Fin 424 Final Exam

Buying a strangle

( + c +p) with different Ks Buy call with K2, buy put with k1 with different k (k1<k2) it is cheaper than straddle, stock price has to move more

Buy calendar spread using puts

( + p2 - p1) buy expensive put, sell the one month put buy put with maturity, T2, sell put with maturity, T1 T2 > T1 P2 > P1 initial outflow (right)

Butterly spread w puts

( p1 -2p2 + p3) buy 1 put with k1 and sell 2 puts w k2 and buy 2 put w k3 K1< k2< k3 p1< P2<p3 initial outflow (right)

Buy calendar spread using calls

(+C2 - C1) ( gives you the tent) buy a call w maturity T2 (december), sell a call with maturity, T1 (november) T2 > T1 C2 > C1 initial outflow (left) 1) buy a long term option ( 2 months ), -C2, selling short term option (1 month) + c

Writing a Covered Call

(+S -C) Buy a stock (going up) (+S) and sell a call (going down) (-C) long a share, short a call option S+ P = B + C +S -C - B = -P short a put, synthetic short a put above 50, sell it at 50 to you, but dont have to buy higher price, still get 5 dollars. for awhile think it is going down (short term), later it will go up (long term) other people willing to pay money! creates a return Exam: best outcome: stock goes up worse outcome: stock goes down, loose money, selling call option will cushion the fall b/c the money you paid me exam: start with put call parity short the stock S + P = B + C -s = + P - C = short a stock, goes up = loose goes down = win

Buying a Straddle

(+c + p) buy a call, and buy a put with same K if stock goes up or down, in the money but it has to go down alot or up a lot before making it back good for uncertainty buying volatility upward v curve

Selling a Straddle

(-c -p) sell a call, sell a put with same k unlimited downside downward v

Problem 12.1. A stock price is currently $40. It is known that at the end of one month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a one-month European call option with a strike price of $39?

(3 x 0.5669 + 0 x 0.4331)e ^-0.08x 0.08333= 1.69

bull spread with calls

(C1- C2), slice up the distrubution Buy call w / k1 (pay C1), sell call w / k2 (recieve c2) K1 < K2 , C1 > C2, so -C1 + C2 < 0 initial outflow left 1) both OTM = stays below $40 2) Bought ITM 3) above 45, loose that dollar bc sold it, win = stock price goes up most loose = -c1 + c2

Upper bonds

1) Call gives holder right to buy one share @ K. Call cannot be more valuable than one share (S). c < S or equal and C < S or equal 2)Put gives holder right to sell one share @ K. Put cannot be more valuable than K. p < K or equal and P < K, or equal 3) At expiration, we know p < K or equal (for European put).Thus, today p < Ke ^ -rT (NPV) (and this is < K). If p > Ke ^ -rT, sell European put today, invest @ r; at expiration, pe ^ rT > K; arbitrage.

Give two reasons that the early exercise of an American call option on a non-dividend-paying stock is not optimal. The first reason should involve the time value of money. The second reason should apply even if interest rates are zero.

1) Delaying exercise delays the payment of the strike price. This means that the option holder is able to earn interest on the strike price for a longer period of time. 2) Delaying exercise also provides insurance against the stock price falling below the strike price by the expiration date. Assume that the option holder has an amount of cash K and that interest rates are zero. When the option is exercised early it is worthT Sat expiration. Delaying exercise means that it will be worth max ( K, ST) at expiration. 1. Delaying exercise delays payment of the strike price. Thus, option holder earns more interest on that money (strike price). 2. Delaying exercise also gives insurance against declines in S.If S declines below K, call OTM, can then keep K!

Early Exercise of American Puts

1) Recall, American call (C) will not be exercised early if no dividend is paid during its life. Thus, C = c. (American call acts like a European call.) 2) American puts (P) are different. It may be wise to exercise American put early(even if no div): 1. If S is low enough; 2. If put is deep ITM. 3. Thus, P > (or equal) p 3)Suppose K = $10; and S + $0. 1. If you exercise, receive (K - S) =$10 today. 2. If you wait, cannot receive more than $10 (S cannot decrease below $0). 3. Receiving $10 now is better than later. 4. Such an option should be exercised early. Like a call, think of put as giving insurance However, unlike a call, it may be optimal: a. To forego this insurance & exercise put early b. To receive $K immediately. early exercise of put is more attractive: If S decreases( put deeper ITM; more intrinsic value ); r increases ( more time value; (K-S) today is more attractive standard deviaton decreases ( less likely for S to decrease further; less extrinsic value ). For European put, p may be < K - S. b. For American put, P may not be < K - S. For American put, a. It is always optimal to exercise early, if S low enough b. Price curve merges into put's intrinsic value, (K - S), as S decreases below A c.For S values to right of point A, P > K - S ( there is extrinsic value! ) d.This extrinsic value increases if: r decreass; (lower interest, so K today less attractive; less time value) ii. standard deviation increases ; (more likely for good things to happen (S↓); more extrinsic value) iii. T increases; (more likely for good things to happen (S↓); more extrinsic value) american put is worth more, can never be less than intrinisic value is american option worth more than european? false for a call if no dividend true for puts, american can never be less than instrinsic value , european can!

risk-neutral valuation

1. On average the stock price grows at the riskfree rate. 2. Setting the probability of an up movement equal to p is the same as assuming the stock earns the riskfree rate. 3. If investors are risk-neutral, require no compensation for risk, and the expected return on all securities is the riskfree rate. 1. Any option can be valued on the assumption that the world is risk-neutral. 2.To value an option, can assume: a. expected return on all traded securities is the riskless rate, r; b. the NPV of expected future cash flows can be valued by discounting at r. 3. The prices we get are correct, not just in a risk-neutral world, but in other worlds as well. 4.Risk-Neutral Valuation works because you can always get a hedge portfolio (that's riskfree) using options, so arbitrageurs force the option value to behave this way.

What is a lower bound for the price of a one-month European put option on a non-dividend-paying stock when the stock price is $12, the strike price is $15, and the risk-free interest rate is 6% per annum?

15e^-0.06 x .0833 (1/12) - 12

stock dividend

20% 120 new shares new k = 5/6 (old k)

stock split

3 for 1 is for 300 shares at new k 1/3 (old k)

contract size

4 expirations, each have 5 strike prices 4x 5 call 4x5 puts = 40 contracts!

What is a lower bound for the price of a two-month European put option on a non-dividend-paying stock when the stock price is $58, the strike price is $65, and the risk-free interest rate is 5% per annum?

65e^ -.05 x 2/12 - 58 = $6.46 p >$65 e -.05 x (2/12) - $58 = $6.46

What is a lower bound for the price of a six-month call option on a non-dividend-paying stock when the stock price is $80, the strike price is $75, and the risk-free interest rate is 10% per annum?

80 - 75e^-.1 x .5= $8.66 c > (or equal)$80 - $75 e-.10 x .5= $8.66

Formula for N-Period Binomial Model

= S * B[ a,N,p' ] - K e^ -r N ΔT * B[ a,N,p ] p = (e ^r ΔT - d) / (u - d) as before, and p' = (u / e ^r ΔT)p ;[ Typically, u > e ^r ∆t ; so p' > p. ] interpretation: its the right price because expected pay off discounted back to the present

#21 Explain why the market maker's bid-offer spread represents a real cost to options investors.

A "fair" price for the option can reasonably be assumed to be half way between the bid and the offer price quoted by a market maker. An investor typically buys at the market maker's offer and sells at the market maker's bid. Each time he or she does this there is a hidden cost equal to half the bid-offer spread.

Explain two ways in which a bear spread can be created?

A bear spread can be created using two call options with the same maturity and different strike prices. The investor shorts the call option with the lower strike price and buys the call option with the higher strike price. A bear spread can also be created using two put options with the same maturity and different strike prices. In this case, the investor shorts the put option with the lower strike price and buys the put option with the higher strike price.

When is it appropriate for an investor to purchase a butterfly spread?

