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A backward Monte Carlo approach to exotic option pricing

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

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.

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

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References
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Albanese C. (2007) Operator methods, Abelian processes and dynamic conditioning. URL: http://ssrn.com/abstract=1018490, 29 March 2017.
Albanese C., Lo H. & Mijatović A. (2009) Spectral methods for volatility derivatives. Quantitative Finance 9 (6), 663692.
Albanese C. & Mijatović A. (2007) Convergence rates for diffusions on continuous-time lattices. URL: http://ssrn.com/abstract=1018609, 29 March 2017.
Anderson D. G. (1965) Iterative procedures for nonlinear integral equations. J. ACM (JACM) 12 (4), 547560.
Andreasen J. & Huge B. (2011) Volatility interpolation. Risk Mag. 24 (3), 76.
Baker G. A. & Graves-Morris P. R. (1996) Padé Approximants. Cambridge University Press, New York.
Bally V. & Pagès G. (2003) A quantization algorithm for solving multidimensional discrete-time optimal stopping problems. Bernoulli 9 (6), 10031049.
Barraquand J. & Martineau D. (1995) Numerical valuation of high dimensional multivariate American securities. J. Financ. Quant. Anal. 30 (3), 383405.
Berkaoui A., Bossy M. & Diop A. (2008) Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence. ESAIM: Probability and Statistics 12, 111.
Björk T. (2009) Arbitrage Theory in Continuous Time. Oxford University Press, Oxford.
Black F. & Scholes M. (1973) The pricing of options and corporate liabilities. J. Political Economy 81 (3), 637654.
Blanes S., Casas F., Oteo J. A. & Ros J. (2009) The Magnus expansion and some of its applications. Phys. Rep. 470 (5–6), 151238.
Bormetti G., Montagna G., Moreni N. & Nicrosini O. (2006) Pricing exotic options in a path integral approach. Quant. Finance 6 (1), 5566.
Bossy M. & Diop A. (2007) An efficient discretization scheme for one dimensional SDEs with a diffusion coefficient function of the form |x| a , a in [1/2,1). Res. Rep. INRIA RR-5396, 44.
Bouchaud J.-P. & Potters M. (2003) Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management. Cambridge University Press, Cambridge.
Callegaro G., Fiorin L. & Grasselli M. (2015) Quantized calibration in local volatility. Risk Mag. 28 (4), 6267.
Carmona R., Del Moral P., Hu P. & Oudjane N. (2012) Numerical Methods in Finance. Springer Proceedings in Mathematics, Vol. 12.
Clewlow L. & Strickland C. (1996) Implementing Derivative Models (Wiley Series in Financial Engineering). John Wiley & Sons, New York.
Coleman T. F., Li Y. & Verma A. (1999) Reconstructing the unknown local volatility function. J. Comput. Finance 2 (3), 77100.
Cox J. (1975) Notes on Option Pricing I: Constant Elasticity of Variance Diffusions. Unpublished note, Stanford University, Graduate School of Business.
Deng G., Mallett J. & McCann C. (2011) Modeling autocallable structured products. J. Derivatives Hedge Funds 17 (4), 326340.
Derman E., Kani I. & Zou J. Z. (1996) The local volatility surface: Unlocking the information in index option prices. Financ. Anal. J. 52 (4), 2536.
Dewynne J. N. & Shaw W. T. (2008) Differential equations and asymptotic solutions for arithmetic Asian options: Black–Scholes formulae for Asian rate calls. Eur. J. Appl. Math. 19 (4), 353391.
Dupire B. (1994) Pricing with a smile. Risk Mag. 7 (1), 1820.
Fritsch F. N. & Carlson R. E. (1980) Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17 (2), 238246.
Gatheral J. (2011) The Volatility Surface: A Practitioner's Guide. John Wiley & Sons, Hoboken, New Jersey.
Glasserman P. (2004) Monte Carlo methods in Financial Engineering. Applications of Mathematics (New York), Vol. 53, Stochastic Modeling and Applied Probability. Springer-Verlag, New York.
Gobet E. (2000) Weak approximation of killed diffusion using Euler schemes. Stoch. Process. Appl. 87 (2), 167197.
