Stochastic finite differences and multilevel monte carlo for a class of SPDEs in finance

SIAM Journal on Financial Mathematics

Volumn 3, Issue 1, 2012, Pages 572-592

  Giles, Michael B ^a   Reisinger, Christoph ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Credit portfolio models; Error and complexity analysis; Multilevel Monte Carlo simulation; Parabolic stochastic partial differential equations

Indexed keywords

EID: 84870887091     PISSN: None     EISSN: 1945497X     Source Type: Journal    
DOI: 10.1137/110841916     Document Type: Article

References (35)

1. A. Bain and D. Crisan (2009), Fundamentals of Stochastic Filtering, Springer, New York.
2. A. Barth, C. Schwab, and N. Zollinger (2010), Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math., 119, pp. 123-161.
3. F. E. Benth and S. Koekebakker (2008), Stochastic modeling of financial electricity contracts, Energ. Econ., 30, pp. 1116-1157. (Pubitemid 351292431)
4. F. Black and J. Cox (1976), Valuing corporate securities: Some effects of bond indenture provision, J. Finance, 31, pp. 351-367.
5. N. Bruti-Liberati and E. Platen (2010), Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin.
6. E. Buckwar and T. Sickenberger (2011), A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods, Math. Comput. Simulation, 81, pp. 1110-1127.
7. K. Bujok and C. Reisinger (2012), Numerical valuation of basket credit derivatives in structural jumpdiffusion models, J. Comput. Finance, 15, pp. 115-158.
8. N. Bush, B. M. Hambly, H. Haworth, L. Jin, and C. Reisinger (2011), Stochastic evolution equations in portfolio credit modelling, SIAM J. Financial Math., 2, pp. 627-664.
9. R. Carmona, J.-P. Fouque, and D. Vestal (2009), Interacting particle systems for the computation of rare credit portfolio losses, Finance Stoch., 13, pp. 613-633.
10. R. Carter and M. B. Giles (2007), Sharp error estimates for discretisations of the 1D convection/diffusion equation with Dirac initial data, IMA J. Numer. Anal., 27, pp. 406-425.
11. K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup (2011), Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients, Comput. Vis. Sci., 14, pp. 3-15.
12. J. Fouque, B. Wignall, and X. Zhou (2008), Modeling correlated defaults: First passage model under stochastic volatility, J. Comput. Finance, 11, pp. 43-78.
13. M. B. Giles (2008a), Multi-level Monte Carlo path simulation, Oper. Res., 56, pp. 981-986.
14. M. B. Giles (2008b), Improved multilevel Monte Carlo convergence using the Milstein scheme, in Monte Carlo and Quasi-Monte Carlo Methods 2006, A. Keller, S. Heinrich, and H. Niederreiter, eds., Springer, Berlin, pp. 343-358.
15. P. Glasserman (2004), Monte Carlo Methods in Financial Engineering, Springer, New York.
16. S. Graubner (2008), Multi-level Monte Carlo Methoden für stochastische partielle Differentialgleichungen, Diplomarbeit, TU Darmstadt, Darmstadt, Germany.
17. I. Gyöngy (1999), Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II, Potential Anal., 11, pp. 1-37.
18. I. Gyöngy and D. Nualart (1997), Implicit schemes for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise, Potential Anal., 7, pp. 725-757.
19. D. Heath, R. Jarrow, and A. Morton (1992), Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation, Econometrica, 60, pp. 77-105.
20. D. J. Higham (2000), Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38, pp. 753-769. (Pubitemid 33778937)
21. J. Hull, M. Predescu, and A. White (2005), The valuation of correlation-dependent credit derivatives using a structural model, J. Credit Risk, 6, pp. 99-132.
22. P. E. Kloeden and E. Platen (1992), Numerical Solution of Stochastic Differential Equations, Springer, Berlin.
23. N. V. Krylov (1994), A Wn 2-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields, 98, pp. 389-421.
24. T. G. Kurtz and J. Xiong (1999), Particle representations for a class of nonlinear SPDEs, Stochastic Process. Appl., 83, pp. 103-126. (Pubitemid 129565341)
25. A. Lang (2010), A Lax equivalence theorem for stochastic differential equations, J. Comput. Appl. Math., 234, pp. 3387-3396.
26. R. Merton (1974), On the pricing of corporate debt: The risk structure of interest rates, J. Finance, 29, pp. 449-470.
27. D. M. Pooley, K. R. Vetzal, and P. A. Forsyth (2003), Remedies for non-smooth payoffs in option pricing, J. Comput. Finance, 6, pp. 25-40.
28. R. D. Richtmyer and K. W. Morton (1967), Difference Methods for Initial-Value Problems, Wiley-Interscience, New York.
29. C. Roth (2002), Difference methods for stochastic partial differential equations, ZAMM Z. Angew. Math. Mech., 82, pp. 821-830. (Pubitemid 44357282)
30. C. Roth (2006), A combination of finite difference and Wong-Zakai methods for hyperbolic stochastic partial differential equations, Stoch. Anal. Appl., 24, pp. 221-240. (Pubitemid 43071156)
31. Y. Saito and T. Mitsui (1996), Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33, pp. 2254-2267. (Pubitemid 126431766)
32. P. J. Schönbucher (2003), Credit Derivatives Pricing Models, Wiley, New York.
33. J. B. Walsh (2005), Finite element methods for parabolic stochastic PDEs, Potential Anal., 23, pp. 1-43. (Pubitemid 40900280)
34. Y. Xia and M. B. Giles (2012), Multilevel path simulation for jump-diffusion SDEs, in Monte Carlo and Quasi-Monte Carlo Methods 2010, H. Wozniakowski and L. Plaskota, eds., Springer, Berlin, pp. 657-670.
35. C. Zhou (2001), An analysis of default correlations and multiple defaults, Rev. Financ. Stud., 14, pp. 555-576.