Efficiently pricing double barrier derivatives in stochastic volatility models. (English) Zbl 1307.91174
This paper uses ideas of a preceding paper of the last two authors [Stat. Probab. Lett. 82, No. 1, 165–172 (2012; Zbl 1229.91310)] to construct converging series in order to represent prices of barrier derivatives. The underlying is modeled as a stochastic volatility diffusion. The technique used involves computing the Laplace transforms of time changes for several contracts of interest. The authors refer to the paper of P. Carr and R. Lee [Math. Finance 19, No. 4, 523–560 (2009; Zbl 1184.91198)] for motivation and for the proof of several pricing formulas.
Reviewer: Gianluca Cassese (Milano)
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
References:
| [1] | Bakshi, G., & Madan, D. (2000). Spanning and derivative security valuation. Journal of Financial Economics, 55(2), 205-238. · doi:10.1016/S0304-405X(99)00050-1 |
| [2] | Ball, C. A., & Roma, A. (1994). Stochastic volatility option pricing. Journal of Financial and Quantitative Analysis, 29, 589-607. · doi:10.2307/2331111 |
| [3] | Barndorff-Nielsen, O., & Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics. Journal of the Royal Statistical Society: Series B, 63(2), 167-241. · Zbl 0983.60028 · doi:10.1111/1467-9868.00282 |
| [4] | Bates, D. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies, 9(1), 69-107. · doi:10.1093/rfs/9.1.69 |
| [5] | Billingsley, P. (2008). Probability and measure. Wiley series in probability. |
| [6] | Bingham, N., & Kiesel, R. (2004). Risk-neutral valuation. Berlin: Springer. · Zbl 1058.91029 · doi:10.1007/978-1-4471-3856-3 |
| [7] | Black, F., & Cox, J. C. (1976). Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance, 31(2), 351-367. · doi:10.1111/j.1540-6261.1976.tb01891.x |
| [8] | Brockwell, P., Chadraa, E., & Lindner, A. (2006). Continuous-time GARCH processes. The Annals of Applied Probability, 16(2), 790-826. · Zbl 1127.62074 · doi:10.1214/105051606000000150 |
| [9] | Carr, P., & Crosby, J. (2010). A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options. Journal of Quantitative Finance, 10(10), 1115-1136. · Zbl 1229.91300 · doi:10.1080/14697680903413605 |
| [10] | Carr, P., & Lee, R. (2009). Put-call symmetry: Extensions and applications. Mathematical Finance, 19(4), 523-560. · Zbl 1184.91198 · doi:10.1111/j.1467-9965.2009.00379.x |
| [11] | Carr, P., & Madan, D. B. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2, 61-73. |
| [12] | Carr, P., Ellis, K., & Gupta, V. (1998). Static heding of exotic derivatives. Journal of Finance, 53(3), 1165-1190. · doi:10.1111/0022-1082.00048 |
| [13] | Carr, P., Geman, H., Madan, D., & Yor, M. (2003). Stochastic volatility for Lévy processes. Mathematical Finance, 13(3), 345-382. · Zbl 1092.91022 · doi:10.1111/1467-9965.00020 |
| [14] | Carr, P., Zhang, H., & Hadjiliadis, O. (2011). Maximum drawdown insurance. International Journal of Theoretical and Applied Finance, 14(8), 1195-1230. · Zbl 1233.91115 · doi:10.1142/S0219024911006826 |
| [15] | Christoffersen, P., Heston, S., & Jacobs, K. (2009). The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Management Science, 55, 1914-1932. · Zbl 1232.91718 · doi:10.1287/mnsc.1090.1065 |
| [16] | Cox, D., & Isham, V. (1980). Monographs on applied probability and statistics: Point processes. London: Chapman & Hall. · Zbl 0441.60053 |
| [17] | Cox, D., & Miller, H. (1965). Theory of stochastic processes. London: Chapman & Hall. · Zbl 0149.12902 |
| [18] | Cox, J., Ingersoll, J., & Ross, S. (1985). A theory of the term structure of interest rates. Econometrica, 53, 187-201. |
| [19] | da Fonseca, J., Grasselli, M., & Tebaldi, C. (2007). Option pricing when correlations are stochastic: An analytical framework. Review of Derivatives Research, 10, 151-180. · Zbl 1174.91006 · doi:10.1007/s11147-008-9018-x |
| [20] | Darling, D., & Siegert, A. (1953). The first passage problem for a continuous Markov process. The Annals of Mathematical Statistics, 24(4), 624-639. · Zbl 0053.27301 · doi:10.1214/aoms/1177728918 |
| [21] | Dassios, A., & Jang, J. W. (2003). Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance and Stochastics, 7, 73-95. · Zbl 1039.91038 · doi:10.1007/s007800200079 |
| [22] | Derman, E., Ergener, D., & Kani, I. (1994). Forever hedged. Risk, 7, 139-145. |
| [23] | Dufresne, D. (2001). The integrated square-root process. Working paper. University of Montreal. · Zbl 1174.91006 |
