Pricing barrier options under stochastic volatility framework. (English) Zbl 1283.91185
Summary: The option pricing problem plays an extremely important role in quantitative finance. In complete markets, the Black-Scholes-Merton theory has been central to the development of financial engineering as both discipline and profession. However, in incomplete markets, there are not any replicating portfolios for those options, and thus, the market traders cannot apply the law of one price for obtaining a unique solution. Fortunately, the authors can get a fair price via local-equilibrium principle. In this paper, the authors apply stochastic control theory to price barrier options and analyze the relationship between the price and the current positions. The authors get the explicit expression for the market price of the risk. The position effect plays a significant role in option pricing, because it can tell the trader how many and which direction to trade with the market in order to reach the local equilibrium with the market.
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 93E03 | Stochastic systems in control theory (general) |
References:
| [1] | Black F and Scholes M, The pricing of options and corporate liabilities, Journal of Political Economy, 1973, 81(3): 637-654. · Zbl 1092.91524 · doi:10.1086/260062 |
| [2] | Rich D, The mathematical foundations of barrier option pricing theory, Advances in Futures Options Research, 1994, 7: 267-312. |
| [3] | Wong H and Kwok Y K, Multi-asset barrier options and occupation time derivatives, Applied Mathematical Finance, 2003, 10(3): 245-366. · Zbl 1089.91031 · doi:10.1080/1350486032000107352 |
| [4] | Gao B, Huang J Z, and Subrahmanyam M, The valuation of American barrier options using the decomposition technique, Journal of Economic Dynamics and Control, 2000, 24(11-12): 1783-1827. · Zbl 1032.91064 · doi:10.1016/S0165-1889(99)00093-7 |
| [5] | Davydov D and Linetsky V, Pricing and hedging path-dependent options under the CEV process, Management Science, 2001, 47(7): 949-965. · Zbl 1232.91659 · doi:10.1287/mnsc.47.7.949.9804 |
| [6] | Apel, T.; Winkler, G.; Wystup, U., Valuation of options in Heston’s stochastic volatility model using finite element methods (2001) |
| [7] | Griebsch S, Exotic option pricing in Heston’s stochastic volatility model, Ph.D. thesis, Frankfurt School of Finance & Mangement, 2008. |
| [8] | Merton R C, Continuous Time Finance, Basil Blackwell, Cambridge, MA, 1992. |
| [9] | Hull J and White A, The pricing of options on assets with stochastic volatilities, Journal of Finance, 1987, 42(2): 281-300. · doi:10.1111/j.1540-6261.1987.tb02568.x |
| [10] | Henderson V, Valuation of claims on non-traded assets using utility maximization, Mathematical Finance, 2002, 12(4): 351-373. · Zbl 1049.91072 · doi:10.1111/j.1467-9965.2002.tb00129.x |
| [11] | Goldman M B, Sosin H B, and Gatto M A, Path-dependent options: “Buy at the low, sell at the high”, Journal of Finance, 1979, 34(5): 1111-1127. |
| [12] | Heston S, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 1993, 6(2): 327-343. · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327 |
| [13] | Yang D, Quantitative Strategies for Derivatives Trading, ATMIF LLC, New Jersey, USA, 2006. |
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