 were entering into new positions. What is the impact of the day's trading on open interest? down by 1200 down by 1400 up by 800 down by 600

Open Interest = Number of contracts outstanding (long and short outstanding) at a particular time. Points to remember 1. Open interest increases by 1 when both sides of the transaction (long and short) enter into a new contract. 2. Open interest decreases by 1 when both sides of the transactions close existing positions. 3. Open interest stays the same when one party enters into a new contract while the other party closes out an existing position. 2000 Buyers 2000 Sellers Close 1400 (short) 1200 (long) New 600 (long) 800 (short) Step 1. According to the first point, Open interest increases by 1 when both sides of the transaction (long and short) enter into a new contract. So, open interest is +600 (because there are 600 new long and 600 new short positions and the other 200 remaing new short posions out of the 800 new short positions are extra) Step. 2 According to the second point, Open interest decreases by 1 when both sides of the transactions close existing positions. So, open interest = -1200 (because there are 1200 closing short positions and 1200 closing long positions and out of 1400 closing short positions 200 are extra) Step 3. According to the third point, 3. Open interest stays the same when one party enters into a new contract while the other party closes out an existing position. In step one there were 200 new short positions and in step two there were 200 closing short positions that were extra so according to the third point 200 parties enter into a new contract and 200 parties close out an existing position so there is no change in open interest. Hence open interest = 0 Now think of it like a math equation +600 - 1200 + 0 = -600 Therefore, the open interest went down by 600.

QUESTION 13 The open interest usually decline during the month preceding the delivery month. True False

Open interest is the number of contracts outstanding. Many traders close out their positions just before the delivery month is reached. This is why the open interest declines during the month preceding the delivery month.

QUESTION 11 Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65, the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and the coefficient of correlation between the two changes is 0.8. A three-month contract is used for hedging.Which of the following is true? The size of the futures position should be 64.2% of the size of the company's exposure in a three-month hedge. The size of the company's exposure should be 64.2% of the size of the futures position in a three-month hedge. The size of the futures position should be 35.8% of the size of the company's exposure in a three-month hedge. The size of the futures position should be 99.7% of the size of the company's exposure in a three-month hedge.

Optimal Hedge Ratio = coefficient of correlation x (standard deviation of quarterly changes in spot price / standard deviation of quarterly changes in future prices) Optimal Hedge Ratio = .8 x (.65 / .81) = .642 = 64.2% The optimal hedge ratio of .642 means that the size of future positions should be 64.2% of the exposure of the company in a 3-month hedge.

1.5 OPTIONS

Options are financial derivatives that give buyers the option, (but not the obligation), to buy or sell an asset at a certain price (strike price, K) on or before a certain date (T, exercise date T, years). Traded both exchanges & over-the-counter market must pay a premium to acquire

Speculation using options

Options speculation allows a trader to hold a leveraged position in an asset at a lower cost than buying shares of the asset. The options strategy is, therefore, 10 times more profitable than directly buying the stock. Options also give rise to a greater potential loss. Investors use options to hedge or reduce the risk exposure of their portfolios. In some cases, the option holder can generate income when they buy call options or become an options writer.

Suppose a 5-year zero rate with continuous compounding is quoted as 5% per annum. This means that $100, if invested for 5 years, grows to:

P x e^(r*n) 100 x exp(.05*5) = 128.4025

1.24. On July 1, 2011, a company enters into a forward contract to buy 10 million Japanese yen on January 1, 2012. On September 1, 2011, it enters into a forward contract to sell 10 million Japanese yen on January 1, 2012. Describe the payoff from this strategy.

Payoff from the contract with long position is ST-K, where ST is the price at maturity and K is the strike price. Payoff from the contract with short position is K-ST. If say ST=30>K=20, then it would be a net profit of 10 for the long forward and a net loss of 10 for the short forward and vice versa. The two would effectively cancel one another out.

Profit from a Long forward Position

Profit from a Long Forward Position (K = delivery price = forward price at time contract is entered into)

Profit from a Short Forward Position

Profit from a Short Forward Position (K = delivery price = forward price at time contract is entered into

Contingent Claims or Derivatives

Provide one party the right but not the obligation to engage in a future transaction on terms agreed upon in advance

High-Yield Bond Funds

Provide the highest yields due to their increased credit risk and are considered speculative investments.

QUESTION 8 A company has derivatives transactions with Banks A, B, and C which are worth +$20 million, −$15 million, and −$25 million, respectively to the company. How much margin or collateral does the company have to provide? The transactions are cleared bilaterally and are subject to one-way collateral agreements where the company posts variation margin, but no initial margin. The banks do not have to post collateral. $20 million $10 million $60 million $40 million $30 million

QUESTION 8 If the transactions are cleared bilaterally, the company has to provide collateral to Banks A, B, and C of $0 million, $15 million, and $25 million, respectively. The total collateral required is 0+15+25=$40 million.

4.1.b LIBOR (London Interbank Offered Rate)

Rate at which the highest credit quality banks borrow from each other in the London interbank market. The rate is reported in 10 currencies

4.1.a.Treasury Rates

Rates on instruments issued by a government in its own currency -rates an investor earns on Treasury bills and Treasury bonds. -used as basis for risk-free rate

10.4 Put-Call Parity; No Dividends Ex: Consider two portfolios: Portfolio A: one European call option plus a zero-coupon bond that provides a payoff of K at time T Portfolio C: one European put option plus one share of the stock.

Relationship between the prices of European put and call options that have the same strike price (K) and time to maturity (T): c + Ke^(-rT) = p + S0 Values of Portfolio A and Portfolio C at time T: ST >K Portfolio A Call option ST - K (exercised) Zero-coupon bond K Total: ST Portfolio C Put option 0 Share ST Total: ST ST < K Portfolio A Call option 0 Zero-coupon bond K Total: K Portfolio C Put option K-ST (will be exercised) Share ST Total: K Extensions of Put-Call Parity American options; D = 0 S0 - K < C - P < S0 - Ke -rT European options; D > 0 c + D + Ke -rT = p + S0 Equation 10.10 p. 249 American options; D > 0 S0 - D - K < C - P < S0 - Ke -rT

Reasonable Basis Suitability

Requires registered rep to conduct due diligence on security and conclude that it is appropriate for at least some investors

Customer-specific suitability

Requires registered rep to have reason to believe that recommendation is appropriate for customer based on customer's specific investment profile

QUESTION 1 A company has a $36 million portfolio with a beta of 1.2. The index futures price is 900. Futures contracts on $250 times the index can be traded. What trade is necessary to reduce beta to 0.9? Long 192 contracts Short 192 contracts Long 48 contracts Short 48 contracts

S&P 500 index = ? S&P 500 futures price = 900 Value of portfolio = $36,000,000 Beta of portfolio = 1.2 As before, VF = 250 x 900 = 225,000 and a complete hedge requires 1.2 x 36,000,000/ 225,000 = 192 contracts to be shorted. To reduce the beta of the portfolio from 1.2 to .9, the number of contracts shorted should be 144 rather than 192. 192-148 = 48 To reduce the beta of the portfolio to .9, a short position in 48 contracts should be taken; and so on.

