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Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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Page 1: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

Fast Convolution Algorithm

Alexander Eydeland, Daniel Mahoney

Page 2: 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

Page 3: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

Objective• Fast algorithm for computing

• Example: backward induction• N – number of nodes in spatial discretization• Straightforward implementation: O(N2)

operations

( )Pr( | )V y z z y dz

Page 4: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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.

Page 5: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

Page 6: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

Basic Concepts

• Consider the case of GBM• k-th time step calculation:

11 1( )Pr( | )k

k k k kzV z e z z dz

1

2 21

( ) /21 2

,1Pr( | )2 k k

k kz z t t

k kt t tz z e

t

Page 7: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

v T u

1kze

0 1 2 1

1 0 1

2 1 0 2

1

1 2 1 0

N

N

a b b bc a bc c a bT

bc c c a

Page 8: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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) )

0 1 2 1 1 2 10N Ny a c c c b b b

, 0U u

1( ( ) ( ))F F U F y

logO N N

v

Page 9: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

FCM operation counts and errors

# Grid Points FCM

Black-Scholes Error

Operation Count, FFT

Operation Count, Matrix

Multiply FFT Ratio Matrix Ratio2 2.8431296 2.1987974 0.644332 1.87E+03 1.01E+034 2.7129767 2.1987974 0.514179 4.01E+03 2.22E+03 2.1454 2.20068 2.4825219 2.1987974 0.283725 8.52E+03 5.38E+03 2.1236 2.4269

16 2.3015936 2.1987974 0.102796 1.81E+04 1.48E+04 2.1259 2.758832 2.2027664 2.1987974 0.003969 3.84E+04 4.60E+04 2.1199 3.103364 2.2007109 2.1987974 0.001914 8.13E+04 1.58E+05 2.1171 3.4230

128 2.1996722 2.1987974 0.000875 1.72E+05 5.77E+05 2.1144 3.6630256 2.1991516 2.1987974 0.000354 3.63E+05 2.20E+06 2.1113 3.8161512 2.1988912 2.1987974 9.38E-05 7.65E+05 8.60E+06 2.1080 3.9038

1024 2.1988179 2.1987974 2.05E-05 1.61E+06 3.40E+07 2.1042 3.95072048 2.1988037 2.1987974 6.27E-06 3.38E+06 1.35E+08 2.1000 3.97514096 2.1987981 2.1987974 6.71E-07 7.09E+06 5.39E+08 2.0959 3.98758192 2.1987977 2.1987974 2.90E-07 1.48E+07 2.0919

16384 2.1987975 2.1987974 9.92E-08 3.10E+07 2.088132768 2.1987974 2.1987974 3.82E-09 6.45E+07 2.084565536 2.1987974 2.1987974 1.83E-09 1.34E+08 2.0811

Page 10: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

Binomial Tree

Nodes ValueBlack-

Scholes ErrorOperation

Count Ratio1024 2.19826 2.1988 -0.000536 3.15E+062048 2.19853 2.1988 -0.000268 1.26E+07 3.9980484096 2.19866 2.1988 -0.000134 5.03E+07 3.9990248192 2.19873 2.1988 -6.68E-05 2.01E+08 3.99951216384 2.19876 2.1988 -3.33E-05 8.05E+08 3.99975632768 2.19878 2.1988 -1.65E-05

Page 11: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

Convergence

.

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

-35

-30

-25

-20

-15

-10

-5

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

FCM

Binomial Tree

Page 12: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

Page 13: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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.

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

time steps

error

Page 14: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1 2 3 4 5 6 7 8 9 10 11 12

Page 15: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

Time Steps Nodes

Crank-Nicolson Value

Black-Scholes Error

Operation Count Ratio

2 10 1.438333 2.198797 0.760465 1.13E+034 20 2.009799 2.198797 0.188998 4.05E+03 3.5795058 40 2.157563 2.198797 0.041234 1.53E+04 3.773939

16 80 2.188694 2.198797 0.010104 5.94E+04 3.88255332 160 2.196291 2.198797 0.002507 2.34E+05 3.94010664 320 2.198176 2.198797 0.000622 9.29E+05 3.969752

128 640 2.198644 2.198797 0.000153 3.70E+06 3.984799256 1280 2.19876 2.198797 3.71E-05 1.48E+07 3.99238512 2560 2.198789 2.198797 8.59E-06 5.90E+07 3.996185

1024 5120 2.198796 2.198797 1.72E-062048 10240 2.198797 2.198797 1.25E-07

Page 16: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

n

SSSA nn

21

11

1

n

nASA nnn

)|Pr(),(),( 11111

1

nnnnnAS

nnrdt

nnn SSSVdSeSAV nn

Page 17: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

5.0,2,1.20 KS

Page 18: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

General case

• An example: swap contracts consisting of futures contracts with different maturities

• FCM can be readily applied to this case as well

))(,),(),(( 211 kmkkk txtxtxFz

))(,),(),(( 1121121 kmkkk txtxtxFz

Page 19: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

))2/()),((exp(2

1)|Pr( 221

2

1 dtdttzMzzdt

zz nnj

nj

nnj

n

hNjzdttzMz nj

nnj

nj )(~),(

Page 20: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

Two-dimensional FCM

• Straightforward implementation: O(N4) operations

• With a little magic N2 two-dimensional

integrals can also be reduced to the Toeplitz form

• The number of operations is O(N2logN)

Page 21: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

))(( 1dwVdtqrSdS

2)( dwVdtVdV

1 , iTiiz

i zzze T

Page 22: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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

1

0

),(),()(),(),()( 11

2),,(

N

j

VDCzziih

jiVDiCzzi

ihz

j

rjj e

i

ede

i

ede

etVzU

Page 23: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

A class of affine jump diffusion models

• Duffie, Pan, and Singleton (2000) or Deng (1999)

• Characteristic function has the form

• Variables Vi 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

...))),(),(),( exp( 2211 DVDVCzi

Page 24: Fast Convolution Algorithm Alexander Eydeland, Daniel Mahoney

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.