Heston Stochastic Volatility Model Research Articles

Analytic Equity Option Pricing with Stochastic Volatility

We consider the Heston stochastic volatility model for equity options and use operator formalism and perturbation expansion techniques to derive a formal analytic expression for the Green's function (pricing kernel). This is expanded explicitly as a perturbation series in powers of the vol of vol up to second order and used to obtain an asymptotically accurate pricing formula for vanilla equity options, replicating results obtained previously by Ito and Malliavin techniques.

Saddlepoint approximations to tail expectations under non-Gaussian base distributions: option pricing applications

The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of the saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We also demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatility model.

Pricing Path-Dependent Payoff Using Quadrature on Wiener Space

Solution to stochastic differential equation that defines the dynamics of asset prices is interpreted as a function of underlying Brownian motion when permissible, and the expectation values of various functionals of the solution – therefore functionals of Brownian motion (Wiener functionals) – are estimated from numerical approximations to high-dimensional Gaussian integrals using quadrature. The simple method is exemplified for a few standard asset price dynamics used in practice for structured and contingent claim pricing, such as local volatility, Heston stochastic volatility, SABR, and LIBOR market model (LMM/BGM). The quadrature in these concrete implementations is performed in m-dimensional space where m is between 10 and 30 depending on the specific model or a payoff structure. The methodology appears to be a broadly viable alternative to Monte Carlo for short- to medium-dated financial products, with potential uses outside those examined in this paper.

A Comparative Analysis of the Black-Scholes- Merton Model and the Heston Stochastic Volatility Model

This paper compares the performance of two different option pricing models, namely, the Black-Scholes-Merton (B-S-M) model and the Heston Stochastic Volatility (H-S-V) model. It is known that the most popular B-S-M Model makes the assumption that volatility of an asset is constant while the H-S-V model considers it to be random. We examine the behavior of both B-S-M and H-S-V formulae with the change of different affecting factors by graphical representations and hence assimilate them. We also compare the behavior of some of the Greeks computed by both of these models with changing stock prices and hence constitute 3D plots of these Greeks. All the numerical computations and graphical illustrations are generated by a powerful Computer Algebra System (CAS), MATLAB.
 GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 127-140

Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option

ABSTRACTThis paper considers an optimal asset-liability management (ALM) problem for an insurer under the mean-variance criterion. It is assumed that the value of liabilities is described by a geometric Brownian motion (GBM). The insurer's surplus process is modeled by a general jump process generated by a marked point process. The financial market consists of one risk-free asset and n risky assets with the risk premium relying on an affine diffusion factor process. By transferring a proportion of insurance risk to a reinsurer and investing the surplus into the financial market, the insurer aims to maximize the expected terminal net wealth and, at the same time, minimize the corresponding variance of the terminal net wealth. By using a backward stochastic differential equation (BSDE) approach, closed-form expressions for both the efficient strategy and efficient frontier are derived. To illustrate the main results, we study an example with the Heston stochastic volatility (SV) model and numerically analyze the economic behavior of the efficient frontier. Finally, a generalization of the Mutual Fund Theorem is obtained.

Saddlepoint Approximations to Tail Expectations Under Non-Gaussian Base Distributions

The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatility model.

A BSDE approach to a class of dependent risk model of mean–variance insurers with stochastic volatility and no-short selling

This paper studies the optimal reinsurance and investment strategy for an insurer with two dependent classes of insurance business, where the claim number processes are correlated through a common shock. It is assumed that the insurer also faces the decision making of investing in a financial market with one risk-free asset and one risky asset following the Heston stochastic volatility (SV) model. The insurer is not allowed to short sell the risky asset. Under the mean–variance criterion, we consider the insurer’s problem of maximizing the expected terminal wealth and, at the same time, minimizing the variance of the terminal wealth. Using the results of stochastic linear–quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), we derive closed-form expressions for the optimal strategies and the efficient frontiers in terms of solutions to the BSDEs. Our approach shows how BSDEs can be used to solve mean–variance problems in insurance applications. Finally, economic behavior of the efficient frontiers is analyzed by using some numerical examples.