| [24] | Dupont, D. (2002). Hedging barrier options: Current methods and alternatives. Working paper. · Zbl 0045.04901 |
| [25] | Eraker, B., Johannes, M., & Polson, N. (2003). The impact of jumps in volatility and returns. Journal of Finance, 58(3), 1269-1300. · doi:10.1111/1540-6261.00566 |
| [26] | Escobar, M., Friederich, T., Seco, L., & Zagst, R. (2011). A general structural approach for credit modeling under stochastic volatility. Journal of Financial Transformation, 32, 123-132. · Zbl 1274.91466 |
| [27] | Feller, W. (1951). Two singular diffusion problems. The Annals of Mathematics, 54(1), 173-182. · Zbl 0045.04901 · doi:10.2307/1969318 |
| [28] | Geman, H., & Yor, M. (1996). Pricing and hedging double-barrier options: A probabilistic approach. Mathematical Finance, 6(4), 365-378. · Zbl 0915.90016 · doi:10.1111/j.1467-9965.1996.tb00122.x |
| [29] | Götz, B. (2011). Valuation of multi-dimensional derivatives in a stochastic covariance framework. Ph.D. Thesis, TU Munich. · Zbl 1092.91022 |
| [30] | He, H., Keirstead, W., & Rebholz, J. (1998). Double lookbacks. Mathematical Finance, 8(3), 201-228. · Zbl 1020.91025 · doi:10.1111/1467-9965.00053 |
| [31] | Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Journal of Finance, 42, 327-343. · Zbl 1384.35131 |
| [32] | Hieber, P., & Scherer, M. (2012). A note on first-passage times of continuously time-changed Brownian motion. Statistics & Probability Letters, 82(1), 165-172. · Zbl 1229.91310 · doi:10.1016/j.spl.2011.09.018 |
| [33] | Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatility. Journal of Finance, 42(2), 281-300. · doi:10.1111/j.1540-6261.1987.tb02568.x |
| [34] | Hurd, T. (2009). Credit risk modeling using time-changed Brownian motion. International Journal of Theoretical and Applied Finance, 12(8), 1213-1230. · Zbl 1182.91188 · doi:10.1142/S0219024909005646 |
| [35] | Kammer, S. (2007). A general first-passage-time model for multivariate credit spreads and a note on barrier option pricing. Ph.D. Thesis, Justus-Liebig Universität Gießen. · Zbl 0940.91026 |
| [36] | Kiesel, R., & Lutz, M. (2011). Efficient pricing of constant maturity swap spread options in a stochastic volatility LIBOR market model. Journal of Computational Finance, 14(4), 37-72. |
| [37] | Kilin, F. (2011). Accelerating the calibration of stochastic volatility models. Journal of Derivatives, 18(3), 7-16. · doi:10.3905/jod.2011.18.3.007 |
| [38] | Klüppelberg, C., Lindner, A., & Ross, M. (2004). A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour. Journal of Applied Probability, 41(3), 601-622. · Zbl 1068.62093 · doi:10.1239/jap/1091543413 |
| [39] | Lin, X. S. (1999). Laplace transform and barrier hitting time distribution. Actuarial Research Clearing House, 1, 165-178. |
| [40] | Lipton, A. (2001). Mathematical methods for foreign exchange. Singapore: World Scientific. · Zbl 0989.91002 · doi:10.1142/4694 |
| [41] | Naik, V. (1993). Option valuation and hedging strategies with jumps in the volatility of asset returns. Journal of Finance, 48(5), 1969-1984. · doi:10.1111/j.1540-6261.1993.tb05137.x |
| [42] | Pelsser, A. (2000). Pricing double barrier options using Laplace transforms. Finance and Stochastics, 4, 95-104. · Zbl 0940.91026 · doi:10.1007/s007800050005 |
| [43] | Pigorsch, C., & Stelzer, R. (2009). A multivariate Ornstein-Uhlenbeck type stochastic volatility model. Working paper. · Zbl 1221.60074 |
| [44] | Raible, S. (2000). Lévy processes in finance: Theory, numerics, and empirical facts. Ph.D. Thesis, Freiburg University. · Zbl 0966.60044 |
| [45] | Reiner, E., & Rubinstein, M. (1991). Breaking down the barriers. Risk 4, 8, 28-35. |
| [46] | Rollin, S., Ferreiro-Castilla, A., & Utzet, F. (2011). A new look at the Heston characteristic function. Working paper. · Zbl 1233.60019 |
| [47] | Schöbel, R., & Zhu, J. (1999). Stochastic volatility with an Ornstein-Uhlenbeck process: An extension. Review of Finance, 3(1), 23-46. · Zbl 1028.91026 · doi:10.1023/A:1009803506170 |
| [48] | Sepp, A. (2006). Extended CreditGrades model with stochastic volatility and jumps. Wilmott Magazine, 50-62. |
| [49] | Stein, E., & Stein, J. (1991). Stock price distributions with stochastic volatility: An analytical approach. Review of Financial Studies, 4, 727-752. · Zbl 1458.62253 · doi:10.1093/rfs/4.4.727 |
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.