5.4. A stock index currently stands at 350. The risk-free interest rate is 8% per annum (with continuous compounding) and the dividend yield on the index is 4% per annum. What should the futures price for a 4-month contract be?

S0=$350, r=.08, DY=.04, T=4/12 Future Price Formula = S0*exp((r-dy)*T) = 350*EXP((0.08-0.04)*4/12) = 350* 1.013423 = 354.6979 = $ 354.70

Long Hedge for Purchase of an Asset

S1 : Spot price at time t1 = $2.5 S2 : Spot price at time t2 $2.00 (asset underlying futures contract) F1 : Futures price at time t1 = $2.20 F2 : Futures price at time t2 = $1.90 b1 : Basis at time t1 = S1 - F1 = 2.5 - 2.20 = .30 b2 : Basis at time t2 = S2 - F2 = 2-1.90 = .10 The effective price that is paid with hedging is therefore S2 + F1 - F2 = F1 + b2 2 + 2.20 - 1.90 = 2.20 + .10 = 2.30 By hedging, a company ensures that the price that will be paid (or received) for the asset: S2 + F1 - F2 = F1 + (S* 2 - F2 ) +(S2 - S*2 ) where: S*2-F2 = basis if asset same as unlerying S2-S*2 = basis from difference between the 2 assets

QUESTION 2 On March 1 a commodity's spot price is $60 and its August futures price is $59. On July 1 the spot price is $64 and the August futures price is $63.50. A company entered into futures contracts on March 1 to hedge its purchase of the commodity on July 1. It closed out its position on July 1. What is the effective price (after taking account of hedging) paid by the company? $59.50 $60.50 $61.50 $63.50

S1 : Spot price at time t1 = $60 S2 : Spot price at time t2 $64 (asset underlying futures contract) F1 : Futures price at time t1 = $59 F2 : Futures price at time t2 = $63.50 b1 : Basis at time t1 = S1 - F1 = 60-59=1 b2 : Basis at time t2 = S2 - F2 = 64-63.50= .50 The effective price that is paid with hedging is therefore S2 + F1 - F2 = F1 + b2 64+59-63.50=59+.50 =59.50

Example 1: A 6-month 50-strike euro call option on a NDPS has a premium, C the current price is $51 and CCRFR is 12% A Find the lower bound, LB for C BIf c=3, Describe an Arb strategy that gives a risk free profit. what is the profit?

SOln: a. S0=51, K=50, T=6/12=.5, r=.12 p.a. LB = max(0,S0-K*exp(-r)(T) for NDPS (D=)) =max(0,51-50*exp(-(.12)(6/12)) =max(0,3.91) =3.91 C.=3.91 otherwise, there is arbitrage! b given c= <LB => ARB(Call is underpriced Strategy: 1. Buy call -> cash outflow of $3 (-3) 2. sell stock@ 51->inflow of $51(+51) cash flow = -3 + 51 = 48 3. invest $48 @ CCRFR of .12 for 6 mos accumulation = $48*exp(.12)(6/12)=50.97 4. at maturity, stock price =ST a. ST<=50 => option is worthless b. repurchase stock on open market for ST c Risk free profit = 50.97-ST>=50.97-ST>=50.97-50 RFP>=.97 Note: if St=48.97, the RFP=2 d ST>50 exercise call option to purchase stock at $50 e. RFP = 50.97 - 50 = .97 thus, in either case, RFP >=.97

Long position payoff (forward contract)

ST - K (If you long a forward on an asset with a delivery price K, and the underlying spot price of the asset at expiry (time T ),)

Break-even point for Calls

ST = Stock price at maturity ie $97 K=strike price ie $100 C=call premium ie $3 B/E = Strike + Premum B/E = $100 + $3 = $103

Break-even point for Puts

ST = Stock price at maturity ie $97 K=strike price ie $100 P=call premium ie $3 B/E = Strike - Premum B/E = $100 - $3 = $100

QUESTION 10 A speculator sells a July 2013 wheat futures contract at 721 cents per bushel. Each futures contract is for 5,000 bushels. The futures price drops to 676 on December 31, 2012 and rises to 712 in May 2013 when she closes the contract. What is the gain or loss for accounting purposes in 2013? $180,000 gain $225,000 gain $45,000 gain $45,000 loss $180,000 loss

Sale price for future contract = 721 cents per bushel Qty of future contract = 5,000 bushels Price on December 31, 2012 = 676 cents Gain to be recognized in 2012 = (721 - 676) * 5000 = 225,000 cents Price in May 2013 = 712 cents Gain or (loss) for accounting purpose to be recognized in 2013 = (676 - 712) * 5000 = ($180,000)

1.4 FUTURES CONTRACTS

Same as forwards except futures contracts are normally traded on an exchange. An agreement between two parties to buy/sell an asset at a certain price at a certain time in the future. Payoff/Profit = ST-F0 (Long) Profit/loss = nx(F1-F0) If the future contract has a loss will have a margin call initial margin = IM ie 5% of actual cost) maintenance margin (MM) (ie 75% of IM Entered using margin accounts .3-15% Down Unlike forwards, you can close out at any time. n= # of contracts x = size of contracts (ie L62,500 for GBP or 5000 bushels for corn) Examples of Futures Contracts: Agreement to: -Buy 100 oz. of gold @ US$1300/oz. in December -Sell £62,500 @ 1.4500 US$/£ in March -Sell 1,000 bbl. of oil @ US$50/bbl. in April

4.1 A bank quotes an interest rate of 14% per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?

Solution: a) continuous compounding exp(r) = (1+ .14/4)^4 r= 4ln(1.035)=13.761% b) annual compounding (1+r) = 1.035)^4 r=(1.035)^4-1=14.752%

12.3 Two-step binomial trees Example: A 6-month Euro Put Option On a Stock Price is 150; The CCRFR is 6% per annum. Using U=1.3 and d=.7, estimate the value of the option using a 2-step binomial tree.

