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Kazufumi Itoy Jari Toivanenz The deterministic numerical valuation of American options under Heston's stochastic volatility model is considered. The prices are given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. A new truncation of the domain is described for small asset values while for large asset values and variance a standard truncation is used. The nite difference discretization is constructed by numerically solving quadratic optimization problem aiming to minimize the truncation error at each grid point. A Lagrange approach is used to treat the linear complementarity problems. Numerical examples demonstrate the accuracy and effectiveness of the proposed approach. Keywords: American option pricing, stochastic volatility model, linear complementarity problem, nite difference method, quadratic programming, multigrid method, Lagrange method, penalty method
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Alos, Elisa and Ewald, Christian-Oliver
We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of the first author [3] in order to derive approximate option pricing formulas in the context of the Heston model. Numerical results are given.
Abstract Theoretical research on option valuation tends to focus on pricing the plain-vanilla European-style derivatives. Duffie, Pan, and Singleton (Econometrica, 2000) have recently developed a general transform method to determine the value of European options for a broad class of the underlying price dynamics. Contrastingly, no universal and analytically attractive approach to pricing of American-style derivatives is yet available. When the underlying price follows simple dynamics, literature suggests using finite difference methods. Simulation methods are often applied in more complicated cases. This paper addresses the valuation of American-style derivatives when the price of an underlying asset follows the Heston model dynamics (Rev.Fin.S., 1993). The model belongs to the class of stochastic volatility models, which have been proposed in the hope of remedying the strike-price biases of the Black–Scholes formula. Option values are obtained by a variant of the Geske–Johnson scheme (JF, 1984), which has been devised in the context of the Black–Scholes model. The scheme exploits the fact that an American option is the limit of a sequence of “Bermudan Discuss this paper
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Jian Wang Abstract The Heston model is a stochastic volatility model. We show that the option price in the Heston model is convex in the underlying asset for convex contract functions. We verify this using the explicit formula for European call options and extend to the general case using an approximation argument. Some other properties of the Heston model are also discussed. Finally, we illustrate the results using numerical methods.
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In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model, by developing efficient transform based pricing methods. This non-affine model gives prices of options on realized variance which allow upward sloping implied volatility of variance smiles. Heston's (1993) model, the benchmark affine stochastic volatility model, leads to downward sloping volatility of variance smiles - in disagreement with variance markets in practice. We show a robust method, using control variates, to express the Laplace transform of the variance call function in terms of the Laplace transform of realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm which gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps. Discuss this paper
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We present a new and general technique for obtaining closed form expansions for prices of options in the Heston model, in terms of Black-Scholes prices and Black-Scholes greeks up to arbitrary orders. We then apply the technique to solve, in detail, the cases for the second order and third order expansions. In particular, such expansions show how the convexity in volatility, measured by the Black-Scholes volga, and the sensitivity of delta with respect to volatility, measured by the Black-Scholes vanna, impact option prices in the Heston model. The general method for obtaining the expansion rests on the construction of a set of new probability measures, equivalent to the original pricing measure, and which retain the affine structure of the Heston volatility diffusion. Finally, we extend our method to the pricing of forward-starting options in the Heston model. Discuss this paper
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Lorella Fatone Francesco Zirilli Abstract. This paper presents a numerical method to price European options on realized variance. A European realized variance option is an option where payoff depends on the time of maturity, on the observed variance of the log-returns of the stock prices in a preassigned sequence of time values ti, i = 0, 1, . . . ,N. The realized variance is the variance observed in the sample of the log-returns considered, so that the value at maturity of the realized variance option depends on the discrete sample of the log-returns of the stock prices observed at the preassigned dates ti, i = 0, 1, . . . ,N. The method proposed to approximate the price of these options is based on the idea of approximating the discrete sum that gives the realized variance with an integral, using as model of the dynamics of the log-return of the stock price the Heston stochastic volatility model. In this way the price of a realized variance option is approximated with the price of an integrated stochastic variance option where payoff depends on the time of maturity and on the integrated stochastic variance. The integrated stochastic variance option is priced with the method of discounted expectations. We derive an integral representation formula for the price of this last kind of options. This integral formula reduces to a one dimensional Fourier integral in the case of the most commonly traded options that have a simple payoff function. The method has been validated on some test problems. The numerical experiments show that the approach suggested in this paper gives satisfactory approximations of the prices of the realized variance options (relative error 10−2, 10−3). This approach also allows substantial savings of computational time when compared with the Monte Carlo method used to evaluate with approximately the same accuracy. The website http://www.econ.univpm.it/recchioni/finance/w4 contains auxiliary material that can help in the understanding of this paper and makes available to the interested users the codes that implement the numerical method proposed here to price realized variance options. The use of these codes on a computing grid has been made user friendly developing a dedicated application using the software Symphony (that is, a Service Oriented Architecture (SOAM) software of Platform Computing Toronto, Canada). The website mentioned above makes this Symphony application available to the users.
