Algorithm for Financial Derivatives Evaluation in Generalized Double-Heston Model

This paper shows how can be estimated the value of an option if we assume the double- Heston model on a message-based architecture. For path trace simulation we will discretize continous model with an Euler division of time.Monte Carlo; algorithms; computational financial engineering; derivatives evaluation; Black�Scholes�Merton model; Heston model; double-Heston model; generalized double-Heston model.

Underlying Dynamics of Typical Fluctuations of an Emerging Market Price Index: The Heston Model from Minutes to Months

We investigate the Heston model with stochastic volatility and exponential tails as a model for the typical price fluctuations of the Brazilian S\~ao Paulo Stock Exchange Index (IBOVESPA). Raw prices are first corrected for inflation and a period spanning 15 years characterized by memoryless returns is chosen for the analysis. Model parameters are estimated by observing volatility scaling and correlation properties. We show that the Heston model with at least two time scales for the volatility mean reverting dynamics satisfactorily describes price fluctuations ranging from time scales larger than 20 minutes to 160 days. At time scales shorter than 20 minutes we observe autocorrelated returns and power law tails incompatible with the Heston model. Despite major regulatory changes, hyperinflation and currency crises experienced by the Brazilian market in the period studied, the general success of the description provided may be regarded as an evidence for a general underlying dynamics of price fluctuations at intermediate mesoeconomic time scales well approximated by the Heston model. We also notice that the connection between the Heston model and Ehrenfest urn models could be exploited for bringing new insights into the microeconomic market mechanics.Comment: 20 pages, 9 figures, to appear in Physica

The Heston stochastic volatility model in Hilbert space

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein-Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein-Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modelling of forward curves in energy markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics

On parameter estimation of stochastic volatility models from stock data using particle filter - Application to AEX index -

We consider the problem of estimating stochastic volatility from stock data. The estimation of the volatility process of the Heston model is not in the usual framework of the filtering theory. Discretizing the continuous Heston model to the discrete-time one, we can derive the exact volatility filter and realize this filter with the aid of particle filter algorithm. In this paper, we derive the optimal importance function and construct the particle filter algorithm for the discrete-time Heston model. The parameters contained in system model are also estimated by constructing the augmented states for the system and parameters. The developed method is applied to the real data (AEX index)

Forecasting the Term Structure of Variance Swaps

Recently, Diebold and Li (2003) obtained good forecasting results for yield curves in a reparametrized Nelson-Siegel framework. We analyze similar modeling approaches for price curves of variance swaps that serve nowadays as hedging instruments for options on realized variance. We consider the popular Heston model, reparametrize its variance swap price formula and model the entire variance swap curves by two exponential factors whose loadings evolve dynamically on a weekly basis. Generalizing this approach we consider a reparametrization of the three-dimensional Nelson-Siegel factor model. We show that these factors can be interpreted as level, slope and curvature and how they can be estimated directly from characteristic points of the curves. Moreover, we analyze a semiparametric factor model. Estimating autoregressive models for the factor loadings we get termstructure forecasts that we compare in addition to the random walk and the static Heston model that is often used in industry. In contrast to the results of Diebold and Li (2003) on yield curves, no model produces better forecasts of variance swap curves than the random walk but forecasting the Heston model improves the popular static Heston model. Moreover, the Heston model is better than the flexible semiparametric approach that outperforms the Nelson-Siegel model.Term structure, Variance swap curve, Heston model, Nelson-Siegel curve, Semiparametric factor model

The Heston stochastic volatility model with piecewise constant parameters - efficient calibration and pricing of window barrier options

The Heston stochastic volatility model is a standard model for valuing financial derivatives, since it can be calibrated using semi-analytical formulas and captures the most basic structure of the market for financial derivatives with simple structure in time-direction. However, extending the model to the case of time-dependent parameters, which would allow for a parametrization of the market at multiple timepoints, proves more challenging. We present a simple and numerically efficient approach to the calibration of the Heston stochastic volatility model with piecewise constant parameters. We show that semi-analytical formulas can also be derived in this more complex case and combine them with recent advances in computational techniques for the Heston model. Our numerical scheme is based on the calculation of the characteristic function using Gauss-Kronrod quadrature with an additional control variate that stabilizes the numerical integrals. We use our method to calibrate the Heston model with piecewise constant parameters to the foreign exchange (FX) options market. Finally, we demonstrate improvements of the Heston model with piecewise constant parameters upon the standard Heston model in selected cases

On refined volatility smile expansion in the Heston model

It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment $s_+$ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: $\sigma_{BS}( k,T)^{2}T\sim \Psi (s_+-1) \times k$ (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Dragulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expansion of the type $\sigma_{BS}( k,T) ^{2}T=( \beta_{1}k^{1/2}+\beta_{2}+...)^{2}$, where all constants are explicitly known as functions of $s_+$, the Heston model parameters, spot vol and maturity $T$. In the case of the "zero-correlation" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim. 61, 3 (2010), 287--315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles: at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of $\log S_{T}$\ (equivalently: Mellin transform of $S_{T}$ ); what matters is that these transforms satisfy ordinary differential equations of Riccati type. Secondly, our analysis reveals a new parameter ("critical slope"), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions