The Quantitative Finance Library

Convergence Heston to SVI By an appropriate change of variables, we prove here that the SVI implied volatility parameterisation proposed in [2] and the large-time asymptotic of the Heston implied volatility derived in [1] do agree algebraically, thus confirming a conjecture proposed by J. Gatheral in [2] as well as proposing a simpler expression for the asymptotic implied volatility under the Heston model. , J.Gatheral, A.Jacquier (2010)

Finite Difference Based Calibration and Simulation In the context of a stochastic local volatility model, we present a numerical solution scheme that achieves full (discrete) consistency between calibration, finite difference solution and Monte-Carlo simulation. The method is based on an ADI finite difference discretisation of the model. , J.Andreasen, B.N.Huge (2010)

Calibrating Option Pricing Models with Heuristics Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston?s stochastic volatility model, and Bates?s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques. Sample Matlab code is provided. , M.Gilli, E.Schumann (2010)

Efficient Simulation of the Double Heston Model Stochastic volatility models have replaced Black-Scholes model since they are able to generate a volatility smile. However, standard models fail to capture the smile slope and level movements. The Double-Heston model provides a more flexible approach to model the stochastic variance. In this paper, we focus on numerical implementation of this model. First, following the works of Lord and Kahl, we correct the analytical call option price formula given by Christoffersen et al. Then, we compare numerically the discretization schemes of Andersen, Zhu and Alfonsi to the Euler scheme. , P. Gauthier, D. Possamai (2009)

Time Dependent Heston Model The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [Hes93] or piecewise constant [MN03]. Hence, using a small volatility of volatility expansion and Malliavin calculus techniques, we derive an accurate analytical formula for the price of vanilla options for any time dependent Heston model (the accuracy is less than a few bps for various strikes and maturities). In addition, we establish tight error estimates. The advantage of this approach over Fourier based methods is its rapidity (gain by a factor 100 or more), while maintaining a competitive accuracy. From the approximative formula, we also derive some corollaries related first to equivalentHestonmodels (extending some work of Piterbarg on stochastic volatility models [Pit05b]) and second, to the calibration procedure in terms of ill-posed problems. , E. Benhamou, E. Gobet, M. Miri (2009)

On the Heston Model with Stochastic Interest Rates In this article we discuss the Heston [17] model with stochastic interest rates driven by Hull-White [18] (HW) or Cox-Ingersoll-Ross [8] (CIR) processes. We define a so-called volatility compensator which guarantees that the Heston hybrid model with a non-zero correlation between the equity and interest rate processes is properly defined. Moreover, we propose an approximation for the characteristic function, so that pricing of basic derivative products can be efficiently done using Fourier techniques [12; 7]. We also discuss the effect of the approximations on the instantaneous correlations, and check the influence of the correlation between stock and interest rate on the implied volatilities. , L. A. Grzelak, K. Oosterlee (2009)

Efficient Option Pricing with Multi-Factor Equity-Interest Rate Hybrid Models In this article we discuss multi-factor equity-interest rate hybrid models with a full matrix of correlations. We assume the equity part to be modeled by the Heston model [Heston-1993] with as a short rate process either a Gaussian two-factor model [Brigo,Mercurio-2007] or a stochastic volatility short rate process of Heston type [Heidari, et al.-2007]. We develop an approximation for the discounted characteristic function. Our approximation scheme is based on the observation that sqrt{sigma_t}, with sigma_t a stochastic quantity of CIR type [Cox, et al.-1985], can be well approximated by a normal distribution. Our approximate hybrid fits almost perfectly to the original model in terms of implied Black-Scholes [Black,Scholes-1973] volatilities for European options. Since fast integration techniques allow us to get European style option prices for a whole strip of strikes in a split second, the hybrid approximation can be directly used for model calibration. , L.A.Grzelak, K.Oosterlee, S.Van Weeren (2009)

Closed Form Convexity and Cross-Convexity Adjustments for Heston Prices 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. , G.G.Drimus (2009)

Gamma Expansion of the Heston Stochastic Volatility Model , P.Glasserman, K-K.Kim (2008)

High order discretization schemes for the CIR process: application to Affine Term Structure and Heston models , A.Alfonsi (2008)

Complex Logarithms in Heston-Like Models , R. Lord, C. Kahl (2008)

Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model , A.V.Haastrecht, A.Pelsser (2008)

Modern Logarithms for the Heston Model , I.Fahrner (2007)

Hedging under the Heston Model with Jump-to-Default , P.Carr, W.Schoutens (2007)

American Options in the Heston Model With Stochastic Interest Rate , S.Boyarchenko, S.Levendorski (2007)

Models with time-dependent parameters using transform methods: application to Heston s model , A.Elices (2007)

Calibration Of the Heston Model with Application in Derivative Pricing and Hedging , C.Bin (2007)

Markovian Projection Onto a Heston Model , V. Piterbarg, A.Antonov, T.Misirpashaev (2007)

Convexity of option prices in the Heston model , J.Wang (2007)

ADI finite difference schemes for option pricing in the Heston model with correlation , K.J.Hout, S.Foulon (2007)

Optimal Fourier inversion in semi-analytical option pricing At the time of writing this article, Fourier inversion is the computational method of choice for a fast and accurate calculation of plain vanilla option prices in models with an analytically available characteristic function. Shifting the contour of integration along the complex plane allows for different representations of the inverse Fourier integral. In this article, we present the optimal contour of the Fourier integral, taking into account numerical issues such as cancellation and explosion of the characteristic function. This allows for robust and fast option pricing for almost all levels of strikes and maturities. , R. Lord, C. Kahl (2006)

Efficient Simulation of the Heston Stochastic Volatility Model , L.Andersen (2006)

The Little Heston Trap , H.Albrecher, P.Mayer, W.Schoutens, J.Tistaert (2006)

Not-so-complex logarithms in the Heston model , C.Kahl, P.Jackel (2006)

The Heston Model: A Practical Approach , N.Moodley (2005)

On the discretization schemes for the CIR (and Bessel squared) processes , A. Alfonsi (2005)

Hedging Exotic Options in Stochastic Volatility and Jump Diffusion Models , K. Detlefsen (2005)

Probability distribution of returns in the Heston model with stochastic volatility 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-Planck equation exactly and, after integrating out the variance, find an analytic formula for the time-dependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow Jones index for time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log-returns with a time-dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow Jones data for 1982-2001 follow the scaling function for seven orders of magnitude. , A. Dragulescu, V.M. Yakovenko (2002)

Option Valuation Using Fast Fourier Transforms , P.Carr, D.Madan (1999)

A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , S. L. Heston (1993)