Probability distribution and option pricing for drawdown in a stochastic volatility environment. (English) Zbl 1203.91299
Summary: This paper studies the probability distribution and option pricing for drawdown in a stochastic volatility environment. Their analytical approximation formulas are derived by the application of a singular perturbation method J.-P. Fouque, G. Papanicolaou and K. R. Sircar [Derivatives in financial markets with stochastic volatility, Cambridge, Cambridge University Press (2000; Zbl 0954.91025)]. The mathematical validity of the approximation is also proven. Then, numerical examples show that the instantaneous correlation between the asset value and the volatility state crucially affects the probability distribution and option prices for drawdown.
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 91B70 | Stochastic models in economics |
| 91G80 | Financial applications of other theories |
Citations:
Zbl 0954.91025References:
| [1] | DOI: 10.1142/S0219024905002767 · Zbl 1100.91040 · doi:10.1142/S0219024905002767 |
| [2] | Cvitanic J., IMA Lecture Notes in Mathematics & Applications 65 pp 77– |
| [3] | Fouque J. P., Derivatives in Financial Markets with Stochastic Volatility (2000) · Zbl 0954.91025 |
| [4] | Fouque J. P., SIAM Journal on Applied Mathematics 63 pp 1648– |
| [5] | DOI: 10.1111/j.1467-9965.1993.tb00044.x · Zbl 0884.90031 · doi:10.1111/j.1467-9965.1993.tb00044.x |
| [6] | DOI: 10.3905/jai.2007.700229 · doi:10.3905/jai.2007.700229 |
| [7] | DOI: 10.1137/S0036139902420043 · Zbl 1099.91062 · doi:10.1137/S0036139902420043 |
| [8] | DOI: 10.1007/978-1-4612-0949-2 · doi:10.1007/978-1-4612-0949-2 |
| [9] | DOI: 10.3905/jai.2002.319040 · doi:10.3905/jai.2002.319040 |
| [10] | Magdon-Ismail M., Risk 17 pp 99– |
| [11] | DOI: 10.1239/jap/1077134674 · Zbl 1051.60083 · doi:10.1239/jap/1077134674 |
| [12] | Pospisil L., Journal of Computational Finance 12 pp 59– · Zbl 1175.91200 · doi:10.21314/JCF.2008.177 |
| [13] | Vecer J., Risk 19 pp 88– · Zbl 1402.91192 |
| [14] | Vecer J., Wilmott 5 |
| [15] | Wilmott P., Mathematics of Financial Derivatives, A Student Introduction (1996) |
| [16] | DOI: 10.1007/s10690-009-9099-z · Zbl 1177.91133 · doi:10.1007/s10690-009-9099-z |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
