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07-Linear2(f) - The Linear Functional Form of Arbitrage...

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Primbs, MS&E 345 1 The Linear Functional Form of Arbitrage
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Primbs, MS&E 345 2 Linear Pricing The basic argument What linear functionals look like A first step toward risk neutrality Girsanov’s Theorem The Big Picture Summary Interpretation as state prices
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Primbs, MS&E 345 3 Linear Pricing The basic argument What linear functionals look like A first step toward risk neutrality Girsanov’s Theorem The Big Picture Summary Interpretation as state prices
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Primbs, MS&E 345 4 Derivative pricing is nothing more than fitting data points with a linear function. Payoff Price x x Bond Stock The stock and bond are our “data points” Price=L(Payoff) The Big Picture
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Primbs, MS&E 345 5 Derivative pricing is nothing more than fitting data points with a linear function. Payoff Price x x Bond Stock Price=L(Payoff) The Big Picture To price a new payoff, we just evaluate our linear function. Call option Payoff x x x Call Price
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Primbs, MS&E 345 6 Linear Pricing The basic argument What linear functionals look like A first step toward risk neutrality Girsanov’s Theorem The Big Picture Summary Interpretation as state prices
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Primbs, MS&E 345 7 Background: X L : We will consider a pricing functional: Price Payoff : L 0 ) ( S S L T = Pricing the stock 0 ) ( B B L T = Pricing the bond We need to define the space of payoffs. Consider the following example... Example:
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Primbs, MS&E 345 8 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 Stock Lattice Probability Space ) , , ( P Think of each ϖ as being a path for the stock (we want the entire path for path dependent derivatives.) The payoff can be a function of the path of the stock. ) ( ϖ T f ϖ Consider
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Primbs, MS&E 345 9 Background: X L : We will consider a pricing functional: Price Payoff : L Our space of payoffs should be: Payoff: ) ( ϖ T f (That is, functions of the path of the stock) ( 29 ) ( 2 P L (Technical condition)
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Primbs, MS&E 345 10 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 An arbitrage is: ) ( )) ( , ( 2 0 P L f f T × ϖ a (price, payoff) pair: with: 0 )) ( , ( 0 - ϖ T f f That is, a positive payoff that has a negative price! You just receive cash flows! 0 0 - f and 0 ) 0 ) ( ( ϖ T f P or
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Primbs, MS&E 345 11 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 The Basic Argument: Linearity: ) ( ) ( ) ( T T T T g L f L g f L + = + If there is a continuous strictly positive linear pricing functional, then there is no arbitrage. Strict Positivity: 0 ) 0 ) ( ( ϖ T f P 0 )) ( ( ϖ T f L 0 ) ( ϖ T f
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Primbs, MS&E 345 12 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 Assume there is a strictly positive linear pricing functional, Consider 0 ) 0 ) ( ( ϖ T f P 0 ) ( ϖ T f with 0 )) ( ( 0 ) 0 ) ( ( 0 = f f L f P T T ϖ ϖ by positivity no arbitrage! The Basic Argument: If there is a continuous strictly positive linear pricing functional, then there is no arbitrage.
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Primbs, MS&E 345 13 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 Consider 0 ) ( = ϖ T f 0 )) ( ( 0 = = f f L T ϖ by linearity The Basic Argument: Assume there is a strictly positive linear pricing functional, no arbitrage!
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