967 words - 4 pages

Option Valuation

Chapter 21

Intrinsic and Time Value

intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option

for a call option: stock price – exercise price for a put option: exercise price – stock price

the intrinsic value for out-the-money or at-themoney options is equal to 0 time value of an option = difference between actual call price and intrinsic value as time approaches expiration date, time value goes to zero

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Determinants of Option Values

Call + – + + + – Put – + + + – +

Stock price Exercise price Volatility of stock price Time to expiration Interest rate Dividend rate of stock

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find the hedge ratio H = (Cu – Cd)/(uS0 – dS0) 3. calculate HdS0, the end-of-year certain value of the portfolio including H shares of the stock and one written call 4. find the present value of HdS0, given the riskfree interest rate r 5. calculate the price of the call using the arbitrage argument: HS0 – C = PV(HdS0)

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Binomial Option Pricing – Example

S0 = 100 d = .75 u = 1.5 X = 120 r = 5% 1. uS0 = 150, dS0 = 75 Cu = uS0 – X = 30, Cd = 0 2. H = (Cu – Cd) / (uS0 – dS0) = 0.4 3. HdS0 = 30 4. PV(HdS0) = HdS0 / (1 + r) = 28.57 5. HS0 = 0.4 ⋅ 100 = 40 C = HS0 – PV(HdS0) = 11.43

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Generalized Binomial Option Pricing

the binomial model can be expanded to more than one period in this case, we would need to find the hedging ratio H at every node in the tree thus, we can construct, at each point in time, a perfectly hedged portfolio – dynamic hedging some of the nodes will be shared by different branches (e.g., the “up and down” scenario would yield the same price as the “down and up” scenario) although numerous and tedious calculations, can “easily” program into a computer 21-9

Black-Scholes Valuation Model

Assumptions

European call option underlying asset does not pay dividends until expiration date both the (riskfree) interest rate r and the variance of the return on the stock σ2 are constant stock prices are continuous (no sudden jumps)

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Black-Scholes Valuation Model (cont.)

Formula

the current price of the call option is C0 = S0 N(d1) – X e–rT N(d2) where:

S0 is the current price of the stock X is the exercise price T is the time...

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