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Unformatted text preview: Black-Scholes Option Pricing ) ( ) ( 2 1 d N Xe d N S C rT-- = where T T r d X S σ σ 29 ( 29 λν( 2 1 2 + + = ln(S /X): percent amount by which option is currently in- or out-of-the-money; σ (T) 1/2 : adjusts amount by which IN- or OUT-of-the-money for volatility of stock price over remaining life of option; and d 2 = d 1- s T ; N(d): The probability that a random draw from a standard normal distribution will be less than d. 1) If N(d) is close to 1.0, then high probability option will be exercised 2)If N(d) is close to 0, then low probability option will be exercised 3) If N(d) between 0 and 1, then call value equals PV of call’s payoff adjusted for the Probability of being IN-the-money e: 2.71828..., the base of the natural log function (ln) r: The annualized, continuously compounded rate of return on a risk-free asset with the same maturity as the expiration of the option. σ : Standard deviation of the annualized continuously compounded rate of return on the stock. Hedge Ratios (change as stock market does) & Black Scholes Option hedge ratio is change in price of an option for a $ change in the stock price. Call option =positive and put=negative ratio. Called options delta (Δ) Single period binomial model equations: ∆ = C u- C d S u- S d and ∆ = P u- P d S u- S d ; call hedge ratio is N(d1), For put it is N(d 1 ) – 1 Risk and Risk Aversion -Real and Nominal I Rates ( r=R – I) r : real interest rate, R : nominal rate, i : inflationrate; exact relationship (when reported rates are compounded annually r = 1 + R 1 + i- 1 but although the approximation is good if the inflation rate is not to high.-Relationship between Real and Nominal rates: Fisher Effect The basic intuition is that investors will require compensation for inflation in order to hold securities whose returns are in nominal terms. The expected real rate is thus the nominal rate minus expected inflation- Risk and Risk Premiums (Holding Period Returns) can quanify uncertainty using probability distributions H P R = E n d i n g p r i c e- B e g i n n i n g p r i c e + C a s h d i v i d e n d...
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- Fall '08
- Volatility, risky portfolio