Question

Suppose the exchange rate is $1.21/C$. Let r $ = 7%, r C$ = 6%, u = 1.20, d = 0.84, and T = 1. Using a 2-step binomial tree, calculate the value of a $1.30-strike American put option on the Canadian dollar? PLEASE DO NOT ANSWER USING EXCEL

Correct answer is $.0.1669

Answer 1

S0 = 1.21; K = 1.30; r = 7%; rf = 6%; u = 1.20; d = 0.84; T = 1; n (number of steps) = 2

Exchange rate tree:

S0 = 1.21

S1u = S0*u = 1.21*1.20 = 1.452

S1d = S0*d = 1.21*0.84 = 1.0164

S2uu = S1u*u = 1.452*1.20 = 1.7424

S20 = S1u*d = 1.452*0.84 = 1.2197

S2dd = S1d*d = 1.0164&0.84 = 0.8538

Option value calculations:

For a put option, the payoff at the final node will be max(K - S2, 0)

For the other nodes, the payoff will be the maximum of max(K-S, 0) and ((Up*S+)+(Dp*S-)*Dr)

where S = exchange rate at the node; Up = probability of Up move; Dp = probability of Down move; Dr = discount factor for one step

T =1 and n = 2 so, discount factor (Dr) = e^(-r*(T/n)) = e^(-7%*0.5) = 0.96561

Up = (a-d)/(u-d) where a = e^((r-rf)*(T/n)) = e^((7%-6%)*(1/2)) = 1.0050

Up = (1.0050 - 0.84)/(1.20 - 0.84) = 45.84%

Dp = 100%-Up = 100%-45.84% = 54.16%

Payoff at node T = 2:
P2uu = max(1.30-1.7424,0) = 0

P200 = max(1.30-1.2197,0) = 0.0803

P2dd = max(1.30-0.8538,0) = 0.4462

Value at node T = 1:

P1uu = max[max(1.30-1.452, 0) and ((45.84%*0) + (54.16%*0.0803))*0.96561]

= max[0, 0,0420] = 0.0420

P1dd = max[max(1.30-1.0164, 0) and ((45.84%*0.0803) + (54.16%*0.4462))*0.96561]

= max[0.2836, 0.2689] = 0.2836

Value at node T = 0:

P0 = max[max(1.30-1.21,0) and ((45.84%*0.0420) + (54.16%*0.2836))*0.96561]

= max[0.090, 0.1669] = 0.1669

American put option value = 0.1669

Option price tree: