MATH 425 EXERCISES G. BERKOLAIKO

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1 MATH 425 EXERCISES G. BERKOLAIKO 1. Definitions and basic properties of options and other derivatives 1.1. Summary. Definition of European call and put options, American call and put option, forward (futures) contract, prepaid forward contract. Their payoffs. Payoff and profit diagrams of option spreads Portfolio lemma, its application to questions such as put-call parity, futures price, estimates max(s Ee rt, 0) C S and max(ee rt S, 0) P Ee rt Exercises. Exercise 1.1. (*) Plot ask prices of SPX or SPY call options as functions of strikes. Use 20 strike points around the money. Make this plot for 3 different expirations one month apart from each other. Exercise 1.2. In class we compared PNL diagrams for 3 long calls (95C@5.50, 100C@2.70, 105C@1.15). Draw PNL diagrams for: short 95C@5.50, 100C@2.70, 105C@1.15; three plots together on Fig.1 long 95P@1.05, 100P@2.50, 105P@5.50; three plots together on Fig.2 short 95P@1.05, 100P@2.50, 105P@5.50; three plots together on Fig.3 Exercise 1.3. Consider two put options with the same expiration but different strikes, E 1 and E 2 (with E 1 < E 2 ). Formulate the conditions on their prices P 1 and P 2 so that the PNL graphs of the two puts (both long) intersect. Justify your answer by drawing all possible types of the graphs arrangement. Hint: there are three arrangements (not counting the borderline cases). Exercise 1.4. One day, Alice and Bob observe the following option prices (assume no bid-ask spread and r = 0): 95C@6.30, 105C@3.10, 110C@1.00. Alice says that one can make some guaranteed profit by buying one 95C, selling three 105C and buying two 110C. But Bob says he read that writing (selling) call options exposes one to the risk of unlimited loss. Draw the PNL diagram for the spread suggested by Alice to see who is correct. Exercise 1.5. Which of the following is a way to profit from a falling price of a stock: buying the stock, short-selling the stock, buying a call on the stock, selling a call on the stock, buying a put on the stock, selling a put on the stock? Point out what is potential risk (limited / large) and reward in each case. Exercise 1.6. Plot the payoff of the butterfly spread: 1 long call with strike E 1 = 90, 1 long call with strike E 3 = 110 and 2 short calls with strike E 2 = 100. Exercise 1.7. Which combination of two calls could have the profit diagram shown in Fig. 1(a)? Which combination of two puts could have the same profit diagram? (You do not have to specify the prices for the options, just short/long and strikes). Exercise 1.8. Argue that it is never profitable to exercise an American call early (assuming the stock pays no dividends). Therefore, its value is identical to that of a European call. (Hint: show that value of an American call is larger than S t E at any time t before expiration). Exercise 1.9. Prove that the call price C t (E, T ) satisfies 1 C(E 2) C(E 1 ) E 2 E 1 0. (1) 1

2 2 G. BERKOLAIKO (a) (b) Figure 1. Profit diagram of two bull spreads. Note the vertical scale difference. (You may assume that r = 0.) Prove that, if C is differentiable with respect to E, then this is equivalent to 1 C 0. (2) E Hints: For the first part you may (a) see Exercise 1.3 and use P-C parity or (b) use the Portfolio Lemma. For the second part, let E 2 = E 1 + E 1 and take the limit E 1 0 to get from (1) to (2). To go from (2) to (1), use Mean Value Theorem. Exercise (*) Prove that the call price C t (E, T ) is a convex function of the strike price E, i.e. where E = α 1 E 1 + α 2 E 2 and α 1 + α 2 = 1. (Hint: consider first α 1 = 1 2 Exercise Prove the inequalities by using Portfolio Lemma. C(E) α 1 C(E 1 ) + α 2 C(E 2 ), (3) and price a butterfly). P t Ee r(t t) S t, (4) P t 0, (5) P t Ee r(t t) (6) Exercise Prove inequalities (4) (6) from the corresponding inequalities for the calls by using Put-Call Parity. Exercise Which combination of calls could have the profit diagram shown in Fig. 1(b)? Which combination of puts could have the same profit diagram? Compare the cost of two spreads using Put-Call Parity (assume r = 0). Exercise You buy a Call and sell a Put with the same strike E = S 0 e rt. (1) Plot the payoff diagram of your spread; calculate the formula for it and simplify it. (2) Derive the price for your spread from first principles (e.g. using the Portfolio Lemma). (3) Derive the price for your spread from the put-call parity. (4) Why is such a spread called synthetic forward? Exercise A bull call spread is composed of a long call at strike E 1 and a short call at strike E 2, with E 2 > E 1. (1) Graph the payoff diagram for the bull call spread. (2) Prove / argue that the setup cost (the price of the spread) must be greater than 0. (3) Hence show that the call price C(S, E, τ) is decreasing as a function of E (other parameters being kept constant).