Solution: q=0(NDPS), r=.06, T=6/12=.5, ΔT = T/2=.25 K=160, S0=150, u=1.3, d=.7 payoff @ T = fT = max(0,k-ST)=MAX(0,160-ST) eg: fud=max(0, 160-136.5) = 23.5 fuu=max(0,160-253.5)=0 Risk neutral probability = P= (exp(r-q)ΔT-d)/(u-d) =(exp(.06).25)-.7)/(1.3-.07) = .5252 1-P = 1-.5252=.4748

Short

Sellers/writers of calls & buyers of puts have a ______ position

Suppose that: The spot price (S0) of oil is US $50 The quoted 1-year futures price (F) of oil is US $40 The 1-year US$ interest rate (r) is 5% per annum The storage costs (u) of oil are 2% per annum Is there an arbitrage opportunity? (refer to 335 2021 01 25 xls ch1)

Solution: Spot price (S0) = $50 Interest rate (r) = .05 Futures price (F) = $40 Storage costs (u) = .02 1. Calculate cost to purchase and store oil on loan = S0*(1+r+u) = 50*(1.07) = 53.5 2. Calculate variance of cost vs future price: 40-53.5 = -13.5 Yes there is an arbitrage opportunity if you sell the oil short and buy futures

Suppose that: The spot price (S0) of oil is US $50 The quoted 1-year futures price (F) of oil is US $60 The 1-year US$ interest rate (r) is 5% per annum The storage costs (u) of oil are 2% per annum Is there an arbitrage opportunity? (refer to 335 2021 01 25 xls ch1)

Solution: Spot price (S0) = $50 Interest rate (r) = .05 Futures price (F) = $60 Storage costs (u) = .02 1. Calculate cost to purchase and store oil on loan = S0*(1+r+u) = 50*(1.07) = 53.5 2. Calculate variance of cost vs future price: 60-53.5 = 6.5 Yes there is an arbitrage opportunity

Consider the table 4.2's list of Treasury zero rates continuously compounded. Calculate the theorectical price of a 2 year treasury bond with a principal of $100 that provides coupons semi-annually at a rate of 6%. Coupons amounts are (100*(.06/2)).

Solution Face Value = 100 Annual Coupon Rate (%): 6% Annual Market Rate/Discount Rate: 5% Years to Maturity: 2 Pymt interval : 2 Semi annual coupon payment $3 Bond Present Values: 1. 3*e^(-.05*.5)+ 2. 3*e^(-.058*1.0)+ 3. 3*e^(-.064*1.5)+ 4. 3*e^(-.068*2) =98.39

Ex: A 6-mo american call on a NDPS has a premium of 2.50; it's strike price is $30. If the current stock price is $29, and the CCRFR is 10% p.a., what is the price range on an equivalent put?

Solution: S0= $29 K=$30 Ca=2.50 T=6/12=.5 r=.1 Given Ca = Ca+K*exp(-rt)-S0<=Pa<=Ca+Kk-S0 2.5+30exp(-(.1)(.5))-29<=Pa<=2.50+30-29 2.04<=Pa<=3.50 note ARB4: Pa<2.04 or Pa>3.50

Exampl: A T-month Euro put option on a DPS has the following properties: S0=$47, K=50, r=12% Dividends of $1 and $2 are paid in 2 and 4 months, respectively. Determine the lower bound on the premium P, if : a T=6m b T=3 mo

Solution: p>=LB = max(0,K*exp(-(r)(T))+D-S0) where S0=47, K$50, r=.12 a. T= 6 months D= PV{D1, D2}=PV{1,2} =D1*exp(-(r)(t1))+D2*exp(-(r)(t2)) = 1*exp(-.12)(2/12)+2*exp(-(.12)(4/12) =2.90P>=max(0, 50*exp(-(.12)(6/12))+2.90-47) =max(0,2.99)=2.99 p>=2.99 b. T=3months D=PV{D1,=PV{1}=1*exp(-.12)(2.12) =.98 p>= max (0,50*exp(-(.12)(3/12))+.98-47) =max(0,2.50)=2.50 p>=2.5 Rule 3. Lower bounds (LB) Premium => LB D=) for NDPS European: call (c): max(0,S0-D-Kexp(-rT) put(p): max(0,K*exp((-(r)(T)+D-S0)

Blue Sky Laws

State laws that regulate the offering and sale of securities for the protection of the public.

Steps to see if a Put is An Arbitrage Opportunity? Suppose that p= 1 S0 = 37 T = 0.5 r =5% K = 40 D = 0 Is there an arbitrage opportunity?

Steps to follow at Put Options: 1. Check the lower bound of the option. See if the option is correctly priced or not. If p < max(Ke^(-rT) - S0, 0)then arbitrage exists and you can continue with following steps. If p ≥ max(Ke^(-rT) - S0, 0)then there is no need to follow any more steps you can conclude with there is no arbitrage possibility. 2. Borrow an amount that is equal to p+S_0. Buy the Stock. 3. At the end of maturity if S_T≤K exercise the option and do not exercise the option if S_T>K and use the proceeds to pay back the loan.

Steps to see if a Call is An Arbitrage Opportunity Suppose: c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0 Is there an arbitrage opportunity?

Steps: 1. Check the lower bound of the option. See if the option is correctly priced or not. If c < max(S0 - Ke^(-rT), 0) then arbitrage exists and you can continue with following steps. If c ≥ max(S0 - Ke -rT, 0) then there is no need to follow any more steps you can conclude with there is no arbitrage possibility 2. Buy the option and short the stock 3. Invest the proceeds until maturity is reached. At the end of maturity if STT≥K exercise the option and do not exercise the option if ST<K

Which two of the following options contracts are subject to automatic exercise at expiration if the market price of ABC is $45.25? I. Long ABC 42 call x II. Long ABC 46 put III. Long ABC 48 call x IV. Long ABC 44 put

Stock options that are in-the-money at the time of expiration will be automatically exercised. For puts, your options are considered in-the-money if the stock price is trading below the strike price. Conversely, call options are considered in-the-money when the stock price is trading above the strike price.

valuing the option

the portfolio that is: long .25 shares short 1 option is worth 4.455 The value of the shares is: 5.000(=25

5.10 Explanation of the Relationship Between Spot and Forward (Figure 5.1)

Suppose that an individual starts with 1,000 units of the foreign currency. There are two ways it can be converted to dollars at time T. One is by investing it for T years at rf and entering into a forward contract to sell the proceeds for dollars at time T. This generates: 1,000e(rf *T)F0 dollars = 1,000s0e(rT) =F0=S0e((r-rf)*T)

1.9 Arbitrageurs

Take offsetting positions in two or more instruments to lock in a profit. Locking in a riskless profit by simultaneously entering into transactions in two or more markets. Usually done by institutional investors, barrier to profit is transaction costs.

common stock

Term used to describe the total amount paid in by stockholders for the shares they purchase.

Nominal Rate

The annual interest rate stated on a financial instrument (a note or bond, for example). Also called face or stated rate.

4.1.c Repo Rate

The annualized percentage difference between lender's purchase and sell back price.

