Question

6.10. Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is six months. a. Calculate , , and for a two step tree b. Value the option using a two step tree.

Answer 1

Part (a)

Current stock price, S₀ = $40

K = the strike price = $40,

the risk-free rate, r_f = 4% per annum,

σ = the volatility = 30% per annum,

and the time to maturity, T = six months = 0.5 year

N = number of steps = 2

Hence,

Δt = T / N = 0.5 / 2 = 0.25

d = 1 / u = 1/1.1618 =  0.8607

Part (b)

At t = 0, Stock price, S₀ = 40

At the end of first step, there are two states of stock prices:

• S_u = u x S₀ = 1.1618 x 40 =  46.47337
• S_d = d x S₀ = 0.8607 x 40 =  34.42832

At the end of second step, there are three states of stock prices:

• S_uu = u² x S₀ = 1.1618² x 40 =   53.99435  
• S_ud = u x d x S₀ = 1.1618 x 0.8607 x 40 =  40
• S_dd = d² x S₀ = 0.8607² x 40 =  29.63273

Stock Price Tree:

At node 2 i.e. on expiration:

Payoff from call option = max (S_T - K, 0)

When S_T =  Suu = 53.99435; payoff from call option, C_uu = max (S_uu - K, 0) = max (53.99435 - 40, 0) = 13.99435

When When S_T =  Sud = 40; payoff from call option, C_ud = max (S_ud - K, 0) = max (40 - 40, 0) = 0

When S_T =  Sdd = 29.63273; payoff from call option, C_dd = max (S_dd - K, 0) = max (29.63273 - 40, 0) = 0

Hence, value of call option at the end of first period at the point where stock price was S_u = C_u

= 6.871376

Hence, value of call option at the end of first period at the point where stock price was S_d = C_d =

= 0

Hence, price of the call option at t = 0 i.e. today

The call option tree can then be mapped as:

Hence the value of the option today = $ 3.373919