Financial Option Valuation: Binomial Model Applications

Problem 1: European Call Option Valuation

A stock price is currently at 40 euros. It is known that at month-end, the price will be either 42 euros or 38 euros. The risk-free interest rate is 8% per annum, continuously compounded. We have a call option with a strike price of 39 euros.

1. Shares for Riskless Portfolio: Calculate the number of shares to buy to create a riskless portfolio. Given Answer: 0.75 shares.
2. Cash for Replication: Determine the cash needed to replicate the portfolio at the end of the month. Given Answer: -57/2 euros.
3. One-Month European Call Value: What is the value of the one-month European Call option? Given Answer: 1.69 euros.
4. Put Option Value: What is the value of a corresponding put option? Given Answer: 0.43 euros.
5. Call-Put Parity Verification: Verify that call-put parity holds.

Problem 2: 5-Step Binomial Asset Price Model

Calculate asset prices for each node in a 5-step binomial model, given an initial stock price (S0) of 100, a risk-free rate (r) of 5%, volatility (σ) of 10%, and a one-year maturity. Additionally, calculate the expected value at the 5th step and verify this with the forward value.

Given Node Values: 110.52, 106.18, 102.02, 98.02, 94.18, and 90.48.

Given Expected Value at 5th Step: 105.13.

Problem 3: European Call and Put Valuation with Parity Test

A stock is currently priced at 100 euros. Over the next year, its price is expected to move up or down by 10%. The risk-free rate is 8% annually, continuously compounded.

1. European Call Value: Determine the value of a one-year European call option with a strike price of 100 euros. Given Answer: 8.46 euros.
2. European Put Value: Similarly, find the value of a European put option. Given Answer: 0.77 euros.
3. Put-Call Parity Verification: Verify that put-call parity holds.

Problem 4: Australian Dollar Call Option Pricing

The Australian dollar (AUD) is currently valued at 0.7 US dollars (USD), with an exchange rate volatility of 5%. The Australian risk-free rate is 8%, and the US risk-free rate is 5%. Assuming a one-step binomial model, what is the price of the one-year corresponding European call option with a strike price (K) of 0.7?

(Note: Consider the US dollar as the domestic currency.)

Given Answer: Call value is $0.006. The calculation steps provided are:

• u = e^(0.05) = 1.05
• d = 1/u
• p = (e^(r_domestic - r_foreign) * T - d) / (u - d) = (e^(0.05 - 0.08) * 1 - 1/1.05) / (1.05 - 1/1.05) ≈ 0.185
• c = e^(-r_domestic * T) * (p * C_up + (1-p) * C_down) = e^(-0.05 * 1) * (0.035 * 0.185) ≈ $0.006

Problem 5: American Put Option Pricing (2-Period)

Calculate the price of an American put option on a non-dividend paying stock, with a 6-month maturity and a strike price (K) of 50, using a 2-period binomial tree (each period is 3 months). The initial price of the underlying stock (S0) at t = 0 is 50. During each period, the price of the underlying can go up or down by 10%. The continuously compounded risk-free rate is 10% annually.