5.6 Known Yield from an Investment Asset

The asset underlying a forward contract provides a known yield rather than a known cash income. Expressed as a percentage (ex: 5% yield)of the asset's price at the time the income is paid. F0 = S0 exp((r-q )T) q = average yield during the life of the contract (expressed with continuous compounding)

Convenience Yields

The benefits from holding the physical asset The greater the possibility that shortages will occur, the higher the convenience yield. If the dollar amount of storage costs is known and has a present value U, then the convenience yield y is defined such that: F0e((yT)=(S0+U)e(rT) If the storage costs per unit are a constant proportion, u, of the spot price, then y is defined so that: F0e(yT) = S0e((r+u)T) or F0=S0e((r+u-y)T)

Normal Backwardation

The condition in futures markets in which futures prices are lower than expected spot prices.

4.9 Convexity

The convexity of a bond is defined as (see image) so that (see image)

Strike price (K)

The cost per share at which an option or a warrant holder may buy or sell the underlying security. Syn. strike price.

Maturity Date (T)

The date on which an investment becomes due for payment.

Bid-offer spread

The difference between the prices at which dealers will buy from a customer (bid) and sell to a customer (offer or ask). It is often used as an indicator of liquidity.

1.1 What is the difference between a long forward position and a short forward position?

The difference is one of buying versus selling. The party that takes the long forward position agrees to buy the underlying asset at a specified future date for a specified price. The other party that assumes the short position agrees to sell the underlying asset at the same specified date for the same price

Suppose that a lender quotes the interest rate on loans as 8% per annum with continuous compounding, and that interest is actually paid quarterly. What is the equivalent rate?

The formula for converting a continuously compounded rate to a periodically compounded rate is Rc = m( e^(Rc/m) − 1) where Rc = continuously compounded interest rate, which is 8% in this question. Rm = periodically compounded interest rate, compounded m times per year. m = compounding times per year, which in this case is 4 for quarterly compounding. e = Euler's number, a constant with a value of roughly 2.71828. ----- SOLUTION ----- Plugging in the values produces Rc = 4(e^.08/4 -1) = .080805 ----- EXCEL STEPS ----- In Excel, the function exp() raises e to a power. In this case, eRc/m is handled as exp(.08/4), which resolves to .02. Note that all the terms in the power component (in this case, Rc/m) must be within the parentheses following the letters "exp". The full equation in Excel is =4*(EXP(0.08/4)-1)

QUESTION 6 It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the December futures contract on a stock index to change beta of the portfolio to 0.5 during the period July 16 to November 16. The index is currently 2,000, and each contract is on $250 times the index. What position should the company take? long 140 contracts short 140 contracts long 240 contracts short 240 contracts

The formula for the number of contracts that should be shorted gives 1.2 *(100,000,000/(2000*250) = 240 Rounding to the nearest whole number, should be shorted. To reduce the beta to (0.5/1.2)*240=100 240-100=140 long contracts required.

QUESTION 5 A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts on a stock index to hedge its risk. The index futures is currently standing at 1080, and each contract is for delivery of $250 times the index. What should the company do if it wants to reduce the beta of the portfolio to 0.6? Rounding to the nearest whole number, 44 contracts should be bought. Rounding to the nearest whole number, 44 contracts should be shorted. Rounding to the nearest whole number, 56 contracts should be shorted. Rounding to the nearest whole number, 89 contracts should be shorted.

The formula for the number of contracts that should be shorted gives 1.2 *(20,000,000/(1080*250) = 88.9 Rounding to the nearest whole number, 89 contracts should be shorted. To reduce the beta to 0.6, half of this position, or a short position in 44 contracts, is required.

QUESTION 9 A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts on a stock index to hedge its risk. The index futures is currently standing at 1080, and each contract is for delivery of $250 times the index. What is the hedge that minimizes risk? Rounding to the nearest whole number, 89 contracts should be bought. Rounding to the nearest whole number, 89 contracts should be shorted. Rounding to the nearest whole number, 62 contracts should be shorted. Rounding to the nearest whole number, 11 contracts should be shorted.

The formula for the number of contracts that should be shorted gives: 1.2 *(20,000,000/(1080*250) = 88.9=89 contracts should be shorted.

Calculate compound interest

The formula to calculate intra-year compound interest with the EFFECT worksheet function is as follows: =P+(P*EFFECT(EFFECT(k,m)*n,n)) The general equation to calculate compound interest is as follows =P*(1+(k/m))^(m*n) where the following is true: P = initial principal k = annual interest rate paid m = number of times per period (typically months) the interest is compounded n = number of periods (typically years) or term of the loan Intra-Year compounding rate Number of compounding periods per year Semiannual 2 Quarterly 4 Monthly 12 Weekly 52 Daily 360 or 365(actual) Equivalent to daily compounding At=Pe^rt A=principal amount e=exponential function t=t years r=rate of interest

4.6 Forward rates

The forward rate is the future zero rate implied by today's term structure of interest rates

QUESTION 1 The forward price on the Swiss franc for delivery in 45 days is quoted as 1.1000. The futures price for a contract that will be delivered in 45 days is 0.9000. Explain these two quotes. Which is more favorable for a trader wanting to sell Swiss francs? The forward market is more attractive for a trader wanting to sell Swiss francs. The futures market is more attractive for a trader wanting to sell Swiss francs. The forward and futures market are equally attractive for a trader wanting to sell Swiss francs. The Swiss frank is expected to decline relative to the US dollar.

The futures market is more attractive for a trader wanting to sell Swiss francs.

QUESTION 4 A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6 correlation with gasoline futures price changes. The company will lose $1 million for each 1 cent increase in the price per gallon of the new fuel over the next three months. The new fuel's price change has a standard deviation that is 50% greater than price changes in gasoline futures prices. (in other words, the sd of the price changes, divided by the sd of the futures price changes = 1.5). Each contract is on 42,000 gallons. If gasoline futures are used to hedge the exposure what should the hedge ratio be? What is the company's exposure measured in gallons of the new fuel? What position measured in gallons should the company take in gasoline futures? How many gasoline futures contracts should be traded? The company has an exposure to the price of 90 million gallons of the new fuel. The number of contracts required, rounding to the nearest whole number is 2143. The company has an exposure to the price of 100 million gallons of the new fuel. The number of contracts required, rounding to the nearest whole number is 2143. The company has an exposure to the price of 100 million gallons of the new fuel. The number of contracts required, rounding to the nearest whole number is 952. The company has an exposure to the price of 40,000 million gallons of the new fuel. The number of contracts required, rounding to the nearest whole number is 952.