Abstract This paper surveys the developments in the finance literature with respect to applying the Fourier transform for option pricing under affine jumpdiffusions. We provide a broad description of the issues and a detailed summary of the main points and features of the models proposed. First, we consider a wide class of affine jump-diffusions proposed for the asset price dynamics: jump-diffusions, diffusions with stochastic volatility, jump-diffusions with stochastic volatility, and jump-diffusions with stochastic volatility and jump intensity. Next we apply the Fourier transform for solving the problem of European option pricing under these price processes. We present two solution methods: the characteristic formula and the Black- Scholes-style formula. Finally, we discuss numerical implementation of pricing formulas and apply the considered processes for modeling the DAX options volatility surface.
Steven L. Heston Abstract I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset Discuss this paper
Description:
ROGER LORD REMMERT KOEKKOEK DICK J.C. VAN DIJK
Abstract: Using an Euler discretisation to simulate a mean-reverting CEV process gives rise to the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the CEV-SV stochastic volatility model, with the Heston model as a special case, where the variance is modelled as a mean-reverting CEV process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the positive bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to recent quasi-second order schemes of Kahl and J Discuss this paper
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by Adrian A Dragulescu1 and Victor M Yakovenko We study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker Discuss this paper
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Gunter Winkler Thomas Apel Uwe Wystup Quoted:
Introduction Due to the smile observed in options markets numerous authors have suggested different models such as generalized Levy processes, fractional Brownian motion, entropy based models [4], jump diffusions and stochastic volatility models. For vanilla options (put and call options) the dependence of the price on the volatility is monotone, whence using the Black-Scholes formula along with a volatility smile matrix is sufficient. Values of exotic options, however, do not always depend on the volatility in a monotone fashion, whence pricing consistently with the smile requires a more sophisticated model. Therefore, it is important to find efficient ways to calculate exotic option values in exotic models.
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SVETLANA BOYARCHENKO SERGEI LEVENDORSKI We consider the Heston model with the stochastic interest rate of the CIR type and more general models with stochastic volatility and interest rates depending on two CIR - factors. Time derivative and infinitesimal generator of the process for factors that determine the dynamics of the interest rate and/or volatility are discretized. The result is a sequence of embedded perpetual options arising in the time - discretization of a Markov - modulated Levy model. Options in this sequence are solved using an iteration method based on the Wiener - Hopf factorization. Typical shapes of the early exercise boundary are shown, and good agreement of option prices with prices calculated with the Longstaff - Schwartz method and Medvedev - Scaillet asymptotic method is demonstrated.
Keywords: Optimal stopping, American options, regime switching,stochastic volatility models, Heston model,stochastic interest rate, CIR process
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Dietmar P.J. Leisen Abstract This paper constructs a sequence of discrete{time models, that converge to stochas- tic volatility models. It generalizes the well{known binomial models from the Black{ Scholes setup to bivariate di usions, applying back{and{forth transformations in the style of Nelson and Ramaswamy (1990). Our guideline in the construction is a general convergence theorem for the weak convergence of processes; this imposes restrictions on the discretization into a grid and the successors of grid points. We discuss the implementation for the Hull and White (1987) model and calculate prices for European{ and American{style put options. Convergence is smooth and fairly accurate with re nements of 20 time steps.
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by Vassilis Galiotos The purpose of this project is to explain to some extent the importance of stochastic volatility models and implied volatility. The model that is studied is the Heston model (1993). Our findings confirm the common belief that the implied volatility smile slopes downwards at the money if the correlation between the spot returns and the volatility is positive. Similarly, if the correlation is negative the implied volatility slopes upwards.
