Question

A stock index currently stands at 300 and has a volatility of 20%. The risk-free interest rate is 8% and the dividend yield on the index is 3%.

1. Use the Black-Scholes-Merton formula to calculate the price of a European call option with strike price 325 and the price of a European put option with strike price of 275. The options will expire in six months.
2. What is the cost of the range forward created using options in Part (a)?
3. Use a two-step binomial tree to evaluate a six-month American put option on the index with a strike price of 300.

Answer 1

a. Black-Scholes-Merton formula
European Call option - C
C= S N(d1) - X^-rt N(d2)

Parameters given                      
s   325       t   0.5   volatility(v)   0.2
r   0.08       x   300      

Find d1 = Ln (S/x)+(r+v^2/2)* t / v 0.5

d1 = 0.9195

d2 = d1 - vt

d2 = 0.778

Now, C = 325*Normaldis(d1) - x ^ -rt * Normaldis(d2)

C = 41.526

b. Range forward price

F = S * e ^ (r*t)

here, e = the mathematical irrational constant approximated by 2.7183

= 325* e^ (0.08*0.5)

F =  338.26



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