The hedge ratio depends on the relationship between changes in spot price and changes in the futures price. Define: Δ S: Change in spot price, S, during a period of time equal to the life of the hedge Δ F: Change in futures price, F, during a period of time equal to the life of the hedge. h* = the minimum variance hedge ratio = Þ (σs / σF) the optimal hedge ratio is the product of the coefficient of correlation between Δ S and Δ F and the ratio of the standard deviation of Δ S to the standard deviation of Δ F. Þ = coefficient of correlation between the two =.6 σs = standard deviation of Δ S σF = std dev of Δ F 1.5 where the slope of the best-fit line from a linear regression of ΔS against F (see Figure 3.2) a) the optimal hedge ratio is: .6 * 1.5 = 90% 0.6 x 1.5 = 0.9 = 90% of the exposure should be in futures contracts b) We can divide the $1,000,000 by each cent per gallon that causes the $1,000,000 loss and get a 1,000,000/.01= 100,000,000 gallon exposure c) 90% x 100,000,000 gallon exposure = 90,000,000 gallons should be the position in futures d) therefore 90,000,000 / 42,000 = 2142.86 = 2143 contracts (rounded up to the nearest integer).

1.5. An investor enters into a short forward contract to sell 100,000 British pounds for US dollars at an exchange rate of 1.4000 US dollars per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is (a) 1.3900 and (b) 1.4200?

The investor at the end of the contract will have to sell 100,000 GBP for 140,000 USD. a. When the spot price is 1.3900 the price of 100,000 GBP is 139,000 USD. Hence, the investor gains 1000 USD for his investment. b. When the spot price is 1.4200, the price of 100,000 GBP is 142,000 USD. Hence, the investor loses 2000 USD.

economic risk

The likelihood that events, including economic mismanagement, will cause drastic changes in a country's business environment that adversely affect the profit and other goals of a particular business enterprise.

political risk

The likelihood that political forces will cause drastic changes in a country's business environment that will adversely affect the profit and other goals of a particular business enterprise.

cost of carry

The net cost of holding an asset, considering both the costs and benefits of holding the asset.

Volume of trading

The number of trades in one day -Trading volume can be greater than both the beginning-of-day and end-of-day open interest. -ex: This was the case for June 2010 gold on May 26, 2010. -This indicates that many traders who entered into positions during the day closed them out before the end of the day

4.4.b Par Yield

The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. Ex: c/2*exp(-.05x0.5)+ c/2*exp(-.058x1)+ c/2*exp(-.064x1.5)+ (100+c/2)*exp(-.068x2) =100 The 2 year par yield is c=6.87 per annum with s.a. compounding In general if m is the number of coupon payments per year, P is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date c=(100-100P)m/A

1.4. Explain carefully the difference between selling a call option and buying a put option.

The party that sells the call option is obliged to buy the stock at the strike price before the expiry date if the buyer of the call option decides to sell. Note that regardless of the buyers exercise of the option, the seller of the call option keeps the premiums. The party that buys the put option has the right, but not the obligation to sell an asset at the strike price before the expiry date. For this right, he pays a premium. An unrelated point is that for a call option, the higher the strike price, the lower the premium. For a put option, the lower the strike price, the lower the premium.

QUESTION 14 What position is equivalent to a long forward contract to buy an asset at K on a certain date and a put option to sell it for K on that date? The equivalent position is a zero coupon bond The equivalent position is a call option The equivalent position is a put option None of the other choices are correct

The payoff of a long forward contract to buy an asset at K on a certain date is S T - K (could be negative), where ST is the asset's price at expiry. The payoff of a put option to sell the asset for K on that same date is max (K - S T , 0). If S T > K, the total payoff of the long forward contract and the put option is S T - K. If S T _ K, the total payoff is 0. Therefore, we can write the total payoff of the long forward and the put option as max (S T - K, 0). Note that this is exactly the payoff of a European call option that gives...

business risk

The possibility of loss (failure) or gain (success) inherent in conducting business

Example 5.2 Consider a 10-month (T) forward contract on a stock when the stock price (S0) is $50. We assume that the risk-free rate of interest (continuously compounded) (r) is 8% per annum for all maturities. We also assume that dividends (I) of $0.75 per share are expected after 3 months, 6 months, and 9 months.

The present value of the dividends, I, is .75exp((-.08*3/12)+.75exp((-.08*6/12)+.75exp((-.08*9/12) = 2.162 The variable T is 10 months, The forward price: F0 =(S0-I)exp(rT) F0=($50-2.162)exp(.08*10/12) F0=$51.14

Settlement price

The price just before the final bell each day -used for the daily settlement process

Sell Put Option

The right to ... Something at a specific future price is called a ie. an investor sells (is bearish) a September put option contract with a strike price of $480. If price stays above $480, investor makes a profit. If stock price falls below $480 the seller takes a loss and the buyer makes a profit.

systemic risk

The risk that a failure by a large financial institution will lead to failures by other large financial institutions and a collapse of the financial system.

emerging market debt fund

The sovereign debt of nondeveloped countries.

5.12 The Cost of Carry (c)

The storage cost + the interest costs - the income earned y=convenience yield c=cost carry For an investment asset F0 = S0e(cT) For a consumption asset F0 = S0e((c-y)T)

Properties of Options

The tables illustrate a number of properties of options. The price of a call option decreases as the strike price increases, while the price of a put option increases as the strike price increases. Both types of option tend to become more valuable as their time to maturity increases.

global income

The total value of world income generated by the production of goods and services.

Suppose K = 100, find the unit profit for a long forward if ST is 110 and if ST = 80

The unit profit is: ST - 100 = { 10 IF ST = 110 -20 IF ST = 80

Value of Options

The value of all call options increases (decreases) as the current stock price (S) increases (decreases). For put options, the value of the put decreases (increases) as S increases (decreases).

•On May 24, 2016, the treasurer of a corporation enters into a long forward contract to buy £1 million in six months (Nov 24, 2016) and wants to hedge against exchange rate of 1.4422 •This obligates the corporation to pay $1,442,200 for £1 million on November 24, 2016 •What are the possible outcomes? (Example (pages 6-7))

There are 3 possible outcomes. 1. spot exchange rate on November 6, 2013, is lower than the forwarding exchange rate of 1.4422. In 1st outcome, let's say the exchange rate on November 6, 2013, is 1.4412. In this case, the corporation will incur a loss because they have to pay $1,442,200 for £1 million whereas in the spot market they can bought £1 million by paying $1,441,200 (£1 million*1.4412). so they will incur loss of $1,000 ($1,442,200 - $1,441,200). 2. spot exchange rate on November 6, 2013, is higher than the forward exchange rate of 1.4422. Let's say exchange rate on November 6, 2013 is 1.4432. in this case the corporation will have a profit because they only have to pay $1,442,200 for £1 million whereas in the spot market they can buy £1 million by paying $1,443,200 (£1 million*1.4432). so they will have profit of $1,000 ($1,443,200 - $1,442,200). 3. spot exchange rate on November 6, 2013, is equal to the forward exchange rate of 1.4422. in 3rd outcome, corporation will neither have profit nor loss because forward rate and spot exchange rate on November 6, 2013 will be the same.