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Francesca Mariani, Graziella Pacelli, Francesco Zirilli Let us suppose that the dynamics of the stock prices and of their stochastic variance is described by the Heston model, that is by a system of two stochastic differential equations with a suitable initial condition. Our aim is to estimate the parameters of the Heston model and one component of the initial condition, that is the initial stochastic variance, from the knowledge of the stock and option prices observed at discrete times. The option prices considered refer to an European call on the stock whose prices are described by the Heston model. The method proposed to solve this problem is based on a filtering technique to construct a likelihood function and on the maximization of the likelihood function obtained. The estimated parameters and initial value component are characterized as being a maximizer of the likelihood function subject to some constraints. The solution of the filtering problem, used to construct the likelihood function, is based on an integral representation of the fundamental solution of the Fokker-Planck equation associated to the Heston model, on the use of the wavelet expansions presented in [1], [2], [3] to approximate the integral kernel appearing in the representation formula of the fundamental solution, on a simple truncation procedure to exploit the sparsifying properties of the wavelet expansions and on the use of the Fast Fourier Transform (FFT). The use of these techniques generates a very efficient and fully parallelizable numerical procedure to solve the filtering problem, this last fact makes possible to evaluate very efficiently the likelihood function and its gradient. As a byproduct of the solution of the filtering problem we have developed a stochastic variance tracking technique that gives very good results in numerical experiments. The maximum likelihood problem used in the estimation procedure is a low dimensional constrained optimization problem, its solution with ad hoc techniques is justified by the computational cost of evaluating the likelihood function and its gradient. We use parallel computing and a variable metric steepest ascent method to solve the maximum likelihood problem. Some numerical examples of the estimation problem using synthetic data obtained with a parallel implementation of the previous numerical method are presented. Very impressive speed up factors are obtained in the numerical examples using the parallel implementation of the numerical method proposed. This website contains two animations and some auxiliary material that helps the understanding of the paper [7] and makes available to the interested users the computer programs used to produce the numerical experience presented here and in [7] .
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We define an equity-interest 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 displaced-diffusion stochastic volatility Libor Market Model [AA02]. We assume a non-zero 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 equity-interest 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. Discuss this paper
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The Heston Model and the Smile, joint with Rafal Weron, Chapter contribution to the book Statistical Tools for Finance and Insurance, eds. Pavel Cizek, Wolfgang Haerdle, Rafal Weron. 2004. (e-book)
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CHRISTIAN-OLIVER EWALD .We implement the Heston stochastic volatility model by using multidimensional Ornstein-Uhlenbeck processes and a special Girsanov transformation, and consider the Malliavin calculus of this model.We derive explicit formulas for theMalliavin derivatives of the Heston volatility and the log-price, and give a formula for the local volatility which is approachable by Monte-Carlo methods
The Derman and Kani Implied Binomial Tree Implied Trinomial Tree of Derman, Kani and Chriss Hull-White Model The Heston Model SABR Model Discuss this paper
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Roger Lord Contents �� The Heston model �� The complex logarithm Joint work with Christian Kahl (University of Wuppertal, ABN Discuss this paper
LEIF B.G. ANDERSEN Banc of America Securities -------------------------------------------------------------------------------- January 23, 2007
Abstract: Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo simulation methods for this class of models. This paper considers several new algorithms for time-discretization and Monte Carlo simulation of Heston-type stochastic volatility models. The algorithms are based on a careful analysis of the properties of affine stochastic volatility diffusions, and are straightforward and quick to implement and execute. Tests on realistic model parameterizations reveal that the computational efficiency and robustness of the simulation schemes proposed in the paper compare very favorably to existing methods.