A butterfly spread involves a position in options with three different strike prices (K1, K2, K3) A butterfly spread should be purchased when the investor considers that the price of the underlying stock is likely to stay close to the central strike price, K2.

#2 What does it mean to assert that the delta of a call option is 0.7? How can a short position in 1,000 options be made delta neutral when the delta of each option is 0.7?

A delta of 0.7 means that, when the price of the stock increases by a small amount, the price of the option increases by 70% of this amount. Similarly, when the price of the stock decreases by a small amount, the price of the option decreases by 70% of this amount. A short position in 1,000 options has a delta of -700 and can be made delta neutral with the purchase of 700 shares

CH7 #1 Companies A and B have been offered the following rates per annum on a $20 million five-year loan Company A requires a floating-rate loan; company B requires a fixed-rate loan. Design a swap that will net a bank, acting as intermediary, 0.1% per annum and that will appear equally attractive to both companies. Company A fixed: 5% and float: Libor + .1 Company B fixed:6.4% float: libor + .6%

A has an apparent comparative advantage in fixed-rate markets but wants to borrow floating. B has an apparent comparative advantage in floating-rate markets but wants to borrow fixed. This provides the basis for the swap. There is a 1.4% per annum differential between the fixed rates offered to the two companies and a 0.5% per annum differential between the floating rates offered to the two companies. The total gain to all parties from the swap is therefore 1.4 - .5 = .9% per annum. Because the bank gets 0.1% per annum of this gain, the swap should make each of A and B 0.4% per annum better off. Because the bank gets 0.1% per annum of this gain, the swap should make each of A and B 0.4% per annum better off. This means that it should lead to A borrowing at LIBOR -.3% and to B borrowing at 6.0%

#13 A company uses delta hedging to hedge a portfolio of long positions in put and call options on a currency. Which of the following would give the most favorable result? a) A virtually constant spot rate b) Wild movements in the spot rate Explain your answer.

A long position in either a put or a call option has a positive gamma. From Figure 17.8, when gamma is positive the hedger gains from a large change in the stock price and loses from a small change in the stock price. Hence the hedger will fare better in case (b)

What is meant by a protective put? What position in call options is equivalent to a protective put?

A protective put consists of a long position in a put option combined with a long position in the underlying shares. It is equivalent to a long position in a call option plus a certain amount of cash. This follows from put-call parity: p + So = c + Ke^ - rt + D

What trading strategy creates a reverse calendar spread?

A reverse calendar spread is created by buying a short-maturity option and selling a long- maturity option, both with the same strike price.

#14 Repeat Problem 17.13 for a financial institution with a portfolio of short positions in put and call options on a currency.

A short position in either a put or a call option has a negative gamma. From Figure 17.8, when gamma is negative the hedger gains from a small change in the stock price and loses from a large change in the stock price. Hence the hedger will fare better in case (a).

#4 Explain what a swap rate is. What is the relationship between swap rates and par yields?

A swap rate for a particular maturity is the average of the bid and offer fixed rates that a market maker is prepared to exchange for LIBOR in a standard plain vanilla swap with that maturity. The swap rate for a particular maturity is the LIBOR/swap par yield for that maturity.

example - building interest rate collar for bank

A) cap borrowing rate @ 4% by buying a put w k = 96 (100-4) must pay 13 bps for this put 13 x 25 = 325 if ED goes up above 4% and Q goes down below 96 and put is ITM cap at 4% if ED goes down below 4% Q goes up above 96 and out is OTM and borrow at < 4% B) if you don't think ED rate will go down below, say 3.25% can recover some of cost by selling a call w k = 96.75 (100 - 3.25) receive 2 bp (50) if ED rates go down below 3.25% q goes up above 96.75 and call ITM - floor at 3.25%

Lower Bound for Euro Call on non-dividend-paying Stock

A: buy one call, -c and k bonds that each pay $1 at T, -Ke^rT B:Buy one share of stock, -S If S increases, both portfolios pay ST; If S decreases, Portfolio A does better (hedged). Thus Portfolio A is worth at least as much as B. c > (or equal) S - Ke-rT European Call should not sell for less than (S - Ke^-rT ).

Lower Bound for Euro Put on non-dividend-paying Stock

A: one put and one share of stock (S+P) protective put B:buy K bonds that each pay $1 at expiration. - Ke^rT If S decreases, portfolios C and D both pay K.If S increases, portfolio C does better than D (hedged) Therefore, portfolio C should be worth more than D: p > Ke^-rT - S Worst outcome, put finishes OTM; So p > max{ (Ke^-rT - S), 0 }

12.10 A stock price is currently $80. It is known that at the end of four months it will be either $85 or $75. The risk-free interest rate is 5%per annum with continuous compounding. What is the value of a 4-monthEuropean put option with a strike price of K = $80?

After four months the value of the put will be either pu = $0 (if the stock price is $85) pd = $5 (if the stock price is $75) u = $85 / $80 = 1.0625 d = $75 / $80 = .9375 p = (e^.05 x (4/12) - .9375) / (1.0625 - .9375) = .6345 (1 - p) = .3655 put value = [ $0 (.6345) + $5 (.3655) ] e^-.05 x (4/12) = $1.80

12.9 A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a 2-monthEuropean call option with a strike price of K = $49?

After two months the value of the call it will be either: cu = $4 (if the stock price is $53) or cd = $0 (if the stock price is $48) u = $53 / $50 = 1.06 d = $48 / $50 = .96 p = (e^.10 x (2/12) - .96) / (1.06 - .96) = .5681 (1 - p) = .4319 call value = [ $4 (.5681) + $0 (.4319) ] e^-.10 x (2/12) = $2.23

Why is an American call option on a dividend-paying stock is always worth at least as much as its intrinsic value. Is the same true of a European call option? Explain your answer.

An American call option can be exercised at any time. If it is exercised its holder gets the intrinsic value. It follows that an American call option must be worth at least its intrinsic value. A European call option can be worth less than its intrinsic value. Consider, for example, the situation where a stock is expected to provide a very high dividend during the life of an option. The price of the stock will decline as a result of the dividend. Because theEuropean option can be exercised only after the dividend has been paid, its value may be less than the intrinsic value today. American call can be exercised anytime. Thus, C ≥ S - K Can't exercise European call early. Thus, c ≤ C . If American, can exercise, & get dividend; If European, can't exercise. Don't get dividend. We know S will decrease after div Thus, European call can be worth less than American call: c < C . European call can even be worth less than intrinsic value: c < S - K since after dividend, we know S will decrease (maybe < current S). This is *only* situation when American call may be exercised early

American vs. European Options

An American option can be exercised at any time during its life A European option can be exercised only at maturity T is good for american, dividend is good, doesnt loose it, rather have it as dividend get dividend european - long term call usually cost more than a short term call, t isnt good, make call worthless dont get dividend, loss intrinsic value Long term European call usually costs more than short term call.But suppose a dividend is paid after short term call expires. Holder of short term European call will get dividend.Holder of long term European call will not.So short term call may be worth more than long term call

"The early exercise of an American put is a trade-off between the time value of money and the insurance value of a put." Explain this statement.

An American put when held in conjunction with the underlying stock provides insurance. It guarantees that the stock can be sold for the strike price, K. If the put is exercised early, the insurance ceases. However, the option holder receives the strike price immediately and is able to earn interest on it between the time of the early exercise and the expiration date.

Chapter 11 Explain how an aggressive bear spread can be created using put options.

An aggressive bull spread using call options is discussed in the text. Both of the options used have relatively high strike prices. Similarly, an aggressive bear spread can be created using put options. Both of the options should be out of the money (that is, they should have relatively low strike prices). The spread then costs very little to set up because both of the puts are worth close to zero. In most circumstances the spread will provide zero payoff. However, there is a small chance that the stock price will fall fast so that on expiration both options will be in the money. The spread then provides a payoff equal to the difference between the two strike prices most circumstances the spread will provide zero payoff

Put-Call Parity

An equation expressing the equivalence (parity) of a portfolio of a call and a bond with a portfolio of a put and the underlying, which leads to the relationship between put and call prices. (S+P = B + C) B and C give some payoff pattern tool to help with financially engineering relationship to build other things

#19 What is the effect of an unexpected cash dividend on (a) a call option price and (b) a put option price?

