UFL Investments: Ch 20 - 23

*Ch 23: Futures, Swaps, and Risk Management* SWAP dealer

An intermediary, usually a bank, who will sync the swap requests out there. Company A issues a 7% coupon fixed-rate bond that wishes to convert into synthetic floating-rate debt. Company B issued a floating-rate bond tied to LIBOR that it wishes to convert into synthetic fixed-rate debt. The dealer would enter a SWAP for the two companies. When combined, the dealer's position is effectively neutral on interest rates: paying LIBOR on one swap and receiving it on another. Paying a fixed rate on one swap and receiving it on another. The dealer finds this activity profitable by charging a bid-ask spread on the transaction.

*Ch 21: Option Valuation* Investor fear gauge

implied volatility correlates with crisis, its is sometimes called an "investor fear gauge"

*Ch 21: Option Valuation* General properties of option prices. Note: some of these have important implications for the effect of stock dividends on option values and the possible profitability of early exercise of an American option.

*1. Restriction on the value of a call option.* - can't be negative - can't impose any liability on its holder - there are upper and lower bound ranges *2. Early exercise and dividends* - proceeds from a sale of the option (at price C) but exceed the proceeds from an exercise (S,t - X) - calls on non-dividend paying stocks are worth more selling than exercising ("worth more alive than dead") *3. early exercise of American puts* - the right to exercise a put option before expiration must have value - The American put must be worth more than the European counterpart because there is some stock price below which early exercise is optimal.

*Ch 21: Option Valuation* Hedge ratios and the black-scholes formula

*Hedge ratio* = An essential tool in portfolio management and control. It is the number of stocks required to hedge against the price risk of holding one option. Also called *the option's delta*. And although dollar movements in options prices are less than dollar movements in the stock price, the rate of return volatility of options remains greater than stock return volatility because options sell at lower prices. The hedge ratio compares the value of a position protected through the use of a hedge with the size of the entire position itself. It is the slope of the curve on a graph, evaluated at the current stock price. That said, the stock price mov't leaves total wealth unaltered, which is exactly what a hedge position is intended to do. So, a call option has a positive hedge ratio and a put option has a negative hedge ratio. Hedge ratio for a call: N(d1) ^ must be positive and less than 1.0 Hedge ratio for a put: N(d1) - 1 ^must be negative and smaller than 1.0.

*Ch 20: Options Markets Intro* Option contract: *Premium*

*Premium* = purchase price of the option ^ compensation the purchaser of the call must pay for the right to exercise the option, only when exercise is desirable. Value at expiration = stock price - exercise price Seller, the one who writes calls, would make a profit of... profit = final value - original investment.

*Ch 20: Options Markets Intro* Name some options strategies

*Protective Put* *Covered Calls* *Straddle* *Spreads* *Collars*

*Ch 21: Option Valuation* What is the elasticity of a put option currently selling for $4 with an exercise price of $120 and hedge ratio -0.4 if the stock price is currently $122?

A $1 increase in stock price is a percentage increase of 1/122 = .82%. The put option will fall by (.4x$1) $0.40, a percentage decrease of $.40/$4 = 10%. Elasticity is -10/.82 = -12.2

*Ch 22: Futures Market* Hedging and Speculation

*Speculator* - uses futures to profit from movement in future prices. - They will take a long position for expected profits or exploit price declines by taking a short position. - They'll buy a futures contract over the asset directly because transaction costs are smaller in futures markets and the leverage that futures trading provides (requires traders to post margin considerably less than the value of the asset underlying the contract). *Hedger* - uses futures to protect against price movement. - short hedge: taking a short futures position to offset risk in the sales price of a particular asset - long hedge: used to eliminate the risk of an uncertain purchase price

*Ch 23: Futures, Swaps, and Risk Management Other interest rate contracts

- Eurodollar contract, which is used by dealers in long-term interest rate swaps as a hedging tool.

*Ch 23: Futures, Swaps, and Risk Management Hedging instruments / type of hedge tools

- foreign exchange futures - stock futures - interest rate futures - swaps - commodity futures

*Ch 20: Options Markets Intro* List some exotic options (imitation option instruments that trade in active over-the-counter markets)

1. Asian Options 2. Barrier Options 3. Lookback Options 4. Currency-Translated Options 5. Digital Options

*Ch 21: Option Valuation* Determinants of Options Values

1. Stock price 2. Exercise price 3. Volatility of stock price 4. time to expiration 5. internrest rate 5. dividend rate of the stock *If ______ increases, the Value of the CALL option ______.* Stock price, S = Increase Exercise price, X = Decrease Volatility, σ = Increases Time to expiration, T = Increase Interest rate, r𝑓 = Increase Dividend payouts = Decrease *If ______ increases, the Value of the American PUT option ______.* Stock price, S = Decrease Exercise price, X = Increase Volatility, σ = Increases Time to expiration, T = Increase Interest rate, r𝑓 = Decrease Dividend payouts = Increase

*Ch 20: Options Markets Intro* Benefits of options (pg 707)

1. The ability they provide to create investment positions with payoffs that depend in a variety of ways on the values of other securities. 2. can be used to custom-design new securities or portfolios with desired pattersn of exposure to the price of an underlying security. "financial engineering" -- such as using an index-linked CD.

