Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-vtfg7 Total loading time: 0.209 Render date: 2022-05-23T21:42:18.762Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Memory-Reduction Method for Pricing American-Style Options under Exponential Lévy Processes

Published online by Cambridge University Press:  28 May 2015

Raymond H. Chan*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong SAR, PR China
Tao Wu*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong SAR, PR China
*
Corresponding author. Email: rchan@math.cuhk.edu.hk
Corresponding author. Email: twu@math.cuhk.edu.hk
Get access

Abstract

This paper concerns the Monte Carlo method in pricing American-style options under the general class of exponential Lévy models. Traditionally, one must store all the intermediate asset prices so that they can be used for the backward pricing in the least squares algorithm. Therefore the storage requirement grows like , where m is the number of time steps and n is the number of simulated paths. In this paper, we propose a simulation method where the storage requirement is only . The total computational cost is less than twice that of the traditional method. For machines with limited memory, one can now enlarge m and n to improve the accuracy in pricing the options. In numerical experiments, we illustrate the efficiency and accuracy of our method by pricing American options where the log-prices of the underlying assets follow typical Lévy processes such as Brownian motion, lognormal jump-diffusion process, and variance gamma process.

Type
Research Article
Copyright
Copyright © Global-Science Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Amin, K.I., Jump diffusion option valuation in discrete time, Journal of Finance, 48 (1993), pp.18331863.CrossRefGoogle Scholar
[2]Barndorff-Nielsen, O., Processes of normal inverse Gaussian type, Finance and Stochastics, 2 (1997), pp. 4168.CrossRefGoogle Scholar
[3]Black, F. and Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), pp. 637654.CrossRefGoogle Scholar
[4]Boyle, P., Option: a Monte Carlo approach, Journal of Financial Economics, 4 (1977), pp. 323338.CrossRefGoogle Scholar
[5]Carr, P., Geman, H., Madan, D., and Yor, M., The fine structure of asset returns: an empirical investigation, Journal of Business, 75 (2002), pp. 305332.CrossRefGoogle Scholar
[6]Chan, R.H., Chen, Y. and Yeung, K.M., A memory reduction method in pricing American options, Journal of Statistical Computation and Simulation, 74 (2004), pp. 501511.CrossRefGoogle Scholar
[7]Chan, R.H., Wong, C.Y. and Yeung, K.M., Pricing multi-asset American-style options bymemory reduction Monte Carlo methods, Applied Mathematics and Computation, 179 (2006), pp. 535544.CrossRefGoogle Scholar
[8]Chapman, S. (1998), Fortran 90/95 for scientists and engineers, McGraw-Hill.Google Scholar
[9]Chaudhary, S.K., American options and the LSM algorithm: quasi-random sequences and Brownian bridges, Journal of Computational Finance, 8 (2005), pp. 101115.CrossRefGoogle Scholar
[10]Cont, R., Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance, 1 (2001), pp. 114.CrossRefGoogle Scholar
[11]Cont, R. and Tankov, P., Financial Modelling with Jump Processes, Chapman & Hall/CRC Press, London, 2004.Google Scholar
[12]Devroye, L., Non-Uniform Random Variate Generation, Springer Verlag, New York, 1986.CrossRefGoogle Scholar
[13]Glasserman, P., Monte Carlo Methods in Financial Engineering, Springer Verlag, New York, 2003.CrossRefGoogle Scholar
[14]Hirsa, A. and Madan, D.B., Pricing American options under variance gamma, Journal of Computational Finance, 7 (2003), pp. 6380.CrossRefGoogle Scholar
[15]Kou, S.G., A jump-diffusion model for option pricing, Management Science, 48 (2002), pp. 10861101.CrossRefGoogle Scholar
[16]Longstaff, F.A. and Schwartz, E.S., Valuing American options by simulation: a simple least-squares approach, The Review of Financial Studies, 14 (2001), pp. 113147.CrossRefGoogle Scholar
[17]Madan, D.B., Carr, P.P., and Chang, E.C., The variance gamma process and option pricing, European Finance Review, 2 (1998), pp. 79105.CrossRefGoogle Scholar
[18]Mandelbrot, B.B., The variation of certain speculative prices, Journal of Business, XXXVI (1963), pp. 392417.Google Scholar
[19]Merton, R.C., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), pp. 125144.CrossRefGoogle Scholar
[20]Moler, C., Numerical Computing with MATLAB, SIAM, Philadelphia, 2004.CrossRefGoogle Scholar
[21]Park, S.K. and Miller, K.W., Random number generators: good ones are hard to find, Communications of the ACM, 31 (1988), pp. 11921201.CrossRefGoogle Scholar
[22]Ribeiro, C. and Webber, N., A Monte Carlo method for the normal inverse Gaussian option valuation model using an inverse Gaussian bridge, working paper, Cass Business School, City University, 2003.Google Scholar
[23]Ribeiro, C. and Webber, N., Valuing path-dependent options in the variance gamma model by Monte Carlo with a gamma bridge, Journal of Computational Finance, 7 (2004), pp. 81100.CrossRefGoogle Scholar
[24]Wilmott, P., Howison, S., and Dewynne, J., The Mathematics of Financial Derivatives, Cambridge University Press, Cambridge, 1998.Google Scholar
[25]Wolfram, S., The Mathematica Book, 4th ed., Cambridge University Press, Cambridge, 1999.Google Scholar
5
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Memory-Reduction Method for Pricing American-Style Options under Exponential Lévy Processes
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Memory-Reduction Method for Pricing American-Style Options under Exponential Lévy Processes
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Memory-Reduction Method for Pricing American-Style Options under Exponential Lévy Processes
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *