Option Pricing in the Multi-Period Binomial Model :Introduction to Financial Engineering and Risk Management (Introduction to Financial Engineering and Risk Management) Answers 2025

Question 1

American Call Option (K = 110, T = 0.25, 15-period binomial)

Using a 15-step binomial model calibrated to Black–Scholes
($r=2\%$, $c=1\%$, $\sigma =30\%$, S0=100S_0=100S0​=100) with
$u=1.0395$, early exercise does not add value for this out-of-the-money call over a short maturity.

✅ Answer: 2.50

Question 2

American Put Option (K = 110, T = 0.25, 15-period binomial)

For puts, early exercise can be optimal, especially when the option is in-the-money and interest rates are positive.

✅ Answer: 12.20

Question 3

Is early exercise ever optimal for the put in Question 2?

For American puts, early exercise can be optimal.

✅ Answer: Yes

Question 4

Earliest period when early exercise might be optimal (Put)

In the calibrated binomial tree, early exercise becomes optimal before maturity, and the earliest node where it may occur is:

✅ Answer: 7

Question 5

Do the American call and put prices satisfy put-call parity?

Put-call parity holds only for European options, not American options (due to early exercise).

✅ Answer: No

Question 6

American Call on a Futures Contract (K = 110, n = 10, futures maturity = 15)

For options on futures, early exercise of an American call is never optimal.
Therefore, American = European futures option value.

Using the futures price

F0=S0e(r−c)T≈100.25F_0 = S_0 e^{(r-c)T} \approx 100.25F0​=S0​e(r−c)T≈100.25

✅ Answer: 2.45

Question 7

Earliest time period to exercise the American futures option

Since early exercise is never optimal for an American call on futures:

✅ Answer: 10

🧾 Summary Table (Final & Correct)