Golub G. H. & Van Loan C. F. (2012) Matrix Computations. JHU Press, USA.
Graf S. & Luschgy H. (2000) Foundations of Quantization for Probability Distributions. Lectures Notes in Mathematics, Vol. 1730, Springer-Verlag, Berlin.
Higham N. J. (2005) The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26 (4), 11791193.
Hui C. H., Lo C.-F. & Yuen P. H. (2000) Comment on ‘Pricing double barrier options using Laplace transforms’ by Antoon Pelsser. Finance Stoch. 4 (1), 105107.
Hull J. (2006) Options, Futures, and Other Derivatives, 9th ed., Pearson Education, India.
Jeanblanc M., Yor M. & Chesney M. (2009) Mathematical Methods for Financial Markets. Springer-Verlag, Berlin.
Joshi M. S. & Leung T. S. (2011) Using Monte Carlo simulation and importance sampling to rapidly obtain jump-diffusion prices of continuous barrier options. J. Comput. Finance 10 (4), 93105.
Kahalé N. (2004) An arbitrage-free interpolation of volatilities. Risk Mag. 17 (5), 102106.
Karatzas I. & Shreve S. (1998) Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, Vol. 113. Springer-Verlag, New York.
Kieffer J. B. (1982) Exponential rate of convergence for Lloyd's method I. IEEE Trans. Inform. Theory 28 (2), 205210.
Kijima M. (1997) Markov Processes for Stochastic Modelling. Stochastic Modeling Series. Chapman & Hall, London.
Kronmal R. A. & Peterson A. V. Jr (1979) On the Alias method for generating random variables from a discrete distribution. Am. Stat. 33 (4), 214218.
Kushner H. & Dupuis P. G. (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.
Labbé C., Rémillard B. & Renaud J. (2011) A simple discretization scheme for nonnegative diffusion processes with applications to option pricing. J. Comput. Finance 15 (2), 3.
Lipton A. & Sepp A. (2011) Filling the gaps. Risk 24 (10), 78.
Merton R. C. (1973) Theory of rational option pricing. BellJ. Econ. Manag. Sci. 4 (1), 141183.
Mitchell A. R. & Griffiths D. F. (1980) The Finite Difference Method in Partial Differential Equations, John Wiley, New York.
Pagès G. (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.
Pagès G., Pham H. & Printems J. (2004) Optimal quantization methods and applications to numerical problems in finance. Handbook of Computational and Numerical Methods in Finance, Birkhauser Boston, 253297.
Pagès G. & Printems J. (2003) Optimal quadratic quantization for numerics: the Gaussian case. Monte Carlo Methods Appl. 9 (2), 135165.
Pagès & Printems J. (2005) Functional quantization for numerics with an application to option pricing. Monte Carlo Methods Appl. 11 (4), 407446.
Pagès G. & Sagna A. (2015) Recursive marginal quantization of the Euler scheme of a diffusion process. Appl. Math. Finance 22 (5), 463498.
Pallavicini A. (2016) A calibration algorithm for the local volatility model. In preparation.
Predota M. (2005) On European and Asian option pricing in the generalized hyperbolic model. Eur. J. Appl. Math. 16 (1), 111144.
Reghai A., Boya G. & Vong G. (2012) Local volatility: Smooth calibration and fast Usage. URL: http://ssrn.com/abstract=2008215, 29 March 2017.
Reiswich D. & Uwe W. (2012) FX volatility smile construction. Wilmott 2012 (60), 5869.
Ren Y., Madan D. & Qian M. Q. (2007) Calibrating and pricing with embedded local volatility models. Risk Mag. 20 (9), 138.
Sidje R. B. (1998) Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Softw. (TOMS) 24 (1), 130156.
Siyanko S. (2012) Essentially exact asymptotic solutions for Asian derivatives. Eur. J. Appl. Math. 23 (3), 395415.
Vecer J. & Xu M. (2004) Pricing Asian options in a semimartingale model. Quant. Finance 4 (2), 170175.
Walker H. F. & Ni P. (2011) Anderson acceleration for fixed-point iterations. SIAM J. Number. Anal. 49 (4), 17151735.
Ward R. C. (1977) Numerical computation of the matrix exponential with accuracy estimate. SIAM J. Numer. Anal. 14 (4), 600610.
Wilmott P., Dewynne J. & Howison S. (1993) Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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