(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (i) (ii) (iii) (iv) The current price of the stock is $60. The call option currently sells for $0.15 more than the put option. Both the call option and put option will expire in 4 years. Both the call option and put option have a strike price of $70.
Calculate the continuously compounded risk-free interest rate. (A) (B) (C) (D) (E) 0.039 0.049 0.059 0.069 0.079
Solution to (1)
The put-call parity formula for a European call and a European put on a nondividendpaying stock with the same strike price and maturity date is C − P = S0 − Ke−rT. We are given that C − P = 0.15, S0 = 60, K = 70 and T = 4. Then, r = 0.039.
Remark 1: If the stock pays n dividends of fixed amounts D1, D2,…, Dn at fixed times t1, t2,…, tn prior to the option maturity date, T, then the put-call parity formula for European put and call options is C − P = S0 − PV0,T(Div) − Ke−rT, where PV0,T(Div) = ∑ Di e − rti is the present value of all dividends up to time T. The
i =1 n
difference, S0 − PV0,T(Div), is the prepaid forward price F0PT ( S ) . , Remark 2: The put-call parity formula above does not hold for American put and call options. For the American case, the parity relationship becomes S0 − PV0,T(Div) − K ≤ C − P ≤ S0 − Ke−rT. This result is given in Appendix 9A of McDonald (2006) but is not required for Exam MFE/3F. Nevertheless, you may want to try proving the inequalities as follows: For the first inequality, consider a portfolio consisting of a European call plus an amount of cash equal to PV0,T(Div) + K. For the second inequality, consider a portfolio of an American put option plus one share of the stock.
(2) Near market closing time on a given day, you lose access to stock prices, but some European call and put prices for a stock are available as follows: Strike Price $40 $50 $55 Call Price $11 $6 $3 Put Price...