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57
Twodimensional Fourier cosine series expansion method for pricing financial options
 SIAM Journal on Scientific Computing
"... the method to higher dimensions, with a multidimensional asset price process. The algorithm can be applied to, for example, pricing twocolor rainbow options but also to pricing under the popular Heston stochastic volatility model. For smooth density functions, the resulting method converges exponen ..."
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the method to higher dimensions, with a multidimensional asset price process. The algorithm can be applied to, for example, pricing twocolor rainbow options but also to pricing under the popular Heston stochastic volatility model. For smooth density functions, the resulting method converges exponentially in the number of terms in the Fourier cosine series summations; otherwise we achieve algebraic convergence. The use of an FFT algorithm, for asset prices modeled by Lévy processes, makes the algorithm highly efficient. We perform extensive numerical experiments.
Consistent modelling of VIX and equity derivatives using a 3/2 plus jumps model
 Applied Mathematical Finance
, 2014
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On CrossCurrency Models with Stochastic Volatility and Correlated Interest Rates
, 2010
"... We construct multicurrency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We first deal with a foreign exchange (FX) model of Hestontype, in which the domestic and foreign interest rates are generated by the shortrate process of Hull ..."
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Cited by 5 (1 self)
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We construct multicurrency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We first deal with a foreign exchange (FX) model of Hestontype, in which the domestic and foreign interest rates are generated by the shortrate process of HullWhite [Hull and White, 1990]. We then extend the framework by modeling the interest rate by a stochastic volatility displaceddiffusion Libor Market Model [Andersen and Andreasen, 2002], which can model an interest rate smile. We provide semiclosed form approximations which lead to efficient calibration of the multicurrency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equityFXinterest rate hybrid payoffs.
An EquityInterest Rate hybrid model with Stochastic Volatility and the interest rate smile
, 2010
"... We define an equityinterest rate hybrid model in which the equity part is driven by the Heston stochastic volatility [Hes93], and the interest rate (IR) is generated by the displaceddiffusion stochastic volatility Libor Market Model [AA02]. We assume a nonzero correlation between the main process ..."
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Cited by 5 (5 self)
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We define an equityinterest rate hybrid model in which the equity part is driven by the Heston stochastic volatility [Hes93], and the interest rate (IR) is generated by the displaceddiffusion stochastic volatility Libor Market Model [AA02]. We assume a nonzero correlation between the main processes. By an appropriate change of measure the dimension of the corresponding pricing PDE can be greatly reduced. We place by a number of approximations the model in the class of affine processes [DPS00], for which we then provide the corresponding forward characteristic function. We discuss in detail the accuracy of the approximations and the efficient calibration. Finally, by experiments, we show the effect of the correlations and interest rate smile/skew on typical equityinterest rate hybrid product prices. For a whole strip of strikes this approximate hybrid model can be evaluated for equity plain vanilla options in just milliseconds.
Extension of Stochastic Volatility Equity Models with HullWhite Interest Rate Process
, 2008
"... We present an extension of the stochastic volatility equity models by a stochastic HullWhite interest rate component. We place this system of stochastic differential equations in the class of affine jump diffusion linear quadratic jumpdiffusion processes (Duffie, Pan and Singleton [11], Cheng and ..."
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Cited by 4 (2 self)
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We present an extension of the stochastic volatility equity models by a stochastic HullWhite interest rate component. We place this system of stochastic differential equations in the class of affine jump diffusion linear quadratic jumpdiffusion processes (Duffie, Pan and Singleton [11], Cheng and Scaillet [8]) so that the pricing of European products can be efficiently done within the Fourier cosine expansion pricing framework [12]. We also apply the model to price some hybrid structured derivatives, which combine the different asset classes: equity and interest rate.
On the application of spectral filters in a Fourier option pricing technique
, 2013
"... When Fourier techniques are employed to specific option pricing cases from computational finance with nonsmooth functions, the socalled Gibbs phenomenon may become apparent. This seriously impacts the efficiency and accuracy of the pricing. For example, the Variance Gamma asset price process gives ..."
