Wavelet optimised PDE valuation of financial derivatives

1
Wavelet optimised PDE valuation of financial
derivatives
AmaMef INRIA 3 February 2006

• M.A.H. DempsterCentre for Financial Research
• Judge Business School
• University of Cambridge
• Cambridge Systems Associates Limited
• B. Carton de WiartCentre for Financial Research
• Exotic Equity Derivatives, CIBC World Markets,
  London

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Outline

• Introduction
• Wavelet transforms
• Wavelet generated computational domain
• Wavelets and PDEs
• Numerical results
• Conclusions and further work

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PDEs in mathematical finance

• PDEs need to be solved fast and accurately in
  practice
• Exotic (i.e. complex options) are hedged using
  many simpler options
• A typical book on a trading desk would include
  hundreds of thousands of options which need to be
  re-valued several times a day

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PDEs in mathematical finance

• Sensitivity of values to input parameters
  essential
• To find the greeks needed to hedge financial
  derivatives
• the pricing equation is solved many times on the
  same domain with slightly different input
• the computational domain adopted can be reused
• Time varying coefficients
• local volatility (diffusion coefficients) means
  that the
• stiffness matrix needs to be redesigned at each
  time step
• non constant interest rates (transport
  coefficients)
• Early redemption (free boundary conditions)
• the solution needs to be projected at each time
  step

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Wavelet transforms in one dimension

• What are wavelets ?
• Wavelets are nonlinear functions which can be
  scaled and translated to form a basis for the
  Hilbert space of square integrable
  functions
• They can be used as building blocks to represent
  other functions
• They can be used to detect local perturbations in
  solution surfaces

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Transforms

• Two families of spaces scaling Vj and wavelet Wj
  
• spaces
• Vj has basis ? (scaling) and Wj has basis ?
  (wavelet)
• and

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• Scaling functions are translations and dilations
  of a mother scaling function ? which solves a
  dilation equation
• Wavelet functions are translations and dilations
  of a mother wavelet ? defined in terms of the
  mother scaling function

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Biorthogonal wavelets

• Four basic function types - two primals and
• and two duals and
• Biorthogonality

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Biorthogonal wavelets

• The biorthogonal wavelet approximation is
  expressed in terms of the primal functions
• with and

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Biorthogonal interpolating wavelet transform of
Donoho (1992)

• The biorthogonal interpolating wavelet transform
  has basis functions of the form
• where d is the Dirac delta function and ?
  is the Deslaurier Dubuc interpolation function

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Deslauriers Dubuc Interpolation
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Interpolating functions
?
? ?
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Fast interpolating wavelet transform algorithm

• The projection of a function f onto a finite
  dimensional scaling function space VJ is given by
• Recall

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Computation of wavelet coefficients

• Vj1 ? Vj and Wj
• Scaling
• Detail
• All we need to know are the values of ? at half
  integer nodes

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Fast interpolating wavelet transform algorithm
0(2J)

• Finest2664
• Coarsest 238
• Example J 6 P 3

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Wavelets in higher dimensions

• Nested triangulation
• Tensor based
• 1 scaling space s and 3 detail spaces a, b and c

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In practice

• Original Transform Transform
  Again
• wrt x wrt
  y

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Processing Lena

• (Reverse pixels)

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Thresholding

• Detail spaces give an idea of local variation
• We can delete points which have tiny coefficients

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Same idea for a function

• We want a
• sparse grid
• with more points in interesting regions
• not to be renewed too often
• on which we can apply the wavelet transform

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Sparse grid generation

• Sparse grid ? thresholding
• Delete all points with small coefficients
• Renewal ? Type I points
• Add extra points next to useful points
• Wavelet transform ? Type II points
• Add all points needed for transformation

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Unitary impulse with full grid (4225pts)
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Threshold (67pts)
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Add Type I (139pts)
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Add Type II (224pts)
5 of original
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Sparse grid

• Smaller grid
• Refined in regions of high gradient plus some
• Wavelet transform available

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Wavelets for PDE evolution

• Parabolic problem
• with L a differential operator defined over a
  domain ? and boundary conditions on ??
• Discretise and localise in time to solve
• The aim is to have a stiffness matrix A which is
  nice to invert and cheap to compute

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Wavelet methods

• Galerkin
• Collocation
• Filter bank methods
• Wavelet optimized finite differences
• Interpolating wavelet optimized finite differences

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Galerkin-Petrov Beylkin (1992) Prosser (1998)
Dempster et al (2000)

• To find the stiffness matrix in wavelet space one
  can often rewrite
• W operator from original domain to wavelet
  space
• A a discretized PDE operator (e.g. finite
  difference)
• and define à WAW-1 to give

