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Chap5 - Chapter 5 Risk-Neutral Pricing We have a...

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Chapter 5. Risk-Neutral Pricing We have a probability space (Ω , F , P ) and a filtration F ( t ), defined for 0 t T . where T is a fixed final time. Suppose further that Z is an almost surely positive random variable with EZ = 1, and we define ˜ P ( A ) = Z A Z ( w ) dP ( w ) (5 . 2 . 1) . We can then define the radon-Nikodym deriva- tive process Z ( t ) = E ( Z |F ( t ) (5 . 2 . 6) . This process is a martingale since E ( Z ( t ) |F ( s )) = E ( E ( Z |F ( t )) |F ( s )) = E ( Z |F ( s )) = Z ( s ) (5 . 2 . 7) for 0 s t T . 1

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Lemma 5.2.1. Let t satisfying 0 t T be given and let Y be an F ( t )-measurable random variable. Then ˜ EY = E ( Y Z ( t )) (5 . 2 . 8) 2
Lemma 5.2.2. Let s and t satisfying 0 s t T be given and let Y be an F ( t )-measurable random variable. Then ˜ E ( Y |F ( s )) = 1 Z ( s ) E ( Y Z ( t ) |F ( s )) (5 . 2 . 9) 3

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Theorem 5.2.3 (Girsanov, one dimension). Let W ( t ) , 0 t T , be a Brownian motion on a probability space (Ω , F , P ), and let F ( t ) , 0 t T be a filtration for this Brownian motion. Let Θ( t ) , 0 t T be an adapted process. Define Z ( t ) = exp {- Z t 0 Θ( u ) du - 1 2 Z t 0 Θ 2 ( u ) du } (5 . 2 . 11) ˜ W ( t ) = W ( t ) + Z t 0 Θ( u ) du (5 . 2 . 12) and assume that E Z T 0 Θ 2 ( u ) Z 2 ( u ) du < (5 . 2 . 13) Set Z = Z ( T ). Then EX = 1 and under the probability measure ˜ P given by (5.2.1), the process ˜ W ( t ) , 0 t T is a Brownian motion. 4
Theorem 5.3.1 (Martingale representation, one dimension). Let W ( t ) , 0 t T be a Brownian motion on a probability space (Ω , F , P ), and let F ( t ) , 0 t T be the filtration gener- ated by this Brownian motion. Let M

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Chap5 - Chapter 5 Risk-Neutral Pricing We have a...

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