*Ch 20: Options Markets Intro* Option Contract

A *Call Option* gives its holder the right to purchase an asset for a specified price, called the *exercise* or *strike* price, on or before a specified expiration date. If the stock price is greater than the exercise price on expiration date, the value of the call option equals the difference between the stock and exercise price. If the stock price is less than the exercise price on expiration date, the call will be worthless. Net profit on the call = value of the option - original price paid VS>>>> A *Put option* gives its holder the right to SELL an asset for a specified exercise or strike price on or before some expiration date. As such, profits on call options increase when assets prices rise, whereas profits on put options increase when asset prices fall. An option is described as *In the money* when its service would produce a positive cash flow. Or, a call is *out of the money* when the asset price is less than the exercise price. Options are *at the money* when the exercise price and asset price are equal.

*Ch 20: Options Markets Intro* Protective Put

A *Put option* gives its holder the right to SELL an asset for a specified exercise or strike price on or before some expiration date. Payoff to put holder = original sell price - option payoff at expiration A protective put is a risk-management strategy that investors can use to guard against the loss of unrealized gains in a stock or other asset. The put option acts like an insurance policy—it costs money, which reduces the investor's potential gains from owning the security but also reduces the risk of losing money if the security declines in value. A protective put is also known as a married put. Suppose strike price is X=$100 and stock is selling at $97 at option expiration. Then the value of your portfolio is $100. The stock is worth $97 and the value of the expiring put option is: X - S,t = $100 - $97 = $3 X=exercise price of an option S,t = stock price Value of a protective put at option expiration: Stock + put = TOTAL Provides a form of portfolio insurance against stock price declines in that it limits losses. The cost of protection is that, in the case of stock price increases, your profit is reduced by the cost of the put, which turned out to be unneeded. This also proves that derivatives can sometimes be used for risk management.

*Ch 22: Futures Market* Futures Contracts

A deferred-delivery sale of some asset with the sales price agreed on now, where the contract protects each party from future price fluctuations. *Margin* = a good-faith deposit made by each trader, to guarantee the contract performance. *Futures price* = the agreed upon price that will be paid at a specified delivery or maturity date Future contracts call for delivery of an "asset" but that rarely happens. Instead, parties to the contract close out their positions before contract maturity, taking gains or losses in cash. They're a zero-sum game, with losses and gains netting out to zero (every long position is offset by a short position), so they shouldn't have a major impact in the spot market for that commodity. The trader taking the *Long Position* commits to purchasing (buy) the commodity on the delivery date and will profit from price increases. If they take the *short position* they commit to deliver (sell) the commodity at contract maturity and will lose with price increases. This buy/sell of an agreement, NOT cash or commodity (literally, less than 1-3% of contracts result in actual delivery of a commodity). Long position profit = Ft - Fo ^^ Spot price at maturity - original futures price (settle price) Short position profit = Fo - Ft ^^ Original futures price (settle price) - spot price at maturity Fo = future price at contract maturity Ft = spot price at contract maturity

*Ch 20: Options Markets Intro* Straddle

A long *straddle* is established by buying both a call and a put on a stock, each with the same exercise price, X, and the same expiration date, T. Straddles are useful strategies for investors who believe a stock will most a lot in price but are uncertain about the direction of the move. Straddle is a bet on volatility -- where big +/- moves are good and stability results in a call/put expiring worthless. *Strips* = two puts and one call on a security with the same exercise price and expiration date. *Straps* = two calls and one put on a security with the same exercise price and expiration date.

*Ch 23: Futures, Swaps, and Risk Management* Neutral market bet

A market-neutral strategy is a type of investment strategy undertaken by an investor or an investment manager that seeks to profit from both increasing and decreasing prices in one or more markets, while attempting to completely avoid some specific form of market risk. Market-neutral strategies are often attained by taking matching long and short positions in different stocks to increase the return from making good stock selections and decreasing the return from broad market movements. The text says you use the neutral market bet to take a position in a stock to capture its alpha (abnormal risk-adjusted expected return) but that market exposure is fully hedged, resulting in a position beta of zero. So, the gains on the (ie.) T-bond itself offset the losses on the bond portfolio overall. Here, the stock's BETA is the key to the hedging strategy. That said, the hedge is imperfect because of slippage. the yield spread does not stay constant overtime.

*Ch 21: Option Valuation* Some techniques to predict changes in volatility

ARCH and stochastic volatility models posit that changes in volatility are partially predictable and analyzing recent levels/trends can improve future predictions.

*Ch 23: Futures, Swaps, and Risk Management Credit risk

An issue but not a big issue, especially when related to the magnitude of notional principle in the markets would suggest. At To, when transaction is initiated, there is zero net present value for both parties because a futures contract has zero value at inception. Both are simply contracts and if one backs out here, it wouldn't cost the other trader anything. Once interest or exchange rates change, then the floating-rate payer would suffer a loss, while the fixed rate payer would enjoy a gain. But the loss is only the difference between the values of the fixed rate and floating-rate obligations, not the total value of payments they were obligated to make.

*Ch 20: Options Markets Intro* Collars

An option strategy that brackets the value of a portfolio between two bounds. Collar option strategies are a protective strategy that is implemented on a long stock position. A collar is also known as hedge wrapper. The put protects the trader in case the price of the stock drops. Writing the call produces income (or offsets the cost of buying the put) and allows the trader to profit on the stock up to the strike price of the call, but not higher. *Max profit if net debit*: call strike - stock purchase price - net premium paid *Max profit if net credit*: call strike - stock purchase price + net premium collected *Max loss if net debit*: put strike - stock purchase price - net premium paid *Max loss if net credit*: put strike - stock purchase price + net premium collected *Breakeven net debit*: stock purchase price + net premium paid *Breakeven net credit*: stock purchase price - net premium paid The strike prices of the call and put are different. The purchased put should have a strike price below the current market price of the stock. The written call should have a strike price above the current market price of the stock. The strategy is used by traders who are mildly bullish, but who also want to protect against a downside move in the stock. They are also willing to give up gains on the stock above the strike price. Therefore, this strategy is not typically used by traders who are very bullish and expect a large price increase.

*Ch 21: Option Valuation* Dynamic hedging

Constant upgrading of hedge positions as market conditions change. By continuously revising the hedge position, the portfolio would remain hedged and earn a risk-less rate of return over each interval, and the resulting option-valuation procedure becomes more precise. Delta hedging (hedge ratio): H = Δ in option value / Δ in stock value

*Ch 20: Options Markets Intro* *Convertible Securities*

Convertible bonds and preferred stock convey options to the holder of the security rather than to the issuing firm, and gives its holder the right to exchange each bond or share of preferred stock for a fixed number of shares of common stock (regardless of market price). EX: A bond with a conversion ratio of 10 allows its holder to convert the bond of par value $1k into 10 shares of common stock (shares are always in sets of $100). Conversion value = the value it would have if you converted it into stock immediately. (pg 702) The convertible bond issued at par value would have a lower coupon rate than a nonconvertible bond issued at par value because the call option is worth less as call protection is expanded. when stock prices are high = bond's price is determined by conversion value and bond becomes "equity" in disguise. When stock prices are low = straight bond's value is lower-bound, and the conversion option is nearly irrelevant.

*Ch 22: Futures Market* The clearinghouse and open interest

Electronic trading platforms: - Eurex (jointly owned by Deutshe Borse and Swiss exchange) - Euronext.liffe (used by Chicago Board of Trade) - Globex (CBOT and CME Group) Once a trade's agreed to, the *clearinghouse* enters the picture. Rather than the buying/selling being between individuals, the clearinghouse acts as the trading partner of each trader. The clearinghouse would be the only one negatively impacted/hurt should either long/short trader fail to fulfill their obligation. Clearinghouse requires all positions to recognize profits as they accrue daily. *open interest* is the number of contracts outstanding. When contracts begin trading, open interest is zero, but it increases as more contracts are entered. Open interest is typically highest in the nearest contract.

*Ch 20: Options Markets Intro* Warrants

Essentially a call options issued by a firm, except that exercised warrants require the firm to issue a new share of stock (so total number of share outstanding increases, vs. remain fixed in the case of an exercise call option). This strategy prefers volatility. Similarities to convertible debt: warrant terms may be tailored to meet the needs of the firm and they're protected against stock splits and dividends.

*Ch 22: Futures Market* Existing Contracts

Futures and forward contracts are traded on a wide variety of goods in four categories: 1. Agricultural commodities 2. Metals and minerals 3. Foreign currencies 4. Financial fixtures You can trade *single-stock futures* on broad stock indexes, individual stocks and narrowly based indexes, such as OneChicago (operates entirely on the electronic market). Note: banks and brokers established their own forward market in foreign exchange. In this forward arrangement, banks simply negotiate contracts for client, as needed. The London market is over $2 trillion currency trades a day!

*Ch 23: Futures, Swaps, and Risk Management* Synthetic positions

Futures represent "synthetic" holdings of the stock market portfolio because stock-index futures can substitute as holding for underlying stocks themselves, thus allowing investors to participate in broad market activity without having to purchase or sell large amounts of individual stock. Here, investors take a long futures position in the index and transaction costs are lower. EX: bills-plus-futures: Investors buy and hold tbills in many long futures positions, not stocks, and simply adjust their futures positions.

*Ch 22: Futures Market* Taxation

Generally, 60% of futures gains or losses are treated as long-term, and 40% are treated as short-term. Taxes are paid at year-end on cumulated profits or losses regardless of whether the position has been closed out.

*Ch 21: Option Valuation* Black-Scholes Option Valuation

If no dividend: Co = SoN(d1) - [Xe^(-rT) x N(d2)] where d1 = [1n(S/X) + (r+.5σ^2)T] / σ√T where d2 = d1 - σ√T Co = current call option value So = current stock price N(d) = the probability that a random draw from a standard normal distribution will be less than d. this equals the area under the normal curve up to d, on a distribution curve. (pg 738) X = exercise price e = the base of the natural log function, 2.71828 r = risk free interest rate T = time to expiration of option, in years 1n = natural logarithm function. In Excel, 1n(x) = LN(x) σ = SD of annualized continuously compounded RR of the stock. The Black-Scholes formula (also called Black-Scholes-Merton) was the first widely used model for option pricing. It's used to *calculate* the theoretical value of European-style options *using* current stock prices, expected dividends, the option's strike price, expected interest rates, time to expiration and expected volatility. The *Black-Scholes model makes certain *assumptions: -- The option is European and can only be exercised at expiration. -- No dividends are paid out during the life of the option. -- Markets are efficient (i.e., market movements cannot be predicted). -- There are no transaction costs in buying the option. -- The risk-free rate and volatility of the underlying are known and constant. -- The returns on the underlying are normally distributed. Key Takeaways --> The Black-Scholes Merton (BSM) model is a differential equation used to solve for options prices. --> The standard BSM model is only used to price European options and does not take into account that U.S. options could be exercised before the expiration date. *Limitations*: Black Scholes model is only used to price European options and does not take into account that U.S. options could be exercised before the expiration date. Moreover, the model assumes dividends, risk-free rates, and volatility are constant, but this may not be true in reality. Moreover, the model assumes that there are no transaction costs or taxes; that the risk-free interest rate is constant for all maturities; that short selling of securities with use of proceeds is permitted; and that there are no risk-less arbitrage opportunities. These assumptions can lead to prices that deviate from the real world where these factors are present.

*Ch 22: Futures Market* Spreads

If the risk free rate is greater than the dividend yield (r𝑓 > d) then the futures price will be higher on longer-maturity contracts. If the risk free rate is lower than the dividend yield (r𝑓 < d) then longer-maturity futures prices will be lower. Futures spread parity: Fo(T2) = Fo(T1)(1+r-d)^(T2-T1)

*Ch 23: Futures, Swaps, and Risk Management* Hedge ratio

In general, the hedge ratio is the number of hedging vehicles (futures contracts) one would establish to offset the risk of a particular unprotected position, in this case against a foreign exchange risk with the firm's export business. H = change in value of unprotected position for a given change in exchange rate / profit derived from one futures position for the same change in exchange rate One interpretation of the hedge ratio is as a ratio of sensitivities to the underlying source of uncertainty. For X amount of swing in the exchange rate, the sensitivity of the operating profits is XYZ. The slope coefficients that reflect this sensitivity are not perfect, but they are still useful indicators of hedge ratios. Hedge ratio = negative of the regression slope because the hedge should offset your original position.

*Ch 23: Futures, Swaps, and Risk Management* Index Arbitrage

Index Arbitrage is an investment strategy that exploits divergences between the actual futures price and it theoretically correct parity value. If futures price is too high, short the futures contracts and buy the stock in the index. If future price is too low, go long in futures and short stocks. Between the two options, you can hedge your position and earn arbitrage profits equal to the mispriced contracts. However, there are some challenges to implement the index arbitrage. First, buying or selling S&P500 share is impractical because of transaction costs (they may outweigh profits made) and timeliness (this index requires all shares from all 500 firms to be sold simultaneously, so any lag in this syncing can thwart the temporary price discrepancies.

*Ch 23: Futures, Swaps, and Risk Management* Foreign Exchange Futures

International interest risk can be hedged through currency futures or forward markets. For example, if you know you'll receive 100k pounds in 90 days, you can sell those pounds forward today in a forward market and lock in an exchange rate equal to today's forward price. The forward market is informal, is not standardized, and there is no marking-to-market, unlike with futures markets. So, each contract is negotiated separately and price/payment execution ONLY takes place at the maturity date (not on a daily basis). Beware of *counterparty risk* where one can't fulfill their contract obligation. Make sure they have a solid creditworthiness. Currency futures, then, trade in formal exchanges (i.e. Chicago Mercantile Exchange or LIFFE), contracts are standardized (by size), daily market-to-marketing exists, and standard clearing arrangements allow traders to enter or reverse positions easily.

