Wavelet optimised PDE valuation of financial derivatives

1. Wavelet optimised PDE valuation of financial derivatives Spectral & Cubature Methods in Finance & Econometrics University of Leicester 18 June 2009 M.A.H. DempsterCentre for Financial Research Statistical Laboratory University of Cambridge & Cambridge Systems Associates Limited B. Carton de WiartExotic Equity Derivatives, Citigroup, London

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

3. 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

4. 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

5. 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

6. Transforms • Two families of spaces: scalingVjand waveletWj spaces • Vj has basis (scaling) and Wj has basis (wavelet) and

7. Scaling functions are translations and dilations of a motherscaling function  which solvesa dilation equation • Wavelet functions are translations and dilations of a mother wavelet defined in terms of the mother scaling function

8. Biorthogonal wavelets • Four basic function types - two primals and and two duals and • Biorthogonality:

9. Biorthogonal wavelets • The biorthogonal wavelet approximation is expressed interms of the primal functions with and

10. Biorthogonal interpolating wavelet transform ofDonoho (1992) • The biorthogonal interpolating wavelet transform has basis functions of the form where δ is the Dirac delta function and is the Deslaurier Dubuc interpolation function

11. Deslauriers Dubuc Interpolation

12. Interpolating functions   … 

13. Fast interpolating wavelet transform algorithm • The projection of a function f onto a finite dimensional scaling function space VJ is given by • Recall

14. Computation of wavelet coefficients • Vj+1VjandWj • Scaling • Detail • All we need to know are the values of  at halfinteger nodes

15. Fast interpolating wavelet transform algorithm 0(2J) Finest:26=64 Coarsest: 23=8 Example: J := 6 P := 3

16. Wavelets in higher dimensions • Nested triangulation • Tensor based 1 scaling space s and 3 detailspaces a, b and c

17. In practice Original Transform Transform Again… wrt x wrt y

18. Processing Lena (Reverse pixels)

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

20. 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

21. “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

22. Unitary impulse with full grid (4225pts)

23. Threshold (67pts)

24. Add Type I (139pts)

25. Add Type II (224pts) 5% of original

26. Sparse grid • Smaller grid • Refined in regions of high gradient … plus some • Wavelet transform available

27. 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 matrixA which is nice to invert and cheap to compute

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

29. Galerkin-PetrovBeylkin (1992) Prosser (1998) Dempsteret 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 Ã:= WAW-1 to give

30. 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

31. Collocation Vasilyev(2000) • Build f' using to give

32. 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

33. 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

34. 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

35. 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

36. WOFD properties • Little overhead • All computations are done in the original domain • Update the grid when needed But… • “Dangling” points

37. Dangling points • Need two neighbouring points…

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

39. 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

40. 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

41. 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

42. Numerical results

43. 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 ||.||

44. American basket put option • Payoff: max(K-S1-S2,0) 1= 2=0.2 =0.2 K=10 • V=0.393931 =10-5min(S1, S2)

45. Fixed-for-floating Libor Bermudan swaption • 3D Gaussian model: • 5 Years, 10 periods, option to enter the swap at each semi-annual period • Vext=0.712930 =10-5min(Xi ) Renew grid at each settlement date

46. Bermudan swaption BGM Model • Each forward LIBOR rate from Ti to Ti+1 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 k=j+1 ((Tk+1- Tk)Lkjkk)/(1 + (Tk+1- Tk)Lk) where Lk is today’s 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 = Black’s 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

47. Bermudan swaption BGM model

48. 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