Pricing generalized variance swaps under the Heston model with stochastic interest rates

Unlike vanilla variance swaps, generalized variance swaps such as gamma, corridor variance and conditional variance swaps are expected to be not free from interest rates because of their weight processes. To examine the impact of stochastic interest rates on the generalized variance swaps, this paper considers discrete sampling times and the Heston stochastic volatility model incorporated by stochastic interest rates driven by the Cox–Ingersoll–Ross process. Based on the explicit calculation of the discounted characteristic function of Duffie et al. (2000), we obtain exact solutions for the fair strike prices of the generalized variance swaps for an affine version of the hybrid model. The solutions are given in closed form expression for the vanilla variance and gamma swaps and in Fourier integral expression for the corridor and conditional variance swaps. We apply the projection techniques of Grzelak and Oosterlee (2011) to the original non-affine model with a generalized correlation structure and obtain affine approximate solutions. We show the effects of stochastic interest rates on the strike prices of the generalized variance swaps.

The volatility target effect in investment-linked products with embedded American-type derivatives

Volatility Target (VolTarget) strategies as underlying assets for options embedded in investment-linked products have been widely used by practitioners in recent years. Available research mainly focuses on European-type options linked to VolTarget strategies. In this paper, VolTarget-linked options of American type are investigated. Within the Heston stochastic volatility model, a numerical study of American put options, as well as American lookback options linked to VolTarget strategies, is performed. These are compared with traditional American-type derivatives linked to an equity index. The authors demonstrate that using a Volatility Target strategy as a basis for an embedded American-type derivative may make any protection fees significantly less dependent of changing market volatilities. Replacing an equity index with the VolTarget strategy may also result in reducing guarantee fees of the corresponding protection features in a highly volatile market environment.

Dual control Monte-Carlo method for tight bounds of value function under Heston stochastic volatility model

The aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution to the HJB equation for power utility due to its homothetic property. It is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the HJB equation for the Heston model. In this paper we propose an efficient dual control Monte-Carlo method for computing tight lower and upper bounds of the value function. We identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions, including power, non-HARA and Yaari utilities. Finally, we perform some numerical tests to see the efficiency, accuracy, and robustness of the method. The numerical results support strongly our proposed scheme.

Happy Xmas (War Is Over)

Have Andersen and Lake put an end to the war to value options under the Heston stochastic volatility model?

Multifactor Heston's stochastic volatility model for European option pricing

AbstractIn this study, we extend the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254] by incorporating a slow varying factor of volatility. The resulting model can be viewed as a multifactor extension of the Heston model with two additional factors driving the volatility levels. An asymptotic analysis consisting of singular and regular perturbation expansions is developed to obtain an approximation to European option prices. We also find explicit expressions for some essential functions that are available only in integral formulas in the work of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254]. This finding basically leads to considerable reduction in computational time for numerical calculation as well as calibration problems. An accuracy result of the asymptotic approximation is also provided. For numerical illustration, the multifactor Heston model is calibrated to index options on the market, and we find that the resulting implied volatility surfaces fit the market data better than those produced by the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254], particularly for long‐maturity call options.

Semi-analytical prices for lookback and barrier options under the Heston model

Under the Heston stochastic volatility model, we derive semi-analytical formulas for the prices of path-dependent options with payoffs linked to the maximum or minimum value of the underlying asset price over a certain period of time. In particular, we obtain prices of lookback and barrier options in the Heston model, but the methodology applies more generally. By conditioning with respect to the variance path, we obtain pricing formulas that can be related to their counterparts in the Black–Scholes model.

On parameter estimation of Heston’s stochastic volatility model: a polynomial filtering method

In this paper, we investigate the problem of estimating the volatility from the underlying asset price for discrete-time observations. This topic has attracted much research interest due to the key role of the volatility in finance. In this paper, we consider the Heston stochastic volatility model with jumps and we develop a new polynomial filtering method for the estimation of the volatility. The method relies on a linear filter which uses a polynomial state-space formulation of the discrete version of the continuous-time model. We demonstrate that a higher-order polynomial filtering method can be efficiently applied in the context of stochastic volatility models. Then, we compare our approach with some, well-established, techniques in the literature.