10.1 Factors affecting option prices

There are six factors affecting the price of a stock option: 1. The current stock price, S0 -Calls (C,c) : increase in value as stock price increases (+) -Puts (P,p) : decreases in value as stock price increases (-) 2. The strike price, K -Calls (C,c) : decrease in value as strike price increases (-) -Puts (P,p) : Increases in value as strike price increases (+) 3. The time to expiration, T -Calls (c) : dividends will cause price to decrease (?) -Calls (C) : increase in value as T increases (+) -Puts (p): dividends will cause price to decrease (?) -Puts (P): Increases in value as T increases (+) 4. The volatility of the stock price, Sigma, σ -Calls (c,C) : increases as volatility increases (+) benefits from price increases but has limited downside risk in the event of price decreases because the most the owner can lose is the price of the option. -Puts (p,P): Increases as volatility increases (+) benefits from price decreases, but has limited downside risk in the event of price increases. 5. The risk-free interest rate, r -Calls (C,c) : when Interests rates rise, the option price tends to rise (+) -Puts (p,P) : When interest rates increase, the option price tends to fall (-). 6. The dividends that are expected to be paid. -Calls (c,C): Increase in dividends tend to decrease the price of a stock option (-) -Puts (p,P): Increases in dividends causes the option to increase (+)

Six Factors Affecting Option Prices

There are six factors that impact the value of an option: S = current stock price K = strike price of the option T = time to expiration of the option r = short-term risk-free interest rate overover TT D = present value of the dividend of the underlying stock σ = expected volatility of stock prices over TT.

Optimal Number of Contracts

To calculate the number of contracts that should be used in hedging, define: QA: Size of position being hedged (units) QF : Size of one futures contract (units) N* : Optimal number of futures contracts for hedging. The futures contracts should be on h*QA units of the asset. The number of futures contracts required is therefore given by N*= (h*QA)/QF

3.5 Hedging Using Index Futures

To hedge the risk in a portfolio the number of contracts that should be shorted is where VA is the value of the portfolio, b is its beta, and VF is the value of one futures contract

Consumption Commodities consumption assets provide no income and can be subject to significant storage costs. F0>(S0+U)e(rT) if storage costs are expressed as a proportion of the spot price, the equivalent result is: F0≤S0e((r+u)T) Investment assets F0<(S0+U)e(rT)

To take advantage of a consumption opportunity, an arbitrageur can implement the following strategy: 1. Borrow an amount S0 + U at the risk-free rate and use it to purchase one unit of the commodity and to pay storage costs. 2. Short a futures contract on one unit of the commodity. If forward contract, no daily settlement, Strategy leads to a profit of: F0 - (S0+U)e(rT) --------------------------------------------------- To take advantage of a investment opportunity, an arbitrageur can implement the following strategy: 1. Sell the commodity , save the storage costs, and invest the proceeeds at the risk-free interest rate. 2. Take a long position in a futures contract. Result is riskless profit at maturity of: (S0+U)e(rT)-F0

QUESTION 11 Trader A enters into futures contracts to buy 1 million euros for 1.1 million dollars in three months. Trader B enters in a forward contract to do the same thing. The exchange rate (dollars per euro) declines sharply during the first two months and then increases for the third month to close at 1.1300. Which of the following in NOT true. It is likely that Trader B has done better because Trader A had to finance its losses during the first two months. The total profit of each trader in dollars is $3,000. Trader B's profit is realized at the end of the three months. Trader A's profit is realized day-by-day during the three months. Substantial losses are made during the first two months and profits are made during the final month.

Trader A futures - margin Qty Euros= 1,000,000 Future Price in 3 months = 1.1 Trader B forward Qty Euros= 1,000,000 Forward Price in 3 months = 1.1 Market price in 3 months= 1.1300 Profit 1.13-1.1=.03*1,00,000 =30,000 If the impact of daily settlement is taken into account Trader B (forward contract) would have done better as his profit is realized at the end of the 3 months and he made no investment in between. Whereas in the case of Trader A (futures) the profit loss is adjust daily and because of the substantial losses in the initial months, he would have of had to invest more in the first 2 months.

1.7 Hedgers

Use derivatives to reduce exposure to adverse movements and risk that they face from potential future movements in a market variable. No guarantee for favorable outcome.

2.8 Types of Traders

Two main types of traders executing trades: 1. Futures commission merchants (FCMs) - following the instructions of their clients and charge a commission for doing so 2. Locals - trading on their own account. categories 1. Hedgers 2. Speculators a. scalpers- watch for short term trends attempt to profit from small changes b. day traders - hold positions for less than one day c.position traders - hold positions for much longer period of time 3. Arbitraugers

Price of a long put (p)

Unit price paid for the right to sell 1 unit of the asset at strike price K, at or before the maturity date, T.

Price of a long call (C)

Unite price paid for the right to buy 1 unit of the asset at the strike price K, at or before maturity date, T.

Futures Price (F0)

What you expect the price of the asset to be at future time, T. It is determined by the laws of supply and demand. If more traders want to go long than to go short, the price goes up; if the reverse is true, then the price goes down. Can be traded at any time the exchange is open does not cost anything up front

5.9 Index Arbitrage

When F0 > S0exp((r-q)T) an arbitrageur buys the stocks underlying the index and sells futures When F0 < S0exp((r-q)T) an arbitrageur buys futures and shorts or sells the stocks underlying the index Index arbitrage involves simultaneous trades in futures and many different stocks Very often a computer is used to generate the trades Occasionally simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold

contango

When a future price is above the spot price; Caused by companies wanting to lock in future rates to match future liabilities

Contango

When a future price is above the spot price; Caused by companies wanting to lock in future rates to match future liabilities E(St)=S0exp(KT)

Margin accounts have the effect of Reducing the risk of one party regretting the deal and backing out Ensuring funds are available to pay traders when they make a profit Reducing systemic risk due to collapse of futures markets All the answers

When an investor can borrow money from broker in order to purchase securities, it is known as margin money.Investor is to make money with investor so it can be profit or loss. It reduces systematic risk and of other party due to fall in future.It ensures that funds will be available for traders when there is profit. The answer is D All of above

5.8 Forward vs Futures Prices

When the maturity and asset price are the same, forward and futures prices are usually assumed to be equal. When interest rates are uncertain they are, in theory, slightly different: A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse Reasonable to assume that forward and futures prices are the same. F0 = both futures and forward price of an asset today.

