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Session+27+l - 1 RISKMANAGEMENTAND SWAPS...

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RISK MANAGEMENT AND SWAPS Options, Futures, and Other Derivatives, 7th Ed., John C. Hull (2008): Chapter 7 1
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2 2 The Concepts Underlying Black Scholes The option price and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk free rate Assumes that the volatility and risk free rate are constant
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The Concepts Underlying Black Scholes Now consider smaller and smaller time steps: Δ t 0 Essentially, this will involve increasing the number of binomial periods Increase the binomial period until each period Then we can invoke the Central Limit Theorem The binomial distribution converges to the normal distribution (CRR) 3
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4 4 The Black Scholes Formulas T d T T r K S d T T r K S d d N S d N e K p d N e K d N S c rT 1 0 2 0 1 1 0 2 2 1 0 ) 2 / 2 ( ) / ln( ) 2 / 2 ( ) / ) ( ) ( ) ( ) ( where
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5 5 The Volatility The volatility is the standard deviation of the continuously compounded rate of return in 1 year The standard deviation of the return in time t is t
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The Black Scholes Formulas Consider an European option on a stock when there are ex dividend dates in two and five months. Dividends are expected to be $0.5, the current share and exercise prices are $40, the stock price volatility is 30% p.a., risk free rate is 9% p.a. and the time to maturity is 6 months. The present value of dividends is: 0.5e 0.09(2/12) + 0.5e 0.09(5/12) =0.9741 Then S 0 = 40 – 0.9741 = $39.0275, K = $40, t = 0.5 d 1 = [ln(39.025) + (0.09 x 0.3 2 /2)0.5 ] = 0.2017, d 2 = 0.0104 N(d 1 ) =0.58 and N(d 2 ) = 0.496 Finally, c 0 = 39.0275(0.58) – 40 e 0.09(0.5) (0.496) = $3.67 6
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Relating DCF to Black Scholes 7
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Relating DCF to Black Scholes 8
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Relating DCF to Black Scholes For a call option NPV q = S/PV(X) If this quotient exceeds 1,the option should be exercised If NPV q < 1, the option is “out of the money” and should not be exercised.
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This note was uploaded on 04/09/2012 for the course FINN 321 taught by Professor Farahsaid during the Spring '12 term at Alvin CC.

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Session+27+l - 1 RISKMANAGEMENTAND SWAPS...

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