An unexpected cash dividend would reduce the stock price on the ex-dividend date. This stock price reduction would not be anticipated by option holders. As a result there would be a reduction in the value of a call option and an increase the value of a put option. (Note that the terms of an option are adjusted for cash dividends only in exceptional circumstances.)

What is the difference between a strangle and a straddle?

Both a straddle and a strangle are created by combining a long position in a call with a long position in a put. In a straddle the two have the same strike price and expiration date. In a strangle they have different strike prices and the same expiration date.

another example

Buy one call, w/ k1 = 40 (c1 = 8) worth more bc cost less (40) sell 2 calls w k2 = 45 (c2 = 5) 1) initial cost = (-1) x 8 + (+2) x 5 = +2 (stock price below 40) if s < k1 = 40 , both options OTM, CV = +2 2) if 40 < S < 45, c1 is ITM value = 2 + 1 (s-k) (coeff = +1) 3) s > $45, c1 and c2 are ITM value = +2 + 1 (s-k1) - (s-K1) coeff = -1 breakeven - goes 7 dollars up from 45 (52), after that unlimited downside.

Protective Put

Buying a stock and a put on the stock to protect the decline of a stock's price; Can be replicated by buying a bond that pays the strike price minus the premium at expiration and a call with the strike price buy a stock (+S) and buy put (+P) another way to get a call option insurance (S+P) insurance - what you pay for it ,never loose $50 what you paid for insurance

Valuing American Calls & Puts

Calls. 1. European calls can be valued with Binomial Model. 2. American calls have same value as European calls if no dividend. (C = c) 3. If dividend, can value American calls using modified Binomial model. This valuation is complicated by the American feature. (ThenC ≥ c) Can assume American call will be exercised early if dividend large enough.Binomial tree can be modified to take this into account. Puts. 1. Put-Call Parity can be used to price European. puts, after you find value of the European call from Binomial Model. 2. Or European puts can be valued directly with Binomial Model, like calls. 3. American puts are different. Will be exercised early if S ↓ (...) (P > p) Must get directly with Binomial trees. Complicated by American feature.Must take into account opportunity to exercise early at each node of tree. american- put always worth more, rather have now, exercise early, put allows you to sell Europe- have to wait till expiration If amount ITM is larger, t hen exercise at node! Carry this value back through the Tree. log: call Atm: 0 Otm: smaller di Itm: di bigger, call is worth more T/F Binominal model use for american and european

#12 Companies A and B face the following interest rates (adjusted for the differential impact of taxes): Company A: floating rate = libor + .5% (US) and fixed rate = 5.0% (canada) Company B: floating rate libor + 1.0% (US) and fixed rate = 6.5% (canada) Assume that A wants to borrow U.S. dollars at a floating rate of interest and B wants to borrow Canadian dollars at a fixed rate of interest. A financial institution is planning to arrange a swap and requires a 50-basis-point spread. If the swap is equally attractive to A and B, what rates of interest will A and B end up paying

Company A has a comparative advantage in the Canadian dollar fixed-rate market. Company B has a comparative advantage in the U.S. dollar floating-rate market. (This may be because of their tax positions.) However, company A wants to borrow in the U.S. dollar floating-rate market and company B wants to borrow in the Canadian dollar fixed-rate market. This gives rise to the swap opportunity. The differential between the U.S. dollar floating rates is 0.5% per annum, and the differential between the Canadian dollar fixed rates is 1.5% per annum. The difference between the differentials is 1% per annum. The total potential gain to all parties from the swap is therefore 1% per annum, or 100 basis points. If the financial intermediary requires 50 basis points, each of A and B can be made 25 basis points better off. Thus a swap can be designed so that it provides A with U.S. dollars at LIBOR 0.25% ( .5- .25)per annum, and B with Canadian dollars at 6.25% per annum (6.5 - .25)

Value Currency SWAP example exam

Company B borrows $ @ 9.4% lends e @ 12% short a bond @ 9.4% and long a bond @ 12% r us = 5% r uk = 10% B has entered Currency SWAP where it: receives 12.0% on e 10 million; .12 (e10) = e1.20 million pays 9.4% on $15 million .094($15) = $1.41 million SWAP will last another 3 years payments once / year Spot exchange rate: $1.5 BD = 1.41 e^-.05 + 1.41 e^-.05x2 + 16.41 e^-.05x3 =$16.74 million BF= 1.20 e^-.10+ 1.20 e^-.10x2 + 11.20 e^-.10x3 = 10.37 e million VSWAP = (1.5) x (10.37) - 16.74 = -$1.19 million

The Lognormal Distribution for ST - square root effect

Consider Annualized Volatility of Lognormal ST - standard deviation (sigma). annualized volatility - volatility depends on how you look! c. "Old economy stocks" - old stocks ranges from .2 to .4 (20% - 40%). d. "New economy stocks" new stocks ranges from .4 to .6 (40% - 60%). sigma(T) ½ is the Std Dev of stock return over time T. Suppose sigma= 30% over 1 year (T = 1), and T = .5 (6 months); the Std Dev of return in 6 months is σ (T) ½ = 30 (.5) ½ = 21.2% The Std Dev of return in 3 months is σ (T) ½ = 30 (.25) ½ = 15.0%; the "square root effect" is important in assessing risk. a. In general, uncertainty about ST increases as the square root of how far ahead we are looking (T).

Effect of Dividends

D = NPV( expected dividends during life of option ).Then all these relations hold, after adjusting for D 1) Lower bound for European call on dividend paying stock:c > (or equal) S - (Ke^-rT + D) 2) Lower bound for European put on dividend paying stock:p > (or equal) (Ke^-rT + D) - S 3)Put-Call Parity; c = S + p - Ke^-rT c = S + p - (Ke^-rT + D)

Position of Equity Holder as being Long a Call

Equity in levered firm is a call option on Value of firm (V) For the usual call option, underlying asset is a share of stock, S. For this illustration, the share of stock is the call option, and underlying asset is the value of the firm, V If at maturity, V > D, this "option" is ITM; Shareholders exercise; pay debt holders D, keep the rest, (V - D) if at maturity, V < D, this "option" is OTM; Shareholders will default; give firm to debt holders. At maturity, shareholder's wealth is: W = max{ V - D, 0 }.--- Same payoff as a call option on V with strike price, K = D!

Chapter 7 Exam Question: Interest rate swaps

First determine margin to be captured. Ex: Diff so 1.2 - .7 = .50 .5 bips! Seconds figure out rate. Ex: each party gets .25 (1/2 of .5) A will pay LIBOR + .3(float) -.25= LIBOR + .05%. B will pay 11.2 (fixed) -.25 = 10.95% Third Construct. A borrows fixed at 10%, B borrows at LIBOR 1%. A pays B LIBOR, B pays A 9.95%. Net borrowing cost: A LIBOR + .05, B 10.95%. Margin captured is .25 and .25

The price of a non-dividend paying stock is $19 and the price of a three-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per annum. What is the price of a three-month European put option with a strike price of $20?

For finding the price of the put option given the price on the call option for the same underlying asset with the same strike price and price is given , we apply the Black-Scholes model of put-call parity to solve the price of the put option. The put-call parity formula: p = K x e^(-rT) + c - St where p = price of put of option K = the strike price = $20 r = Applied Interest rate = 4% T = time to maturity denominated in year = 3/12 = 0.25 c= call option price St = spot price of underlying asset = $19 Thus p = 20 x e^(-0.04 x 0.25) + 1 - 19 = $1.80

#16 The treasurer of a corporation is trying to choose between options and forward contracts to hedge the corporation's foreign exchange risk. Discuss the advantages and disadvantages of each.

Forward contracts lock in the exchange rate that will apply to a particular transaction in the future. Options provide insurance that the exchange rate will not be worse than some level. The advantage of a forward contract is that uncertainty is eliminated as far as possible. The disadvantage is that the outcome with hedging can be significantly worse than the outcome with no hedging. This disadvantage is not as marked with options. However, unlike forward contracts, options involve an up-front cost.

