Heston Stochastic Volatility Model Research Articles

The rough Hawkes Heston stochastic volatility model

AbstractWe study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes‐type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier‐Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati‐Volterra equations, which can be efficiently approximated using a multi‐factor approximation technique. We calibrate a parsimonious specification of our model characterized by a power kernel and an exponential law for the jumps. We show that our parsimonious setup is able to simultaneously capture, with a high precision, the behavior of the implied volatility smile for both S&P 500 and VIX options. In particular, we observe that in our setting the usual shift in the implied volatility of VIX options is explained by a very low value of the power in the kernel. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self‐exciting jumps in order to capture the joint evolution of the S&P 500 and VIX.

Reduced-Order Modeling for Heston Stochastic Volatility Model

In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

Covariance and Correlation Swaps in Energy Markets

Many valuations of variance and volatility derivatives already exist, especially in discrete time, but continuous-time valuations of covariance and correlation swaps do not currently exist for assets following the Heston stochastic volatility model. Energy commodity markets contain some of the broadest and most liquid options markets. Thus, the development of covariance and correlation swap valuation will be of significant use to investors looking to hedge covariance or correlation. In this paper, we derive approximations for an arbitrage-free valuation of natural gas and crude oil covariance and correlation swaps in the Heston model using a continuous-time regime. The approximations are obtained through the use of successive Talyor approximations on otherwise intractable terms. We find that the first order approximations of covariance and correlation swap fair strikes are reasonably effective. When the approximations are taken to the second order, a significant problem with the ML-ARCH approximation of the GARCH(1,1) process creates a large error in the valuation and its error bounds. Leveraging the properties of the Lagrange error bound, we refine our approximation to avoid the use of a frequently miscalibrated parameter, leading to a better approximation of the fair strikes of the two swaps.

APPRAISAL OF STOCHASTIC PRICING FORMULA FOR HYBRID EQUITY WARRANTS

An evaluator of warrant models is essential for enhancing the current procedures used in the country's stock market and economy. This would provide additional insight for investors to make informed investment decisions and help to better understand the dynamics of warrant pricing in Malaysia. It has been established that the conventional process for valuing equity warrants using call option pricing models, like the Black-Scholes model, has numerous flaws, including the presumption of constant volatility and interest rates. As a matter of fact, existing alternative models were more concerned with providing techniques for pricing than with conducting empirical tests of warrant pricing models. Consequently, it is crucial to have a mathematical model that takes into account stochastic volatility and stochastic interest rates when pricing and analyzing equity warrants. Recent research by Roslan, Ibrahim, Jameel, and Ibrahim (2020) and Roslan, Ibrahim, Karim (2020) examined the pricing of hybrid equity warrants in the presence of stochastic factors. There was no comparative analysis conducted in these studies to ascertain the relative efficacy of the two methods used. This study's main goal is to empirically investigate how well equity warrant pricing models perform in the Malaysian market. This evaluation will take into account the impact of stochastic interest rates, using the Cox-Ingersoll-Ross (CIR) stochastic interest rate model, and stochastic volatility, using the Heston stochastic volatility model. To perform the analysis, the study will make use of a number of mathematical techniques, such as partial differential equations, Fourier transform approaches, and the change of measure method. The research methodology is divided into three main phases. It is anticipated that the research's findings will provide new perspectives that go beyond the domain of warrant pricing, creating chances for more widespread applications in fields like foreign exchange, insurance, and mortgage contracts.

A financial modeling approach to industry exchange-traded funds selection

This study uses a comprehensive approach to optimize the portfolio allocation to equity sector Exchange Traded Funds. We combine data on the market prices of options written on the funds, the Heston stochastic volatility model, risk premium transformation, copulas, and optimization with stochastic dominance constraints. This comprehensive strategy provides significant performance out-of-sample gains relative to the passive and active alternative strategies, both before and after accounting for risk and transaction costs. Our findings point at market inefficiencies that can be exploited using sector funds, past public data, and blending multiple methods.

