Fourier inversion formulas for multiple-asset option pricing

Bruno Feunou 1  and Ernest Tafolong 2
  • 1 Bank of Canada, 234, Wellington Street, Ottawa, ON, K1A 0G9, Canada
  • 2 National Bank of Canada, 1155 Metcalfe Street, Montreal, QC H3B 4S9, Canada
Bruno Feunou and Ernest Tafolong

Abstract

Plain vanilla options have a single underlying asset and a single condition on the payoff at the expiration date. For this class of options, a well known result of Duffie, Pan, and Singleton (Duffie, D., J. Pan, and K. Singleton. 2000. “Transform Analysis and Asset Pricing for Affine Jump-Diffusions.” Econometrica 68: 1343–1376. http://dx.doi.org/10.1111/1468-0262.00164.) shows how to invert the characteristic function to obtain a closed-form formula for their prices. However, multiple-asset and multiple-condition derivatives such as rainbow options cannot be priced within this framework. This paper provides an analytical solution for options whose payoffs depends on two or more conditions. We take the advantage of the inversion of the Fourier transform, resorting to neither Black and Scholes’s framework, nor the affine models’s settings. Numerical experiments based on the aforementioned class of derivatives are provided to illustrate the usefulness of the proposed approach.

    • Supplemental_Data_and_Code
  • Ahn, D., and B. Gao. 1999. “A Parametric Nonlinear Model of Term Structure Dynamics.” Review of Financial Studies 12: 721–762. http://rfs.oxfordjournals.org/content/12/4/721.abstract.

    • Crossref
    • Export Citation
  • Andersen, L., and J. Andreasen. 2000. “Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing.” Review of Derivatives Research 4: 231–262. http://dx.doi.org/10.1023/A%3A1011354913068.

    • Crossref
    • Export Citation
  • Andersen, T. G., T. Bollerslev, F. X. Diebold, and H. Ebens. 2001. “The Distribution of Realized Stock Return Volatility.” Journal of Financial Economics 61: 43–76. http://www.sciencedirect.com/science/article/pii/S0304405X01000551.

  • Bakshi, G., C. Cao, and Z. Chen. 1997. “Empirical Performance of Alternative Option Pricing Models.” The Journal of Finance 52: 2003–2049. http://dx.doi.org/10.1111/j.1540-6261.1997.tb02749.x.

    • Crossref
    • Export Citation
  • Bates, D. 1996. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” Review of Financial Studies 9: 69–107. http://rfs.oxfordjournals.org/content/9/1/69.abstract.

    • Crossref
    • Export Citation
  • Buraschi, A., P. Porchia, and T. Fabio. 2010. “Correlation Risk and Optimal Portfolio Choice.” The Journal of Finance 65: 393–420. http://dx.doi.org/10.1111/j.1540-6261.2009.01533.x.

    • Crossref
    • Export Citation
  • Buraschi, A., P. Porchia, and T. Fabio. 2014. “When Uncertainty Blows in the Orchard: Comovement and Equilibrium Volatility Risk Premia.” The Journal of Finance 69: 101–137. http://dx.doi.org/10.1111/jofi.12095.

    • Crossref
    • Export Citation
  • Buss, A., and G. Vilkov. 2012. “Measuring Equity Risk with Option-Implied Correlations.” Review of Financial Studies 25: 3113–3140. http://rfs.oxfordjournals.org/content/25/10/3113.abstract.

  • Chang, C.-C., S.-L. Chung, and M.-T. Yu. 2006. “Loan Guarantee Portfolios and Joint Loan Guarantees with Stochastic Interest Rates.” The Quarterly Review of Economics and Finance 46: 16–35. http://www.sciencedirect.com/science/article/pii/S1062976903000917.

  • Christoffersen, P., S. Heston, and K. Jacobs. 2006. “Option Valuation with Conditional Skewness.” Journal of Econometrics 131: 253–284. http://www.sciencedirect.com/science/article/pii/S0304407605000126.

    • Crossref
    • Export Citation
  • Christoffersen, P., B. Feunou, K. Jacobs, and N. Meddahi. 2014. “The Economic Value of Realized Volatility: Using High-Frequency Returns for Option Valuation.” Journal of Financial and Quantitative Analysis 49: 663–697. http://journals.cambridge.org/article_S0022109014000428.

    • Crossref
    • Export Citation
  • Davies, R. B. 1973. “Numerical Inversion of a Characteristic Function.” Biometrika 60: 231–262.

    • Crossref
    • Export Citation
  • Duffie, D., J. Pan, and K. Singleton. 2000. “Transform Analysis and Asset Pricing for Affine Jump-Diffusions.” Econometrica 68: 1343–1376. http://dx.doi.org/10.1111/1468-0262.00164.

    • Crossref
    • Export Citation
  • Duffie, D., D. Filipović, and W. Schachermayer. 2003. “Affine Processes and Applications in Finance.” Annals of Applied Probability 13: 984–1053.

    • Crossref
    • Export Citation
  • Duffy, D. J. 2009. “Numerical Analysis of Jump Diffusion Models: A Partial Differential Equation Approach.” Wilmott Magazine.

  • Dufresne, D., J. Garrido, and M. Morales. 2009. “Fourier Inversion Formulas in Option Pricing and Insurance.” Methodology and Computing in Applied Probability 11: 359–383. http://dx.doi.org/10.1007/s11009-007-9049-z.

