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28 Apr 2018

(Binomial Option Pricing Model) Consider a binomial world in which the current stock
price of 80 can either go up by 10 percent or down by 8 percent. The risk-free rate is 4
percent. Assume a one-period world. Answer questions below about a call with an exercise
price of 80.
a. What would be the call's price if the stock goes up?
b. What would be the call's price if the stock goes down?
c. What is the hedge ratio?
d. What is the theoretical value of the call?

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Sixta Kovacek
Sixta KovacekLv2
30 Apr 2018

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Question 3

Consider a stock currently worth $80 (S = 80) that can go up or down by 25 percent per period. The exercise price is $80 (X = $80) and the risk-free rate is 5 percent (r = .05). The option will expire at the next period (t = 1). You try to find a theoretically fair value of the call using a one-period binomial option pricing model. Answer the following questions. (Hint: See Ch 16 Teaching Notes and Excel Spreadsheet)

Compute the u and d in the binomial option pricing model such that the stock prices at the next period are Su and Sd.

What are two possible stock prices at the next period?

What are the intrinsic values at expiration of a European call option if the stock goes up and down?

Compute two values at expiration of a European call option (Cu and Cd)?

Compute the binomial probability (p)?

Find the theoretical fair value of the call option today using the one-period binomial pricing model.

Compute the riskless hedge ratio (h).

In order to construct a riskless hedge position by buying stocks and selling 1,000 calls using the riskless hedge ratio (h). How many stocks should you buy at what price? At what price should you sell the calls?

Compute the initial value of the riskless hedge position (V0).

Compute the market values of the riskless hedge position at the next period if the stock goes up (Vu) and goes down (Vd).

Find the rate of return on the riskless hedge position (R).

XYZ is currently selling at a price of $35. In nine months, the price will be either $27.50 or $46. The annual risk-free rate is 7 percent. Using binomial option pricing model, compute the price of a call option on XYZ stock. Assume the option

Consider a stock with a current price of P = $27. Suppose that over the next 6 months the stock price will either go up by a factor of 1.41 or down by a factor of 0.71. Consider a call option on the stock with a strike price of $25 which expires in 6 months. The risk-free rate is 6%.

1. Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option?

2. Suppose you write 1 call option and buy Ns shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (which is called a hedge portfolio)? What is the payoff of the portfolio?

d. 3. What is the present value of the hedge portfolio’s riskless payoff? What is the value of the call option?

d. 4. What is a replicating portfolio? What is arbitrage?

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