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*Learning objectives: Describe how the value calculated using a binomial model converges as time periods are added. Explain how the binomial model can be altered to price options on: stocks with dividends and stock indices.*

**Questions:**

812.1. The NASDAQ-100 stock index is currently 7,300.0 and has a volatility of 40.0% and a dividend yield of 1.0%. The risk-free rate is 3.0%. If we employ a two-step binomial tree, which is

**nearest**to the value of a European 6-month call option with a strike price of 7,500.0; i.e., the call is out-of-the-money by exactly 200? Note: this is a variation on Hull's example 13.1

a. $714.77

b. $734.20

c. $756.93

d. $777.51

812.2. Martha used a three-step binomial model to value a (long-term) put option with three years to maturity; i.e., each time step is one year. While the risk-free rate is 4.0%, the underlying asset's volatility is 28.40%. Using these assumptions, she was pleasantly surprised to see that the risk-neutral probability of up movement in her model as 50.0%; i.e., p = d = 0.50. However, she forgot to include the assumption that the asset will pay a continuous dividend of 3.0% per annum. By how much will this assumption change her mode's risk-neutral probability of a down (d) movement?

a. Decrease probability of down movement, (d), by about 10.79%

b. Decrease probability of down movement, (d), by about 5.33%

c. Increase probability of down movement, (d), by about 5.33%

d. Increase probability of down movement, (d), by about 10.79%

812.3. Peter initially values a one-year European put option on a non-dividend-paying (q = 0%) stock with the following assumptions:

- One-year put option maturity with twelve steps in the binomial model (one for each month): Δt = 1/12
- The strike price is equal to the stock's current price of $40.00; i.e., at-the-money
- Volatility of stock, σ = 25.0% per annum
- Riskfree rate is 3.0% per annum with continuous compounding, r = 3.0%

**EXCEPT**which is false?

a. In switching from monthly to weekly steps, the number of terminal stock prices increases from 13 to 51

b. In switching from monthly to weekly steps, the SMALLEST terminal stock price (in the binomial tree) decreases from $16.82 (when Δt = 1/12) to $6.83 (when Δt = 1/50)

c. In switching from monthly to weekly steps, the LARGEST terminal stock price (in the binomial tree) increases from $95.10 (when Δt = 1/12) to $234.31 (when Δt = 1/50)

d. In switching from monthly to weekly steps, the probability that the terminal stock price equals EXACTLY $40.00 (i.e., same as the initial price) approximately doubles because SQRT(50/12) ≈ 2.0

**Answers here:**

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