    • Crossref
    • Export Citation
  • Engle, R. 2000. “Dynamic Conditional Correlation – A Simple Class of Multivariate Garch Models.” Journal of Business and Economic Statistics 20: 339–350.

    • Crossref
    • Export Citation
  • Feunou, B., and J.-S. Fontaine. 2010. “Discrete Choice Term Structure Models: Theory and Applications.” Working Paper, Duke University and Bank of Canada.

    • Crossref
    • Export Citation
  • Feunou, B., and N. Meddahi. 2009. “Generalized Affine Model, Provides a General Framework that Characterizes Infinite Order Affine Model.” Working Paper, Duke University.

    • Crossref
    • Export Citation
  • Forsberg, L., and T. Bollerslev. 2002. “Bridging the Gap Between the Distribution of Realized (ecu) Volatility and Arch Modelling (of the euro): The Garch-nig Model.” Journal of Applied Econometrics 17: 535–548. http://dx.doi.org/10.1002/jae.685.

    • Crossref
    • Export Citation
  • Gay, G. D., and S. Manaster. 1984. “The Quality Option Implicit in Futures Contracts.” Journal of Financial Economics 13: 353–370. http://www.sciencedirect.com/science/article/pii/0304405X84900047.

    • Crossref
    • Export Citation
  • Gerber, H. U., and E. S. W. Shiu. 1994. “Option Pricing by Esscher Transforms.” Transactions of the Society of Actuaries 46: 99–191.

  • Glasserman, P. 2003. Monte Carlo Methods in Financial Engineering. 1st ed. New York, NY, USA: Springer.

    • Crossref
    • Export Citation
  • Gourieroux, C. 2006. “Continuous Time Wishart Process for Stochastic Risk.” Econometric Reviews 25: 177–217. http://dx.doi.org/10.1080/07474930600713234.

    • Crossref
    • Export Citation
  • Gourieroux, C. and R. Sufana. 2010. “Derivative Pricing with Wishart Multivariate Stochastic Volatility.” Journal of Business and Economic Statistics 28: 438–451. http://dx.doi.org/10.1198/jbes.2009.08105.

    • Crossref
    • Export Citation
  • Hazewinkel, M. 2002. Encyclopaedia of Mathematics. 1st ed. Berlin Heidelberg New York, NY, USA: Springer-Verlag.

  • Heston, S. 1993. “A closed-Form Solution for Options with Stochastic Volatility with Applications To Bond and Currency Options.” Review of Financial Studies, 6: 327–343. http://rfs.oxfordjournals.org/content/6/2/327.abstract.

    • Crossref
    • Export Citation
  • Johnson, H. 1987. “Options on The Maximum or The Minimum of Several Assets.” Journal of Financial and Quantitative Analysis 22: 277–283. http://journals.cambridge.org/article_S002210900001262X.

    • Crossref
    • Export Citation
  • Kempf, A., O. Korn, and S. Saßning. 2014. “Portfolio Optimization Using Forward-Looking Information.” Review of Finance. DOI: 10.1093/rof/rfu006. http://rof.oxfordjournals.org/content/early/2014/03/07/rof.rfu006.abstract.

    • Crossref
    • Export Citation
  • Margrabe, W. 1978. “The Value of An Option to Exchange One Asset for Another.” The Journal of Finance 33: 177–186. http://dx.doi.org/10.1111/j.1540-6261.1978.tb03397.x.

    • Crossref
    • Export Citation
  • Martzoukous, S. H. 2001. “The Option on n Assets with Exchange Rate and Exercise Price Risk.” Journal of Multinational Financial Management 11: 1–15.

    • Crossref
    • Export Citation
  • Navatte, P., and C. Villa. 2000. “The Information Content of Implied Volatility, Skewness and Kurtosis: Empirical Evidence From Long-term cac 40 Options.” European Financial Management, 6: 41–56. http://dx.doi.org/10.1111/1468-036X.00110.

    • Crossref
    • Export Citation
  • Qu, D. 2010. “Pricing Basket Options with Skew.” Wilmott Magazine.

  • Rota, G.-C. 1964. “On the Foundations of Combinatory Theory of Mö,bius Functions.” Wahrscheinlichkeitstheorie und Verw 2: 340–368.

    • Crossref
    • Export Citation
  • Shephard, N., and K. Sheppard. 2010. “Realising the Future: Forecasting with High frequency-Based Volatility (heavy) Models.” Journal of Applied Econometrics 25: 197–231. http://dx.doi.org/10.1002/jae.1158.

    • Crossref
    • Export Citation
  • Stoll, R. M. 1969. “The Relationship Between Put and Call Option Prices.” Journal of Finance 24: 801–824.

    • Crossref
    • Export Citation
  • Stulz, R. M. 1982. “Options on The Minimum or The Maximum of Two Risky Assets: Analysis and Applications.” Journal of Financial Economics 10: 161–185. http://www.sciencedirect.com/science/article/pii/0304405X82900113.

    • Crossref
    • Export Citation
  • van Binsbergen, J., M. Brandt, and R. Koijen. 2012. “On the Timing and Pricing of Dividends.” American Economic Review 102: 1596–1618. http://www.aeaweb.org/articles.php?doi=10.1257/aer.102.4.1596.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

SNDE recognizes that advances in statistics and dynamical systems theory can increase our understanding of economic and financial markets. The journal seeks both theoretical and applied papers that characterize and motivate nonlinear phenomena. Researchers are required to assist replication of empirical results by providing copies of data and programs online. Algorithms and rapid communications are also published.

Search