derivatives09-4 - Option Valuation

derivatives09-4 - Option Valuation - Session 4 Option...

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Session 4 Option Valuation
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Section Outline Valuing Stock Options by The Black- Scholes Model Options on Stock Indices and Currencies
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The Black-Scholes Random Walk Assumption Consider a stock whose price is S In a short period of time of length t the return on the stock ( S / S ) is assumed to be normal with mean μ∆ t and standard deviation μ is expected return and σ is volatility t σ
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The Lognormal Property These assumptions imply ln S T is normally distributed with mean: and standard deviation : Because the logarithm of S T is normal, S T is lognormally distributed T σ T S ) 2 / ( ln 2 0 σ - μ +
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The Lognormal Property continued where φ [ m , v ] is a normal distribution with mean m and variance v [ ] [ ] T T S S T T S S T T 2 2 0 2 2 0 , ) 2 ( ln , ) 2 ( ln ln σ σ - μ φ σ σ - μ + φ or
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The Lognormal Distribution E S S e S S e e T T T T T ( ) ( ) ( ) = = - 0 0 2 2 2 1 var μ μ σ
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The Expected Return The expected value of the stock price is S 0 e μ T The expected return on the stock with continuous compounding is μ σ 2 /2 The arithmetic mean of the returns over short periods of length t is μ The geometric mean of these returns is μ σ 2 /2
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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 If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day? t σ
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Estimating Volatility from Historical Data 1. Take observations S 0 , S 1 , . . . , S n at intervals of τ years 2. Define the continuously compounded return as: 3. Calculate the standard deviation, s , of the u i ´s 4. The historical volatility estimate is: u S S i i i = - ln 1 τ = σ s ˆ
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Nature of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed For this reason time is usually measured in “trading days” not calendar days when options are valued
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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
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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 rT σ - = σ σ - + = σ σ + + = - - - = - = - - 1 0 2 0 1 1 0 2 2 1 0 ) 2 / 2 ( ) / ln( ) 2 / 2 ( ) / ln( ) ( ) ( ) ( ) ( where
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Rearrangement of d1, d2 T T Ke S d T T Ke S d rT rT σ 2 1 ) ln( 2 1 ) ln( 2 1 - = + = - -
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Properties of B-S formula When S/Ke -rT increases, the chances of exercising the call option increase, from the formula, d1 and d2 increase and N(d1)
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derivatives09-4 - Option Valuation - Session 4 Option...

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