X5Black_Scholes

X5Black_Scholes - The Black-Scholes Formula Nobel Prize in...

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Slide 12-1 The Black-Scholes Formula Nobel Prize in 1997 & The Option Greeks Chapter 12 – 12.3
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Slide 12-2 Black-Scholes Formula European Call Options: European Put Options: where and d 1 ! ln( S/K ) " ( r #$ " 1 2 % 2 ) T T d d T 21 ! # % C S,K, ,r,T, Se N d Ke N d - T -rT ( ) = ( ) ( ) %$ $ 12 # P S,K, ,r,T, Ke N d Se N d -rT - T ( ) = ( ) ( ) $ # # # Intuition : Call = long stock and short bond Put = long bond and short stock
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Slide 12-3 Formula Inputs N ( x ) – Standard normal distribution function, bounded between 0 and 1. S – spot price of underlying asset K – strike price r – risk-free rate (continuously compounded) T – time to maturity $ – dividend yield of underlying asset % – Volatility of continuously compounded returns Units : The time periods of the inputs have to be consistent Why does the expected stock return not show up in the Black Scholes formula?
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Slide 12-4 The Black-Scholes formula and replicating a call The Black-Scholes formula shows how to replicate a call option. Recall the formula: & S + B Se - $ T N(d 1 ) – Ke -rT N(d 2 ) & = e - $ T N(d 1 ) e - $ T N(d 1 ) is referred to as the option’s delta . B = -Ke -rT N(d 2 ) d 1 and d 2 change at each moment in time. Since N(d 1 ) and N(d 2 ) are continually changing, you need to continually rebalance your hedge portfolio - “delta-hedging”. A call is like a levered position in the underlying stock, assuming the position is continuously rebalanced.
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Slide 12-5 Black-Scholes Assumptions Assumptions about the stock return distribution Continuously compounded returns on the stock are normally distributed and independent over time (no “jumps”). The volatility of continuously compounded returns is known and constant. Future dividends are known, either as a dollar amount or as a fixed dividend yield. Assumptions about the economic environment The risk-free rate is known and constant There are no transaction costs or taxes It is possible to short-sell costlessly, and to borrow at the risk-free rate. Arbitrage is ruled out. The pricing formula is independent of investors’ risk preferences, utility functions, or expected returns.
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Slide 12-6 Example 12.1 Let S=$41, K=$40, % =0.3, r=8%, T=0.25, and $ =0. BS call: $41 e # 0 0.25 N ln 41 40 " (0.08 # 0 " 0.3 2 2 )0.25 0.3 0.25 ( * + + + , - . . . # $40 e # 0.08 0.25 N ln 41 40 " (0.08 # 0 # 0.3 2 2 )0.25 0.3 0.25 ( * + + + , - . . . ! $41 e # 0 0.25 N (0.373) # $40 e # 0.08 0.25 N (0.223) ! $41 e # 0 0.25 0.644 # $40 e # 0.08 0.25 0.587 ! $3.39
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Slide 12-7 Proof of BS using Risk-Neutral Pricing* C ! e # rT CE (max{ 0, S T # K }) ! e # E *(max{0, S T # K then , )) (ln( let and d distribute y lognormall is If : Lemma 2 w V Var V !
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This note was uploaded on 10/25/2009 for the course 15 15.402 taught by Professor Bergman during the Fall '09 term at MIT.

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X5Black_Scholes - The Black-Scholes Formula Nobel Prize in...

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