3 MATH 425 EXERCISES 3 Exercise From Options Industry Council: A long call condor consists of four different call options of the same expiration. The strategy is constructed of 1 long in-the-money call, 1 short higher middle strike in-the-money call, 1 short middle out-of-money call, 1 long highest strike out-of-money call. Assume the spot price is S 0 = 60. Consider the following spread: 1 long , 1 short , 1 short , 1 long (1) Sketch the payoff and the profit diagrams. (2) Does buying the above spread represent a bet that the stock price will grow, fall or stay flat? (3) Create a spread with the identical payoff but using only puts. Exercise (Based on a real story) You observe that almost every time a certain Twitter account issues a tweet, the market moves strongly. Sometimes it moves up and sometimes it moves down. To make money from either move, you write a program that tracks the updates to this Twitter account and buys a straddle spread on SPY. A straddle consists of one (long) call and one (long) put with the same strike prices. SPY is a fund that tracks the stock-market index. For your strike you use the current (spot) price of SPY. At 4pm on , the twitter account becomes active and your program makes a purchase. The spot price is S 0 = 237, so your program buys 100 calls with strike E = 237 and expiration in 3 days at the premium $0.73 per call and also 100 puts with the same strike and expiration at the premium $0.72 per put. (1) Draw the payoff diagram of your spread. (2) Draw the profit diagram of your spread. Label your axes and choose the appropriate scale for them. For simplicity, take r = 0. (3) By how much does the SPY price have to swing so that your purchase generates profit. Does the direction of the swing matter? (4) Is it a coincidence that the option prices quoted above are so close? Support your answer with equations. Exercise You purchase a long call ladder spread by buying one 90 Call for $12.35, selling one 100 Call for $3.15 and selling a 110 Call for $1.20. (1) Plot the payoff diagram of your spread. Label your axes! (2) Plot the profit diagram of your spread (assume, for simplicity, that r = 0). (3) Would the spread be worth buying if the 90 Call was priced at $15.00 while the other prices remained the same? Exercise The so-called box spread consists of four options: long E 1 call, short E 1 put, short E 2 call and a long E 2 put. (1) Calculate the payoff from a box spread at expiration, in terms of E 1 and E 2. (2) Use put-call parity to calculate the price of the box spread at time τ = T t before expiration, if the risk-free rate is r > 0. You do not need to know the prices of the individual options to price the box spread! (3) Give an explanation for your answer from the no-arbitrage point of view. (4) Profits from option trading are often taxed at a reduced rate (because the investor undertakes risk), when compared to tax rate on wage or (risk-free) interest earnings. Why do you think the IRS takes a dim view of using box spreads? 2. Binomial Trees, Basic 2.1. Summary. 1-level binomial tree for a call, fair price and the hedging procedure 1-level binomial tree for a general derivative, fair price and the hedging procedure many-level binomial tree for a European option; hedging

