Question

A share of stock is currently worth $90 and has a volatility of 20%. The domestic risk-free interest rate is 5% and the stock does not pay any dividend. Use a two-step binomial tree to derive a) the value of a European four-month call option written on a 100 shares of stock with a strike price of $91 per share, and b) the position in shares of stock (long or short) which will hedge a short position in the European call option today.

Answer 1

All financials are in $

S0 = 90, K = 91, σ = 20% = 0.20, t = 4 months / 2 steps = 2 months = 2 / 12 year = 1/6 year; r = risk free rate = 5% = 0.05

Please see the tree diagram below. Stock price, call value and probability are all shown in the same diagram. Formula to calculate each one is also shown. The stock price at various node is shown in yellow colo

r and the call value at different node is shown in blue color.

Expected Value of the one call option at the end of period 2 = Sum of [joint probability x call value] =  0.2817 x 14.97 +  0.4981 x 0 + 0.22 x 0 = 4.2162

Value of one call option today, C₀ = Value at the end of 2 period x e^-2rt = 4.2162 x e^{-0.05 x 2 x 1/6} =  4.1465

Hence, the value of a European four-month call option written on a 100 shares of stock with a strike price of $91 per share = 100 x C₀ = 100 x 4.1465 = 414.65

In order to hedge the short position in the call option, we need to take long position in the stock.

Δ = (Cu - Cd) / (Su - Sd) = (6.66 - 0) / (97.66 - 82.94) =  0.4524

Hence, take long position in 100 x Δ = 100 x 0.4524 = 45.24 number of shares.

(You may round it off to 45 or 46 shares)