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Unformatted text preview: ACTL3004: Weeks 910 Financial Economics for Insurance and Superannuation: Week 910 Continuous Time Derivative Valuation 1 / 43 ACTL3004: Weeks 910 Measure Theory RadonNikodym Derivative Change of Measure Eg. Consider the two step random walk tree. With each possible path we can attach a P probability : Path PProbability 0,1,2 p 1 p 2 1 0,1,0 p 1 ( 1 p 2 ) 2 0,1,0 ( 1 p 1 ) p 3 3 0,1,2 ( 1 p 1 )( 1 p 3 ) 4 If we were to define another measure Q on this process, we can set Path QProbability 0,1,2 q 1 q 2 1 0,1,0 q 1 ( 1 q 2 ) 2 0,1,0 ( 1 q 1 ) q 3 3 0,1,2 ( 1 q 1 )( 1 q 3 ) 4 2 / 43 ACTL3004: Weeks 910 Measure Theory RadonNikodym Derivative we can encode the differences between the two measures by the ratios i i We call this dQ dP the "Radon Nikodym derivative" of Q with respect to P, up to time 2. Using dQ dP , we can immediately derive Q from P. The only concern is if with p or q is zero or one. 3 / 43 ACTL3004: Weeks 910 Measure Theory Equivalent Measures Equivalent Measures The probability measure Q is sometimes called an equivalent martingale measure for the realworld probability measure P . Definition: Two measures P and Q are equivalent if: P ( A ) > Q ( A ) > 0 for any outcome A . In words, this means that all the universe of outcomes for both measures are the same. 4 / 43 ACTL3004: Weeks 910 Measure Theory Expectations and the RadonNikodym Derivative Expectations and dQ dP The Radon Nikodym derivative acts in a very natural way when we consider expectations. We know that E P [ X ] = X i i x i . So, E Q [ X ] = X i i x i = X i i i i x i = E P dQ dP X 5 / 43 ACTL3004: Weeks 910 Measure Theory Expectations and the RadonNikodym Derivative Example 5 What is E P h dQ dP i ? 6 / 43 ACTL3004: Weeks 910 Measure Theory RadonNikodym as a Process RadonNikodym as a Process At the moment we have only defined the Radon Nikodym derivative as a random variable (on maturity T). How about a process? We can do this by defining ( t ) to be the RadonNikodym derivative up to time t. In fact, there is another representation: ( t ) = E P dQ dP F t 7 / 43 ACTL3004: Weeks 910 Measure Theory RadonNikodym Summary RadonNikodym Summary Given P and Q equivalent measures and a time horizon T, we can define a random variable dQ dP defined on the P possible paths, taking positive values, such that E Q [ X ( T )] = E P dQ dP X ( T ) for a claim known at time T. We also have for 0 s t T , E Q [ X ( t ) F s ] =  1 ( s ) E P [ ( t ) X ( t ) F s ] where ( t ) = E P dQ dP F t . 8 / 43 ACTL3004: Weeks 910 Measure Theory RadonNikodym Summary Black Scholes Formula Main result that we will want to show: The value of the call option V ( S ( ) , ) is V ( S ( ) , ) = S ( )( d 1 ) Ke r ( T ) ( d 2 ) d 1 = ln ( S ( t ) / K ) + ( r + 1 2 2 ) ( T ) T d 2 = d 1 T and more interestingly it is also V ( S ( ) , ) = E Q h e rT ( S ( T ) K ) + i ....
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This note was uploaded on 08/07/2011 for the course ACTL 3004 at University of New South Wales.
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