Question

In the following, S0 is the stock price in dollars as of today, K is the strike price in dollars, r is the continuously-compounded risk-free interest (as a decimal), q is the continuous dividend yield (as a decimal), sigma is the volatility (as a decimal) and T is the time to maturity in years. Compute option prices in dollars for the following types of options and the following parameter values with a three step binomial tree.

A. S0 = 100, K = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1. European call. What is the price in dollars today?

B. S0 = 100, K = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1. European put. What is the price in dollars today?

C. S0 = 100, K = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1. American call. What is the price in dollars today?

D. S0 = 100, K = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1. American put. What is the price in dollars today?

We define the “Early exercise premium” to the difference between the American option price and the corresponding European option price. For the case of call options, and S0 = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1.

E. What is the Early exercise premium when K = 100?

F. What is the Early exercise premium when K = 125?

G. What is the Early exercise premium when K = 75?

H. What is the Early exercise premium when K = 60?

I. In the case that S0 = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1, and for an American call option with strike K = 60, what is the price in dollars today?

J. In the case that S0 = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1, and for an American call option with strike K = 60, at what time would a holder of the option optimally exercise?

All the questions above set the dividend yield to q = 0.05. Change the dividend yield to q = 0.10 and answer the questions corresponding to questions E, F, G and H above.

K. What is the Early exercise premium when the dividend yield q = 0.10 and K = 100?

L. What is the Early exercise premium when the dividend yield q = 0.10 and K = 125?

M. What is the Early exercise premium when the dividend yield q = 0.10 and K = 75?

N. What is the Early exercise premium when the dividend yield q = 0.10 and K = 60?

Now set the dividend yield q = 0.10 and the strike K = 75. . For the case of the European call, q = 0.10 and K = 75, what is the Delta (position in stock to make portfolio of the stock and a short position in one option riskless).

P. For the case of the American call, q = 0.10 and K = 75, what is the Delta (position in stock to make portfolio of the stock and a short position in one option riskless).

Answer 1

a) we can use black scholes hypothesis

So here we have S0 = 100, K = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1

Now , d1 = [ln(100/100) + (0.07-0.05 + 0.32/2)1]/0.3 = 0.065/0.3 = 0.22

d2 = 0.22 - 0.3 = -0.08

So N(d1) = 0.5871

N(d2) = 0.4681

C = 100 e-0.05 x1x0.5871 - 100 e-0.07 x 1 x 0.4681 = 100 x 0.5871 x 0.95 - 100 x 0.93 x 0.4681 = 55.7745 - 43.5333 = 12.24

b) P (put) = C + Xe-rT - Se-T = 12.24 + 100e-0.07 x 1 - 100e-0.05x1 = 12.24 + 93 - 95 = 10.24