Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T17:29:02.017Z Has data issue: false hasContentIssue false
  • English
  • Français

A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

Published online by Cambridge University Press:  05 January 2021

Puneet Pasricha Open the ORCID record for Puneet Pasricha [Opens in a new window]  and
Anubha Goel Open the ORCID record for Anubha Goel [Opens in a new window]
Puneet Pasricha
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India E-mails: [email protected]; [email protected]
Anubha Goel
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India E-mails: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

Keywords

double stochastic volatilityexchange optionsFeynman–Kac theoremHeston's stochastic volatility
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 3 , July 2022 , pp. 606 - 615
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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

Alòs, E. & Rheinländer, T. (2017). Pricing and hedging margrabe options with stochastic volatilities. In Economic Working Papers 1475. Department of Economics and Business, Universitat Pompeu Fabra.Google Scholar
Antonelli, F., Ramponi, A., & Scarlatti, S. (2010). Exchange option pricing under stochastic volatility: a correlation expansion. Review of Derivatives Research 13(1): 45–73.CrossRefGoogle Scholar
Bates, D.S. (2000). Post-’87 crash fears in the S&P 500 futures option market. Journal of Econometrics 94(1): 181–238.CrossRefGoogle Scholar
Blenman, L.P. & Clark, S.P. (2005). Power exchange options. Finance Research Letters 2(2): 97–106.CrossRefGoogle Scholar
Cheang, G.H. & Garces, L.P.D.M. (2020). Representation of exchange option prices under stochastic volatility jump-diffusion dynamics. Quantitative Finance 20(2): 291–310.CrossRefGoogle Scholar
Christoffersen, P., Jacobs, K., Ornthanalai, C., & Wang, Y. (2008). Option valuation with long-run and short-run volatility components. Journal of Financial Economics 90(3): 272–297.CrossRefGoogle Scholar
Da Fonseca, J., Grasselli, M., & Tebaldi, C. (2008). A multifactor volatility Heston model. Quantitative Finance 8(6): 591–604.CrossRefGoogle Scholar
Fischer, S. (1978). Call option pricing when the exercise price is uncertain, and the valuation of index bonds. The Journal of Finance 33(1): 169–176.CrossRefGoogle Scholar
Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6(2): 327–343.CrossRefGoogle Scholar
Kim, J.-H. & Park, C.-R. (2017). A multiscale extension of the Margrabe formula under stochastic volatility. Chaos, Solitons & Fractals 97: 59–65.CrossRefGoogle Scholar
Margrabe, W. (1978). The value of an option to exchange one asset for another. The Journal of Finance 33(1): 177–186.CrossRefGoogle Scholar
Pasricha, P. & Goel, A. (2019). Pricing vulnerable power exchange options in an intensity based framework. Journal of Computational and Applied Mathematics 355: 106–115.CrossRefGoogle Scholar
Pasricha, P. & Goel, A. (2021). Pricing power exchange options with Hawkes jump diffusion processes. Journal of Industrial & Management Optimization 17(1): 133–149.CrossRefGoogle Scholar
Wang, X. (2016). Pricing power exchange options with correlated jump risk. Finance Research Letters 19: 90–97.CrossRefGoogle Scholar
Wang, X., Song, S., & Wang, Y. (2017). The valuation of power exchange options with counterparty risk and jump risk. Journal of Futures Markets 37(5): 499–521.CrossRefGoogle Scholar
Xu, X. & Taylor, S.J. (1994). The term structure of volatility implied by foreign exchange options. Journal of Financial and Quantitative Analysis 29(1): 57–74.CrossRefGoogle Scholar
Xu, G., Shao, X., & Wang, X. (2019). Analytical valuation of power exchange options with default risk. Finance Research Letters 28: 265–274.CrossRefGoogle Scholar