*Ch 20: Options Markets Intro* Levered Equity and Risk Debt

Investors holding stock in INC firms are protected by limited liability, which means if the firm can't pay its debts, the firm's creditors may attach only the firm's assets, not sue the corp's equity-holders for further payment. The equity-holders have a put option to transfer their ownership claims on the firm to the creditors in return for the face value of the firm's debt. The equity-holders also retain a call option because they can "buy back" the firm for a specified price. ^^soo... analysts can value corporate bonds using option-pricing techniques.

*Ch 20: Options Markets Intro* Collateralized Loans

Many loan arrangements require that the borrower put up collateral to guarantee the loan will be paid back. A *nonrecourse loan* gives the lender no recourse beyond the right to the collateral and this arrangement gives an implicit call option to the borrower. Here, the borrower can turn over the collateral to the lender but retain the right to reclaim it by paying off the loan. Also, the ability to "sell" the collateral for a price of L (L = exercise price of the option) represents a put option, which guarantees the borrower can raise enough money to satisfy the loan simply by turning over the collateral.

*Ch 22: Futures Market* The Margin Account and Marketing to Market

Marketing to market = the daily settling of the profits/losses that are accrued on a daily basis --- it means that the maturity date of the contract doesn't govern realization of profit or loss, instead profits are accrued to trader's margin acct immediately. At each initial execution of trade, each trader establishes a margin acct. The margin is a security acct consisting of cash or near-cash securities, such as Tbills, to ensure the trader can satisfy the obligations of the futures contract. Initial margin is set between 5-15% of total value of the contract, but contracts written on assets with more volatile prices require higher margins. *maintenance/variation margin* = An established value below which a trader's margin cannot fall. Reaching the maintenance margin triggers a margin call. *convergence property* is the convergence of future (stock) prices and spot prices at the maturity of the futures contract. If this isn't met, then an arbitrage opportunity would exist and investors will rush to purchase X from the cheap source in order to sell X in the high-priced market.

*Ch 20: Options Markets Intro* Options Trading

OTC (over the counter trades) can offer the advantage that the terms of the option contract can be tailored to the needs of the traders, but this has a higher exchange-traded option. - sold in 100's and are listed on the exchanges because exchanges offer two benefits: ease of trading and a liquid secondary market. - sold on the iNternational Security Exchange - *American Option* allows its holder to exercise the right to purchase (if a call) or sell (if a put) the underlying asset on or before the expiration date. - *European options* allow for exercise of the option only on the expiration date. - *options clearing corporation (OCC)* The clearing house for options tarding where, after buyers and sellers strike a price, OCC becomes the middle-man and guarantees contract performance. OCC requires option writers to post margin to guarantee they can fulfill their contract obligations. - *Index options* a call or put based on a stock market index such as the S&P 500, NASDAQ, S&P 100, Dow Jones. Unlike OCC, index options don't require the call writer to actually "deliver the index" - instead, the cash settlement procedure is used. - *futures options* Give their holders the right to buy or sell a specified future contract, using as a futures price the exercise price of the option. - *Foreign currency option* A type of currency futures options are foreign exchange futures options, which provide payoffs that depend on the difference between the exercise price and the exchange rate at expiration. currency options offer the right to buy or sell a quantity of foreign currency for a specified amount of domestic currency. - *Interest rate options* i-options are traded on Treasury notes and bonds, t-bills, and gov't bonds. These are contracts on T-bonds, T-notes, fed funds, LIBOR, Euribor and Eurodollar futures.

*Ch 21: Option Valuation* Two-State Option Pricing

Options may be priced relative to the underlying stock price using a simple two-period, two-state pricing model. As the number of periods increases, the binomial model can approximate more realistic stock price distributions, thus overcoming a limitation of the valuation model (see pg 735). Binomial Model: u = exp(σ √Δt) d = exp(-σ √Δt) p = [exp(r Δt)-d] / u - d u = price increase "Up" d = Price decrease "Down" p = Present value of the payoff σ = volatility (estimate of SD of stock's continuously compounded annualized RR) Δt = length of each sub-period ^^this shows that both higher σ and longer holding periods Δt make future stock prices more uncertain. Note: the Black-Scholes formula may be seen as a limiting case of the binomial option model as the holding period is divided into progressively smaller sub-periods when the interest rates and stock volatility are constant.

*Ch 23: Futures, Swaps, and Risk Management* Hedging interest rate risk

Price value of a basis point (PVBP) = the sensitivity of the dollar value of the portfolio to changes in interest rates. PVBP = Change in portfolio value / predicted change in yield IE. you own $10M in bond portfolio with modified duration of 9 years. If market interest rate and bond yield's rise by $0.10 (10 basis points), the fund will suffer a capital loss of D x Δy, or 9 x .10 = .90%. So, $10M x .90 = $90,000 loss. The PVBP would be = $90,000 / .10 = $9,000 per basis point. One could offset or hedge this position by taking a position in an interest rate futures contract (T-bond contract). If the PVBP of one futures contract is $90 for a change in 1 basis point, then the hedge ratio is: H = PVBP of portfolio / PVBP of hedge vehicle H = $9000 / $90 H = 100 contracts Sooo... 100 T-bond futures contracts will offset the portfolio's exposure to interest rate fluctuations. This is another example of a market-neutral strategy, which used a T-bond contract to drive the interest rate exposure of a bond position to zero.

*Ch 20: Options Markets Intro* Options VS. Stock Investments

Purchasing call options is a bullish strategy because the calls provide profits when the stock prices increase; however writing call options is bearish. Purchasing puts, by contrast, is a bearish strategy; however writing puts is bullish. So, why might you purchase a call option rather than buy shares of stock directly? Take this example: 6 month maturity call option with exercise price of $100, currently sells for $10. Interest rate is 3%. Investing sum of $10k. The firm won't pay dividends past 6 months. Strategy A: Invest entirely in stock. Buy 100 shares, each selling for $100. Strategy B: Invest entirely in at-the-money call options. Buy 1,000 calls, each selling for $10. (this would require 10 contracts, each for 100 shares). Strategy C: Purchase 100 call options for $1000. Invest your remaining $9,000 in 6-month T-bills, to earn 3% interest. The bills will grow in value from $9k to (9k * 1.03) $9,270. So, these strategies would look like this when the option expires: A (value of stock price): @$95 -- $9500; @$100 -- $; @$105 -- $10,500, ... @$120 -- $12k A (rates of return): @$95 -- -5.0%; @$100 -- 0.0%; @$105 -- 5.0%, ... *@$120 -- 20.0%* B (value of stock price): @$95 -- $0; @$100 -- $0; @$105 -- $5k, ... @$120 -- $20k B (rates of return): @$95 -- -100.0%; @$100 -- -100%; @$105 -- -50%, ... *@$120 -- 100%* C (value of stock price): @$95 -- $9,270; @$100 -- $9270; @$105 -- $10,270k, ... @$120 -- $11,270k C (rates of return): @$95 -- -7.3%; @$100 -- -7.3%; @$105 -- -2.3%, ... *@$120 -- 12.7%* So, as you can see, *options offer leverage* (levered investment on the stock). Their values respond more than proportionality to changes in the stock value. Additionally, *options offer a potential insurance value*, where investors can use them to tailor their risk exposures in creative ways. By incorporating a calls-plus-bills strategy, once can limit the downside risk of being all-in.

*Ch 20: Options Markets Intro* Put Options (value of options at expiration)

Recall, a put option is the right to sell an asset at the exercise price. The holder won't exercise option unless the asset is worth LESS than the exercise price. Payoff to put holder = original sell price - option payoff at expiration

*Ch 20: Options Markets Intro* Call options (Value of options at expiration)

Recall, call options give you the right to purchase a security at the exercise price. The payoff here cannot be negative, because you either exercise when you'll make a profit or you don't exercise and, in which case, you lose/gain nothing. However, the loss to the option holder (seller) would equal the price originally paid for the option. Profit to option holder = option payoff at expiration - original purchase price

*Ch 20: Options Markets Intro* Derivative securities (or simply, derivatives)

Securities whose prices are determined by, or "derive from", the prices of other securities. *Types of derivatives*: -Options -Future contracts - SWAPs Their payoffs depend on the value of other securities, as such they can be powerful tools for both *hedging* (investing in an asset to reduce the overall risk of a portfolio) and *speculation* (undertaking risky investments with the objective of earning a greater profit than an investment in a risk-free alternative).

*Ch 22: Futures Market* The Spot-Futures Parity Theorem

Spot-futures parity is a parity condition whereby, if an asset can be purchased today and held until the exercise of a futures contract, the value of the future should equal the current spot price adjusted for the cost of money, dividends, "convenience yield" and any carrying costs (such as storage). Spot-future parity is an application of the law of one price. Fo(T) = So (1+r-d)^T For gold futures, we would set the dividend yield to zero. For bond contracts, we would let the coupon income on the bond play the role of dividend payments.

*Ch 23: Futures, Swaps, and Risk Management* SWAPs

Swaps are multiperiod forward contracts that trade over the counter. Foreign exchange swap = would call for an exchange of currencies on several future dates Interest rate swap = call for the exchange of a series of cash flows proportional to a given interest rate for a corresponding series of cash flows proportional to a floating interest rate. SWAPs provide a quick, cheap and anonymous way to restructure the balance sheet.

