@article{fouque_garnier_nachbin_solna_2005, title={Time-reversal refocusing for point source in randomly layered media}, volume={42}, ISSN={["1878-433X"]}, DOI={10.1016/j.wavemoti.2005.03.001}, abstractNote={This paper demonstrates the interest of a time-reversal method for the identification of source in a randomly layered medium. An active source located inside the medium emits a pulse that is recorded on a small time-reversal mirror. The wave is sent back into the medium, either numerically in a computer with the knowledge of the medium, or physically into the real medium. Our goal is to give a precise description of the refocusing of the pulse. We identify and analyze a regime where the pulse refocuses on a ring at the depth of the source and at a critical time. Our objective is to find the location of the source and we show that the time-reversal refocusing contains information which can be used to this effect and which cannot be obtained by a direct arrival-time analysis. The time-reversal technique gives a robust procedure to locate and characterize the source also in the case with ambient noise created by other sources located at the surface.}, number={3}, journal={WAVE MOTION}, author={Fouque, JP and Garnier, J and Nachbin, A and Solna, K}, year={2005}, month={Sep}, pages={238–260} }
@article{fouque_papanicolaou_sircar_solna_2004, title={Maturity cycles in implied volatility}, volume={8}, ISSN={["1432-1122"]}, DOI={10.1007/s00780-004-0126-7}, abstractNote={The skew effect in market implied volatility can be reproduced by option pricing theory based on stochastic volatility models for the price of the underlying asset. Here we study the performance of the calibration of the S&P 500 implied volatility surface using the asymptotic pricing theory under fast mean-reverting stochastic volatility described in [8]. The time-variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. By extending the mathematical analysis to incorporate model parameters which are time-varying, we show this behaviour can be explained in a manner consistent with a large model class for the underlying price dynamics with time-periodic volatility coefficients.}, number={4}, journal={FINANCE AND STOCHASTICS}, author={Fouque, JP and Papanicolaou, G and Sircar, R and Solna, K}, year={2004}, month={Nov}, pages={451–477} }
@article{vigo_fouque_garnier_nachbin_2004, title={Robustness of time reversal for waves in time-dependent random media}, volume={113}, ISSN={["1879-209X"]}, DOI={10.1016/j.spa.2004.04.002}, abstractNote={This paper addresses the impact of time fluctuations of a random medium on refocusing during a time-reversal experiment. Even in the presence of moderate time perturbations a coherent refocused pulse is observed. The theory predicts the level of recompression observed as well as the conditions for the loss of statistical stabilization. It is shown that the statistical properties of the refocused pulse depend on a simple set of parameters that describe the correlation degree of the medium. The refocused pulse has in general a random shape that can be described in terms of a system of stochastic transport equations driven by a single Brownian motion. Pulse stabilization is also demonstrated for some particular configurations, and the convolution kernel that describes the pulse reshaping is explicitly computed. Numerical simulations are presented and show a very good agreement with the theoretical predictions, thus providing a clear illustration of the robustness of time reversal.}, number={2}, journal={STOCHASTIC PROCESSES AND THEIR APPLICATIONS}, author={Vigo, DGA and Fouque, JP and Garnier, J and Nachbin, A}, year={2004}, month={Oct}, pages={289–313} }
@article{fouque_garnier_nachbin_2004, title={Shock structure due to stochastic forcing and the time reversal of nonlinear waves}, volume={195}, ISSN={["1872-8022"]}, DOI={10.1016/j.physd.2004.05.003}, abstractNote={This paper is concerned with the study of the deformation of a nonlinear pulse traveling in a random medium. We consider shallow water waves with a spatially random depth. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear conservation law is regularized leading to a viscous shock profile. This enables us to perform time-reversal experiments beyond the critical time for shock formation.}, number={3-4}, journal={PHYSICA D-NONLINEAR PHENOMENA}, author={Fouque, JP and Garnier, J and Nachbin, A}, year={2004}, month={Aug}, pages={324–346} }
@article{cotton_fouque_papanicolaou_sircar_2004, title={Stochastic volatility corrections for interest rate derivatives}, volume={14}, number={2}, journal={Mathematical Finance: An International Journal of Mathematics, Statistics, and Financial Theory}, author={Cotton, P. and Fouque, J. P. and Papanicolaou, G. and Sircar, R.}, year={2004}, pages={173–200} }
