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Markov-modulated jump-diffusions for currency option pricing. (English) Zbl 1231.91425
Summary: We introduce dynamic models for the spot foreign exchange rate with capturing both the rare events and the time-inhomogeneity in the fluctuating currency market. For the rare events, we use a compound Poisson process with log-normal jump amplitude to describe the jumps. As for the time-inhomogeneity in the market dynamics, we particularly stress the strong dependence of the domestic/foreign interest rates, the appreciation rate and the volatility of the foreign currency on the time-varying sovereign ratings in the currency market. The time-varying ratings are formulated by a continuous-time finite-state Markov chain. Based on such a spot foreign exchange rate dynamics, we then study the pricing of some currency options. Here we will adopt a so-called regime-switching Esscher transform to identify a risk-neutral martingale measure. By determining the regime-switching Esscher parameters we then get an integral expression on the prices of European-style currency options. Finally, numerical illustrations are given.
91G20Derivative securities
91G30Interest rates (stochastic models)
60J75Jump processes
[1]Aït-Sahalia, Y.: Telling from discrete data whether the underlying continuous-time model is a diffusion, Journal of finanance 57, 2075-2112 (2002)
[2]Andersen, T. G.; Benzoni, L.; Lund, J.: An empirical investigation of continuous-time equity return models, Journal of finanance 57, 1239-1284 (2002)
[3]Bates, D. S.: Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options, Review of financial studies 9, 69-107 (1996)
[4]Biger, N.; Hull, J.: The valuation of currency options, Financial management 12, 24-28 (1983)
[5]Carr, P.; Geman, H.; Madan, D.; Yor, M.: The fine structure of asset returns: an empirical investigation, Journal of business. 75, 305-332 (2002)
[6]Cox, J.; Ingersoll, J.; Ross, S.: A theory of the term structure of interest rates, Econometrica 53, 385-407 (1985)
[7]Di Graziano, G.; Rogers, L. C. G.: Equity with Markov-modulated dividends, Quantitative finance 9, 19-26 (2009) · Zbl 1171.91337 · doi:10.1080/14697680802036168
[8]Elliott, R. J.; Chan, L.; Siu, T. K.: Option pricing and esscher transform under regime switching, Annals of finance 1, 423-432 (2005) · Zbl 1233.91270 · doi:10.1007/s10436-005-0013-z
[9]Elliott, R. J.; Siu, T. K.; Chan, L.; Lau, J. W.: Pricing options under a generalized Markov-modulated jump-diffusion model, Stochastic analysis and applications 25, 821-843 (2007) · Zbl 1155.91380 · doi:10.1080/07362990701420118
[10]Elliott, R. J.; Osakwe, C. J.: Option pricing for pure jump processes with Markov switching compensators, Finance and stochastics 10, 250-275 (2006) · Zbl 1101.91034 · doi:10.1007/s00780-006-0004-6
[11]Garman, M.; Kohlhangen, S.: Foreign currency option values, Journal of international money and finance 2, 239-253 (1983)
[12]Gerber, H. U.; Shiu, E. S. W.: Option pricing by esscher transforms (with discussions), Transactions of socieity of actuaries 46, 99-191 (1994)
[13]Gerber, H. U.; Shiu, E. S. W.: Actuarial bridges to dynamic hedging and option pricing, Insurance: mathematics and economics 18, 183-218 (1996) · Zbl 0896.62112 · doi:10.1016/0167-6687(96)85007-4
[14]Harrison, J. M.; Pliska, S. R.: Martingales and stochastic integrals in the theory of continuous trading, Stochastic processes and their applications 11, 215-280 (1981) · Zbl 0482.60097 · doi:10.1016/0304-4149(81)90026-0
[15]Harrison, J. M.; Pliska, S. R.: A stochastic calculus model of continuous trading: complete markets, Stochastic processes and their applications 15, 313-316 (1983) · Zbl 0511.60094 · doi:10.1016/0304-4149(83)90038-8
[16]Heston, S. L.: A closed-form solution for options with stochastic volatility with applications to Bond and currency options, Review of financial studies 6, 327-343 (1993)
[17]Heston, S.L., 1999. A simple new formula for options with stochastic volatility. Washington University of St. Louis Working Paper.
[18]Hubaleka, F.; Sgarra, C.: On the esscher transforms and other equivalent martingale measures for barndorff–Nielsen and shephard stochastic volatility models with jumps, Stochastic processes and their applications 119, 2137-2157 (2009) · Zbl 1177.60068 · doi:10.1016/j.spa.2008.10.005
[19]Jobert, A.; Rogers, L. C. G: Option pricing with Markov-modulated dynamics, SIAM journal on control and optimization 44, 2063-2078 (2006) · Zbl 1158.91380 · doi:10.1137/050623279
[20]Johannes, M.: The statistical and economic role of jumps in continuous-time interest rate models, Journal of finanance 59, 227-260 (2004)
[21]Johnson, M.; Schneewels, T.: Jump-diffusion processes in the foreign exchange markets and the release of macroeconomic news, Computational economics 7, 309-329 (1994) · Zbl 0824.90028 · doi:10.1007/BF01299458
[22]Jorion, P.: On jump processes in the foreign exchange and stock markets, Review of financial studies 1, 427-445 (1988)
[23]Liu, J.; Pan, J.; Wang, T.: An equilibrium model of rare-event premia and its implication for option smirks, Review of financial studies 18, 132-164 (2005)
[24]Melino, A.; Turnbull, S. M.: Pricing foreign currency options with stochastic volatility, Journal of econometrics 45, 239-265 (1990)
[25]Merton, R. C.: Option pricing when underlying stock returns are discontinuous, Journal of finance and economics 3, 125-144 (1976) · Zbl 1131.91344 · doi:10.1016/0304-405X(76)90022-2
[26]Protter, P.: Stochastic integration and differential equations, (1990)
[27]Shastri, K.; Wetheyavivorn, K.: The valuation of currency options for alternate stochastic processes, Journal of financial research 10, 283-293 (1987)
[28]Siu, T. K.; Yang, H.; Lau, J. W.: Pricing currency options under two-factor Markov-modulated stochastic volatility models, Insurance: mathematics and economics 43, 295-302 (2008) · Zbl 1152.91550 · doi:10.1016/j.insmatheco.2008.05.002