Question

A share of WTB stock sells for $50 and has a standard deviation of returns of 20 percent per year. The current risk free rate is 5% and a call option with a strike price of $50 expires in 3 months.

a. Using the Black-Scholes formula, what is the value of the call option?

b. If you found this option describe above selling for $1.50 more then what you calculated in part (a) above, would you want to buy it or sell it? Explain.

c. If you found a different option that had an exercise price of $55 and expired in 2 months, would you expect this option to have a higher or lower price than what you found in part (a). Explain without using any calculations.

Answer 1

(a) Black Scholes Formula for pricing options is as given below:

C = S x N(d1) - K x e^(-r x t) x N(d2) where K is the option strike price, S is the price of the underlying asset, r is the CC interest rate and t is the option maturity.

d1 = [ln (S/K) + (r + (s^(2))/2) x t] / s x (t)^(1/2) and d2 = d1 - s x (t)^(1/2) where s is the underlying asset's standard deviation.

Therefore, in this case, C = 50 x 0.56946 - 50 x 0.529893 x e^(- 0.05 x 0.25) = $ 2.307499 or $ 2.3075 approximately.

(b) If the option price is $1.5 above the no arbitrage BSM calculated price one would sell the higher price option now to gain (2.3075 + 1.5) = $ 3.8075

Lend the money for three months at the CC rate of 5 % per annum to gain back (3.8075 x e^(0.05 x 0.25)) = $ 3.8554

The actual option price after three months =(2.3075 x e^(0.05 x 0.25)) = $ 2.3665

The short position in the option can be covered by buying back the option at the then prevalent option price pf $ 2.3665 with the lending proceeds of $ 3.8554, thereby generating a riskless profit of (3.8554 - 2.3665) = $ 1.4889 approximately.

(c) As the option is out of money in case the strike price is $ 55 and has less time to reach the strike price (because this option's maturity is 2 months instead of 3 months) it has less probability of being in the money at expiry and rendering a profit to the option holder (as compared to the previous option from part(a)). Hence, this option would be priced lower than the one in part(a).