4 4 G. BERKOLAIKO many-level binomial tree for an American option; hedging short-cut formula for the tree 2.2. Exercises. Exercise 2.1. Repeat the derivation (from the first principles) of the 1-level tree formula for the price P of a put. Assume, as usual, that S d < S 0 e rt < S u and S d < E < S u. If you wrote a put and want to hedge, do you buy or sell stock? Exercise 2.2. Obtain the formula for the value of a put by choosing the appropriate values of V u and V d in the formula for the value of a general derivative. Show that it agrees with the answer you got in Exercise 2.1. Exercise 2.3. You assumed the 1-level tree model, priced and sold a call with strike E, S d < E < S u, followed the hedging procedure but the market ended up at S 1, S d < S 1 < S u. Did you lose or make money? You may assume r = 0 for simplicity. Exercise 2.4. You assumed the 1-level tree model, priced and sold a call with strike E, S d < E < S u, followed the hedging procedure but the market ended up at S 1, S 1 < S d. Did you lose or make money? What if the market ended up at S 1 > S u? How do your answers fit with the rule of thumb selling options is selling volatility? Look up Volatility (Finance) on Wikipedia, if necessary. You may assume r = 0 for simplicity. Exercise 2.5. (*) Assume the 1-level tree model with r = 0 and S d < S 0 < S u and plot the call price C = (S 0 S d ) S u E S u S d for a range of strike prices E that starts below S d and ends above S u. Observe that our call price violates the bound we derived earlier C max(s 0 E, 0). This suggests the above formula is not valid for E < S d or E > S u. Derive the appropriate formulas for the call prices in these ranges of E. Exercise 2.6. For the European Put with the parameters S 0 = 100, E = 110, r = 0, expiration in 3 months, use the 3-level tree model ( t = 1/12) with u = 1.1, d = 0.9. Calculate the option price and deltas. Generate two random paths through the tree and describe hedging procedure and results. Exercise 2.7. For the American Call with the parameters S 0 = 100, E = 120, r = 0.05, expiration in 3 years, use the tree model with t = 1, u = 1.2, d = 0.8. Calculate the option price. At every node also calculate the payoff from the early exercise and confirm that it is less than the recursively calculated option value. Exercise 2.8. For the American Put with the parameters S 0 = 100, E = 120, r = 0.05, expiration in 3 months, use the tree model with t = 1/12, u = 1.1, d = 0.9. Calculate the option price and deltas. Consider the price path: Up, Down, Up and describe hedging procedure and results. Exercise 2.9. Price the following American put option using the tree model: stock price now is S 0 = 100, strike is E = 115, interest rate is 5%, time to expiration 3 months. (1) Construct the 3-level tree (1 level per month) if the stock price is expected to either go up by 10% or go down by 5%. Price the above option using this tree. (2) Describe the hedging procedure undertaken by a writer of the option who seeks to eliminate risk. You must assume that the holder of the option will do what is best for them (i.e. exercise early if optimal). Consider the price paths: (a) Up, Down, Up

5 (b) Down, Up, Up MATH 425 EXERCISES 5 Exercise You go to lunch with a financial guru. In strictest confidence, they tell you that in 1 year s time the E&R 500 stock index will be either at 2000 or at 2300 points. The index is currently at Acting upon this information and using the 1-level binomial tree model, you price and sell 100 calls for the index at the strike E = In the call price, you include a $10 mark-up. Assume, for simplicity, that you have access to interest-free borrowing, i.e. r = 0. (1) What is the price at which you sell the calls (including the mark-up)? (2) Describe the hedging procedure you undertake. What is you total profit/loss if the index ends up at 2000? What is you total profit/loss if the index ends up at 2300? (3) If the guru is wrong and the index ends up at 2320, what is your total profit/loss? Exercise Calculate the prices for the spread consisting of one call and one put using the following settings: 3 level tree (one level per day), daily move is u = 1.01 or d = 0.99, S 0 = 237, E = 237, r = 0. Assume the options are European. (Hint: you can compute prices of two options separately and add together, but since you know the payoff for the whole spread you can use those values and make it a single calculation). Suppose the stock moves up, then up again and then down. Describe the delta-hedging procedure the writer undertakes, calculate cash balances at every node. What is the final balance? (If you round to cents every number you use, numerical error is not more than a few cents). 3. Random Variables 4. Asset Price Model and choosing tree parameters 5. Ito Formula and Stochastic Calculus 6. Black-Scholes-Merton equation