Text of Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney
Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney
Fast Convolution Method (FCM) Eydeland (1994) Eydeland, Mahoney (2001, 2002) Computational efficiency Flexible numerical set-up Wide range of applications
Objective Fast algorithm for computing Example: backward induction N number of nodes in spatial discretization Straightforward implementation: O(N 2 ) operations
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.
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
Basic Concepts Consider the case of GBM k-th time step calculation:
Basic Concepts Use finite element approximation for. It can be shown that the integrals for all z k can be computed exactly as a product of a Toeplitz matrix and a vector: Toeplitz matrix
Basic Concepts Let Then is the first N coordinates of F is the Fast Fourier Transform (FFT) operator This calculation requires operations (not O(N 2 ) )
FCM operation counts and errors
Convergence. Log-Log Plot of Convergence Rates. The binomial tree exhibits linear convergence, while the FCM has convergence closer to quadratic.
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
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.
Total error~ O(h 2 /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
Comparison with Finite Difference Methods Crank-Nicolson Finite Difference Solution of Black- Scholes Equation. Error ~ O(h 2,k). Need to increase time resolution together with space resolution
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
Asian Options: Results Averaging period: 1 year one sample per day Grid (N S =256, N A =200) : 0.3087 Inverse Laplace Transform: 0.3062 Curran: 0.3066 Monte Carlo: 0.3060
General case An example: swap contracts consisting of futures contracts with different maturities FCM can be readily applied to this case as well
The case of non-constant means Transition probability By choosing the grid points for z n to be such that the valuation can be put into Toeplitz form Also need projection (interpolation) on the regular grid after intergation
Two-dimensional FCM Straightforward implementation: O(N 4 ) operations With a little magic N 2 two-dimensional integrals can also be reduced to the Toeplitz form The number of operations is O(N 2 logN)
Extension of FCM to Stochastic Vol Heston Payoff is a piece-wise function for some set of grid points z i, i = 0, 1, , N1
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
A class of affine jump diffusion models Duffie, Pan, and Singleton (2000) or Deng (1999) Characteristic function has the form Variables V i are 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
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.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.