08-Exotics3(f) - Applications of the Linear Functional Form...

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Primbs, MS&E 345 1 Applications of the Linear Functional Form: Pricing Exotics
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Primbs, MS&E 345 2 Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows
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Primbs, MS&E 345 3 The Black-Scholes formula: ] ) [( 0 + - - = K S E e c T Q rT - = - K S K S T rT T T KdQ dQ S e ] ) [( ] [ 0 + - - - = = K S E e c E e c T Q rT T Q rT where Sdz rSdt dS σ + = under Q Basically, we just have to calculate the expectation by brute force! - = - K S T rT T dQ K S e ) ( This time we use risk neutral pricing for a European call option:
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Primbs, MS&E 345 4 The Black-Scholes formula: - = - K S K S T rT T T KdQ dQ S e c 0 = K S K S T T dQ K KdQ ) ( K S KQ T = ) Pr( K S K T = under Q So: ) Pr( K S K T ) Pr( ) ( 0 2 2 1 K e S K T z T r = + - σ σ ) Pr( 0 ) ( 2 2 1 S K e K T z T r = + - σ σ - - = T T r S K T z K T σ σ ) ( ) ln( Pr 2 2 1 0 But ) 1 , 0 ( ~ N T z T - + = T T r K S T z K T σ σ ) ( ) ln( Pr 2 2 1 0 so by symmetry ) ( 2 d KN = T T r K S d σ σ ) ( ) ln( 2 2 1 0 2 - + = where where N( ) is the distribution function for a standard Gaussian. T z T r T e S S σ σ + - = ) ( 0 2 2 1 Now, note that under Q we have:
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Primbs, MS&E 345 5 The Black-Scholes formula: - = - K S T rT T d KN dQ S e c ) ( 2 0 K S T T dQ S + - = K S z T r T T dQ e S σ σ ) ( 0 2 2 1 Let’s also rewrite this in terms of z T . We actually just did that calculation: 2 d T z T - - + - = 2 2 2 1 ) ( 0 d T z z T r T T dQ e S σ σ T d T z T Z z T r dz z q e S T T - + - = 2 2 2 1 ) ( ) ( 0 σ σ )) 2 /( ( 2 1 2 ) ( T z T T Z T e z q - = π where corresponds to Brownian motion. T d T z T z T z T r dz e e S T T T - - + - = 2 2 2 2 1 )) 2 /( ( 2 1 ) ( 0 π σ σ T d T z T T z T rT dz e e S T T - - - = 2 2 2 ) ( 2 1 0 σ π This looks like a Gaussian with mean σ T .
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Primbs, MS&E 345 6 The Black-Scholes formula: - = - K S T rT T d KN dQ S e c ) ( 2 0 T d T z T T z T rT dz e e S T T - - - = 2 2 2 ) ( 2 1 0 σ π This looks like a Gaussian with mean σ T . T d T z T T z T rT dz e e S T T - - - = 2 2 2 ) ( 2 1 0 σ π ) Pr( 2 0 d T z e S T rT - = where z T is Gaussian with mean σ T and variance T . - - - = T T d T T T z e S T rT σ σ 2 0 Pr T T z T σ - is standard Gaussian ) ( 1 0 d N e S rT = where T d d σ + = 2 1 + - = T T d T T T z e S T rT σ σ 2 0 Pr by symmetry
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Primbs, MS&E 345 7 The Black-Scholes formula: ( 29 ) ( ) ( 2 1 0 0 d KN d N e S e c rT rT - = - ) ( ) ( 2 1 0 0 d N Ke d N S c rT - - = T T r K S d σ σ ) )( ( ) / ln( 2 2 1 0 1 + + = T d d σ - = 1 2 where: ) ( N distribution function for a standard Normal (i.e. N(0,1) )
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Primbs, MS&E 345 8 dz rcdt dc ) ( + = under Q or Question: Where does the Black-Scholes partial differential equation come from in this framework???? But we know that ) , ( t S c c t = where Sdz rSdt dS σ + = under Q So by Ito’s lemma: dz Sc dt c S rSc c dc S SS S t σ σ + + + = ) ( 2 2 2 1 Hence: rc c S rSc c SS S t = + + 2 2 2 1 σ The Black-Scholes pde! 0
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