Skip to main content
    • Aa
    • Aa

A backward Monte Carlo approach to exotic option pricing

  • G. BORMETTI (a1), G. CALLEGARO (a2), G. LIVIERI (a3) and A. PALLAVICINI (a4) (a5)

We propose a novel algorithm which allows to sample paths from an underlying price process in a local volatility model and to achieve a substantial variance reduction when pricing exotic options. The new algorithm relies on the construction of a discrete multinomial tree. The crucial feature of our approach is that – in a similar spirit to the Brownian Bridge – each random path runs backward from a terminal fixed point to the initial spot price. We characterize the tree in two alternative ways: (i) in terms of the optimal grids originating from the Recursive Marginal Quantization algorithm, (ii) following an approach inspired by the finite difference approximation of the diffusion's infinitesimal generator. We assess the reliability of the new methodology comparing the performance of both approaches and benchmarking them with competitor Monte Carlo methods.

Hide All

GB and GL acknowledge research support from the Scuola Normale Superiore Grant SNS_14_BORMETTI.

Hide All
C. Albanese , H. Lo & A. Mijatović (2009) Spectral methods for volatility derivatives. Quantitative Finance 9 (6), 663692.

D. G. Anderson (1965) Iterative procedures for nonlinear integral equations. J. ACM (JACM) 12 (4), 547560.

V. Bally & G. Pagès (2003) A quantization algorithm for solving multidimensional discrete-time optimal stopping problems. Bernoulli 9 (6), 10031049.

J. Barraquand & D. Martineau (1995) Numerical valuation of high dimensional multivariate American securities. J. Financ. Quant. Anal. 30 (3), 383405.

A. Berkaoui , M. Bossy & A. Diop (2008) Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence. ESAIM: Probability and Statistics 12, 111.

F. Black & M. Scholes (1973) The pricing of options and corporate liabilities. J. Political Economy 81 (3), 637654.

S. Blanes , F. Casas , J. A. Oteo & J. Ros (2009) The Magnus expansion and some of its applications. Phys. Rep. 470 (5–6), 151238.

J.-P. Bouchaud & M. Potters (2003) Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management. Cambridge University Press, Cambridge.

R. Carmona , P. Del Moral , P. Hu & N. Oudjane (2012) Numerical Methods in Finance. Springer Proceedings in Mathematics, Vol. 12.

E. Derman , I. Kani & J. Z. Zou (1996) The local volatility surface: Unlocking the information in index option prices. Financ. Anal. J. 52 (4), 2536.

J. N. Dewynne & W. T. Shaw (2008) Differential equations and asymptotic solutions for arithmetic Asian options: Black–Scholes formulae for Asian rate calls. Eur. J. Appl. Math. 19 (4), 353391.

F. N. Fritsch & R. E. Carlson (1980) Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17 (2), 238246.

P. Glasserman (2004) Monte Carlo methods in Financial Engineering. Applications of Mathematics (New York), Vol. 53, Stochastic Modeling and Applied Probability. Springer-Verlag, New York.

E. Gobet (2000) Weak approximation of killed diffusion using Euler schemes. Stoch. Process. Appl. 87 (2), 167197.

N. J. Higham (2005) The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26 (4), 11791193.

C. H. Hui , C.-F. Lo & P. H. Yuen (2000) Comment on ‘Pricing double barrier options using Laplace transforms’ by Antoon Pelsser. Finance Stoch. 4 (1), 105107.

M. Jeanblanc , M. Yor & M. Chesney (2009) Mathematical Methods for Financial Markets. Springer-Verlag, Berlin.

M. S. Joshi & T. S. Leung (2011) Using Monte Carlo simulation and importance sampling to rapidly obtain jump-diffusion prices of continuous barrier options. J. Comput. Finance 10 (4), 93105.

J. B. Kieffer (1982) Exponential rate of convergence for Lloyd's method I. IEEE Trans. Inform. Theory 28 (2), 205210.

M. Kijima (1997) Markov Processes for Stochastic Modelling. Stochastic Modeling Series. Chapman & Hall, London.

H. Kushner & P. G. Dupuis (2001) Numerical Methods for Stochastic Control Problems in Continuous Time. Applications of Mathematics (New York), 2nd ed., Vol. 24, Stochastic Modeling and Applied Probability, Springer-Verlag, New York.

C. Labbé , B. Rémillard & J. Renaud (2011) A simple discretization scheme for nonnegative diffusion processes with applications to option pricing. J. Comput. Finance 15 (2), 3.

R. C. Merton (1973) Theory of rational option pricing. BellJ. Econ. Manag. Sci. 4 (1), 141183.

G. Pagès (2015) Introduction to vector quantization and its applications for numerics. CEMRACS 2013 – modelling and simulation of complex systems: Stochastic and deterministic approaches. ESAIM: Proc. Surv. 48, 2979.

G. Pagès , H. Pham & J. Printems (2004) Optimal quantization methods and applications to numerical problems in finance. Handbook of Computational and Numerical Methods in Finance, Birkhauser Boston, 253297.

G. Pagès & J. Printems (2003) Optimal quadratic quantization for numerics: the Gaussian case. Monte Carlo Methods Appl. 9 (2), 135165.

& J. Printems (2005) Functional quantization for numerics with an application to option pricing. Monte Carlo Methods Appl. 11 (4), 407446.

G. Pagès & A. Sagna (2015) Recursive marginal quantization of the Euler scheme of a diffusion process. Appl. Math. Finance 22 (5), 463498.

M. Predota (2005) On European and Asian option pricing in the generalized hyperbolic model. Eur. J. Appl. Math. 16 (1), 111144.

D. Reiswich & W. Uwe (2012) FX volatility smile construction. Wilmott 2012 (60), 5869.

R. B. Sidje (1998) Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Softw. (TOMS) 24 (1), 130156.

S. Siyanko (2012) Essentially exact asymptotic solutions for Asian derivatives. Eur. J. Appl. Math. 23 (3), 395415.

H. F. Walker & P. Ni (2011) Anderson acceleration for fixed-point iterations. SIAM J. Number. Anal. 49 (4), 17151735.

R. C. Ward (1977) Numerical computation of the matrix exponential with accuracy estimate. SIAM J. Numer. Anal. 14 (4), 600610.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 28 *
Loading metrics...

Abstract views

Total abstract views: 222 *
Loading metrics...

* Views captured on Cambridge Core between 12th April 2017 - 17th October 2017. This data will be updated every 24 hours.