An improved convolution algorithm for discretely sampled Asian options

Abstract

We suggest an improved FFT pricing algorithm for discretely sampled Asian options with general independently distributed returns in the underlying. Our work complements the studies of Carverhill and Clewlow [Risk, 1990, 3(4), 25–29], Benhamou [J. Comput. Finance, 2002, 6(1), 49–68], and Fusai and Meucci [J. Bank. Finance, 2008, 32(10), 2076–2088], and, if we restrict our attention only to log-normally distributed returns, also Večeř [Risk, 2002, 15(6), 113–116]. While the existing convolution algorithms compute the density of the underlying state variable by moving forward on a suitably defined state space grid, our new algorithm uses backward price convolution, which resembles classical lattice pricing algorithms. For the first time in the literature we provide an analytical upper bound for the pricing error caused by the truncation of the state space grid and by the curtailment of the integration range. We highlight the benefits of the new scheme and benchmark its performance against existing finite difference, Monte Carlo, and forward density convolution algorithms.

Keywords:

• Asset pricing
• Incomplete markets
• Performance evaluation
• Path-dependent options

Acknowledgements

We would like to thank Gianluca Fusai and Jan Večeř for helpful discussions. We are also grateful to two anonymous referees for helpful comments and suggestions. The work of Ioannis Kyriakou was supported by the Cass Business School Doctoral Programme and the EPSRC.

Notes

†See Linetsky (2004) and references therein for transform methods and their numerical implementation.

‡All the above PDEs, apart from Večeř (2002), suffer from instability under standard (explicit, implicit or Crank–Nicolson) finite difference schemes. The instability occurs because the drift may dominate the diffusion coefficient in some regions of the grid. The instability can be remedied by ‘upwind’ differences, at the cost of lower convergence speed in the spatial dimension (Kushner and Dupuis 2001). Therefore, of the PDE schemes surveyed here, only Večeř (2002) is numerically competitive.

†We do not require λ to be constant or of the same sign and we do not need variables Z to be identically distributed.

†CZT pre-dates FrFT by more than 20 years (Rabiner et al. 1969). It is more general in form but has the same computational cost. It can be evaluated at the cost of three standard DFTs, using the so-called Bluestein (1968) algorithm. For more details on the speed and implementation in the context of option pricing, see Černý (2004).

†We do not use Matlab's built-in PDE solver but rather we design a customized parabolic PDE solver in which we implement an efficient sparse LU decomposition to speed up matrix inversion in the Crank–Nicolson scheme. Furthermore, we extrapolate the resulting prices quadratically in time and space step to increase accuracy, which is particularly important for high volatility levels.