**Abstract:**

The calculation of the fair value and the sensitivity parameters of a financial derivative requires special numerical methods, which are often computationally very demanding. In this chapter we discuss the design and implementation of efficient GPU solvers for the partial differential equations (PDEs) of derivative pricing problems.

For derivatives on a single asset like a stock or an index we consider a massively parallel PDE solver which simultaneously prices a large collection of similar or related derivatives with finite difference schemes. We achieve a speedup of a factor of 25 on a single GPU and up to a factor of 40 on a dual GPU configuration against an optimized CPU version.

Often derivatives are written on multiple underlying assets, e.g. baskets, or the future asset price evolution is modeled with additional risk factors, like for instance stochastic volatilities. The resulting PDE is defined on a multidimensional state space. For these kind of derivatives it is not necessary to pool multiple pricing calculations: alternating direction implicit (ADI) schemes for PDEs on two or more state variables have enough parallelism for an efficient GPU implementation. We benchmark a specific ADI solver for the Heston stochastic volatility model against a fully multi-threaded and optimized CPU implementation. On a recent C2050 Fermi GPU we attain a speedup of a factor of 70 and more for a sufficiently large problem size.

Our results demonstrate the importance of the effective use of GPU resources such as fast on-chip memory and registers.

For more information visit:

**GPU Computing Gems Jade Edition (Applications of GPU Computing Series), Wen-mei W. Hwu (Editor), 2012, p. 309-322**