An Extension of the Black-Scholes and Margrabe Formulas to a Multiple Risk Economy
Author(s)
Werner Hürlimann
ABSTRACT
We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. For the simple class of log-normally distributed instantaneous rates of return, we construct an explicit state-price deflator. Since this includes the Black-Scholes and the Vasicek (Ornstein-Uhlenbeck) return models, the considered deflator is called Black-Scholes- Vasicek deflator. Besides a new elementary proof of the Black-Scholes and Margrabe option pricing formulas a validation of these in a multiple risk economy is achieved.
We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. For the simple class of log-normally distributed instantaneous rates of return, we construct an explicit state-price deflator. Since this includes the Black-Scholes and the Vasicek (Ornstein-Uhlenbeck) return models, the considered deflator is called Black-Scholes- Vasicek deflator. Besides a new elementary proof of the Black-Scholes and Margrabe option pricing formulas a validation of these in a multiple risk economy is achieved.
Cite this paper
nullW. Hürlimann, "An Extension of the Black-Scholes and Margrabe Formulas to a Multiple Risk Economy," Applied Mathematics, Vol. 2 No. 4, 2011, pp. 427-432. doi: 10.4236/am.2011.24053.
nullW. Hürlimann, "An Extension of the Black-Scholes and Margrabe Formulas to a Multiple Risk Economy," Applied Mathematics, Vol. 2 No. 4, 2011, pp. 427-432. doi: 10.4236/am.2011.24053.
References
[1] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 8, 1973, pp. 141-183. Reprinted in [2]. doi:10.2307/3003143 [2] R. C. Merton, “Continuous-Time Finance,” Basil Blackwell, 1990. [3] D. Duffie, “Dynamic Asset Pricing Theory,” Princeton University Press, New Jersey, 1992. [4] R. Rebonato and P. J?ckel, “The Most General Methodology to Create a Valid Correlation Matrix for Risk Management and Option Pricing Purposes,” Journal of Risk, Vol. 2, No. 2, 2000, pp. 17-27. [5] W. Hürlimann, “Méthodes Stochastiques D'évaluation du Rendement,” Proceedings of the 3rd International AFIR Colloquium, Rom, Vol. 2, 1993, pp. 629-649. [6] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-59. Reprinted in [7-9]. doi:10.1086/260062 [7] M. C. Jensen (Editor), “Studies in the Theory of Capital Market,” Praeger, New York, 1972. [8] D. L. Luskin (Editor), “Portfolio Insurance: A Guide to Dynamic Hedging,” John Wiley, New York, 1988. [9] L. Hugston (Editor), “Options: Classic Approaches to Pricing and Modelling,” Risk Books, London, 1999. [10] W. Margrabe, “The Value of an Option to Exchange one Asset for Another,” Journal of Finance, Vol. 33, No. 1, pp. 177-186, 1978. Reprinted in [9]. doi:10.2307/2326358
[1] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 8, 1973, pp. 141-183. Reprinted in [2]. doi:10.2307/3003143 [2] R. C. Merton, “Continuous-Time Finance,” Basil Blackwell, 1990. [3] D. Duffie, “Dynamic Asset Pricing Theory,” Princeton University Press, New Jersey, 1992. [4] R. Rebonato and P. J?ckel, “The Most General Methodology to Create a Valid Correlation Matrix for Risk Management and Option Pricing Purposes,” Journal of Risk, Vol. 2, No. 2, 2000, pp. 17-27. [5] W. Hürlimann, “Méthodes Stochastiques D'évaluation du Rendement,” Proceedings of the 3rd International AFIR Colloquium, Rom, Vol. 2, 1993, pp. 629-649. [6] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-59. Reprinted in [7-9]. doi:10.1086/260062 [7] M. C. Jensen (Editor), “Studies in the Theory of Capital Market,” Praeger, New York, 1972. [8] D. L. Luskin (Editor), “Portfolio Insurance: A Guide to Dynamic Hedging,” John Wiley, New York, 1988. [9] L. Hugston (Editor), “Options: Classic Approaches to Pricing and Modelling,” Risk Books, London, 1999. [10] W. Margrabe, “The Value of an Option to Exchange one Asset for Another,” Journal of Finance, Vol. 33, No. 1, pp. 177-186, 1978. Reprinted in [9]. doi:10.2307/2326358