@article{fouque_garnier_nachbin_2004, title={Time reversal for dispersive waves in random media}, volume={64}, ISSN={["1095-712X"]}, DOI={10.1137/S0036139903422371}, abstractNote={Refocusing for time reversed waves propagating in disordered media has recently been observed experimentally and studied mathematically. This surprising effect has many potential applications in domains such as medical imaging, underwater acoustics, and wireless communications. Time refocusing for one-dimensional acoustic waves is now mathematically well understood. In this paper the important case of one-dimensional dispersive waves is addressed. Time reversal is studied in reflection and in transmission. In both cases we derive the self-averaging properties of time reversed refocused pulses. An asymptotic analysis allows us to derive a precise description of the combined effects of randomness and dispersion. In particular, we study an important regime in transmission, where the coherent front wave is destroyed while time reversal of the incoherent transmitted wave still enables refocusing.}, number={5}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Fouque, JP and Garnier, J and Nachbin, A}, year={2004}, pages={1810–1838} }
@article{haider_mehta_fouque_2004, title={Time-reversal simulations for detection in randomly layered media}, volume={14}, ISSN={["0959-7174"]}, DOI={10.1088/0959-7174/14/2/007}, abstractNote={Abstract A time-reversal mirror is, roughly speaking, a device which is capable of receiving an acoustic signal in time, keeping it in memory and sending it back into the medium in the reversed direction of time. In this paper, we employ an accurate numerical method for simulating waves propagating in complex one-dimensional media. We use numerical simulations to reproduce the time-reversal self-averaging effect which takes place in randomly layered media. This is done in the regime where the inhomogeneities are smaller than the pulse, which propagates over long distances compared to its width. We show numerical evidence for possible use of an expanding window time-reversal technique for detecting anomalies buried in the medium.}, number={2}, journal={WAVES IN RANDOM MEDIA}, author={Haider, MA and Mehta, KJ and Fouque, JP}, year={2004}, month={Apr}, pages={185–198} }
@article{fouque_han_2004, title={Variance reduction for Monte Carlo methods to evaluate option prices under multi-factor stochastic volatility models}, volume={4}, ISSN={["1469-7696"]}, DOI={10.1080/14697680400020317}, abstractNote={We present variance reduction methods for Monte Carlo simulations to evaluate European and Asian options in the context of multiscale stochastic volatility models. European option price approximations, obtained from singular and regular perturbation analysis [Fouque J P, Papanicolaou G, Sircar R and Solna K 2003 Multiscale stochastic volatility asymptotics, SIAM J. Multiscale Modeling and Simulation 2], are used in importance sampling techniques, and their efficiencies are compared. Then we investigate the problem of pricing arithmetic average Asian options (AAOs) by Monte Carlo simulations. A two-step strategy is proposed to reduce the variance where geometric average Asian options (GAOs) are used as control variates. Due to the lack of analytical formulas for GAOs under stochastic volatility models, it is then necessary to consider efficient Monte Carlo methods to estimate the unbiased means of GAOs. The second step consists in deriving formulas for approximate prices based on perturbation techniques, a...}, number={5}, journal={QUANTITATIVE FINANCE}, author={Fouque, JP and Han, CH}, year={2004}, month={Oct}, pages={597–606} }
@article{fouque_papanicolaou_sircar_solna_2003, title={Multiscale stochastic volatility asymptotics}, volume={2}, ISSN={["1540-3459"]}, DOI={10.1137/030600291}, abstractNote={In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical Black--Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index, say, and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena make it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work (see, for instance, [J. P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000]), we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the so-called term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the implied volatility surface. In particular, the introduction of the slow factor gives a much better fit for options with longer maturities. We use option data to illustrate our results and show how exotic option prices also can be approximated using our multiscale perturbation approach.}, number={1}, journal={MULTISCALE MODELING & SIMULATION}, author={Fouque, JP and Papanicolaou, G and Sircar, R and Solna, K}, year={2003}, pages={22–42} }
@article{fouque_han_2003, title={Pricing Asian options with stochastic volatility}, volume={3}, ISSN={["1469-7696"]}, DOI={10.1088/1469-7688/3/5/301}, abstractNote={Abstract In this paper, we generalize the recently developed dimension reduction technique of Vecer for pricing arithmetic average Asian options. The assumption of constant volatility in Vecer's method will be relaxed to the case that volatility is randomly fluctuating and is driven by a mean-reverting (or ergodic) process. We then use the fast mean-reverting stochastic volatility asymptotic analysis introduced by Fouque, Papanicolaou and Sircar to derive an approximation to the option price which takes into account the skew of the implied volatility surface. This approximation is obtained by solving a pair of one-dimensional partial differential equations.}, number={5}, journal={QUANTITATIVE FINANCE}, author={Fouque, JP and Han, CH}, year={2003}, month={Oct}, pages={353–362} }
