Question

In: Finance

Value of a stock is currently at $40. Volatility of that stock is 30% per year...

Value of a stock is currently at $40. Volatility of that stock is 30% per year and risk-free interest rate with
continuous compounding is at 5% per year. Suppose you are planning to value a 3-month European call
option with strike price at $41 using a two-step binomial model. Answer the following using this
information.

What is the value of the option at present?

Solutions

Expert Solution

S(0) = $40

Vol = 30%, therefore, size(up) = 1 + 0.3 = 1.3 and size(down) = 1 - 0.3 = 0.7

r(f) = 5%

Strike price = X = $41

each step time period = 3/2 = 1.5 month = 0.125 year

Prob(up) = (ert - size(down))/(size(up) - size(down))

Prob(up) = (e(0.05*0.125) - 0.7)/(1.3 - 0.7) = 0.5104 = 51.04%

Prob(down) = 1 - Prob(up) = 1 - 51.04% = 48.96%

At t = 0 t = 1.5months t = 3months
52 * 1.3 = $67.6

max(67.6 - 41, 0)

value of call = $26.6

40 * 1.3 = $52

Value of call = $13.4920

$40

Value of call = $6.8436

52 * 0.7 = $36.4

max(36.4 - 41, 0)

value of call = $0

40 * 0.7 = $28

Value of call = $0

28 * 0.7 = $19.6

max(19.6 - 41, 0)

value of call = $0

value of call (u,u) = 26.6

value of call (u,d) = 0

value of call (d,d) = 0

Therefore,

value of call (u) = ((0.5104 * 26.6) + (0.4896 * 0)) * e-(0.05 * 0.125)

Value of call (u) = 13.5766 * 0.9938 = 13.4920

Value of call (d) = 0

Value of call (t=0) = ((0.5104 * 13.4920) + (0.4896 * 0)) * e-(0.05 * 0.125)

Value of call (t=0) = 6.8863 * 0.9938 = 6.8436

Hence, value of call option at present = $6.84


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