zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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
Full Text: DOI
[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) · Zbl 1274.91447
[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) · Zbl 1126.91374
[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) · Zbl 0694.60047
[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