1.3. What is the difference between entering into a long forward contract when the forward price is $50 and taking a long position in a call option with a strike price of $50?

When you enter into a long forward contract with a forward price of $50, you are obliged to buy at the contract maturity date that is pre-specified. When you enter into a long position in a call option with a strike price of $50, you have the option but not the obligation of buying the underlying assets by a certain maturity date. Lastly, though there is no cost to entering a forward or futures contract, while there is a cost to aquiring an option.

Beta

When β=1 the return tends to mirror the return on the index β = 2 = necessary to use twice as many contracts to hedge the portfolio with

1.8 Speculators (speculation)

Wish to take a position in the market. Use derivatives to bet on the future direction of a market variable

Suppose you consider yourself to be good at picking stocks that will outperform the market. You own a single stock or a small portfolio of stocks. You do not know how well the market will perform over the next few months, but you are confident that your portfolio will do better than the market. What should you do?

You should short BVA/VF index futures contracts, where B is the beta of your portfolio, VA is the total value of the portfolio, and VF is the current value of one index futures contract. If your portfolio performs better than a well-diversified portfolio with the same beta, you will then make money. Consider an investor who in April holds 20,000 IBM shares, each worth $100. The investor feels that the market will be very volatile over the next three months but that IBM has a good chance of outperforming the market. The investor decides to use the August futures contract on the S&P 500 to hedge the market's return during the three-month period. The B of IBM is estimated at 1.1. Suppose that the current futures price for the August contract on the S&P 500 is 900. Each contract is for delivery of $250 times the index. In this case, VA = 20,000 x 100 = 2,000,000 and VF = 900 x 250 = 225,000. The number of contracts that should be shorted is therefore 1.1 x 2,000,000 / 225,000 = 9.78 Rounding to the nearest integer, the investor shorts 10 contracts, closing out the position in July. Suppose IBM falls to $90 and the futures price of the S&P 500 falls to 750. The investor loses 20,000 x ($100 - $90) = $200,000 on IBM, while gaining 10 x 250 x (900 -750) = $375,000 on the futures contracts. The overall gain to the investor in this case is $175,000 because IBM did not go down by as much as a well-diversified portfolio with a B of 1.1. If the market had gone up and IBM went up by more than a portfolio with a B of 1.1 (as expected by the investor), then a profit would be made in this case as well.

a call option

a 3-month call option on the stock has a strike price of 21 stock price = 20 stock price = 22 stock price

4.1 Types of Rates

a Treasury Rates b LIBOR Rates c Repo Rates

A zero- coupon bond

a bond that makes no coupon payments and is thus initially priced at a deep discount

A REIT

a company that owns income-producing real estate

Variable Rate Demand Obligation (VRDO)

a debt security whose interest rate is regularly re-set and which can be "put" or sold back to the issuer or a designated third party for the par value plus accrued interest

CFTC (Commodity Futures Trading Commission)

a government agency that regulates commodity, futures, and options markets in the United States.

Treasury bond

a government bond that can be issued for as long as 30 years

capital appreciation bond (CAB)

a long-term, high-interest-paying bond that pays off both principal and interest in one lump sum when the bond reaches maturity

stock beta

a measure of volatility, or systematic risk of a security r a portfolio in comparison to the market as a whole (Covariance of stock and market) / (Variance of market)

capital asset pricing model

a model that relates the required rate of return on a security to its systematic risk as measured by beta the capital asset pricing model gives: Expected return on portfolio - Risk-free interest rate = 1.5 x (Return on index - Risk-free interest rate)

Money market fund

a savings-investment plan offered by investment companies, with earnings based on investments in various short-term financial instruments

Stock index

a statistic that tracks how the prices of a specific set of stocks have changed weight of a stock in the portfolio = proportion of portfolio invested in the stock at that time.

Bank CD

a way to invest money in a bank

futures contract

an agreement to buy or sell at a specific date in the future at a predetermined price traded on an exchange

1.6. A trader enters into a short cotton futures contract when the futures price is 50 cents per pound. The contract is for the delivery of 50,000 pounds. How much does the trader gain or lose if the cotton price at the end of the contract is (a) 48.20 cents per pound and (b) 51.30 cents per pound?

a) The trader sells for 50 cents per pound something that is worth 48.20 cents per pound. Gain = ($0.5000 − $0.4820) × 50,000 = $900.b) The trader sells for 50 cents per pound something that is worth 51.30 cents per pound. Loss = ($0.5130 − $0.5000) × 50,000 = $650.

QUESTION 8 It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. Suppose that the company decides to increase the beta of the portfolio from 1.2 to 1.5 during the period July 16 to November 16. The index is currently 2,000, and each contract is on $250 times the index. What position should the company take? long 140 contracts short 140 contracts long 60 contracts short 60 contracts

a. calculation of number of position =[(stock value*(beta1-Beta2))/(Contractprice*indexvalue)] =((100000000*(1.2-1.5))/(250*2000)) 60 contracts short increase - strengthening - short

Arbitrage strategies

arbitrage arises if one of the following conditions occurs: 1. lower bound is violated ex: option premium < LB or C<LB or p<LB 2. Put-call parity (PCP) is violated: either: i. call is overpriced (put underpriced) ii. call is underpriced (put overpriced) Basic arbitrage strategy: Buy low sell, high case 1: premium < LB a c < LB => call is underpriced strategy: Buy call, sell stock b. p<LB _> put is underpriced strategy: Buy put, buy stock

Commodity Funds

are a significant factor in the movement of commodity prices

Example 5.3 Consider a 6-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once during a 6-month period. The risk-free rate of interest (with continuous compounding) is 10% per annum. The asset price is $25.

asset price (S0) = 25, r =0.10, T = 0.5 The yield is 4% per annum with semiannual compounding Equivalent rate with continuous compounding is = ln(1+r) or ln(1.04)=3.92% Rc=mln(1+((Rm)/m)) where: Rc=rate of interest with continuous compounding Rm= equivalent rate with compounding=.04 per annum m=times per annum=2 semi-annual Rc=2ln(1.02)=0.03960 The Forward Price is then given by: F0=25exp((.10-.0396)x.5)=$25.77

American Options (A)

can be exercised at any time, t, at or before expiration date, T, at price K

Hybrid Funds

consist of both stock and bond securities

Tailing the hedge

corrects for possibility of over hedging take # of contracts*(new S_P/new future price)

Over the counter derivatives

customized private contracts between counterparties; not standardized nor backed by a clearinghouse, greater risk of default

Cyclical stock

has a market value that tends to reflect the state of the economy

convex

having an outline or surface curved like the exterior of a circle or sphere.

