An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

Takahashi, Akihiko; Takehara, Kohta

doi:10.1007/s10690-007-9054-9

Cited by 25 publications

(26 citation statements)

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“…Due to limitation of space, only the structure of the stochastic differential equations of our model is described here. For details of the underlying model, see Takahashi and Takehara [53].…”

Section: Currency Option Under a Libor Market Model Of Interest Ratesmentioning

confidence: 99%

“…[68] applies a formula derived more generally by the asymptotic expansion of small diffusion processes. Thereafter, the asymptotic expansion have been applied to a broad class of problems in finance: See [47], [48], [49], [50], Kunitomo and Takahashi [16], [17], [18], [19], Kawai [11], Matsuoka, Takahshi and Uchida [31], Takahashi and Matsushima [51], Takahashi and Saito [52], Takahashi and Yoshida [57], [58], Kobayashi, Takahashi and Tokioka [13], Muroi [33], Osajima [38], Takahashi and Uchida [56], Kunitomo and Kim [14], Kawai and Jäckel [12], and [53], [54], [55].…”

Section: Introductionmentioning

confidence: 99%

“…In the application of the asymptotic expansion based on Watanabe theory, they calculated certain conditional expectations which appear in their expansions and play a key role in computation, by the formulas up to the third order given explicitly in [46], [47] and [53]. In many applications, these formulas give sufficiently accurate approximation, but in some cases, for example in the cases with long maturities or/and with highly volatile underlying variables, the approximation up to the third order may not provide satisfactory accuracies.…”

Section: Introductionmentioning

confidence: 99%

“…[46], [47] and [53] provide explicit formulas of conditional expectations necessary for the asymptotic expansion up to the third order. In general, the crucial step in practical applications of the expansion is calculation of conditional expectations for a certain kind of Wiener functionals.…”

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Computation in an Asymptotic Expansion Method

2009

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An asymptotic expansion scheme in finance initiated by Kunitomo and Takahashi [15] and Yoshida[68] is a widely applicable methodology for analytic approximation of the expectation of a certain functional of diffusion processes. [46], [47] and [53] provide explicit formulas of conditional expectations necessary for the asymptotic expansion up to the third order. In general, the crucial step in practical applications of the expansion is calculation of conditional expectations for a certain kind of Wiener functionals. This paper presents two methods for computing the conditional expectations that are powerful especially for high order expansions: The first one, an extension of the method introduced by the preceding papers presents a general scheme for computation of the conditional expectations and show the formulas useful for expansions up to the fourth order explicitly. The second one develops a new calculation algorithm for computing the coefficients of the expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations. To demonstrate their effectiveness, the paper gives numerical examples of the approximation for λ-SABR model up to the fifth order and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate up to the fourth order.

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Section: Currency Option Under a Libor Market Model Of Interest Ratesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Computation in an Asymptotic Expansion Method

2009

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“…Bates (1996) proposed a jump-diffusion stochastic volatility model based on the Heston model. Takehara and Yamazaki (2006), Takahashi and Takehara (2009) incorporated a term-structure model of interest rates into the jump-diffusion stochastic volatility model. Daal and Madan (2005) studied a variance-gamma-jump model in currency option pricing.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium valuation of currency options under a jump-diffusion model with stochastic volatility

Xing

Yang

2015

Journal of Computational and Applied Mathematics

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Please cite this article as: Y. Xing, X. Yang, Equilibrium valuation of currency options under a jump-diffusion model with stochastic volatility, Journal of Computational and Applied Mathematics (2014), http://dx. AbstractIn this paper, we continue to investigate the model related to Bakshi and Chen (1997). In our model, both of the money supplies in the two countries are assumed to follow jump-diffusion processes with stochastic volatility. In the set of the two-country economy, we obtain the equilibrium price of the nominal exchange rate. With the help of Fourier transform to solve a partial integro-differential equation (PIDE), we get a closedform solution to the PIDE for a European call currency option. We also do Monte Carlo simulations to verify the correctness of the derived formula. Our model contains some existing currency option models as special cases, for example the stochastic-volatility jumpdiffusion (SVJD) model in Bates(1996), in which the jump of the exchange rate is driven by one Poisson process. We also provide some numerical analysis to show that our model is effective to the foreign currency option market.

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Asymptotic Expansion Approach in Finance

Takahashi

2015

Springer Proceedings in Mathematics &Amp; Statistics

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This paper provides a survey on an asymptotic expansion approach to valuation and hedging problems in finance. The asymptotic expansion is a widely applicable methodology for analytical approximations of expectations of certain Wiener functionals. Hence not only academic researchers but also practitioners have been applying the scheme to a variety of problems in finance such as pricing and hedging derivatives under high-dimensional stochastic environments. The present note gives an overview of the approach.

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An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

Abstract: Asymptotic expansion, Currency options, Libor market model, Malliavin calculus, Stochastic volatility,

Cited by 25 publications

References 40 publications

Computation in an Asymptotic Expansion Method

Computation in an Asymptotic Expansion Method

Equilibrium valuation of currency options under a jump-diffusion model with stochastic volatility

Asymptotic Expansion Approach in Finance

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