Hedge portfolio

Hedge Portfolio: Buy Δ shares of stock for each call written f= c = value of call today; fu = cu = value of call if S increases fd = cd = value of call if S decreases In our example, Δ = ½ ; fu = max { 75 - 50, 0 } = $25; fd = $0; f = $15. To be a riskless "hedge portfolio,"cash flows at end of period must be same. Δ = ( fu - fd ) / (Su - Sd) ($25 - $0) / ($75 - $25) = ½ Delta: The ratio of the change in the price of an option to the change in the price of the underlying stock This is the hedge ratio, Δ; The number of shares we should hold for each call written, in hedge portfolio. this is delta hedging. the hedge portfolio should yield the riskless rate;(riskless investment) * erT = riskless outcome

Explain carefully the arbitrage opportunities in Problem 10.14 if the European put price is $3.

If the put price is $3.00, it is too high relative to the call price. An arbitrageur should buy the call, short the put and short the stock. This generates −2 + 3 +29 =$30 in cash which is invested at 10%. Regardless of what happens a profit with a present value of 3.00 - 2.51 = $0.49 is locked in. If the stock price is above $30 in six months, the call option is exercised, and the put option expires worthless. The call option enables the stock to be bought for $30, or 30 e ^ -.10 x 6/12 = $28.54 in present value terms. The dividends on the short position cost 0.5e ^-0.1 x 2/12 + 0.5e ^- 0.1 x 5/12 = $.97 in present value terms so that there is a profit with a present value of 30 - 28.54 −0.97 = $0.49. If the stock price is below $30 in six months, the put option is exercised and the call option expires worthless. The short put option leads to the stock being bought for $30, or 30e ^-.10 x 6/12 = $28.54 in present value terms. The dividends on the short position cost 0.5e ^ -.1 x 2/12 + 0.5e ^ -.1 x 5/12 = $.97 in present value terms so that there is a profit with a present value of 30 - 28.54 −0.97 = $0.49. or the Put price should be $2.51 (from put-call parity).If the put price is $3.00, it is too high relative to the call price. Arbitrageur should sell this put for $3.00, and buy synthetic put for $2.51, as follows: p = c + (Ke^-rT + D) - S = $2 + ($30e^-.10x.5 + D) - $29:= $2.51 buy the call for $2, invest the present value of K ($30e^-.10x.5) and D, and short the stock, to receive $29. This synthetic put costs a total of $2.51 Then sell the actual put for $3. Now you're long & short the same put!Regardless of what happens to S, you'll earn profit of $3 - $2.51 = $0.49 2 is the price of a european call

Put-Call Parity

If we know the price of a European put, can determine the price of a European call. -c = -S - p + Ke^-rT or c = S + p - Ke^-rT

#9 Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised?

Ignoring the time value of money, the holder of the option will make a profit if the stock price at maturity of the option is greater than $105. This is because the payoff to the holder of the option is, in these circumstances, greater than the $5 paid for the option. The option will be exercised if the stock price at maturity is greater than $100. Note that if the stock price is between $100 and $105 the option is exercised, but the holder of the option takes a loss overall.

#10 Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under what circumstances will the seller of the option (the party with the short position) make a profit? Under what circumstances will the option be exercised?

Ignoring the time value of money, the seller of the option will make a profit if the stock price at maturity is greater than $52.00. This is because the cost to the seller of the option is in these circumstances less than the price received for the option. The option will be exercised if the stock price at maturity is less than $60.00. Note that if the stock price is between $52.00 and $60.00 the seller of the option makes a profit even though the option is exercised.

#15 Why is the expected loss from a default on a swap less than the expected loss from the default on a loan with the same principal?

In an interest-rate swap a financial institution's exposure depends on the difference between a fixed-rate of interest and a floating-rate of interest. It has no exposure to the notional principal. In a loan the whole principal can be lost.

A one-month European put option on a non-dividend-paying stock is currently selling for$2.50. The stock price is $47, the strike price is $50, and the risk-free interest rate is 6% per annum. What opportunities are there for an arbitrageur?

In this case the present value of the strike price is 50 e ^ -.06 x 1/12 =49.75 because 2.5 < 49.75 - 47.00 the condition in equation (10.5) is violated. An arbitrageur should borrow $49.50 at 6% for one month, buy the stock, and buy the put option. This generates a profit in all circumstances. If the stock price is above $50 in one month, the option expires worthless, but the stock can be sold for at least $50. A sum of $50 received in one month has a present value of $49.75 today. The strategy therefore generates profit with a present value of at least $0.25. If the stock price is below $50 in one month the put option is exercised and the stock owned is sold for exactly $50 (or $49.75 in present value terms). The trading strategy therefore generates a profit of exactly $0.25 in present value terms.

#3 "Calculate the delta of an at-the-money six-month European call option on a non-dividend-paying stock when the risk-free interest rate is 10% per annum and the stock price volatility is 25% per annum."

In this case, S0=K r=.10 σ=.25 T=.50 In this case, So=K d1 = Ln (So/K) + (( 0.1 + .25^2)/2) x .5 / 0.25x Square root of .50 = .3712 The delta of the option is N(d1) or 0.64

Position of Debt Holder as Short a Put

Long a bond (make a loan) that pays D at maturity; Short a put option on the value of the firm, with K = D. If value of firm (V) increases, a. Shareholders' long call position [ max { (V - D), 0 } ] is more valuable. b. Bondholders' short put position [ max { (D - V), 0 } ] is smaller liability. (Now bondholders are short a put that is worth less, since OTM; sell for D!) c. Obviously, both parties want to see V increase.

What is a lower bound for the price of a 4-month call option on a non-dividend-paying stock when the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% per annum?

Lower bound = Stock Price - Strike Price*exp[-rt] = 28 - 25*e^[-8%*4/12] = 3.66 Therefore, The lower bound for the price of a 4-month call option on a non-dividend-paying stock is 3.66

OTM and ITM

OTM less likely option will be exercised so less will be required margin for selling ITM more likely option will be exercised so initial margin will be greater

#5 A stock option is on a February, May, August, and November cycle. What options trade on (a) April 1, and (b) May 30?

On April 1, options trade with expiration months of April, May, August, and November. On May 30, options trade with expiration months of June, July, August, and November

one period

S = $50; K = $50, call expires in one year; ST will either increase or decrease by 50% •$75(i.e., ST = either Su = $75 or Sd = $25,(Cu=$25)where u = 1.5 and d = 0.5) r = .25 call itm by 25 bucks (75) cu= 75 call otm by 25 bucks (25) cd=0 hedge: Sell 2 calls +2c 0 -50 Buy 1 share -50 +25 +75 Total:+2c-50 +25 +25 (riskless) value of a call investment) * (1+r) = outcome [-2c+50] * (1.25) = $25 -2c+50 = $25 / 1.25 c=15 Borrow $20 sell 2 calls, buy 1 share owe $25 (have $25) arb: c = 20 need to borrow $10 owe 12.50(have 25) keep the rest do this until c =15 imagine a portfolio that is long ∆ shares & short 1 call. If S ↑ to Su, the value of ∆ shares ↑, & the value of short call (C) ↓; If S ↓to Sd, the value of ∆ shares ↓, & the value of short call (C) ↑. c = [ fu p + fd (1-p) ] e^-rT where p = (e^rT - d) / (u-d) Interpretation: For a risk-neutral investor (be careful)*, p is probability that stock will increase. Thus, in a risk-neutral world, call value is expected payoff of the option discounted back at the risk free rate. c = [ $25 (.75) + $0 (.25) ] / 1.25 = $18.75 / 1.25 = $15 Put value = [ fu p + fd (1-p) ] e^-rT where p = (e^rT - d) / (u-d)

Payoff return for any combination example

Strip: buy one call and two puts with same k k = 50 c = 5 p = 6 1) initial cost = (-1) x 5 + (-2) x 6 = -$17 cost me this s = k = 50 both options are ATM, CV = -17 2) if S > 50, call ITM CV = -17 + 1 (S-K) Slope = +1 if S gos to $5, $1 dollar in the money breakeven = has to go up to 17, $67 3) if S < 50, put ITM, CV = -17 + 2 (k-s) if price goes down $1 I make 2 dollars slope = -2 50 - 8.50 = make 17 dollars, breakeven

How can a forward contract on a stock with a particular delivery price and delivery date be created from options?