Simulation schemes for the Heston model with Poisson conditioning

Exact simulation schemes under the Heston stochastic volatility model (e.g., Broadie–Kaya and Glasserman–Kim) suffer from computationally expensive modified Bessel function evaluations. We propose a new exact simulation scheme without the modified Bessel function, based on the observation that the conditional integrated variance can be simplified when conditioned by the Poisson variate used for simulating the terminal variance. Our approach also enhances the low-bias and time discretization schemes, which are suitable for pricing derivatives with frequent monitoring. Extensive numerical tests reveal the good performance of the new simulation schemes in terms of accuracy, efficiency, and reliability when compared with existing methods.

Risk management with local least squares Monte Carlo

AbstractThe least squares Monte Carlo method has become a standard approach in the insurance and financial industries for evaluating a company’s exposure to market risk. However, the non-linear regression of simulated responses on risk factors poses a challenge in this procedure. This article presents a novel approach to address this issue by employing an a-priori segmentation of responses. Using a K-means algorithm, we identify clusters of responses that are then locally regressed on their corresponding risk factors. The global regression function is obtained by combining the local models with logistic regression. We demonstrate the effectiveness of the proposed local least squares Monte Carlo method through two case studies. The first case study investigates butterfly and bull trap options within a Heston stochastic volatility model, while the second case study examines the exposure to risks in a participating life insurance scenario.

Optimal Defined Contribution Pension Management with Jump Diffusions and Common Shock Dependence

This work deals with an optimal asset allocation problem for a defined contribution (DC) pension plan during its accumulation phase. The contribution rate is assumed to be proportional to the individual’s salary. The salary follows a Heston stochastic volatility model with jumps, and there exists common shock dependence between the salary and the volatility. Since the time horizon of pension management is quite long, the influence of inflation is considered in the given context. The aim of the pension plan described in this paper is to reduce fluctuations in terminal wealth by investing in the bond and the stock. Through the dynamic programming principle, the Hamilton–Jacobi–Bellman equation is shown. The explicit expression of the investment decision is derived by solving the Hamilton–Jacobi–Bellman equation. In the last part, a numerical analysis is shown to illustrate the impacts of different parameters on the optimal investment policy.

Analytically pricing European options under a hybrid stochastic volatility and interest rate model with a general correlation structure

AbstractIn this paper, an additional factor is introduced into the Heston–Hull–White (HHW) hybrid model, which originally combines the Heston stochastic volatility model and the Hull–White stochastic interest rate model, to capture the correlation between the underlying price and the interest rate, while at the same time preserve the analytical tractability. With the analytical solution to the characteristic function of the underlying price being successfully derived, a closed‐form pricing formula for European options under the two‐factor HHW hybrid model is presented. Numerical experiments are also carried out to show the effect of the newly introduced factor. To further demonstrate the importance of introducing the correlation, we have also conducted an empirical study, comparing the performance of our model and that of the HHW model, based on European options written on S&P 500 index.

Conditional expectation strategy under the long memory Heston stochastic volatility model

This article deals with an European option pricing via proportional transaction costs in the incomplete environment with and without arbitrage opportunities under two long memory versions of the Heston model. Observing and introducing a traded proxy for the volatility in the modern market, we use the conditional expectation and the delta hedging strategies and present the generalized fractional Ito formula to obtain the option price partial differential equations (PDEs). To solve these PDEs, we apply the finite difference method and employ the K-antithetic variates algorithm based on the Monte-Carlo simulation as a benchmark for this method. Finally, we provide numerical results to illustrate the effectiveness of the proposed model.

The Heston stochastic volatility model has a boundary trace at zero volatility

We establish boundary regularity results in Hölder spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) mathbb {H} = mathbb {R}times (0,infty )subset mathbb {R}^2. Starting with nonsmooth initial data u_0in H, we take advantage of smoothing properties of the parabolic semigroup textrm{e}^{-t{mathcal {A}}}:,Hrightarrow H, tin mathbb {R}_+, generated by the Heston model, to derive the smoothness of the solution u(t) = textrm{e}^{-t{mathcal {A}}} u_0 for all t>0. The existence and uniqueness of a weak solution is obtained in a Hilbert space H = L^2(mathbb {H};mathfrak {w}) with very weak growth restrictions at infinity and on the boundary partial mathbb {H}= mathbb {R}times { 0}subset mathbb {R}^2 of the half-plane mathbb {H}. We investigate the influence of the boundary behavior of the initial data u_0in H on the boundary behavior of u(t) for t>0.

