By Aurélien Alfonsi

This booklet provides an outline of affine diffusions, from Ornstein-Uhlenbeck strategies to Wishart approaches and it considers a few similar diffusions reminiscent of Wright-Fisher approaches. It specializes in various simulation schemes for those approaches, in particular second-order schemes for the susceptible mistakes. It additionally offers a few versions, generally within the box of finance, the place those tools are appropriate and offers a few numerical experiments.

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E. 37) is affine with respect to x. This result is well-known in the literature and can be found for example in Filipovi´c [53]. uX xT /jFt D exp. T t/Xtx /; since X is a time homogeneous diffusion. Ft /martingale for t 2 Œ0; T that we denote Mt . Xtx /dW t : We see that Mt is a Doléans-Dade exponential and cannot vanish. s. e. t; Xt /, we get by letting t ! T / for some constant c 2 R. This immediately gives that u is constant and thus u Á u from the initial condition. T / for any T > 0, and both sides are necessarily constant.

Getting fast pricing method is really important in practice, especially to calibrate the parameters to market data, since calibration usually requires an intensive use of the pricing routines. Thus, these formulas partly explain why these affine models are widely used in finance. Maybe, other diffusions would have been more relevant for modelling the short rate, but the lack of explicit formula for basic financial products has made them not suitable for a practical use. Chapter 2 An Introduction to Simulation Schemes for SDEs Let us start this chapter by a general motivation for having simulation schemes.

E. 30 1 Real Valued Affine Diffusions Xi D K for some K > 0). 44) This rate is called the swap rate. With the same kind of argument as for the simply compounded rate, any other choice would lead to an arbitrage for the borrower or for the lender. When the rate is fixed at time t < T0 , the contract above is a Forward Rate Agreement since both parts agrees for a rate on the period ŒT0 ; Tl before T0 . When the rate is fixed at the beginning of the contract T0 , this is a bond. Ti 1 ; Ti / in the contract described above.