Suppose the delivery price is K, and the delivery date is T. A forward contract can be created by buying a European call (+C),and selling a European put (-P), with the same (K) and (T). If ST↑, the call is ITM; will buy at K; If ST↓, the put is ITM; will sell at K. This portfolio provides a payoff of (ST - K) under all circumstances. Suppose the forward price (F0) is set to = delivery price (K). Then the forward contract that is created has zero value at start. This shows that the price of a call (C) equals the price of a put (P) when the forward price (F0) equals the strike price (K).

#1 Explain how a stop-loss trading rule can be implemented for the writer of an out-of-the-money call option. Why does it provide a relatively poor hedge?

Suppose the strike price is 10.00. The option writer aims to be fully covered whenever the option is in the money and naked whenever it is out of the money. The option writer attempts to achieve this by buying the assets underlying the option as soon as the asset price reaches 10.00 from below and selling as soon as the asset price reaches 10.00 from above. The trouble with this scheme is that it assumes that when the asset price moves from 9.99 to 10.00, the next move will be to a price above 10.00. (In practice the next move might back to 9.99.) Similarly it assumes that when the asset price moves from 10.01 to 10.00, the next move will be to a price below 10.00. (In practice the next move might be back to 10.01.) The scheme can be implemented by buying at 10.01 and selling at 9.99. However, it is not a good hedge. The cost of the trading strategy is zero if the asset price never reaches 10.00 and can be quite high if it reaches 10.00 many times. A good hedge has the property that its cost is always very close the value of the option

Problem 14.1.What does the Black-Scholes-Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the continuously compounded rate of return on the stock during the year?

The Black-Scholes-Merton option pricing model assumes that the probability distribution of the stock price in 1 year (or at any other future time) is lognormal. It assumes that the continuously compounded rate of return on the stock during the year is normally distributed.

#8 A corporate treasurer is designing a hedging program involving foreign currency options. What are the pros and cons of using (a) the NASDAQ OMX and (b) the over-the-counter market for trading?

The NASDAQ OMX offers options with standard strike prices and times to maturity. Options in the over-the-counter market have the advantage that they can be tailored to meet the precise needs of the treasurer. Their disadvantage is that they expose the treasurer to some credit risk. Exchanges organize their trading so that there is virtually no credit risk.

#16 A bank finds that its assets are not matched with its liabilities. It is taking floating-rate deposits and making fixed-rate loans. How can swaps be used to offset the risk?

The bank is paying a floating-rate on the deposits and receiving a fixed-rate on the loans. It can offset its risk by entering into interest rate swaps (with other financial institutions or corporations) in which it contracts to pay fixed and receive floating.

#24 A financial institution has the following portfolio of over-the-counter options on sterling: A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8. a. What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral? b. What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral? Assume that all implied volatilities change by the same amount so that vegas can be aggregated.

The delta of the portfolio is -1,000 x .50-500 x .80-2,000x (-.40)-500x .70 =-450 The gamma of the portfolio is -1,000x 2.2 -500x .6 - 2,000 x 1.3 -500 x 1.8 = -6,000 The vega of the portfolio is -1,000 x 1.8 -500 x .2 -2,000 x .7 -500 x 1.4 = -4,000 a) A long position in 4,000 traded options will give a gamma-neutral portfolio since the long position has a gamma of 4,000 x 1.5 = +6,000 The delta of the whole portfolio is then 4,000 x .6 - 450 = 1,950 Hence, in addition to the 4,000 traded options, a short position of 1,950 in sterling is necessary so that the portfolio is both gamma and delta neutral. b) A long position in 5,000 traded options will give a vega-neutral portfolio since the long position has a vega of 5,000 x .80 = +4,000 the delta of the whole portfolio is then 5,000 x .6 - 450 = 2,550 Hence, in addition to the 5,000 traded options, a short position of 2,550 in sterling is necessary so that the portfolio is both vega and delta neutral.

Give an intuitive explanation of why the early exercise of an American put becomes more attractive as the risk-free rate increases and volatility decreases.

The early exercise of an American put is attractive when the interest earned on the strike price is greater than the insurance element lost. When interest rates increase, the value of the interest earned on the strike price increases making early exercise more attractive. When volatility decreases, the insurance element is less valuable. Again, this makes early exercise more attractive.

#18 "If most of the call options on a stock are in the money, it is likely that the stock price has risen rapidly in the last few months." Discuss this statement.

The exchange has certain rules governing when trading in a new option is initiated. These mean that the option is close-to-the-money when it is first traded. If all call options are in the money, it is therefore likely that the stock price has risen since trading in the option began.

#17 Explain how you would value a swap that is the exchange of a floating rate in one currency for a fixed rate in another currency.

The floating payments can be valued in currency A by (i) assuming that the forward rates are realized, and (ii) discounting the resulting cash flows at appropriate currency A discount rates. Suppose that the value is Va. The fixed payments can be valued in currency B by discounting them at the appropriate currency B discount rates. Suppose that the value is Vb.If Q is the current exchange rate (number of units of currency A per unit of currency B), the value of the swap in currency A is Va , Qvb. Alternatively, it it is Va, Q, Vb, in currency B.

#13 Explain why an American option is always worth at least as much as a European option on the same asset with the same strike price and exercise date

The holder of an American option has all the same rights as the holder of a European option and more. It must therefore be worth at least as much. If it were not, an arbitrageur could short the European option and take a long position in the American option.

#14 Explain why an American option is always worth at least as much as its intrinsic value

The holder of an American option has the right to exercise it immediately. The American option must therefore be worth at least as much as its intrinsic value. If it were not an arbitrageur could lock in a sure profit by buying the option and exercising it immediately.

Chapter 17: the greeks #10 Suppose that a stock price is currently $20 and that a call option with an exercise price of $25 is created synthetically using a continually changing position in the stock. Consider the following two scenarios: a) Stock price increases steadily from $20 to $35 during the life of the option. b) Stock price oscillates wildly, ending up at $35. Which scenario would make the synthetically created option more expensive? Explain your answer.

The holding of the stock at any given time must be N(d1). Hence the stock is bought just after the price has risen and sold just after the price has fallen. (This is the buy high sell low strategy referred to in the text.) In the first scenario the stock is continually bought. In second scenario the stock is bought, sold, bought again, sold again, etc. The final holding is the same in both scenarios. The buy, sell, buy, sell... situation clearly leads to higher costs than the buy, buy, buy... situation. This problem emphasizes one disadvantage of creating options synthetically. Whereas the cost of an option that is purchased is known up front and depends on the forecasted volatility, the cost of an option that is created synthetically is not known up front and depends on the volatility actually encountered.

14.6 What is implied volatility? How can it be calculated?

The implied volatility is the volatility that makes the Black-Scholes-Merton price of an option equal to its market price. It is calculated using an iterative procedure.

#2 An investor sells a European call on a share for $4. The stock price is $47 and the strike price is $50. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised?

The investor makes a profit if the price of the stock is below $54 on the expiration date. If the stock price is below $50, the option will not be exercised and the investor makes a profit of $4. If the stock price is between $50 and $54, the option is exercised and the investor makes a profit between $0 and $4.

Chapter 9 Homework Problems #1 An investor buys a European put on a share for $3. The stock price is $42 and the strike price is $40. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised?

The investor makes a profit if the price of the stock on the expiration date is less than $37. In these circumstances the gain from exercising the option is greater than $3. The option will be exercised if the stock price is less than $40 at the maturity of the option.

A four-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in one month. The risk-free interest rate is 12% per annum for all maturities. What opportunities are there for an arbitrageur?

