L9 - Lecture 9 Forward Risk Adjusted Probability Measures...

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Unformatted text preview: Lecture 9 Forward Risk Adjusted Probability Measures and Fixed-income Derivatives 9.1 Forward risk adjusted probability measures This section is a preparation for valuation of fixed-income derivatives. For a F -measurable con- tingent claim Y ( ), its time t value is given by V ( t ) = E Q [ Y ( ) B ( t ) /B ( ) | 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 ( ) E Q [ Y ( ) | 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 T , let M ( ) > 0 be a F -measurable random variable satisfying E Q M ( ) = 1. Define a new probability measure P by P ( ) = M ( ; ) Q ( ) . (9.3) Obviously P is a probability measure and P ( ) > 0 for all . Let E ( ) denote the expectation corresponding to P . Define a Q-martingale M = { M ( t ) : t = 0 , 1 ,..., } by M ( t ) = E Q [ M ( ) | F t ] . Now we state a basic result of changing martingale measures. Proposition 9.1 MY = { M ( t ) Y ( t ) : t = 0 , 1 ,..., } is a Q-martingale if and only if Y = { Y ( t ) : t = 0 , 1 ,..., } is a P -martingale. Proof A major step is to show that E [ M ( t ) Y ( ) | F t ] = E Q [ M ( ) Y ( ) | F t ] t . (9.4) To verify (9.4), take an arbitrary event A F t . Then M ( t ) is constant on A and M ( t ; ) = E Q [ M ( ) | A ] = A M ( ; ) Q ( ) /Q ( A ) = P ( A ) /Q ( A ) A. 1 Therefore, E [ M ( t ) Y ( ) | A ] = M ( t ) E [ Y ( ) | A ] = P ( A ) /Q ( A ) A Y ( ; ) M ( , ) Q ( ) /P ( A ) = A Y ( ; ) M ( , ) Q ( ) /Q ( A ) = E Q [ M ( ) Y ( ) | A ] . Note that MY is a Q-martingale if and only if M ( t ) Y ( t ) = E Q [ M ( ) Y ( ) | F t ] for all t , which is equivalent to Y ( t ) = E [ Y ( ) | F t ] for all t by (9.4), i.e. Y is a P -martingale. This completes the proof of Proposition 9.1. To apply Proposition 9.1 to term structure models, we first let M ( ) = 1 B (0 , ) B ( ) . (9.5) Note that E Q [ M ( )] = E Q [1 /B ( )] B (0 , ) = 1. Hence M ( t ) = E Q [1 /B ( ) | F t ] B (0 , ) = B ( t, ) B (0 , ) B ( t ) . (9.6) Next, define the forward risk adjusted probability measure (or called the - forward measure ) P ( ) = M ( ; ) Q ( ) = Q ( ) B (0 , ) B ( ; ) . (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 ( ) | F t ] = S ( t ) B ( t, ) , (9.8) which is the time t forward price of the security to be delivered at maturity...
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L9 - Lecture 9 Forward Risk Adjusted Probability Measures...

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