American option pricing under double Heston stochastic volatility model: simulation and strong convergence analysis

ABSTRACTIn this work, we investigate the double Heston model dynamics which is defined by two independent variance processes with non-Lipschitz diffusions. Next, it is analysed the strong convergence of the volatility processes of the double Heston model. Then, we examine the LSM algorithm to determine the American style option price in the double Heston model. Besides, by performing the antithetic simulation in the original LSM algorithm, we aim to accelerate the algorithm. The accuracy and speed of two algorithms are studied and at last, by some numerical results illustrate the accuracy of the proposed algorithms and examine the effect of the different parameters of the model on the value of the option.

Pricing Options and Computing Implied Volatilities using Neural Networks

This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods. With ANNs being universal function approximators, this method trains an optimized ANN on a data set generated by a sophisticated financial model, and runs the trained ANN as an agent of the original solver in a fast and efficient way. We test this approach on three different types of solvers, including the analytic solution for the Black-Scholes equation, the COS method for the Heston stochastic volatility model and Brent’s iterative root-finding method for the calculation of implied volatilities. The numerical results show that the ANN solver can reduce the computing time significantly.

Anchoring-Adjusted Option Pricing Models

Relying on a useful starting point and attempting to adjust it appropriately is a robust human decision-making heuristic. Evidence suggests that underlying stock volatility is such a starting point, which is scaled up to estimate call option volatility. The author adjusts the Black-Scholes, Heston, and Bates models for reliance on this starting point. The adjustment mechanism captures several option-return puzzles. The adjusted Black-Scholes generates implied-volatility skew. The adjusted Heston stochastic-volatility model matches the same data better, does so at more plausible parameter values, and generates a steep short-term skew. Furthermore, 2 novel predictions are empirically tested and strongly supported in the data.

A Sensitivity Analysis of the Long-Term Expected Utility of Optimal Portfolios

This paper discusses the sensitivity of the long-term expected utility of optimal portfolios for an investor with constant relative risk aversion. Under an incomplete market given by a factor model, we consider the utility maximization problem with long-time horizon. The main purpose is to find the long-term sensitivity, that is, the extent how much the optimal expected utility is affected in the long run for small changes of the underlying factor model. The factor model induces a specific eigenpair of an operator, and this eigenpair does not only characterize the long-term behavior of the optimal expected utility but also provides an explicit representation of the expected utility on a finite time horizon. We conclude that this eigenpair therefore determines the long-term sensitivity. As examples, explicit results for several market models such as the Kim-Omberg model for stochastic excess returns and the Heston stochastic volatility model are presented.

The Impact of Jumps on American Option Pricing: The S&P 100 Options Case.

This paper analyzes the importance of asset and volatility jumps in American option pricing models. Using the Heston (1993) stochastic volatility model with asset and volatility jumps and the Hull and White (1987) short rate model, American options are numerically evaluated by the Method of Lines. The calibration of these models to S&P 100 American options data reveals that jumps, especially asset jumps, play an important role in improving the models' ability to fit market data. Further, asset and volatility jumps tend to lift the free boundary, an effect that augments during volatile market conditions, while the additional volatility jumps marginally drift down the free boundary. As markets turn more volatile and exhibit jumps, American option holders become more prudent with their exercise decisions, especially as maturity of the options approaches.

Geometric Asian Options Pricing under the Double Heston Stochastic Volatility Model with Stochastic Interest Rate

This paper presents an extension of double Heston stochastic volatility model by incorporating stochastic interest rates and derives explicit solutions for the prices of the continuously monitored fixed and floating strike geometric Asian options. The discounted joint characteristic function of the log‐asset price and its log‐geometric mean value is computed by using the change of numeraire and the Fourier inversion transform technique. We also provide efficient approximated approach and analyze several effects on option prices under the proposed model. Numerical examples show that both stochastic volatility and stochastic interest rate have a significant impact on option values, particularly on the values of longer term options. The proposed model is suitable for modeling the longer time real‐market changes and managing the credit risks.