Question

10. A stock is currently trading at $80. After one period, the price will either increase by 20% or decrease by 20% and the stock does not pay any dividend. The risk-free rate per period is 2%. Consider a put option on the stock with strike price of $70. Set up a replicating portfolio and price the option.

Answer 1

Please round off your answers as per your requirement.

Let the replicating portfolio be:

• Short (Sell) N number of stocks
• And lend an amount L at risk free rate

Current stock price, S₀ = $ 80;

Strike Price, K = $ 70;

Up state:

Stock price after one period in up state, S_1u = S₀ x (1 + 20%) = 80 x (1 + 20%) = $ 96

Hence payoff from Put option, P_u = Max (K - S_1u, 0) = Max (70 - 96, 0) = 0

Payoff from replicating portfolio = - N x S_1u + L x (1 + r_f) = - N x 96 + L x (1 + 2%) = -96N + 1.02L

Payoff from the replicating portfolio = Payoff from put option

Hence, -96N + 1.02L = 0; Hence, 1.02L = 96N -----------Equation (1)

Down state:

Stock price after one period in down state, S_1d = S₀ x (1 - 20%) = 80 x (1 - 20%) = $ 64

Hence payoff from Put option, P_u = Max (K - S_1d, 0) = Max (70 - 64 0) = 6

Payoff from replicating portfolio = - N x S_1d + L x (1 + r_f) = - N x 64 + L x (1 + 2%) = -64N + 1.02L

Payoff from the replicating portfolio = Payoff from put option

Hence, -64N + 1.02L = 6; Hence, 1.02L = 6 + 64N -------------Equation (2)

From equation (1) & (2)

6 + 64N = 96N

Hence, N = 6 / (96 - 64) = 6 / 32 = 0.1875

From equation (1): 1.02L = 96N = 96 x 0.1875 =  18.00

Hence, L = 18 / 1.02 =  17.6471

Hence, the replicating portfolio should be:

• Short (Sell) 0.1875 number of stocks
• And lend $ 17.6471 at risk free rate

And price of the option = Value of the replicating portfolio at period 0 = -N x S₀ + L = - 0.1875 x 80 + 17.6471 =  $ 2.6471