The present value of the strike price is 60e^ -.12 x 4/12 = $57.65 The present value of the dividend is .80 e ^-.12 x 1/12 = .79 because 5 < 64 - 57.65 - .79 the condition in equation (10.8) is violated. An arbitrageur should buy the option and short the stock. This generates 64 - 5 = $59. The arbitrageur invests $0.79 of this at 12% for one month to pay the dividend of $0.80 in one month. The remaining $58.21 is invested for four months at 12%. Regardless of what happens a profit will materialize. If the stock price declines below $60 in four months, the arbitrageur loses the $5 spent on the option but gains on the short position. The arbitrageur shorts when the stock price is $64, has to pay dividends with a present value of $0.79, and closes out the short position when the stock price is $60 or less. Because $57.65 is the present value of $60, the short position generates at least 64 - 57.65 - 0.79 = $5.56 in present value terms. The present value of the arbitrageur's gain is therefore at least 5.56 - 5.00 = $0.56. If the stock price is above $60 at the expiration of the option, the option is exercised. The arbitrageur buys the stock for $60 in four months and closes out the short position. The present value of the $60 paid for the stock is $57.65 and as before the dividend has a present value of $0.79. The gain from the short position and the exercise of the option is therefore exactly 64 - 57.65 − 0.79 = $5.56. The arbitrageur's gain in present value terms is 5.56 - 5.00 = $0.56.

14.3 Explain the principle of risk-neutral valuation

The price of an option or other derivative when expressed in terms of the price of the underlying stock is independent of risk preferences. Options therefore have the same value in a risk-neutral world as they do in the real world. We may therefore assume that the world is risk neutral for the purposes of valuing options. This simplifies the analysis. In a risk-neutral world all securities have an expected return equal to risk-free interest rate. Also, in a risk-neutral world, the appropriate discount rate to use for expected future cash flows is the risk-free interest rate

#9 Companies X and Y have been offered the following rates per annum on a $5 million 10-year investment: Company X requires a fixed-rate investment; company Y requires a floating-rate investment. Design a swap that will net a bank, acting as intermediary, 0.2% per annum and will appear equally attractive to X and Y. Company x: fixed is 8% and float is LIBOR Company y fixed is 8.8% and float is LIBOR

The spread between the interest rates offered to X and Y is 0.8% per annum on fixed rate investments and 0.0% per annum on floating rate investments. This means that the total apparent benefit to all parties from the swap is 0.8% per annum. Of this 0.2% per annum will go to the bank. This leaves 0.3% per annum for each of X and Y. In other words, company X should be able to get a fixed-rate return of 8.3% per annum while company Y should be able to get a floating-rate return LIBOR + 0.3% per annum. The required swap is shown in Figure S7.1. The bank earns 0.2%, company X earns 8.3%, and company Y earns LIBOR + 0.3%.

An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction. Identify six different strategies the investor can follow and explain the differences among them. Possible strategies are: Strangle Straddle Strip Strap Reverse calendar spread Reverse butterfly spread

The strategies all provide positive profits when there are large stock price moves. A strangle is less expensive than a straddle, but requires a bigger move in the stock price in order to provide a positive profit. Strips and straps are more expensive than straddles but provide bigger profits in certain circumstances. A strip will provide a bigger profit when there is a large downward stock price move. A strap will provide a bigger profit when there is a large upward stock price move. In the case of strangles, straddles, strips and straps, the profit increases as the size of the stock price movement increases. By contrast in a reverse calendar spread and a reverse butterfly spread there is a maximum potential profit regardless of the size of the stock price movement. strip - more money if price goes down strap - more money if price goes up

#6 A company declares a 2-for-1 stock split. Explain how the terms change for a call options with a strike price of $60.

The strike price is reduced to $30, and the option gives the holder the right to purchase twice as many shares.

Chap 7 Interest rate swaps another way of thinking of it

Through bank: A borrows fixed at 10%, B borrows at LIBOR at 1.0%. A pays banks LIBOR, Bank pays A 9.90%, Bank pays B LIBOR, B pays bank 10%. Margin is .20% for A and B and 10% for bank.

7 What are the formulas for u and d in terms of volatility?

U and d are the up factor and down factor movement of the stock price

Mechanics of Currency SWAPS Exam Question

USA 8% $ and 11.6% e UK 10% $ and 12% e Diff: 2.0% and .4% inflation is lower in US 1) Margin 2-.4 = 1.6 2) Bank takes .4% so both A and B get .6% A : 11.6 - .6 = A will pay 11% B: 10-.6 = B will pay 9.4% so A borrows at 8%, B borrows at 12%, Through Bank A will lend $15m and B will lend 10m e. A will get 8% for $ and will pay 11% for e B will pay 9.4% for $ and gets 12% for e. so A and B will get .6% and the bank will get .4 if exchange rates are stable.

Value Currency SWAPS

VSwap = S bf (bond fc) -B d (bond$) S = spot rate exchange ($/FC)

Chapter 12 Binomial Model

Value of a Call = c = f(S, K, T, r, σS, D). Call Option is more valuable if: a. The underlying stock price (S) increases; b. You have right to buy at a lower strike price (K decreases); c. The time to maturity (T) increases; d. The risk-free interest rate (r) increases; e. The underlying stock price is more volatile (σS increases)

Valuing Swaps

Vswap = Bfix - Bfl Bfl = L when SWAP is entered Bfl = L = at par Fixed: PV of future cash flows at time SWAP is entered Floating: immediately after a payment date Bfl = L K* is floating rate payment (LIBOR) to be made at t1

#7 Employee stock options are different from regular exchange-traded stock options because they can change the company's capital structure." Explain this statement.

When an employee stock option is exercised, the company issues new shares and sells them to the employee for the strike price. This increases the company's equity and therefore changes its capital structure.

#4 Explain why brokers require margin when clients write options but not when they buy options.

When an investor buys and options, cash must be paid up front. There is no possibility of future liabilities and therefore no need for a margin account. When an investor sells an option, there are potential future liabilities. To protect against the risk of a default, margins are required.

Explain why the arguments leading to put-call parity for European options cannot be used to give a similar result for American options.

When early exercise is not possible, we can argue that two portfolios that are worth the same at time T must be worth the same at earlier times. When earlier exercise is possible, the argument falls down. Suppose that P + S > C + Ke^(-rT) This does NOT lead to an arbitrage opportunity. If we buy the call, short the put, and short the stock, we cannot be sure of the result because we do not know when the put will be exercised.

#15 Explain carefully the difference between writing a put option and buying a call option

Writing a put gives a payoff of min(St-K,0) Buying a call gives a payoff of max(St-K,0) In both cases the potential payoff is St-K. The difference is that for a written put the counter party chooses whether you get the payoff (and will allow you to get it only when it is negative to you). For a long call you decide whether you get the payoff (and you choose to get it when it is positive to you.

#2 Design a swap that will net a bank, acting as intermediary, 50 basis points per annum. Make the swap equally attractive to the two companies and ensure that all foreign exchange risk is assumed by the bank. Company X yen - 5% dollars 9.6% Company y yen 6.5% dollars 10%

X has a comparative advantage in yen markets but wants to borrow dollars. Y has a comparative advantage in dollar markets but wants to borrow yen. This provides the basis for the swap. There is a 1.5% per annum differential between the yen rates and a 0.4% per annum differential between the dollar rates. The total gain to all parties from the swap is therefore 1.5% - .4% = 1.1 % per annum.The bank requires 0.5% per annum, leaving 0.3% per annum for each of X and Y. The swap should lead to X borrowing dollars at 9.6% - .3% = 9.3% per annum and to Y borrowing yen at 6.5 - .3 = .2% per annum.

