BUSI 588, 2007 MAC Final Exam Solutions

BUSI 588, 2007 MAC Final Exam Solutions - Introduction to...

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Unformatted text preview: Introduction to derivatives, Fall 2008 BUSI 588, 2007 MAC final exam solutions MAC 866, Final Exam Solutions, Fall 2007 Problem 1 - Oratio Dominica (20 points) 1. The Black—Scholes formula for a call option is C = SN($) — fimx — (Ix/T) where 1 loge) — Iog<K/(1 + MT) 10 r“ __ a __.T +2 T. The term N (a: — ax/T) can be interpreted as the risk-neutral probability of the stock ending up above K, or the value today of a security that promises (1 + rf)T if and only if ST > K. IL‘ 2. Discuss whether you agree with the following two statements (keep the discussions separate, and use less than 50 words for each). (a) Disagree. “A put is equivalent to call, coupled with a short position in the underlying asset and lending.” (b) Disagree. “A call is equivalent to buying the underlying asset partially financed with borrowing, buying more of the underlying asset as the stock price goes up, using borrowing to finance these purchases (when the stock goes down you ought to sell the underlying and pay off part of the borrowed amount).” Problem 2 - Sevilla 1987 (30 points) It is 1987, and Gari Kasparov is a well-known financial trader in Sevilla, a beautiful city in Southern Spain. Gari Kasparov knows the prices of two forward contracts on aluminum, one with a 6 month maturity (t = 0.5) and one with a one year maturity (t = 1). The price of the former is F05 = 65.89 and the later is F1 = 67.84. Gari further knows the term structure is flat, and that the volatility of aluminum is 35%. Furthermore, he estimates storage costs and/ or convenience yields for aluminum to be negligible, 1. The future price formula Ft 2 + T‘f)t yields two equations with two unknowns: 6589 = 3(1 + rf)°-5; 67.84 = 5(1 + rf) which can be solved to find S = 64 and rf = 6%. Note that the only assumptions we really make here is that trading aluminum is possible and cheap (taking long and short positions does not incur in transaction costs), as well as the fact that traders can borrow/lend at the risk-free rate (on top of the storage/convenience yield assumptions made above). 2. Using the put—call parity one has K 65 P_C-S+——_5—64+1.060,5 = 4.13 (1 + T'f)T © Diego Garcia, Kenan—Flagler Business School Page 1 of 4 Introduction to derivatives, Fall 2008 BUSI 588, 2007 MAC final exam solutions 3. Both set of options seem to violate the put—call parity, although one needs to be careful, since the options trades are associated with non—zero bid-ask spreads. Using the put-call parity one sees the calls are slightly overpriced with respect to the puts. The following table shows how one could create an arbitrage trading the options with K = 60. Today ST < K ST 2 K Sell call 7.75 0 —(ST — 60) Put put -1.95 60 — ST 0 Buy asset , -64 ST ST Borrow PV(60) 58.28 —60 —60 Net +0.0772 0 0 Problem 3 - 50-cents (30 points) 1. The risk-neutral probabilities in this binomial world were 13“ = (1.01 —0.6) /(1.4~0.6) = 0.5125 and 13.1 = 0.4875. Working backwards through the put option tree one has that the price of the put is $8.53. 8.53 3.45 0.00 0.00 14.05 7.14 0.00 21.60 14.80 29.20 2. The replicating portfolios, composed in each node to by Aw shares of the stock and Ba, dollars in cash, can be solved at each point and state by using the recursion qu “ de Sum ‘" Swd where w is a node in the tree, and um and and denote the up and down nodes following it, X denotes the values of the derivative in each of thestates, and S the value of the underlying asset. Aw: Since we know the value of the put from the previous question, we can back out the cash account via Pu, = Awa + B“; => B“, = Pu, — Awa. The following tree gives the number of shares of the stock necessary to replicate the payoffs of the put 0.27 -0.13 0.00 ‘ -0.60 —0.44 —1.00 whereas the one following shows the amount of cash to be held in the replicating portfolio: 21.79 12.38 0.00 32.13 25.64 39.60 Problem 4 — Takacs Quartet (40 points) © Diego Garcia, Kenan—Flagler Business School * Page 2 of 4 Introduction to derivatives, Fall 2008 BUSI 588, 2007 MAC final exam solutions 1. The price of the three options are $12.7540, $10.4242 and $8.4891, where you should note the daily volatility can be converted into an annual volatility number (for consistency with the other Black—Scholes inputs, since we are measuring time and the risk-free rate in years) of on = \/2520.03 = 47.6%. 