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A backward Monte Carlo approach to exotic option pricing†

Published online by Cambridge University Press:  12 April 2017

G. BORMETTI ,
G. CALLEGARO ,
G. LIVIERI  and
A. PALLAVICINI
G. BORMETTI
Affiliation:
Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy, email: [email protected]
G. CALLEGARO
Affiliation:
Department of Mathematics, University of Padova, via Trieste 63, 35121 Padova, Italy, email: [email protected]
G. LIVIERI
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy email: [email protected]
A. PALLAVICINI
Affiliation:
Department of Mathematics, Imperial College, London SW7 2AZ, UK, email: [email protected] Banca IMI, Largo Mattioli 3, 20121 Milano, Italy
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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.

Keywords

Monte Carlo methods (65C05)Markov chains (60J10)Computational methods in Markov chains (60J22)Derivative securities (91G20)Numerical methods (including Monte Carlo methods) (91G60)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 1 , February 2018 , pp. 146 - 187
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

†

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

References

Albanese, C. (2007) Operator methods, Abelian processes and dynamic conditioning. URL: http://ssrn.com/abstract=1018490, 29 March 2017.CrossRefGoogle Scholar
Albanese, C., Lo, H. & Mijatović, A. (2009) Spectral methods for volatility derivatives. Quantitative Finance 9 (6), 663–692.CrossRefGoogle Scholar
Albanese, C. & Mijatović, A. (2007) Convergence rates for diffusions on continuous-time lattices. URL: http://ssrn.com/abstract=1018609, 29 March 2017.CrossRefGoogle Scholar
Anderson, D. G. (1965) Iterative procedures for nonlinear integral equations. J. ACM (JACM) 12 (4), 547–560.CrossRefGoogle Scholar
Andreasen, J. & Huge, B. (2011) Volatility interpolation. Risk Mag. 24 (3), 76.Google Scholar
Baker, G. A. & Graves-Morris, P. R. (1996) Padé Approximants. Cambridge University Press, New York.CrossRefGoogle Scholar
Bally, V. & Pagès, G. (2003) A quantization algorithm for solving multidimensional discrete-time optimal stopping problems. Bernoulli 9 (6), 1003–1049.CrossRefGoogle Scholar
Barraquand, J. & Martineau, D. (1995) Numerical valuation of high dimensional multivariate American securities. J. Financ. Quant. Anal. 30 (3), 383–405.CrossRefGoogle Scholar
Berkaoui, A., Bossy, M. & Diop, A. (2008) Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence. ESAIM: Probability and Statistics 12, 1–11.CrossRefGoogle Scholar
Björk, T. (2009) Arbitrage Theory in Continuous Time. Oxford University Press, Oxford.Google Scholar
Black, F. & Scholes, M. (1973) The pricing of options and corporate liabilities. J. Political Economy 81 (3), 637–654.CrossRefGoogle Scholar
Blanes, S., Casas, F., Oteo, J. A. & Ros, J. (2009) The Magnus expansion and some of its applications. Phys. Rep. 470 (5–6), 151–238.CrossRefGoogle Scholar
Bormetti, G., Montagna, G., Moreni, N. & Nicrosini, O. (2006) Pricing exotic options in a path integral approach. Quant. Finance 6 (1), 55–66.CrossRefGoogle Scholar
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.Google Scholar
Bouchaud, J.-P. & Potters, M. (2003) Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Callegaro, G., Fiorin, L. & Grasselli, M. (2015) Quantized calibration in local volatility. Risk Mag. 28 (4), 62–67.Google Scholar
Carmona, R., Del Moral, P., Hu, P. & Oudjane, N. (2012) Numerical Methods in Finance. Springer Proceedings in Mathematics, Vol. 12.CrossRefGoogle Scholar
Clewlow, L. & Strickland, C. (1996) Implementing Derivative Models (Wiley Series in Financial Engineering). John Wiley & Sons, New York.Google Scholar
Coleman, T. F., Li, Y. & Verma, A. (1999) Reconstructing the unknown local volatility function. J. Comput. Finance 2 (3), 77–100.CrossRefGoogle Scholar
Cox, J. (1975) Notes on Option Pricing I: Constant Elasticity of Variance Diffusions. Unpublished note, Stanford University, Graduate School of Business.Google Scholar
Deng, G., Mallett, J. & McCann, C. (2011) Modeling autocallable structured products. J. Derivatives Hedge Funds 17 (4), 326–340.CrossRefGoogle Scholar
Derman, E., Kani, I. & Zou, J. Z. (1996) The local volatility surface: Unlocking the information in index option prices. Financ. Anal. J. 52 (4), 25–36.CrossRefGoogle Scholar
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), 353–391.CrossRefGoogle Scholar
