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HIGH ORDER DISCRETIZATION SCHEMES FOR THE CIR PROCESS: APPLICATION TO AFFINE TERM STRUCTURE AND HESTON MODELS
"... Abstract. This paper presents weak second and third order schemes for the CoxIngersollRoss (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya a ..."
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Abstract. This paper presents weak second and third order schemes for the CoxIngersollRoss (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya and Victoir. Combine both these results, this allows us to propose a second order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models.
Gamma Expansion of the Heston Stochastic Volatility Model
"... We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We ..."
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Cited by 15 (1 self)
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We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We give an explicit representation of this quantity in terms of infinite sums and mixtures of gamma random variables. The increments of the variance process are themselves mixtures of gamma random variables. The representation of the integrated conditional variance applies the PitmanYor decomposition of Bessel bridges. We combine this representation with the BroadieKaya exact simulation method and use it to circumvent the most timeconsuming step in that method.
TIME DEPENDENT HESTONMODEL
, 2009
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 12 (0 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Efficient, almost exact simulation of the Heston stochastic volatility model
 International Journal of Theoretical and Applied Finance
, 2010
"... Efficient, almost exact simulation of the Heston stochastic volatility model A. van Haastrecht1 2 and A.A.J. Pelsser3. ..."
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Efficient, almost exact simulation of the Heston stochastic volatility model A. van Haastrecht1 2 and A.A.J. Pelsser3.
A simple and exact simulation approach to Heston model
, 2008
"... In this paper we will propose a simple approach to simulating Heston model efficiently and accurately. All existing simulation schemes so far directly work with the meanreverting square root process of the variance in Heston model, instead we transform the variance to an equivalent volatility which ..."
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Cited by 6 (0 self)
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In this paper we will propose a simple approach to simulating Heston model efficiently and accurately. All existing simulation schemes so far directly work with the meanreverting square root process of the variance in Heston model, instead we transform the variance to an equivalent volatility which follows a meanreverting OrnsteinUhlenbeck process. We will show it is more convenient to simulate the transformed volatility process than the original variance process since the new OrnsteinUhlenbeck process does not have any term of square root, and is not restricted to any parameter restriction. Based on the transformed volatility process, we suggest a simple and exact scheme for the simulation of Heston model. Numerical examples show that the new scheme and Andersen’s QE scheme perform very closely, and outperform other schemes such as lognormal scheme. While QE scheme suffers from the problem of "leaking correlation", transformed volatility scheme does not, and therefore, provides a highquality alternative to the existing simulation schemes for Heston model.
A Fourierbased valuation method for Bermudan and barrier options under Heston’s model
 SIAM Journal of Financial Mathematics
, 2011
"... We develop an efficient Fourierbased numerical method for pricing Bermudan and discretely monitored barrier options under the Heston stochastic volatility model. The twodimensional pricing problem is dealt with by a combination of a Fourier cosine series expansion, as in [9, 10], and highorder qu ..."
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We develop an efficient Fourierbased numerical method for pricing Bermudan and discretely monitored barrier options under the Heston stochastic volatility model. The twodimensional pricing problem is dealt with by a combination of a Fourier cosine series expansion, as in [9, 10], and highorder quadrature rules in the other dimension. Error analysis and experiments confirm a fast error convergence. 1
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.
High order discretization schemes for stochastic volatility models
, 2009
"... In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous onedimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô’s formula, we get rid, in the asset price dynamics, ..."
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Cited by 5 (3 self)
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In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous onedimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô’s formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, a scheme, based on the NinomiyaVictoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an OrsteinUhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a,b].
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 3 (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.
Exact Scenario Simulation for Selected Multidimensional Stochastic Processes
, 2009
"... Accurate scenario simulation methods for solutions of multidimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for instance, in finance. They have been playing a crucial role as standard models in vario ..."
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Cited by 2 (2 self)
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Accurate scenario simulation methods for solutions of multidimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for instance, in finance. They have been playing a crucial role as standard models in various areas and dominate often the communication and thinking in a particular field of application, even that they may be too simple for more advanced tasks. Various discrete time simulation methods have been developed over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and numerical instabilities. Therefore, it is valuable to identify multidimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multidimensional solutions of stochastic differential equations including OrnsteinUhlenbeck, square root, squared Bessel, Wishart and Lévy type processes.