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SMG FE 449 Chapter Notes - Chapter 21: Valuation Of Options, Put Option, Call Option

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School
Boston University
Department
Finance
Course
SMG FE 449
Professor
Buffa Andrea

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Document Summary

Binomial option pricing model - the model prices options by making the simplifying assumption that at the end of the next period, the stock price has only two possible values. A two-state single-period model: we will value the option by first constructing a replicating portfolio, a portfolio of other securities that has exactly the same value in one period as the option. Today, this kind of replication strategy is called a dynamic trading strategy. The black-scholes option pricing model: black-scholes option pricing model can be derived from the binomial option pricing. Model by making the length of each period, and the movement of the stock price per period, shrink to zero and letting the number of periods grow infinitely large. P = pv(k )[1 - n(d2)] - s[1 - n(d1)] Implied volatility: of the five required inputs in the black-scholes formula, four are directly observable: s, The replicating portfolio: black-scholes replicating portfolio of a call option.

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Related Questions

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).

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?

Stock of Apple is currently selling at $143. Suppose in the next six months it can increase by 10% or decrease by 5%. The risk-free interest rate is 1% per six month period. Use the one period binomial option pricing model to value a European call option with strike price of $140 and time to maturity of six months.