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thesis by Sensen Lin Stochastic volatility is an interesting area in financial mathematics. Parabolic partial differential equations with mixed differentiation terms are the focus of numerical solution of Heston model. This document covers the numerical methods to Heston model. Chapter 1 is an introduction to the problem and my main interest. Chapter 2 is an overview of Heston model and its closed form solution. The closed form solution is a benchmark to test the numerical methods Chapter 3 talks about the explicit scheme which is a straightforward method in solving Heston model. The result and restriction of this model are illustrated. Chapter 4 discusses the ADI method dealing with special equations like Heston PDE. The details of this method are covered and comparison between schemes is given. Discuss this paper
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by Sergei Mikhailov, Ulrich Nogel The paper discusses theoretical properties, shows the performance and presents some extensions of Heston Discuss this paper
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Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli Abstract
In [1] we study the problem of obtaining accurate estimates of the parameters, of the initial stochastic variance and of the risk premium parameter of the risk neutral measure of the Heston stochastic volatility model from the observation at discrete times of the stock log-returns and of the prices of a European call option on the stock. This problem is an inverse problem known in the literature as calibration problem. As a byproduct of the solution of the calibration problem we develop a tracking procedure that can be used to forecast the stochastic variance and the stock price. From a mathematical point of view the problem considered is formulated as a constrained optimization problem where the objective function is the logarithm of the likelihood associated to the parameter, the initial stochastic variance and the risk premium parameter values given the observed stock log-returns, option prices and observation times, this function is called (log-)likelihood function. The evaluation of the (log-)likelihood function associated to a given choice of the parameter, of the initial stochastic variance and of the risk premium parameter values requires the solution of a filtering problem for the Heston model. An accurate and computationally efficient solution of this filtering problem is necessary for a satisfactory solution of the calibration problem for the Heston model. A similar problem has been considered in [2] and [3]. The aim of this paper is to extend and improve the results obtained in [2] with respect to the formulation of the problem, the accuracy of the solution obtained and the computational efficiency of the solution method. In particular in comparison with [2] we reformulate the problem adding to the quantities that must be estimated the risk premium parameter and to the observations the option price at the initial time. The addition of the risk premium parameter to the quantities that must be estimated makes the problem considered more realistic and makes interesting the analysis of time series of real financial data using the method proposed. The addition of the option price at the initial time to the data used in the solution of the problem is natural and improves significantly the estimate of the initial stochastic variance. Moreover we simplify the expression of some formulae used in [2] in the computation of the (log-)likelihood function and we improve the optimization method employed to solve the maximum likelihood problem introducing some ad hoc preliminary optimization steps. Finally a new, easy to compute, formula that gives the ``Heston" option price is derived. The use of this new formula reduces substantially the computational cost of evaluating the (log-)likelihood function when compared to the cost of the (log-)likelihood function evaluation in [2]. Some numerical examples of the calibration problem using synthetic and real data are presented. As real data we use the 2005 historical data (precisely the data of the period Jan. 3, 2005, May 11, 2005) of the US S&P500 index and of the corresponding option prices. Very good results are obtained forecasting future values of the S&P500 index and of the corresponding option price using the solution of the calibration problem and the forecasting and tracking procedure mentioned above. In this website some auxiliary material useful in the understanding of [1] including some animations and some numerical experiments is shown. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
The Heston stochastic volatility model with volatility jumps can serve as a good tool for pricing and risk-managing derivatives on realized volatility and variance. In this paper, we derive analytical and approximate solutions for the values of contingent claims on realized variance and volatility under the Heston model with volatility jumps. By employing generalized Fourier transform we obtain analytical solutions (up to numerical inversion of Fourier integral) for swaps on realized volatility and variance with floor and cap protections, and for options on realized variance and volatility swaps. We also consider pricing forward-start claims on realized variance and volatility, including options on VIX, and obtain a closed-form solution and an accurate convexity adjustment formula for pricing these claims. Our solutions allow to unify pricing and risk managing of many volatility-dependent claims into one single framework. In addition, we derive a log-normal approximation to the density of the realized variance in the Heston model and obtain accurate approximate solutions for volatility- and variance-dependent claims with longer maturities. Discuss this paper
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Nimalin Moodley ABSTRACT This document covers various aspects the Heston model. The structure and topics covered is as follows: Chapter 1 introduces the model and provides theoretical and graphical motivation for its robustness and hence popularity. It also discusses pricing using the Partial Differential Equation and Equivalent MartingaleMeasure techniques Chapter 2 discusses how the different components of the model can be evaluated computationally and how this can be achieved with different methods. These methods are then compared to each other. Chapter 3 addresses the calibration problem. Different methods are presented as well as practical implementation, results thereof, and comparisons. All the MATLAB code required to implement the model is provided Discuss this paper
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