*Ch 22: Futures Market* Basis Risk and Hedging

The *basis* is the difference between the futures and the stock price (which is zero, at maturity). *Basis risk* is risk attributable to uncertain movements in the spread between a futures price and spot price (before maturity, where they don't necessarily equal zero and can vary widely). Note: a long spot-short futures position will profit when the basis narrows. Some investors will take a strategy called a *calendar spread*, where the investor takes a long position in a futures contract of one maturity and a short position in a contract on the same commodity, but with a different maturity.

*Ch 20: Options Markets Intro* Put-Call Parity Relationship

The *put-call parity theorem* is an equation representing the proper relation between put and call prices. Violation of parity allows arbitrage opportunities. Applies only to European put and call options. *P = C - So + PV(X) + PV(dividends)* P = Put value C = Call option value (call price) So = stock price PV(X) = present value of the option's exercise price PV(dividends) = present value of the dividends OR.... *C + [X / (1+r𝑓)^T] = So + P* Stock price = $110 Call price (X=1.05) = $17 Put price (X=1.05) = $5 r𝑓 interest rate = 5% per year C + [X / (1+r𝑓)^T] = So + P 17 + (1.05 / 1.05) = 115 117 (does not)= 115 This result, a violation of parity, indicates mispricing. To exploit this and earn arbitrage profits, you buy the relatively cheap portfolio (stock-plus-put on right) and sell the relatively expensive portfolio (call-plus-bond on left). 2nd example: February expiration call with exercise price $195. Time-to-expiration of 28 days costs $3.65, while the corresponding put option costs $5. IBM was selling for $194.47, and the annualized short-term interest rate on this date was 0.1%. IBM was expected to pay a dividend of $0.85 with an ex-dividend date of February 8, 18 days hence. According to parity, we should find that: *P = C + PV(X) - So + PV(dividends) 5 = 3.65 + [195/(1.001)^28/365] - 194.47 + [.85/(1.001)^18/365] 5 = 3.65 + 194.985 - 194.47 + .85 5 = 5.015 So arbitrage opportunity is only $0.15 and probably not worth going through the hassle to exploit.

*Ch 20: Options Markets Intro* *Option-Like Securities: Callable Bonds*

The bond issuer holds a call option with exercise price equal to the price at which the bond can be repurchased, which is essentially a sale of a *straight bond* (a bond with no option features such as callability or convertibility) to the investor and the insurance of a call option by the investor to the bond-issuing firm. A callable bond is similar to a covered call strategy in that the covered call strategy would consist of a straight bond with a call written on the bond. Pg 701

*Ch 23: Futures, Swaps, and Risk Management* Cross hedging

The hedge vehicle is a different asset than the one being hedged. Cross-hedges can eliminate a large fraction of the total risk of the unprotected portfolio, but they typically are far from risk-free positions.

*Ch 22: Futures Market* Big difference between options vs. futures and forwards

The holder of an option isn't compelled to buy or sell; however, holders of a future or forward contract are obligated to go through with the agreed-upon transaction. Forwards are ONLY a commitment today to transact in the future, they are NOT an investment in the sense of paying for an asset. Futures and forwards differ in that they call for a daily settling up of any gains or losses on the contract; by contract, no money changes hands in forward contracts until the delivery date. Also, marketing to market is the major difference between the two, besides contract standardization. Futures follow this pay (or receive) as you go method via marketing2market. Forward contracts are simply held until maturity, and no funds are transferred until that date, although the contracts may be traded.

*Ch 21: Option Valuation* Explain why the put-call relationship is valid only for European options on non-dividend paying stocks. If the stock pays no dividends, what inequality for American options would correspond to the parity theorem?

The parity relationship assumes that all options are held until expiration and that there are no cash flows until expiration. These assumptions are valid only in the special case of European options on non-dividend-paying stocks. If the stock pays no dividends, the American and European calls are equally valuable, whereas the American put is worth more than the EUropean put Therefore, although the parity theorem for European options states that P = C - So + PV(X), in fact P will be GREATER than this value if the put is American. Put-call Parity: P = C + PV(X) - So + PV(dividends)

*Ch 20: Options Markets Intro* Why might one characterize both buying calls and writing puts as "bullish" strategies? What's the diff between them?

The payoffs and profits to both buying calls and writing puts generally are higher when the stock price is higher. In this sense, both positions are bullish. Both involve potentially taking delivery of the stock. However, the call holder will CHOOSE to take delivery when the stock price is high, while the put writer is OBLIGATED to take delivery when the stock price is low.

*Ch 20: Options Markets Intro* Why might one characterize both buying puts and writing calls as "bearish" strategies? What's the difference between them?

The payoffs and profits to both writing calls and buying puts generally are higher when the stock price is lower. In this sense, both positions are bearish. Both involve potentially making delivery of the stock. However, the put holder will CHOOSE to make delivery when the stock price is low, while the call writer is OBLIGATED to make delivery when the stock price is high.

*Ch 21: Option Valuation* Portfolio Insurance

The protective put is a simple and convenient way to achieve portfolio insurance, which means it limits the worst-case portfolio rate-of-return. They minimize trading costs by buying or selling stock index futures as a substitute for sale of the stocks themselves.