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Cited by 3 (0 self)
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When Fourier techniques are employed to specific option pricing cases from computational finance with nonsmooth functions, the socalled Gibbs phenomenon may become apparent. This seriously impacts the efficiency and accuracy of the pricing. For example, the Variance Gamma asset price process gives rise to algebraically decaying Fourier coefficients, resulting in a slowly converging Fourier series. We apply spectral filters to achieve faster convergence. FilteringiscarriedoutinFourierspace; theseries coefficients arepremultipliedbyadecreasing filter, which does not add significant computational cost. Tests with different filters show how the algebraic index of convergence is improved.
ROBUST PRICING OF EUROPEAN OPTIONS WITH WAVELETS AND THE CHARACTERISTIC FUNCTION
"... Abstract. We present a novel method for pricing European options based on the wavelet approximation (WA) method and the characteristic function. We focus on the discounted expected payoff pricing formula, and compute it by means of wavelets. We approximate the density function associated to the unde ..."
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Abstract. We present a novel method for pricing European options based on the wavelet approximation (WA) method and the characteristic function. We focus on the discounted expected payoff pricing formula, and compute it by means of wavelets. We approximate the density function associated to the underlying asset price process by a finite combination of jth order Bsplines, and recover the coefficients of the approximation from the characteristic function. Two variants for wavelet approximation will be presented, where the second variant adaptively determines the range of integration. The compact support of a Bsplines basis enables us to price options in a robust way, even in cases where Fourierbased pricing methods may show weaknesses. The method appears to be particularly robust for pricing longmaturity options, fat tailed distributions, as well as staircaselike density functions encountered in portfolio loss computations. 1.
IMPLIED LÉVY VOLATILITY
, 2008
"... This paper introduces the concept of implied Lévy volatility, hereby extending the intuitive BlackScholes implied volatility into a more general context. More precisely, Lévy implied time and space volatility are introduced and a study of the shape of implied Lévy volatilities is made. Model perfor ..."
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This paper introduces the concept of implied Lévy volatility, hereby extending the intuitive BlackScholes implied volatility into a more general context. More precisely, Lévy implied time and space volatility are introduced and a study of the shape of implied Lévy volatilities is made. Model performance is studied by analyzing deltahedging strategies for the Normal Inverse Gaussian and the Meixner model, both qualitatively and on historical timeseries of the S&P500. It is shown that under such parameter settings the model performs systematically better.
On the Efficacy of Fourier Series Approximations for Pricing European Options
, 2014
"... This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semiclosed form. The algorithms investigated here are the halfrange Fourier cos ..."
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This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semiclosed form. The algorithms investigated here are the halfrange Fourier cosine series, the halfrange Fourier sine series and the fullrange Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closedform solution. The results suggest that the halfrange sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the halfrange cosine series and the fullrange Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the fullrange Fourier series is at least as good as the halfrange Fourier cosine series, and outperforms the latter in pricing outofthemoney call options, in particular with maturities of three months or less. Second, the computational time required by the halfrange Fourier cosine series is uniformly longer than that required by the fullrange Fourier series for an interval of fixed length. Taken to
Estimating the Parameters of Stochastic Volatility Models Using Option Price Data. Unpublished Working Paper
, 2012
"... Abstract This paper describes a maximum likelihood method for estimating the parameters of Heston's model of stochastic volatility using data on an underlying market index and the prices of options written on that index. Parameters of the physical measure (associated with the index) and the pa ..."
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Abstract This paper describes a maximum likelihood method for estimating the parameters of Heston's model of stochastic volatility using data on an underlying market index and the prices of options written on that index. Parameters of the physical measure (associated with the index) and the parameters of the riskneutral measure (associated with the options) are identified including the equity and volatility risk premia. The estimation is implemented using a particle filter. The computational load of this estimation method, which previously has been prohibitive, is managed by the effective use of parallel computing using Graphical Processing Units. The efficacy of the filter is demonstrated under simulation and an empirical investigation of the fit of the model to the S&P 500 Index is undertaken. All the parameters of the model are reliably estimated and, in contrast to previous work, the volatility premium is well estimated and found to be significant.