Dupire, B. (1994) Pricing with a smile. Risk Mag. 7 (1), 18–20.Google Scholar
Fritsch, F. N. & Carlson, R. E. (1980) Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17 (2), 238–246.CrossRefGoogle Scholar
Gatheral, J. (2011) The Volatility Surface: A Practitioner's Guide. John Wiley & Sons, Hoboken, New Jersey.Google Scholar
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.Google Scholar
Gobet, E. (2000) Weak approximation of killed diffusion using Euler schemes. Stoch. Process. Appl. 87 (2), 167–197.CrossRefGoogle Scholar
Golub, G. H. & Van, Loan C. F. (2012) Matrix Computations. JHU Press, USA.Google Scholar
Graf, S. & Luschgy, H. (2000) Foundations of Quantization for Probability Distributions. Lectures Notes in Mathematics, Vol. 1730, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Higham, N. J. (2005) The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26 (4), 1179–1193.CrossRefGoogle Scholar
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), 105–107.CrossRefGoogle Scholar
Hull, J. (2006) Options, Futures, and Other Derivatives, 9th ed., Pearson Education, India.Google Scholar
Jeanblanc, M., Yor, M. & Chesney, M. (2009) Mathematical Methods for Financial Markets. Springer-Verlag, Berlin.CrossRefGoogle Scholar
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), 93–105.CrossRefGoogle Scholar
Kahalé, N. (2004) An arbitrage-free interpolation of volatilities. Risk Mag. 17 (5), 102–106.Google Scholar
Karatzas, I. & Shreve, S. (1998) Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, Vol. 113. Springer-Verlag, New York.CrossRefGoogle Scholar
Kieffer, J. B. (1982) Exponential rate of convergence for Lloyd's method I. IEEE Trans. Inform. Theory 28 (2), 205–210.CrossRefGoogle Scholar
Kijima, M. (1997) Markov Processes for Stochastic Modelling. Stochastic Modeling Series. Chapman & Hall, London.CrossRefGoogle Scholar
Kronmal, R. A. & Peterson, A. V. Jr (1979) On the Alias method for generating random variables from a discrete distribution. Am. Stat. 33 (4), 214–218.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Lipton, A. & Sepp, A. (2011) Filling the gaps. Risk 24 (10), 78.Google Scholar
Merton, R. C. (1973) Theory of rational option pricing. BellJ. Econ. Manag. Sci. 4 (1), 141–183.CrossRefGoogle Scholar
Mitchell, A. R. & Griffiths, D. F. (1980) The Finite Difference Method in Partial Differential Equations, John Wiley, New York.Google Scholar
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, 29–79.CrossRefGoogle Scholar
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, 253–297.CrossRefGoogle Scholar
Pagès, G. & Printems, J. (2003) Optimal quadratic quantization for numerics: the Gaussian case. Monte Carlo Methods Appl. 9 (2), 135–165.CrossRefGoogle Scholar
Pagès & Printems, J. (2005) Functional quantization for numerics with an application to option pricing. Monte Carlo Methods Appl. 11 (4), 407–446.Google Scholar
Pagès, G. & Sagna, A. (2015) Recursive marginal quantization of the Euler scheme of a diffusion process. Appl. Math. Finance 22 (5), 463–498.CrossRefGoogle Scholar
Pallavicini, A. (2016) A calibration algorithm for the local volatility model. In preparation.Google Scholar
Predota, M. (2005) On European and Asian option pricing in the generalized hyperbolic model. Eur. J. Appl. Math. 16 (1), 111–144.CrossRefGoogle Scholar
Reghai, A., Boya, G. & Vong, G. (2012) Local volatility: Smooth calibration and fast Usage. URL: http://ssrn.com/abstract=2008215, 29 March 2017.Google Scholar
Reiswich, D. & Uwe, W. (2012) FX volatility smile construction. Wilmott 2012 (60), 58–69.CrossRefGoogle Scholar
Ren, Y., Madan, D. & Qian, M. Q. (2007) Calibrating and pricing with embedded local volatility models. Risk Mag. 20 (9), 138.Google Scholar
Sidje, R. B. (1998) Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Softw. (TOMS) 24 (1), 130–156.CrossRefGoogle Scholar
Siyanko, S. (2012) Essentially exact asymptotic solutions for Asian derivatives. Eur. J. Appl. Math. 23 (3), 395–415.CrossRefGoogle Scholar
Vecer, J. & Xu, M. (2004) Pricing Asian options in a semimartingale model. Quant. Finance 4 (2), 170–175.CrossRefGoogle Scholar
Walker, H. F. & Ni, P. (2011) Anderson acceleration for fixed-point iterations. SIAM J. Number. Anal. 49 (4), 1715–1735.CrossRefGoogle Scholar
Ward, R. C. (1977) Numerical computation of the matrix exponential with accuracy estimate. SIAM J. Numer. Anal. 14 (4), 600–610.CrossRefGoogle Scholar
Wilmott, P., Dewynne, J. & Howison, S. (1993) Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford.Google Scholar