3 + 30e 0.10(3/12) > <

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1 Millersville University Department of Mathematics MATH 472, Financial Mathematics, Homework 06 November 8, 2011 Please answer the following questions. Partial credit will be given as appropriate, do not leave any problem blank. Each problem is worth ten points. Your completed assignment will be due at class time on Tuesday, November 15, Suppose the price of a non-dividend-paying stock is $31, three-month options on the stock have a strike price of $30, and the risk-free, continuously compounded interest rate is 10%. (a) If the price of a European call option is $3 and the price of a European put option is $2.25, describe the arbitrage opportunity that is present. Recall the Put-Call Parity Formula C e + e rt = P e + S where in this situation e 0.10(3/12) < < An investor can buy the call and sell the put and the stock. Thus the initial cash flow is P e + S(0) C e = At expiry: if S(T) > 30 the call will be exercised and the stock purchased for $30 to close out the short position in the stock. The put will expire unused. if S(T) < 30 the call will expire unused and the put option will be exercised. The investor purchases the stock for $30. The final cash flow is = 0.25 > 0. Thus a risk-less positive profit has been generated with no initial investment. (b) If the price of a European call option is $3 and the price of a European put option is $1, describe the arbitrage opportunity that is present. Recall the Put-Call Parity Formula C e + e rt = P e + S where in this situation e 0.10(3/12) > > 32. An investor can buy the put and the stock and sell the call. Thus the initial cash flow is C e P e S(0) = 29 which must be borrowed at the risk-free interest rate. At expiry: if S(T) > 30 the call will be exercised and the stock sold for $30. The put will expire unused. if S(T) < 30 the call will expire unused and the put option will be exercised. The investor sells the stock for $30. The final cash flow is 30 29e 0.10(3/12) = > 0. Thus a risk-less positive profit has been generated with no initial investment. 2. What is the lower bound for the price of a three-month European call option on a nondividend-paying stock when the stock price is $40, the strike price is $38, and the risk-free, continuously compounded interest rate is 6%?

2 Using the following inequality C e S(0) e rt 40 38e 0.06(3/12) = Show that a lower bound for the price of a European put option on a non-dividend-paying stock whose current price is S(0) is P e e rt S(0). The strike price is, the risk-free, continuously compounded interest rate is r and the strike time is T. Starting with the Put-Call Parity Formula we see that C e + e rt = P e + S(0) e rt P e + S(0) P e e rt S(0). 4. What is the lower bound for the price of a four-month European put option on a non-dividendpaying stock when the stock price is $40, the strike price is $42, and the risk-free, continuously compounded interest rate is 8%? Using the following inequality P e e rt S(0) = 42e 0.08(4/12) 40 = European call and put options on a stock each have a strike price of $20 and expire in three months. Each type of option costs $3. The risk-free, continuously compounded interest rate is 10%. The current stock price is $19 and a single dividend of $1 will be paid in one month. Describe the arbitrage opportunity open to an investor. According to the Put-Call Parity Formula for a stock which pays a single dividend. In this case C e + e rt = P e + S De rt e 0.10(3/12) > (1)e 0.10(1/12) > An investor should purchase the stock and the put and sell the call. The funds to create this portfolio can be borrowed at the risk-free rate. The investor will collect the dividend. Thus the initial cash flow is C e P e S(0) = 19. At expiry:

3 if S(T) > 20 the call will be exercised and the stock sold for $20. The put will expire unused. if S(T) < 20 the call will expire unused and the put option will be exercised. The investor sells the stock for $20. The final cash flow is e 0.10(3/12) = > 0. Thus a risk-less positive profit has been generated with no initial investment. 6. Assume that the price of a share of stock is S(0) and that the strike price of a call option on the stock is = S(0). Draw graphs showing the profit or loss at expiry of the following portfolios. (a) Long position in one share of stock and short position in one call option. The profit/loss from the short position in the call option is C e (S(T) ) +. The profit/loss from the long position in the stock is S(T). These profits/losses are shown as dashed curves in the figure below. The net profit/loss is S(T) +C e (S(T) ) + = C e ( S(T)) +. The net profit/loss is shown as the solid thick curve below. C (b) Long position in two shares of stock and short position in one call option. The profit/loss from the short position in the call option is C e (S(T) ) +. The profit/loss from the long position in the stocks is 2(S(T) ). The profit/loss is C e ( S(T)) + + S(T). C

4 (c) Long position in one share of stock and short position in two call options. The profit/loss from the short position in the call option is 2(C e (S(T) ) + ). The profit/loss from the long position in the stock is S(T). The net profit/loss is 2(C e ( S(T)) + ) S(T) +. 2 C (d) Long position in one share of stock and short position in three call options. The profit/loss from the short position in the call option is 3(C e (S(T) ) + ). The profit/loss from the long position in the stock is S(T). The net profit/loss is 3(C e ( S(T)) + ) 2S(T) C 7. A diagonal spread is created by buying a call option with strike price 2 and exercise date T 2 and selling a call option with strike price 1 and exercise date T 1. Assume that T 1 < T 2. Draw graphs showing the profit or loss at expiry T 2 when (a) 1 < 2, There are two cases to consider here. C( 2 ) > C( 1 ): in this case the long call is worth more than the short call as may happen when expiry T 2 is much later than expiry T 1. The profit/loss from the short call is C( 1 ) (S(T 1 ) 1 ) +. The profit/loss from the long call is 1 2 +C( 2 ). The net profit/loss is C( 1 ) + C( 2 ) (S(T 1 ) 1 ) +.

5 1 2 C( 2 ) < C( 1 ): in this case the long call is worth less than the short call. The profit/loss from the short call is C( 1 ) (S(T 1 ) 1 ) +. The profit/loss from the long call is C( 2 ). The net profit/loss is C( 1 ) + C( 2 ) (S(T 1 ) 1 ) (b) 2 < 1. In this situation the long call must be worth more than the short call since the long call has the lower strike price and matures later than the short call. The profit/loss from the short call is C( 1 ) (S(T 1 ) 1 ) +. The profit/loss from the long call is C( 2 ). The net profit/loss is C( 1 ) + C( 2 ) (S(T 1 ) 1 )