Fast Convolution Algorithm

1. Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

2. Fast Convolution Method (FCM) • Eydeland (1994) • Eydeland, Mahoney (2001, 2002) • Computational efficiency • Flexible numerical set-up • Wide range of applications

3. Objective • Fast algorithm for computing • Example: backward induction • N – number of nodes in spatial discretization • Straightforward implementation: O(N2) operations

4. FCM: Numerical Characteristics • Efficiency. The numerical complexity of the method is almost linear (N logN) in the number of nodes • Accuracy and flexibility. FCM does not require time steps to be small to arrive to an accurate solution. The only requirement is that time steps are in agreement with the cashflow or exercise schedules. Therefore, in choosing time steps one is guided solely by the nature of the problem and not by numerical considerations.

5. Other Properties Generality • The method can handle a range of processes governing the evolution of prices of underlying assets, such as Brownian motion and its numerous offshoots, some jump-diffusion and stochastic volatility processes. • The method is able to price a wide range of exotic options

6. Basic Concepts • Consider the case of GBM • k-th time step calculation:

7. Basic Concepts • Use finite element approximation for . It can be shown that the integrals for all zk can be computed exactly as a product of a Toeplitz matrix and a vector: • Toeplitz matrix

8. Basic Concepts • Let • Then is the first N coordinates of • F is the Fast Fourier Transform (FFT) operator • This calculation requires operations (not O(N2) )

9. FCM operation counts and errors

10. Binomial Tree

11. Convergence . Log-Log Plot of Convergence Rates. The binomial tree exhibits linear convergence, while the FCM has convergence closer to quadratic.

12. Convergence • The integrals are computed exactly on the subspace of functions (finite elements) • Therefore the problem of numerical convergence of the method is replaced with the problem of interpolation accuracy (hence near quadratic convergence for piecewise linear FE) • This also explains why the time step can be arbitrary

13. Multiple time steps • Dependence of FCM Error on Time Discretization. Fixed number of grid points (1024); the number of time steps varies. The error is linear in number of time steps.

14. Total error~ O(h2/k), where k is the time step and h is step size of the grid in the log-price space • Convergence Rate of FCM with Both Time and Grid Size:log-log plot of the error as both the grid size and number of time steps are doubled

15. Comparison with Finite Difference Methods • Crank-Nicolson Finite Difference Solution of Black-Scholes Equation. Error ~ O(h2,k). Need to increase time resolution together with space resolution

16. Asian options • These integrals can be computed by FCM • A payoff based on the maximum or minimum (i.e., a lookback option) can be treated similarly

17. Asian Options: Results • Averaging period: 1 year • one sample per day • Grid (NS=256, NA=200) : 0.3087 • Inverse Laplace Transform: 0.3062 • Curran: 0.3066 • Monte Carlo: 0.3060

18. General case • An example: swap contracts consisting of futures contracts with different maturities • FCM can be readily applied to this case as well

19. The case of non-constant means • Transition probability • By choosing the grid points for zn to be such that • the valuation can be put into Toeplitz form • Also need projection (interpolation) on the regular grid after intergation

20. Two-dimensional FCM • Straightforward implementation: O(N4) operations • With a little magic N2two-dimensional integrals can also be reduced to the Toeplitz form • The number of operations is O(N2logN)

21. Extension of FCM to Stochastic Vol • Heston • Payoff is a piece-wise function • for some set of grid points zi, i = 0, 1, …, N–1

22. Stochastic Vol • Backward induction step • These integrals are no harder to evaluate numerically than those in the original Heston formula • Most importantly, at the end we again have a Toeplitz matrix and can use FCM

23. A class of affine jump diffusion models • Duffie, Pan, and Singleton (2000) or Deng (1999) • Characteristic function has the form • Variables Viare associated stochastic processes • C and D may not have analytic form (may be solutions of ODE) • Still at the end we have Toeplitz matrix and can use FCM

24. References • Deng, Shijie, 1999, “Stochastic Models of Energy Commodity Prices and Their Applications: Mean-reversion with Jumps and Spikes,” PWP-073, available at www.ucei.berkeley.edu/ucei/pubs-pwp.html. • Duffie, Darrell, Pan, Jun, and Singleton, Kenneth, 2000, “Transform Analysis and Asset Pricing for Affine Jump-Diffusions,” Econometrica, 68, 1343-1376. • Eydeland, A., 1994, “A Fast Algorithm for Computing Integrals in Function Spaces: Financial Applications,” Comp. Econ., 7, 277-285. • Eydeland, A., and Mahoney, D. J., 2001, “The Grid Model for Derivative Pricing,” Mirant Technical Report. • Duffie, Darrell, Pan, Jun, and Singleton, Kenneth, 2000, “Transform Analysis and Asset Pricing for Affine Jump-Diffusions,” Econometrica, 68, 1343-1376.