Principal Protected Notes (B+C)

a fixed-income security structured product, comprised of a bond and an option component that promise a minimum return equal to the investor's initial investment if held to maturity buy bond (+B) and buy call (+C) if you buy a zero coupon, discount bond, the initial outlay (B) is small (if r is high) if volatility of S is low, call (C) is cheap, than the initial cost (B+C) may be set = k (PPN) then your principal is protected (worse outcome S<K, call OTM, get to keep bond payoff (k)

4 naked puts

a) 400 x (5 + .2 (38)-0) = $5,040 investor puts up $5,040 for this short position

#20 Options on General Motors stock are on a March, June, September, and December cycle. What options trade on (a) March 1, (b) June 30, and (c) August 5?

a) March, April, June and September b) July, August, September, December c) August, September, December, March. Longer dated options may also trade.

#17 Consider an exchange-traded call option contract to buy 500 shares with a strike price of $40 and maturity in four months. Explain how the terms of the option contract change when there is a) A 10% stock dividend b) A 10% cash dividend c) A 4-for-1 stock split

a) the option contract becomes one to buy 550 (500 x.1) shares with an exercise price 36.36 (40/1.1) b)there is no effect. the terms of an options contract are not normally adjusted for cash dividends c)The option contract becomes one to buy 2000 (500x4) shares with an excerise price of $10 (40/4)

Problem 12.15. Calculate , , and when a binomial tree is constructed to value an option on a foreign u d p currency. The tree step size is one month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.

a= e ^(0.05-.08)x 1/12= 0.9975 u = e ^ 0.12 sqrt 1/12 = 1.0352 d = 1/u = .9660 p = 0.9975-.9660/ 1.0352-.9660 = .4553

12.11A stock price is currently $40. It is known that at the end of three months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three-month European put option on the stock with an exercise price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

at the end of three months the value of the option is either $5 (if the stock price is $35) or $0 (if the stock price is $45). Consider a portfolio consisting of: 45p + 35(1-p)= 40x 1.02 10p=58 p=.58 The expected value of the option in a risk-neutral world is: 0 x 0.58 + 5 x 0.42 = 2.10 pv: 2.10 /1.02 =2.06

Chapter 13: Black / Scholes Model

binomial model coverges to black schloes model 1. Stock price (ST) follows the lognormal distribution; S follows a random walk; If ST has lognormal distribution, ln(ST ) & ln(ST /S) have normal distribution, can get negative returns!! d1= how far call is ITM N ( negative infinity) = 0 N (0) = .5 N ( positive infinity) 1 Thus, we can use same intuition as the Binomial Model. Call value is the expected payoff of the option discounted back to the present at the riskless rate. c = S * N(d1) - Ke^-r T * N(d2) p = Ke^-r T * N(-d2) - S * N(-d1) thus, the Black/Scholes formula can be interpreted as: [ Stock Price multiplied by hedge ratio ( x #shares you'll own) ] minus [ NPV (K) multiplied by probability that you'll pay K {P(call ITM)} ]. Black/Scholes Model: c = S0* N(d1) - Ke^-rT * N(d2) Thus, value of call today is expected future value of call, discounted back. So, call value = NPV of expected value of purchase if option exercised. 13.13 and 13.14 will not be one exam When ST gets large, call will likely be exercised. When ST gets large, put will not be exercised

S will rise

buy a call if right, s > k, buy at K, sell @s,worth (S-K) most you loose is what you bought for or sell a put right s > k keep price of put, wrong s<k will be exercised, must buy at k, sell at s , lose k-s

4 naked calls example

c = 5 k = 40 s = 38 OTM = s-k = -2 sale proceeds = 5 x 400 (c x 4 naked calls) = $2,000 calls = keep this if stays at 40 a) 400 x ( 5 +.2 (38) - 2) = $4,240 b) 400 x (5 + .1(38))= $3,520 investor puts up 4,240 for this short position, need another 2,240 to get into my margin account bc of sale proceeds of 2,000

Covered call example

c = 7 s = 63 k = 60 itm = s - k = $3 buy 200 shares and write 2 calls cost of shares = 63 x 200 = $12,600 margin allowed on stock purchase = -$6,300 ( 50% of cost of shares) price recieved for 2 calls= 7 x 200 = -$1,400 (can use sale proceeds to pay) minimal initial investment is 12,600 - 6,300 - 1,400 = $4,900

Two Period

c = { fuu p^2 + fud (1-p)p + fud p(1-p) + fdd (1-p)^2 } e ^-r 2ΔT The same interpretation: Call value is expected payoff over 2 periods discounted (twice) at (1-pd) riskfree rate. volatility of S appears directly in the formula. Thus, in 2-period model, σ2 = 2 p (1-p).

Lower Bound for Euro Call on non-dividend-paying Stock

c > (or equal) S - Ke^-rT If r higher, Ke^ -rT lower; call is more valuable; Then pay less today for bond that promises K at expiration; If r increases, don't have to tie up as much $ today. if interest rates increase don't have to put up as much money, call is more valuable American call will not be exercised early (if no dividend) C > (or equal) S - Ke-rT > S - K. want out ? Can exercise American call early, & receive S - K; (10 bucks) Or can sellAmerican call & receive C [ > S - K ] (11 bucks) Will never exercise American call early (if no dividend) American call acts like European call (if no dividend) American and European call are worth same: C = c. exam -exercise an american option early

Butterfly spread w calls

c1 - 2c2 + c3 , k1<k2<k3, c1>c2>c3 1) right buy 40, right to sell at 50 Buy 1 call w/ k1 and sell 2 calls w/ k2 buy 1 call w k3 initial outflow - left go up a lot, or down a lot = unlimited downside, limited downside

bear spread w calls

c2 - c1 sell call w/ k1 (receive c1), buy call with k2 (pay c2) 1) sell expense call 40, buy cheap call at 45, initial inflow on the left 2) sold is ITM 3) 45, ITM

Using options on ED futures to build floors, caps, and collars

call option - right to lend put option - right to sell Lender: want higher rates, buy ED in future, to hedge risk of loss w falling rate 1) buy ED futures, rates go down lock in min lending rate ( hedged) if rates go up, opportunity loss ( could have loaned at higher rates) 2) buy call option on ED futures, if rates go down lock in min lending rate, if rates go up lend at higher rates, call is OTM, interest risk FLOOR Borrow: want to lower rates, want to sell ED in future, to hedge risk off loss with rising rates 1) sell ED if rates go up, lock in max, borrowing rate (hedged) if rates go down opportunity loss ( could of borrowed at lower rates) 2) buy put option on ED futures, if rates go up, lock in max, borrowing rate, if rates go down, borrow at lower rates, put is OTM, interest rate CAP combining call and put on ED future gives COLLAR

Determination of u, d, & p

get the mean,variance 1) observe riskfree rate 2) pick volatility 3) define maturity know how high and low price can get if sigma is bigger, u is higher, d lower, option will be more valuable first condition for u, d, & p depends on the mean. second condition for u, d, & p depends on the variance of S over Δt: third condition is also often imposed:(3)u = 1 / d . observe r ; - pick σ & ∆t ; - get er Δt,u, d, & p

What if Stock Pays a Dividend?

if stock will pay a dividend with NPV = D, then expectS to ↓ by D when D is paid. The Binomial Tree can be modified to account for the expected decline in S, at the time of the dividend. Note: If S is adjusted down. by the amount D, then theTree does not re-combine following the dividend. This feature makes the analysis somewhat more cumbersome. But this is a simple task for computers.

Problem 13.22

if the volatility of a stock is σ = 18% per annum (i.e., over 1 year), estimate the standard deviation of the stock return over: (a)One day σ(1 day) = 18% x Sqrt( 1/252 ) = 1.1% (b)One week σ(1 week) = 18% x Sqrt( 1/52 ) = 2.5% (c)One monthσ(1 month) = 18% x Sqrt( 1/12 ) = 5.2%

#3 A $100 million interest rate swap has a remaining life of 10 months. Under the terms of the swap, six-month LIBOR is exchanged for 7% per annum (compounded semiannually). The average of the bid offer rate being exchanged for six-month LIBOR in swaps of all maturities is currently 5% per annum with continuous compounding. The six-month LIBOR rate was 4.6% per annum two months ago. What is the current value of the swap to the party paying floating? What is its value to the party paying fixed?

in four months .5 x.07 x 100 million = 3.5 million will be received and .5 x 0.046 x 100 million = 2.3 million will be paid in 10 months, 3.5 million will be received, and Libor rate prevailing in four months time will be paid. fixed: 3.5 e ^ -.05 x 4/12 +103.5 e ^ - .05 x 10/12 = 102. 718 million floating: (100 + 2.3)e^ -.05 x 4/12 = 100.609 million value of paying floating = 102.718 - 100.609 = 2.109 value paying fixed = - 2.109

2 Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.

in the no-arbitrage approach, we set up a riskless portfolio consisting of a position in the option and a position in the stock. By setting the return on the portfolio equal to the risk-free interest rate, we are able to value the option. When we use risk-neutral valuation, we first choose probabilities for the branches of the tree so that the expected return on the stock equals the risk-free interest rate. We then value the option by calculating its expected payoff and discounting this expected payoff at the risk-free interest rate.