Example: Ley Y=-2X+5, with Nx=3 and std devx=2 find variance and std dev y

i. Ny=E[y]=-2E[X]+5=-2Nx+5=-2(3)+5=-1 ii. std dev^2y=V[y]=-2)^2v[x]=4stddev^2x=4(2^2)=16 iii.stddevy=SD[y]=|-2|stddevx=2(2)=4 or=sqrt(V[Y])=sqrt(16)=4

intrinsic value

is the amount by which the strike price of an option is profitable or in-the-money as compared to the stock's price in the market. If the strike price of the option is not profitable as compared to the price of the stock, the option is said to be out-of-the-money.

A speculative investor has a strong bearish outlook on ABC stock. Which of the following positions is most suitable for this investor? A) Short ABC put B) Long ABC put C) Long ABC call D) Short ABC call

long put

Converting equivalent rate/annual effective rate (Rm) to continuously compounded rate (Rc)

m = times compounded per period Continuously compounded rate = Rc = m *ln ( 1 + Rm/m) ex: 10% semi-annual=2*LN(1+.1/2) ln(1+annual effective rate) Equivalent Rate/Annual effective rate = exp(continuously compounded rate)-1 Rm = m( e^(Rc/m) − 1)

Consider an interest rate that is quoted as 10% per annum with semiannual compounding. Convert to continuous compounding.

m=2 Rm=.1 Rc = m *ln ( 1 + Rm/m) Rc = 2ln(1+(.1/2) =2*LN(1.05) =.09758

2.8 Types of Orders

market order - request that a trade be carried out immediately at the best price available in the market. Limit -specifies particular price Stop-loss - becomes an order to sell if price falls below stop loss order Stop-limit - it is an order triggered on a stop that converts to a limit - by simultaneously limit the amount of slippage you will incur if the market moves quickly if liquidity is low Market-if touched -a stop order in reverse:it is designed to get an investor into a position rather than out of one Discretionary -is traded as a market order except that execution may be delayed at the broker's discretion in an attempt to get a better price. Time of day- specifies a particular period of time during the day when the order can be executed. Open -order or goodtill- canceled order is in effect until executed or until the end of trading in the particular contract. Fill or kill - as its name implies, must be executed immediately on receipt or not at all.

The put option buyer has unlimited profit potential, and the put option seller has ___________ _____________ potential.

maximum loss

The maximum loss of the call option buyer is the _____ _______ of the ______ ________

maximum profit of the call option seller

The maximum loss of the put option buyer is the

maximum profit of the put option seller.

Income stock

pays higher-than-average dividends compared to other stock issues

Calculating the Minimum Variance Hedge Ratio Optimal Hedge Ratio (page 59)

ratio depends on relationship between changes in spot price and changes in the futures price. Define: Δ S: Change in spot price, S, during a period of time equal to the life of the hedge Δ F: Change in futures price, F, during a period of time equal to the life of the hedge. h* = the minimum variance hedge ratio = Þ (σs / σF) the optimal hedge ratio is the product of the coefficient of correlation between Δ S and Δ F and the ratio of the standard deviation of Δ S to the standard deviation of Δ F. Þ = coefficient of correlation between the two σs = standard deviation of Δ S σF = std dev of Δ F where the slope of the best-fit line from a linear regression of ΔS against F (see Figure 3.2)

market risk

risk that affects all companies in the stock market

Private activity municipal bonds

securities issued by or on behalf of local governments that bear tax-exempt interest for regular income tax and are used to provide debt financing for private projects. Private Activity Municipal Bond income is an AMT preference item

closed-end fund

shares may not be redeemed, but instead are traded at prices that can differ from net asset value

commercial paper

short-term unsecured debt issued by large corporations

1.13. Suppose that a March call option to buy a share for $50 costs $2.50 and is held until March. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a long position in the option depends on the stock price at maturity of the option.

since the holder paid 2.50 to have the right to buy, then he will only make a gain if the price is above 52.50 in march.

Example 2: A 3 month 50 strike euro put option on a NDPS has a premium, P. The current stock price is 47 and the CCRFR is 10% p.a. a. Fina lower bound for P b. if p=1.3655, describe the arbitrage strategy that gives a risk free profit. what is the profit?

solution: S0=47, K=50, T=3/12=.25, r=.10, D=0 a. For a euro put on a NDPS LB=max(0,K*exp(-(r)(T)-S0) =max(0,50*exp(-(.10)(.25))-47)=1.7655 p>=1.7655, b

In state AAA municipal bonds

tax-exempt bonds issued by state and local governments

QUESTION 10 Sixty futures contracts are used to hedge an exposure to the price of silver. Each futures contract is on 5,000 ounces of silver. At the time the hedge is closed out, the basis is $0.20 per ounce. What is the effect of the basis on the hedger's financial position if (a) the traderis hedging the purchase of silver. gain of $60,000 gain of$12,000 loss of $12,000 loss of $60,000

the excess of the spot over the futures at the time the hedge is closed out is $0.20 per ounce. a). if the trader is hedging the purchase of silver, the price paid is the futures price plus the basis. the trader therefore loses 60x5,000x.20=$60,000. b). if the trader is hedging the sales of silver, the price received is the futures price plus the basis. the trader therefore gains $60,000.

the law of one price

the notion that a good should sell for the same price in all markets

Forward Price

the price that both parties agree to buy/sell the underlying asset for at any point T, in the future. -no payment required K = delivery price in the contract (fixed) =same as Forward price measured from contract date F0=E[ST] Expected price at maturity Forward/futures price today (@t=0) ST= spot price of underlying at maturity

Credit Risk

the probability that the borrower will fail to pay some of the interest or principal

risk-free rate

the rate of return that can be earned with certainty Financial institutions have traditionally used LIBOR rates as risk-free rates

Hedge ratio

the ratio of the size of the position taken in futures contracts to the size of the exposure.

Put Option

the right to sell an asset at a specified exercise price (K) on or before a specified expiration date (T)

reinvestment rate risk

the risk that a decline in interest rates will lead to lower income when bonds mature and funds are reinvested

valuing the portfolio (risk-free rate is 4%)

the riskless portfolio

blue-chip stock

the stock of a large, well-established and financially sound company that has operated for many years

Open Interest

the total number of contracts outstanding -equal to number of long positions or number of short positions

the underlying

trades in the cash or spot markets and its price is call the cash or spot price

Arguments against Hedging

•Shareholders are usually well diversified and can make their own hedging decisions •It may increase the risk to hedge when competitors do not •Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult.

3.6 Stack and Roll

•We can roll futures contracts forward to hedge future exposures •Initially we enter into futures contracts to hedge exposures up to a time horizon •Just before maturity we close them out and replace them with a new contract reflect the new exposure

MA 343 / FI 645

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