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Unformatted text preview: Lecture 9 Forward Risk Adjusted Probability Measures and Fixedincome Derivatives 9.1 Forward risk adjusted probability measures This section is a preparation for valuation of fixedincome derivatives. For a F τmeasurable con tingent claim Y ( τ ), its time t value is given by V ( t ) = E Q [ Y ( τ ) B ( t ) /B ( τ )  F t ] (9.1) with a risk neutral probability measure Q . (9.1) is particularly convenient when the bank account process B is deterministic, because it simply becomes V ( t ) = B ( t ) /B ( τ ) E Q [ Y ( τ )  F t ] . (9.2) In the case of stochastic interest rate r , an alternative method is developed based on the following change of measures. For a fixed τ ≤ T , let M ( τ ) > 0 be a F τmeasurable random variable satisfying E Q M ( τ ) = 1. Define a new probability measure P τ by P τ ( ω ) = M ( τ ; ω ) Q ( ω ) ∀ ω ∈ Ω . (9.3) Obviously P τ is a probability measure and P τ ( ω ) > 0 for all ω . Let E τ ( · ) denote the expectation corresponding to P τ . Define a Qmartingale M = { M ( t ) : t = 0 , 1 ,...,τ } by M ( t ) = E Q [ M ( τ )  F t ] . Now we state a basic result of changing martingale measures. Proposition 9.1 MY = { M ( t ) Y ( t ) : t = 0 , 1 ,...,τ } is a Qmartingale if and only if Y = { Y ( t ) : t = 0 , 1 ,...,τ } is a P τmartingale. Proof A major step is to show that E τ [ M ( t ) Y ( τ )  F t ] = E Q [ M ( τ ) Y ( τ )  F t ] ∀ t ≤ τ. (9.4) To verify (9.4), take an arbitrary event A ∈ F t . Then M ( t ) is constant on A and M ( t ; ω ) = E Q [ M ( τ )  A ] = ∑ ω ′ ∈ A M ( τ ; ω ′ ) Q ( ω ′ ) /Q ( A ) = P τ ( A ) /Q ( A ) ∀ ω ∈ A. 1 Therefore, E τ [ M ( t ) Y ( τ )  A ] = M ( t ) E τ [ Y ( τ )  A ] = P τ ( A ) /Q ( A ) ∑ ω ∈ A Y ( τ ; ω ) M ( τ,ω ) Q ( ω ) /P τ ( A ) = ∑ ω ∈ A Y ( τ ; ω ) M ( τ,ω ) Q ( ω ) /Q ( A ) = E Q [ M ( τ ) Y ( τ )  A ] . Note that MY is a Qmartingale if and only if M ( t ) Y ( t ) = E Q [ M ( τ ) Y ( τ )  F t ] for all t ≤ τ , which is equivalent to Y ( t ) = E τ [ Y ( τ )  F t ] for all t ≤ τ by (9.4), i.e. Y is a P τmartingale. This completes the proof of Proposition 9.1. To apply Proposition 9.1 to term structure models, we first let M ( τ ) = 1 B (0 ,τ ) B ( τ ) . (9.5) Note that E Q [ M ( τ )] = E Q [1 /B ( τ )] B (0 ,τ ) = 1. Hence M ( t ) = E Q [1 /B ( τ )  F t ] B (0 ,τ ) = B ( t,τ ) B (0 ,τ ) B ( t ) . (9.6) Next, define the forward risk adjusted probability measure (or called the τ forward measure ) P τ ( ω ) = M ( τ ; ω ) Q ( ω ) = Q ( ω ) B (0 ,τ ) B ( τ ; ω ) ∀ ω ∈ Ω . (9.7) Let S ( t ) denote the time t price of a security (e.g. stock, bond, or contingent claim). Based on (6.3), define Y ( t ) = FO ( t ) = S ( t ) E Q [ B ( t ) /B ( τ )  F t ] = S ( t ) B ( t,τ ) , (9.8) which is the time t forward price of the security to be delivered at maturity...
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This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.
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