True or false

long term option is worth more than short term? can be either, True American call is worth atleast european C > c or equal it is european plus more! American call gives all rights in European call, plus right to exercise early. Thus, American call must be at least as valuable as European call: C > c or equal

Chap 9 finishing slides problems

options clearing house - guarantees option writer will honor obligations purchased - funds deposited with OCC, buyer pay in full by morning of next business day exercise - notifies buyer, OCC randomly selects member with outstanding short position call - writer must sell @ k put - writer must buy @ k taxation - unless hedger, all gains taxed as capital gains, losses can be used to offset capital gains. long - close out with offsetting gains - realized then allowed to expire (out of the money) unexercised - realized then call is exercised - writer sold stock (K+C) used as a basis for later, buyer bought stock (K+C) gains/losses put exercised - buyer sold stock (K-P) writer bought stock (K-P) commissions are always deductible The Wash Sale Rule: buy stock at 60 for long term investment S decreases $40, may want to sell to realize tax loss and then buy back @ s = 40 gains PREVENTED by this rule cant deduct on tax bill within 30 days means any tax loss on the stock sale is disallowed

2.12 A stock price is currently $50. Over each of the next two three-month periods it is expected to go up by 6% (u=1.06) or down by 5% (d=.95). The risk-free interest rate is 5%per annum. What is the value of a six-month European call option with a strike price of $51

p = [e^.05 x (3 / 12) - .95 ] / [1.06 - .95] = .5689 1 - p = .4311 c = { $5.18 x .5689^2 + 2 x $0 x .5689 x .4311 + $0 x .4311^2 } e -.05 x (6 / 12) == $1.635

2.13 A stock price is currently $50. Over each of the next two three-month periods it is expected to go up by 6% (u=1.06) or down by 5% (d=.95). The risk-free interest rate is 5%per annum. What is the value of a six-month European put option with a strike price of $51?

p = [e^.05 x (3 / 12) - .95 ] / [1.06 - .95] = .5689 1 - p = .4311 put = { $0 x .5689^2 + 2 x $0.65 x .5689 x .4311 + $5.875 x .4311^2 } e^ -.05 x (6/12)=$1.376 Or Put - Call Parity: S+P = $50+$1.376 = $51.376 B+C = $51e^-.05 x (6/12) + $1.635 = $51.376

bull spread w puts

p1 - p2 k1 < k2 p1 < p2 -p1 + p2 > 0 1) buy at 40, sell expense one 45, give someone else the right to sell at 45 2) ITM, sold at k2, k2 to k1, loose a dollar 3) ITM, below 40 puts - selling expense one to buy cheap one calls - buying expense one to sell covers myself, insurance, if it goes below k1 don't loose k1 = most you will loose

Bear spread w puts

p2 - p1 k1 < k2 P1 < P2 + P1 -P2 <0 1) buy at 40, gives their right to sell at 40 2) below 45, one I bought is ITM 3) below 40, stop make money

The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and again in five months. Risk-free interest rates for all maturities are 10%. What is the price of a European put option that expires in six months and has a strike price of $30?

p= 2 + 30 e ^ -.1 x 6/12 + (.5e ^-0.1 x 2/12 + .5 e ^ -0.1 x 5/12) - 29 = 2.51 the put price is $2.51 Let D = NPV(expected dividends during life of option). Put-Call Parity; S + p = c + Ke&-rT : with D S + p = c + (Ke^-rT + D) p = c +(Ke^-rT + D) - S c + K e^ -r T+ D -S

Valuing SWAPS example:

receive fixed at 8%, semiannual, pay floating 6 month libor notional price L = 100 million remaining life = 1.25 years (15 months) payments in 3,9, and 15 months LIBOR zero rates= r1 = 10% r2= 10.5% and r3=11% 6 month libor on the last payment was 10.2% with semi annual comp k = $4.0 million (= ½ of 8.0% fixed coupon) k* = $5.1 million (= ½ of 10.2% last LIBOR) B fixed= 4 e ^ -.10 x .25 + 4 e ^ -.105 x .75 + 104 e ^-.11 x 1.25= $98.24 Million B floating = = 5.1 e ^ -.10 x .25 + 100 e ^-.10 x .25 = $102.51 million because you only know r1 and t1!!! VSWAP = Bfixed - Bfl = $98.24 - $102.51 = - $4.27 negative! loosing money! Receive fixed & pay floating; Today, fixed rate = 8% < floating rate = 10.2%; losing money!

Total value

s increases call option becomes more valuable, more expensive change in the variable not the axis, red curve shifts red curve Changes in S represent a movement along the red curve. Changes in K,T,r,σS,D mean a shift in the curve.

s will fall

sell a call, right s <k, keep price of call, wrong, s > k will be exercised might buy at s, sell at k, loose s-k, unlimited downside!! or buy a put, right, s < k, buy at s, sell at k, worth (k-s) wrong s > k loose price of input price can only go down to zero!

Problem 14.2. The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?

sigma = .3 assuming 252 trading days 1/252 = .004 .3 x .004 = 1.9%

Total value

sum of intrinsic ( can get right now), but worth more b/c chance that is is worth a lot TV = E + I

#22 A bank's position in options on the dollar-euro exchange rate has a delta of 30,000 and a gamma of -80,000 . Explain how these numbers can be interpreted. The exchange rate (dollars per euro) is 0.90. What position would you take to make the position delta neutral? After a short period of time, the exchange rate moves to 0.93. Estimate the new delta. What additional trade is necessary to keep the position delta neutral? Assuming the bank did set up a delta-neutral position originally, has it gained or lost money from the exchange-rate movement?

the delta indicates that when the value of the euro exchange rate increases by $0.01, the value of the bank's position increases by 0.01×30,000 = $300. The gamma indicates that when the euro exchange rate increases by $0.01 the delta of the portfolio decreases by 0.01 x 80,000= 800. For delta neutrality 30,000 euros should be shorted. When the exchange rate moves up to 0.93, we expect the delta of the portfolio to decrease by (0.93-.90) x 80,000 = 2,400 so that it becomes 27,600. To maintain delta neutrality, it is therefore necessary for the bank to unwind its short position 2,400 euros so that a net 27,600 have been shorted. As shown in the text (see Figure 17.8), when a portfolio is delta neutral and has a negative gamma, a loss is experienced when there is a large movement in the underlying asset price. We can conclude that the bank is likely to have lost money.

#5 "What is meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is highly negative and the delta is zero?"

the gamma of an option position is the rate of change of the delta of the position with respect to the asset price. For example, a gamma of 0.1 indicates that, when the asset price increases by a certain small amount, delta increases by 0.1 times this amount. When the gamma of an option writer's position is highly negative and the delta is zero, the option writer will lose money if there is a large movement (either up or down) in the asset price.

Chapter 10 List the six factors affecting stock option prices.

the six factors affecting stock option prices are the stock price, strike price, risk-free interestrate, volatility, time to maturity, and dividends. stock option price , stock price goes up good for call option strike price, k right to buy at a lower strike price , less valuable right, pay at 40 but dont have to pay as all if it was 100 instead of 40 longer maturity is better interest rate is good more volalititliy is good dividends decrease the stock price, if a european option on friday dont get dividend