Monte Carlo methods for pricing financial options
Tóm tắt
Pricing financial options is amongst the most important and challenging problems in the modern financial industry. Except in the simplest cases, the prices of options do not have a simple closed form solution and efficient computational methods are needed to determine them. Monte Carlo methods have increasingly become a popular computational tool to price complex financial options, especially when the underlying space of assets has a large dimensionality, as the performance of other numerical methods typically suffer from the ‘curse of dimensionality’. However, even Monte-Carlo techniques can be quite slow as the problem-size increases, motivating research in variance reduction techniques to increase the efficiency of the simulations. In this paper, we review some of the popular variance reduction techniques and their application to pricing options. We particularly focus on the recent Monte-Carlo techniques proposed to tackle the difficult problem of pricing American options. These include: regression-based methods, random tree methods and stochastic mesh methods. Further, we show how importance sampling, a popular variance reduction technique, may be combined with these methods to enhance their effectiveness. We also briefly review the evolving options market in India.
Tài liệu tham khảo
Ahamed T P I, Borkar V S, Juneja S 2004 Adaptive importance sampling technique for Markov chains using stochastic approximation.Oper. Res. (to appear)
Andersen L, Broadie M 2001 A primal-dual simulation algorithm for pricing multi-dimensional American options. Working Paper, Columbia Business School, New York
Bertsekas D P, Tsitsiklis J N 1996Neuro-dynamic programming (Belmont, MA: Athena Scientific)
Black F, Scholes M 1973 The pricing of options and corporate liabilities.J. Polit. Econ. 81: 637–654
Bolia N, Glasserman P, Juneja S 2004 Function-approximation-based importance sampling for pricing American options.Proceedings of the 2004 Winter Simulation Conference (New York: IEEE Press)
Bossaerts P 1989 Simulation estimators of optimal early exercise. Working Paper, Carnegie Mellon University, Pittsburg
Boyle P, Broadie M, Glasserman P 1997 Monte Carlo methods for security pricing.J. Econ. Dynam. Contr. 21: 1267–1321
Bratley P, Fox B L, Schrage L 1987A guide to simulation (New York: Springer-Verlag)
Broadie M, Glasserman P 1997 Pricing American-style securities by simulation.J. Econ. Dynam. Contr. 21: 1323–1352
Broadie M, Glasserman P 2004 A stochastic mesh method for pricing high-dimensional American options.J. Comput. Finan. 7: 35–72
Carriere J F 1996 Valuation of the early-exercise price for derivative securities using simulations and splines.Insurance: Math. Econ. 19: 19–30
Cox J C, Ross S A, Rubinstein M 1979 Option pricing: A simplified approach.J. Finan. Econ. 7: 229–263
Duffie D 1996Dynamic asset pricing theory (Princeton, NJ: University Press)
Esary J D, Proschan F, Walkup D W 1967 Association of random variables, with applications.Ann. Math. Stat. 38: 1466–1474
Garcia D 2003 Convergence and biases of Monte Carlo estimates of American option prices using a parametric exercise rule.J. Econ. Dynam. Contr. 27: 1855–1879
Glasserman P 2004Monte Carlo methods in financial engineering (New York: Springer-Verlag)
Glynn P W, Iglehart D L 1989 Importance sampling for stochastic simulations.Manage. Sci. 35: 1367–1392
Glynn P W, Whitt W 1992 The asymptotic efficiency of simulation estimators.Oper. Res. 40: 505–520
Grant D, Vora G, Weeks D 1996 Simulation and the early-exercise option problem.J. Finan. Eng. 5: 211–227
Harrison J M, Kreps D M 1979 Martingales and arbitrage in multiperiod securities markets.J. Econ. Theor. 20: 381–408
Harrison J M, Pliska S R 1981 Martingales and stochastic integrals in the theory of continuous trading.Stochast. Process. Appl. 11: 215–260
Haugh M B, Kogan L 2001 Pricing American options: A duality approach.Oper. Res. 52: 258–270
Heidelberger P 1995 Fast simulation of rare events in queueing and reliability models.ACM Trans. Modeling Comput. Simul. 5: 43–85
Jamshidian F 2003 Minimax optimality of Bermudan and American claims and their Monte-Carlo upper bound approximation (preprint)
Juneja S 2001 Importance sampling and the cyclic approach.Oper. Res. 49: 900–912
Juneja S, Shahabuddin P 2005 Rare event simulation techniques: An introduction and recent advances.Handbook on simulation (eds) S Henderson, B Nelson (to appear)
Karatzas I, Shreve S 1991Brownian motion and stochastic calculus, (New York: Springer-Verlag)
Kemna A G Z, Vorst A C F 1990 A pricing method for options based on average asset values.J. Banking and finance 14: 113–129
Law A M, Kelton W D 1991Simulation modeling and analysis (New York: McGraw-Hill)
Lawson C L, Hanson R J 1974Solving least squares problems (Englewood Cliffs, NJ: Prentice-Hall) 385
Li B, Zhang G 1996 Hurdles removed. Working Paper, Merrill Lynch
Longstaff F A, Schwartz E S 2001 Valuing American options by simulation: A simple least-squares approach.Rev. Finan. Stud. 14: 113–147
Nelsen R B 1999An introduction to copulas (New York: Springer-Verlag)
Oksendal B 2003Stochastic differential equations: An introduction with applications (New York: Springer-Verlag)
Shah A, Thomas S 2000Equity derivatives in India: The state of the art in derivatives markets in India 2003 (ed.) S THOMAS (New Delhi: Tata McGraw-Hill)
Shah A, Thomas S 2003 Derivatives in India: Frequently asked questions from http://www.igidr.ac.in/ susant/sue-resli4.html#x5-9000
Shirayev A 1978Optimal stopping rules (New York: Springer-Verlag)
Sundaram R K 1999A first course in optimization theory (Cambridge: University Press)
Tsitsiklis J, Van Roy B 2001 Regression methods for pricing complex American-style options.IEEE Trans. Neural Networks 12: 694–703
Williams D 1991Probability with Martingales (Cambridge: University Press)
Wilmott P, Dewynne J, Howison S 1993Option pricing: Mathematical models and computation (Oxford: Oxford Financial Press)
Wu R, Fu M 2003 Optimal exercise policies and simulation-based valuation for American-Asian options.Oper. Res. 51: 52–66