10.12 Let S = $100, K = $95, r = 8% (continuously compounded), σ = 30%, δ = 0, T = 1 year, and n = 3. a. Verify that the binomial option price for an American call option is $18.283. Verify that there is never early exercise; hence, a European call would have the same price. b. Show that the binomial option price for a European put option is $5.979. Verify that put-call parity is satisfied. c. Verify that the price of an American put is $6.678.

American Put Value

N. down moves 3 2 1 0

0 0 0 1.09077967 6.677901

1 0 2.0623567169 11.7087201

2 3.8993348796 20.4035172625

3 30.5715733249

Explanation:

Let S = $100, K = $95, r = 8% (continuously compounded), σ = 30%, δ = 0, T = 1 year, and n = 3. a. Verify that the binomial option price for an American call option is $18.283. Verify that there is never early exercise; hence, a European call would have the same price. b. Show that the binomial option price for a European put option is $5.979. Verify that put-call parity is satisfied. c. Verify that the price of an American put is $6.678.

Let S0 = $100, K = $95, r = 8% (continuously compounded), σ = 30%, δ = 0, T = 1 year, and n = 3.

a. Confirm that the binomial option price for an American call option is$18.283. Hint: there is no early exercise; therefore, a European call would have the same price.

S0 100

K 95

σ 0.3

δ 0

u 1.2212461202

d 0.8636925537

r 0.08

T 1

n 3

h 0.3333333333

p * 0.4568066592

American Call Value

N. down moves 3 2 1 0

0 87.1417860953 56.6440624107 33.1493175 18.28255

1 33.8147423997 15.0403285537 6.6897296

2 0 0

3 0

European Call Value

N. down moves 3 2 1 0

0 87.1417860953 56.6440624107 33.1493175 18.28255

1 33.8147423997 15.0403285537 6.6897296

2 0 0

3 0

The American option is never exercised early, and the American and European values are the s.

b. Demonstrate that the binomial option price for a European put option is 5.979%. Verify that put-call parity is satisfied.

European Put Value

N. down moves 3 2 1 0

0 0 0 1.09077967 5.978605

1 0 2.0623567169 10.3865484

2 3.8993348796 17.903663451

3 30.5715733249

Put call parity:

C - P: 12.3039470933

S-Ke^ ( - (r-5) T) 12.3039470933

c. Confirm that the price of an American put is $6.678

American Put Value

N. down moves 3 2 1 0

0 0 0 1.09077967 6.677901

1 0 2.0623567169 11.7087201

2 3.8993348796 20.4035172625

3 30.5715733249