To formulate and prove 57 we will introduce a family

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Unformatted text preview: that E (Mn+1 Mn |Av ) = Mn E (Xn+1 1|Av ) = 0 The reason for a multiplicative model is that changes in stock prices are thought to be proportional to its value. Also, in contrast to an additive model, we are guaranteed that prices will stay positive. The last example generalizes easily to give: Example 5.6. Exponential martingale. Let Y1 , Y2 , . . . be independent and identically distributed with (✓) = E exp(✓Y1 ) < 1. Let Sn = S0 + Y1 + · · · + Yn . Then Mn = exp(✓Sn )/ (✓)n is a martingale with respect to Yn . In particular, if (✓) = 1 then (✓Sn ) is a martingale. Proof. If we let Xi = exp(✓Yi )/ (✓) then Mn = M0 X1 · · · Xn with EXi = 1 and this reduces to the previous example. Having introduced a number of examples, we will now derive some basic properties. Lemma 5.6. If Mn is a martingale and is a convex function then (Mn ) is a submartingale. If Mn is a submartingale and is a nondecreasing convex function then (Mn ) is a submartingale. Proof. Using Lemma 5.2 and the definition of a martingale, we have E ( (Mn+1 )|Av ) (E (Mn+1 |Av )) = (Mn ) In the proof of the second statem...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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