2. The options with a strike of $50 have a delta of 0.6333. Therefore, if we would like to be hedged, we would need to buy 716 such that 40000 + nc0.6333 = 0 i.e. short 63163 calls. 3. Let n50 denote the number of calls with a strike of $50 that the Takacs trade, and let n45 denote the number of calls with a strike of $45. The delta of their portfolio will be 40000 + 715006333 + 714507129 Furthermore, the gamma of their portfolio will be ' 715000158 + 714500143 Making the above two equations equal to zero will result in a portfolio that is hedged against movements in the underlying asset, and which does not need much rebalancing (at least for small movements in the underlying asset price). My calculations suggest 1150 = 259033 and 1145 = —286224 would achieve a delta and gamma neutral position. 4. One can verify that the portfolio from the previous question is already vega-neutral, i.e. if volatility changes the value of the portfolio will not change. Therefore the portfolio in which one buys n50 = 259033 options with a strike K = 50 and sells 7145 = -286224 options with a strike of K = 45 should satisfy Andras. Problem 5 - Torero bonds (30 points) 1. The risk—free rate could be readily estimated from the forward prices: 25.617 = 25(1+ rf)” so that T): = 5%. Using this into the Black-Scholes formula to find the implied volatilities of the three options listed in the problem yielded estimates for the volatility of olive oil of 32%, 30% and 28% (the options showed a strong volatility skew). 2. The payoffs from the bond can be replicated using plain-vanilla options as follows: S<2O SE(20,30) S>30 Torero bonds 100 100 + 6-20 110 Lend PV 100 100 100 100 BuycallK=20 0 3—20 8—20 Sell call K = 30 O O —(S — 30) Net 100 100 + g — 20 110 © Diego Garcia, Kenan—Flagler Business School Page 3 of 4 Introduction to derivatives, Fall 2008 BUSI 588, 2007 MAC final exam solutions The table shows how trading in risk-free bonds, and buying a bull spread on olive oil generates the same cash flows as those from the Torero bonds. 3. One needs to price the call options with a strike of $20 and those with a strike of $30 in order to come up with the value of the Torero bonds. Since there is a significant smile in the option prices, one can estimate that the implied volatility of options with K = 20 should be around 34%, and that of options with K = 30 should be around 26% (since the options at 22.5, 25 and 27.5 have implied volatilities 0f 32%, 30% and 28% respectively). The call with a strike of $20 has a Black-Scholes price of 6.84, whereas the one with a strike of $30 has a price of $1.34. The value of the Torero bonds is therefore 100 =— .4— .4: 1 .7 V 105—4—68 13 $00 3 Problem 6 - David Beckham (30 points) 1. The profits from this project, letting Pd denote the price of diamonds, is given by 7r 2 10(2500 — Pd) — 12000 = 13000 — 10Pd = 10(1300 — Pd) Since we do not produce if the profits are negative, we have that 7r = 10 max(1300 — Pd, 0). The payoffs from the project are therefore identical to those one could get from 10,000 put options on diamonds with a strike price of $1300. By LOOP, the value of the project and the value of these options should be the same. 2. If we produce right away we get profits of $1m. If we wait, we are holding 10,000 American put options on diamonds with a strike of $1300, which are valued at $351.5 each, i.e. by waiting our project is worth $3.5m. We should therefore wait. 3. Profits are now given by 10 7r = 10(1000 + £3, — Pd) — 12000 = $13,, ~ 2000 = F (Pd — 1200) or more precisely 7r = 10/ 6 max(Pd — 1200, O), i.e. profits are 10 / 6 American calls on diamonds with a strike price of $1200. In order to price these American calls, we first note that since diamonds are a non-dividend paying asset, the prices of American and European calls should be the same, so the problem reduces to finding the price of European calls. Further note that this implies will will wait until time 2 to produce (if and only if the price of diamonds at time 2 is above $1200). Using the put-call parity we have that the price of a European call with a strike of $1200 is $404.6, so the value of the project will be $674,441. © Diego Garcia, Kenan—Flagler Business School Page 4 of 4 ...
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BUSI 588, 2007 MAC Final Exam Solutions - Introduction to...

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