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Unformatted text preview: Chapter 5. RiskNeutral 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 radonNikodym 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 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 Theorem 5.2.3 (Girsanov, one dimension). Let W ( t ) , ≤ t ≤ T , be a Brownian motion on a probability space (Ω , F ,P ), and let F ( t ) , ≤ t ≤ T be a filtration for this Brownian motion. Let Θ( t ) , ≤ t ≤ T be an adapted process. Define Z ( t ) = exp { Z t Θ( u ) du 1 2 Z t Θ 2 ( u ) du } (5 . 2 . 11) ˜ W ( t ) = W ( t ) + Z t Θ( u ) du (5 . 2 . 12) and assume that E Z T 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 ) , ≤ t ≤ T is a Brownian motion....
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 Fall '11
 Peng,Liang
 Probability

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