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Galerkin-Petrov properties

• A good property for is that it is nicer to
  invert (IF the wavelet basis has suitable
  properties) Cohen 2003
• If the operator is integro-differential (e.g.
  from jumps in the stochastic process of the
  underlying) the operator can be
  thresholded Matache et al
  2003
• Multigrid and thresholding come naturally
• But
• Computations are done in wavelet space so that
  boundary conditions and free boundaries are
  expensive to apply
• The stiffness matrix is expensive to compute and
  is very expensive for non-constant coefficients

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Collocation Vasilyev(2000)

• Build f' using
• to give

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Collocation properties

• Can easily work on a sparse domain
• Computations are done in the original domain so
  that boundary conditions, nonlinear terms and
  free boundary conditions are easily applied
• But
• Must transform to and from wavelet space on a
  sparse domain to obtain derivatives

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Filter bank Walden 2000

• Transform and threshold working in a sparse
  domain
• Use a finite difference filter to compute
  derivatives
• Start on a coarse scale and refine if necessary

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Filter bank properties

• Very similar to collocation but with different
  and less constraining differentiation filters
• Works with original domain
• But
• Transforms to and from wavelet space every time
  it applies the operator

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Wavelet optimized finite-difference (WOFD)
Jameson (1998)

• Use wavelets to define an irregular grid that is
  updated from time to time
• Then apply local finite difference operators with
  unequal step sizes
• e.g.
• becomes

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WOFD properties

• Little overhead
• All computations are done in the original domain
• Update the grid when needed
• But
• Dangling points

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Dangling points

• Need two neighbouring points

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IWOFD

• Use wavelets to interpolate missing points
• Algorithm
• Apply operator at all regular points
• Interpolate all others (typically 4)

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IWOFD summary

• Little overhead
• All computations are done in the original domain
• Grid can be updated when needed
• Can be applied to any function
• Can easily be applied to equations with non
  constant coefficients
• But
• Loses second order accuracy of the finite
  difference scheme in some regions

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Method of Lines

• Once the space operator has been discretised we
  apply several ODE methods to solve the time
  evolution described by a stiff ODE
• Crank-Nicolson
• Several solvers for linear algebra
• LU decomposition in one dimension
• SOR
• Krylov method Bi-conjugate gradient stabilized
• Backward differentiation
• Dufort-Frankel

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American Options

• Solve a free boundary problem
• Cannot compare functions in wavelet space
• No problem for IWOFD which solves solvable points
  and interpolates others in the original domain
• Alternatives are solving the LCP using the PSOR
  algorithm with or without pre-solve using a
  direct solver
• Other LCP methods -- e.g. ADI-LP -- are hard to
  implement in several dimensions

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Numerical Results
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Vanilla American Put option

• Stock 100 Strike 100 IR 10 ? 30
  Maturity 1 Year
• Vext 8.33845 ? 10-5?S ?S 2-N(Smax- Smin)
• Time T is in 1/100ths of a second on a 2.4GHz
  Dual P4 Xeon
• Error is at the money as a proxy for .?

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American basket put option

• Payoff max(K-S1-S2,0) ?1 ?20.2 ?0.2 K10
• V0.393931 ?10-5min(?S1, ?S2)

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Fixed-for-floating Libor Bermudan swaption

• 3D Gaussian model
• 5 Years, 10 periods, option to enter the swap at
  each semi-annual period
• Vext0.712930 ?10-5min(?Xi ) Renew grid at each
  settlement date

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Bermudan Swaption BGM Model

• Each forward LIBOR rate from Ti to Ti1 is
  modelled using
• dLi -?I(t) dt ? i(t) Li dWi (t)
• The drift is adjusted in order to value the
  product in the terminal measure
• ?N-1 0
• ?j(t) -? I ?kj1 ((Tk1- Tk)Lk?jk ?k)/(1
  (Tk1- Tk)Lk)
• where Lk is todays forward value and ?jk is the
  correlation between the rates 
• The pricing equation is then the usual
  convection-diffusion equation
• Product 1 1year caplet (in which case BGM
  Blacks model)
• Product 2 3year fixed for floating Bermudan
  swaption
• Annual fixed rate 5.5 Notional 100
• T1 1y T2 2y T3 3y
• L0 0.02433306 L1 0.03281384 L2
  0.03931690
• ? L_1 24.73 ?L_2 22.45 ? L_1,L_2
  e-0.1

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Bermudan swaption BGM model

•  

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Conclusion

• IFWOD is a flexible wavelet method for solving
  PDEs in up to 3 dimensions
• Little overhead to the algorithm
• Can price both European and American contracts
• Can handle non-constant coefficients
• So far up to 3 times faster than alternatives
  without significant loss in precision