fast convolution algorithm alexander eydeland, daniel mahoney
Post on 19-Dec-2015
321 views
TRANSCRIPT
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(N2)
operations
( )Pr( | )V y z z y dz
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:
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
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
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
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
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
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
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.
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
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
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
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
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
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
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 )(~),(
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)
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
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
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
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.