*Ch 20: Options Markets Intro* Covered Calls

The purchase of a share of stock with a simultaneous sale of a call option on that stock. The call is "covered" because the potential obligation to deliver the stock can be satisfied using the stock held in the portfolio. In contrast, writing an option without an offsetting stock position is called a *naked option writing* *Value of call position at expiration* = stock value - value of the call This is a popular strategy among institutional investors. Pg 693 Assume a pension fund holds 1k shares of stock, with a current price of $100/share. Suppose the manager intends to sell all shares if the price hits $110, and a call expiring in 60 days with an exercise price of $110 currently sells for $5. By writing 10 call contracts (for 100 shares ea) the fund can pick up $5k in extra income. The fund would lose its share of profits from any movement of the stock price above $110 per share, but given that it would have sold its shares at $110, it would not have realized those profits anyway.

*Ch 21: Option Valuation* gamma (of the option) vs. vega (of the option)

The sensitivity of the delta to the stock price. Option gammas are analogous to bond convexity. The curvature of an option pricing function (as a function of the value of the underlying asset) *Delta* = An essential tool in portfolio management and control. It is the number of stocks required to hedge against the price risk of holding one option. Also called *the Hedge Ratio*. vs. Vega = the sensitivity of an option price to changes in volatility.

*Ch 22: Futures Market* Expectations hypothesis

The simplest theory of futures pricing. It states that the futures price equals the expected value of the futures spot price: Fo = E(P,T) Here, the Expected P would equal zero, which relies on a notion of risk neutrality.

*Ch 23: Futures, Swaps, and Risk Management* Interest Rate Parity Relationship (Theorem) Or... Covered Interest Arbitrage Relationship

The spot-futures exchange rate relationship that prevails in well-functioning markets. Should this IRP be violated, arbitrageurs will make risk-free profits in foreign exchange markets with zero-net-investment. *Interest Rate Parity Fo = Eo [ (1 + r𝑓US) / 1 + r𝑓UK)^T* Eo = Current exchange rate between the two currencies Fo = Forward price (the # of $ agreed to today to purchase one pound at time T) r𝑓US = US risk-free rates r𝑓UK = UK risk-free rates SPECIAL NOTE:... if r𝑓US < r𝑓UK and Fo must be less than Eo, so then money invested in US will grow at a slower rate than money invested in the UK. However, there is a forward premium on US dollars where, despite the dollar growing slower than pound investments in the US, each dollar may be worth more pounds in the forward market than the spot market, which offsets the advantage of the higher UK interest rate. but.... if r𝑓US > r𝑓UK then the forward price must be greater than the current exchange rate. The net proceeds to the arbitrage portfolio are risk-free and given by Eo(1 + rUS) - Fo(1 + rUK). IF this value is positive, borrow in the UK, lend in the US, and enter a long futures position to eliminate foreign exchange risk. IF this value is negative, borrow US, lend in UK, and take a short position in pound futures.

*Ch 23: Futures, Swaps, and Risk Management* Indirect vs. Direct quotes

US and euro dollars are typically expressed as direct quotes. Other countries, including Japanese Yen and Swiss franc, are expressed as indirect quotes, that is, units of foreign currency per dollar. Direct = A direct quote is a foreign exchange rate involving a quote in fixed units of foreign currency against variable amounts of the domestic currency. Indirect = An indirect quote is also known as a "quantity quotation," since it expresses the quantity of foreign currency required to buy units of the domestic currency. In other words, the domestic currency is the base currency in an indirect quote, while the foreign currency is the counter currency. NOTE: If the interest rate is higher in the US than Japan, the dollar will sell in the forward market at a lower price (will buy fewer yen) than in the spot market.

*Ch 23: Futures, Swaps, and Risk Management* Contracts (Stock-Index Futures)

Unlike most futures contracts, stock index contract are settled by a cash amount equal to the value of the index on the contract maturity date, multiplied by the scale/size of the contract. Examples of traded stock futures: - S&P 500 - Dow Jones Indust. Av - Russell 2000 - NASDAQ 100 - Ftse 100 - DAX-30 - CAC-40

*Ch 21: Option Valuation* Dividends and Call Option Valuation

When an option has dividends, you can still use the Black-Scholes equation, but you'd simply adjust the "So" factor to being So - PV(dividends). If dividend... use: Co = [So - PV(dividends)]N(d1) - [Xe^(-rT) x N(d2)]

*Ch 21: Option Valuation* Delta Neutral

When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, your portfolio is said to be delta neutral, meaning that the portfolio has no tendency to either increase or decrease in value when the stock price fluctuates. Although delta-neurtal option hedges might eliminate exposure to risk from fluctuations in the value of the underlying asset, they don't eliminate volatility risk.