By Domingo Tavella

This ebook turns out to were written for mathematical finance experts...but then what is the aspect? when you already recognize the stuff, why hassle paying for a booklet you realize every thing approximately?

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89) where we made use of the fact that var[W(t)] = t. 87. 94) A case of importance in finance is when a and b are constant. 95) Moments of SDE Solutions Often we are interested in computing the moments of the solutions of SDEs. One way to do this is to solve the SDE and then compute the moments. It turns out, however, that it is much easier to get ordinary differ- Fundamentals of Stochastic Calculus 29 ential equations for the moments that can then be solved either analytically or numerically. 98) We can remark the following about this ordinary differential equation.

As it turns out, these probabilities are determined by the drift of the processes involved (not by their volatility). 131) where Z(T) is called the Radon-Nikodym derivative. 125. 136) where ␪(t) = ay . 127. 139) It is also possible to derive a multidimensional version of the Girsanov theorem, in which case W and ␪ are vector processes. For a detailed derivation of the Girsanov theorem, refer to Oksendal (1995). MARTINGALE REPRESENTATION THEOREM We will use the martingale representation theorem in the next chapter to discuss the existence of a hedging process.

122) ˆ ( t ) is also a Wiener process? The answer is yes. 123) ˜ ( t ) is a Wiener process where a˜ is any (reasonable) drift we want, and W under that measure. 34 QUANTITATIVE METHODS IN DERIVATIVES PRICING Why is this important? To illustrate the implication of this statement, consider a case where we have two processes, X(t) and Y(t), and assume that we are interested in some computation that is only valid if process Y(t) is a martingale. 125) where Y(t) is not a martingale. If we now transform the measure (distort ˆ (t), the probability distribution) of process Y(t) such that dY(t) = by dW ˆ where W ( t ) is a Wiener process in this new measure, and then appropriately carry this distortion over to process X(t), we will have both processes in the measure that makes Y(t) a martingale.

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