By Aurélien Alfonsi

This ebook provides an outline of affine diffusions, from Ornstein-Uhlenbeck methods to Wishart procedures and it considers a few comparable diffusions akin to Wright-Fisher procedures. It makes a speciality of varied simulation schemes for those tactics, particularly second-order schemes for the vulnerable mistakes. It additionally provides a few versions, regularly within the box of finance, the place those equipment are correct and gives a few numerical experiments.

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**Example text**

Rt /dt C Wt : 0 The parameters k; Â; and r0 are assumed to be positive. Â; 2k /. We know that r is a Gaussian process. This gives in particular that rt can take negative values with some positive probability, while interest rates are in practice usually positive. t/: Now, we would like to calculate the price of a floorlet. 45), it boils down to calculate the price of a call option on zero coupon bond. Without loss of generality, we assume that the current time is t D 0. rT0 ; T1 We now observe that the function r 7!

T t/ as a reward, where the rate X is fixed at time t. We exclude here and in the sequel any kind of default. The market practice is to quote the rate X which is fair for both parts. 43) Any other rate would give an arbitrage for the borrower or for the lender. T t// D 0: We now introduce some other standard products such as bonds, swaps, caps and floors. e. t Ä T0 . We consider a contract between two parts that is defined as follows. The borrower receives a unit of cash at time T0 , that he has to get back at time Tl .

Xtx ; t 2 Œ0; T /. Xtx ; t 2 Œ0; T /. Wt ; t 2 Œ0; T /, and we can define it intrinsically. XOt ; t 2 Œ0; T / matters. Getting accurate schemes for the weak error is then sufficient for this use. 2 Strong Approximations The main way to construct strong discretization schemes is to use iterated stochastic Taylor expansion. To fix the ideas, let us assume for a while that the dimension d D 1 and that the coefficients b and are smooth. 5) can be seen as the approximation where Xsx is replaced by Xtxi .