Brazilian Journal of Probability and Statistics

Option pricing with bivariate risk-neutral density via copula and heteroscedastic model: A Bayesian approach

Lucas Pereira Lopes, Vicente Garibay Cancho, and Francisco Louzada

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Multivariate options are adequate tools for multi-asset risk management. The pricing models derived from the pioneer Black and Scholes method under the multivariate case consider that the asset-object prices follow a Brownian geometric motion. However, the construction of such methods imposes some unrealistic constraints on the process of fair option calculation, such as constant volatility over the maturity time and linear correlation between the assets. Therefore, this paper aims to price and analyze the fair price behavior of the call-on-max (bivariate) option considering marginal heteroscedastic models with dependence structure modeled via copulas. Concerning inference, we adopt a Bayesian perspective and computationally intensive methods based on Monte Carlo simulations via Markov Chain (MCMC). A simulation study examines the bias, and the root mean squared errors of the posterior means for the parameters. Real stocks prices of Brazilian banks illustrate the approach. For the proposed method is verified the effects of strike and dependence structure on the fair price of the option. The results show that the prices obtained by our heteroscedastic model approach and copulas differ substantially from the prices obtained by the model derived from Black and Scholes. Empirical results are presented to argue the advantages of our strategy.

Article information

Braz. J. Probab. Stat., Volume 33, Number 4 (2019), 801-825.

Received: August 2018
Accepted: April 2019
First available in Project Euclid: 26 August 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Option pricing heterocedastic copula Bayesian inference


Pereira Lopes, Lucas; Garibay Cancho, Vicente; Louzada, Francisco. Option pricing with bivariate risk-neutral density via copula and heteroscedastic model: A Bayesian approach. Braz. J. Probab. Stat. 33 (2019), no. 4, 801--825. doi:10.1214/19-BJPS445.

Export citation


  • Abanto-Valle, C. A., Lachos, V. H. and Dey, K. (2015). Bayesian estimation of a skew-student-$t$ stochastic volatility model. Methodology and Computing in Applied Probability 17, 721–738.
  • Ausin, M. C. and Lopes, H. F. (2010). Time-varying joint distribution through copulas. Computational Statistics & Data Analysis 54, 2383–2399.
  • Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654.
  • Cherubini, U. and Luciano, E. (2002). Bivariate option pricing with copulas. Applied Mathematical Finance 9, 69–85.
  • Chiou, S. C. and Tsay, R. S. (2008). A copula-based approach to option pricing and risk assessment. Journal of Data Science 6, 273–301.
  • Delatola, E. I. and Griffin, J. E. (2011). Bayesian nonparametric modelling of the return distribution with stochastic volatility. Bayesian Analysis 6, 901–926.
  • Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300, 463–520.
  • Duan, J. C. (1995). The GARCH option pricing model. Mathematical Finance 5, 13–32.
  • Embrechts, P., McNeil, A. and Straumann, D. (2002). Correlation and dependence in risk management: Properties and pitfalls. Risk Management: Value at Risk and Beyond.
  • Fonseca, T. C., Migon, H. S. and Ferreira, M. A. (2012). Bayesian analysis based on the Jeffreys prior for the hyperbolic distribution. Brazilian Journal of Probability and Statistics, 327–343.
  • Forbes, K. J. and Rigobon, R. (2002). No contagion, only interdependence: Measuring stock market comovements. The Journal of Finance 57, 2223–2261.
  • Franses, P. H. and Van Dijk, D. (2000). Non-linear Time Series Models in Empirical Finance. Cambridge: Cambridge University Press.
  • French, K. R., William, S. G. and Stambaugh, R. F. (1987). Expected stock returns and volatility. Journal of Financial Economics 19, 3–29.
  • Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 169–193. London: Oxford University Press.
  • Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas, Vol. 2. New York: McGraw-Hill.
  • Hull, J. (1992). Introduction to Futures and Options Markets. Englewood Cliffs, NJ: Prentice Hall.
  • Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance 42, 281–300.
  • Hürlimann, W. (2004). Fitting bivariate cumulative returns with copulas. Computational Statistics & Data Analysis 45, 355–372.
  • Johnson, H. and David, S. (1987). Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis, 143–151.
  • Kang, T. and Brorsen, B. W. (1993). GARCH option pricing with asymmetry. In Proceedings of the NCR-134 Conference on Applied Commodity, Forecasting, and Market Risk Management.
  • Klugman, S. A. and Parsa, R. (1999). Fitting bivariate loss distributions with copulas. Insurance Mathematics & Economics 24(1–2), 139–148.
  • Leão, W. L., Abanto-Valle, C. A. and Chen, M. H. (2017). Bayesian analysis of stochastic volatility-in-mean model with leverage and asymmetrically heavy-tailed error using generalized hyperbolic skew student’s $t$-distribution. Statistics and its Interface 10, 529–541.
  • Lopes, L. P. and Pessanha, G. R. G. (2018). Análise de dependência entre mercados financeiros: Uma abordagem do modelo copula-GARCH. Revista de Financas e Contabilidade da Unimep.
  • Madan, D. B. and Milne, F. (1991). Option pricing with VG martingale components. Mathematical Finance 1, 39–55.
  • Margrabe, W. (1978). The value of an option to exchange one asset for another. The Journal of Finance 33, 177–186.
  • Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 141–183.
  • Nelsen, R. B. (2006) An Introduction to Copulas, 2nd ed. New York: Springer Science Business Media.
  • Rosenberg, J. V. (2002). Nonparametric pricing of multivariate contingent claims. Technical Report.
  • Rossi, J. L., Ehlers, R. S. and Andrade, M. G. (2012). Copula-GARCH Model Selection: A Bayesian Approach. University of São Paulo, Technical Report 88.
  • Sanfins, M. A. and Valle, G. (2012). On the copula for multivariate extreme value distributions. Brazilian Journal of Probability and Statistics 26, 288–305.
  • Schmidt, T. (2007). Coping with Copulas. Copulas-from Theory to Application in Finance. Risk Books.
  • Sharifonnasabi, Z., Alamatsaz, M. H. and Kazemi, I. (2018). A large class of new bivariate copulas and their properties. Brazilian Journal of Probability and Statistics 32, 497–524.
  • Shimko, D. C. (1994). Options on futures spreads: Hedging, speculation, and valuation. The Journal of Futures Markets 14, 183–213.
  • Sklar, A. (1959). Fonctions de repartition a n dimensions et leures marges. Publications de L’Institut de Statistique de L’Université de Paris 8, 229–231.
  • Spiegelhalter, D. J., et al. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B, Statistical Methodology 64, 583–639.
  • Stulz, R. (1982). Options on the minimum or the maximum of two risky assets: Analysis and applications. Journal of Financial Economics 10(2), 161–185.
  • Zhang, J. and Guegan, D. (2008). Pricing bivariate option under GARCH processes with time-varying copula. Insurance Mathematics & Economics 42, 1095–1103.