@article{fouque_papanicolaou_sircar_solna_2003, title={Singular perturbations in option pricing}, volume={63}, ISSN={["1095-712X"]}, DOI={10.1137/S0036139902401550}, abstractNote={After the celebrated Black--Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics motivated numerous works during the 1980s and 1990s. In particular, a lot of attention has been paid to stochastic volatility models in which the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000; Internat. J. Theoret. Appl. Finance, 13 (2000), pp. 101--142] that, in the presence of a separation of time scales between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas. From the point of view of PDEs, this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff function inherent to option pricing. We present the case of call options for which the payoff function forms an angle at the strike price. This case is important since these are the typical instruments used in the calibration of pricing models. We establish the pointwise accuracy of the corrected Black--Scholes price by using an appropriate payoff regularization which is removed simultaneously as the asymptotics is performed.}, number={5}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Fouque, JP and Papanicolaou, G and Sircar, R and Solna, K}, year={2003}, pages={1648–1665} }
@article{fouque_solna_2003, title={Time-reversal aperture enhancement}, volume={1}, ISSN={["1540-3467"]}, DOI={10.1137/S1540345902414443}, abstractNote={Time-reversal refocusing for waves propagating in inhomogeneous media have recently been observed and studied experimentally in various contexts (ultrasound, underwater acoustics, ...); see, for instance, [M. Fink, Scientific American, November (1999), pp. 63--97]. Important potential applications have been proposed in various fields, for instance in imaging or communication. However, the full mathematical analysis, meaning both modeling of the physical problem and derivation of the time-reversal effect, is a deep and complex problem. Two cases that have been considered in depth recently correspond to one-dimensional media and the parabolic approximation regime where the backscattering is negligible. In this paper we give a complete analysis of time-reversal of waves emanating from a point source and propagating in a randomly layered medium. The wave transmitted through the random medium is recorded on a small time-reversal mirror and sent back into the medium, time-reversed. Our analysis enables us to contrast the refocusing properties of a homogeneous medium and a random medium. We show that random medium fluctuations actually enhance the spatial refocusing around the initial source position. We consider a regime where the correlation length of the medium is much smaller than the pulse width, which itself is much smaller than the distance of propagation. We derive asymptotic formulas for the refocused pulse which we interpret in terms of an enhanced effective aperture. This interpretation is, in fact, comparable to the superresolution effect obtained in the other extreme regime corresponding to the parabolic approximation. However, as we discuss, the mechanism that generates the superresolution is very different in these two extreme situations.}, number={2}, journal={MULTISCALE MODELING & SIMULATION}, author={Fouque, JP and Solna, K}, year={2003}, pages={239–259} }
@article{fouque_nachbin_2003, title={Time-reversed refocusing of surface water waves}, volume={1}, ISSN={["1540-3467"]}, DOI={10.1137/S1540345902412110}, abstractNote={A time-reversal mirror is, roughly speaking, a device which is capable of receiving a signal in time, keeping it in memory, and sending it back into the medium in the reversed direction of time. A brief mathematical review of the time-reversal (in reflection) theory is presented in the context of the linear shallow water equations. In particular, an explicit expression is given for the refocused pulse in the simplest time-reversal case. The explicit expression for the power spectral density of the reflection process is used to construct the highpass filter, which controls the refocusing process. Time-reversal numerical experiments in the (effectively) linear regime are used to validate the nonlinear shallow water code. The numerically refocused pulse is compared with the theoretical predicted shape. Further numerical experiments illustrate the robustness of the theory, in particular the time-reversal refocusing with smaller cutoff windows, the self-averaging property, and finally refocusing when the nonlinear term is small but not negligible.}, number={4}, journal={MULTISCALE MODELING & SIMULATION}, author={Fouque, JP and Nachbin, A}, year={2003}, pages={609–629} }
@book{fouque_papanicolaou_sircar_2000, title={Derivatives in financial markets with stochastic volatility}, ISBN={0521791634}, publisher={Cambridge; New York: Cambridge University Press}, author={Fouque, J.-P and Papanicolaou, G. and Sircar, K. R.}, year={2000} }