Arbitrage-Free Neural-SDE Market Models

Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This paper develops a nonparametric model for the European options book respecting underlying financial constraints and while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a model is learnt from discrete time series data of stock and option prices. We use neural networks as function approximators for the drift and diffusion of the modelled SDE system, and impose constraints on the neural nets such that no-arbitrage conditions are preserved. In particular, we give methods to calibrate neural SDE models which are guaranteed to satisfy a set of linear inequalities. We validate our approach with numerical experiments using data generated from a Heston stochastic local volatility model.

A Stackelberg reinsurance-investment game under Heston's stochastic volatility model

<p style='text-indent:20px;'>This paper studies the optimal reinsurance-investment problem under Heston's stochastic volatility (SV) in the framework of Stackelberg stochastic differential game, in which an insurer and a reinsurer are the two players. The aim of the insurer as the follower in the game is to find the optimal reinsurance strategy and investment strategy such that its mean-variance cost functional is maximized, while the aim of the reinsurer as the leader is to maximize its mean-variance cost functional through finding its optimal premium pricing strategy and investment strategy. To overcome the time-inconsistency problem in the game, we derive the optimization problems of the two players as embedded games and obtain the corresponding extended Hamilton–Jacobi–Bellman (HJB) equations. Then we present the verification theorem and get the equilibrium reinsurance-investment strategies and the corresponding value functions of both the insurer and the reinsurer. Finally, we provide some numerical examples to draw the findings.</p>

Pricing vulnerable fader options under stochastic volatility models

<p style='text-indent:20px;'>In this paper, we incorporate default risk into Heston's stochastic volatility model and focus on the valuation of vulnerable fader options. Fader options are path-dependent derivatives, depending on the time the underlying asset price stays inside a given range. We obtain an explicit pricing formula of vulnerable fader options, including fader options and (vulnerable) European options as special cases. Finally, we illustrate the effect of stochastic volatility and default risk on fader option prices. Specially, an inverted U-shaped curve is observed when we keep initial levels of the default intensity constant, but change the relative proportions of two factors in the default intensity.</p>

Pricing Variance Swaps Under Heston Stochastic Volatility Model with Stochastic Interest Rate and Stochastic Dividend Yield

Pricing Variance Swaps Under Heston Stochastic Volatility Model with Stochastic Interest Rate and Stochastic Dividend Yield

On the calibration of stochastic volatility models to estimate the real-world measure used in option pricing

It is widely noted that the Heston stochastic volatility model fails to capture the fat tails often observed in daily equity returns. Adding random jumps improves the model’s ability to capture extreme events. This extension is known as the Bates stochastic volatility jump (SVJ) model. The model parameters for the Heston and Bates SVJ models are generally calibrated to option prices inducing the so-called risk-neutral measure. However, in the absence of a sufficiently liquid options market, one has to resort to calibration under the realworld measure. In this paper, we calibrate the Heston and Bates SVJ models to historical equity returns in the United States and South Africa using the efficient method of moments(EMM). We then show how a real-world stochastic volatility model can be used in practice to test a simple volatility targeting strategy. Our findings suggest that stochastic volatility and jumps are both required to characterise equity returns in South Africa. Furthermore, volatility targeting is an effective strategy that allows investors to manage the downside risk of a portfolio.

An economic scenario generator for embedded derivatives in South Africa

It is well known that interest rate risk is a dominating factor when pricing long-dated contingent claims. The Heston stochastic volatility model fails to capture this risk as the model assumes a constant interest rate throughout the life of the claim. To overcome this, the risk-free interest rate can be modelled by a Hull-White short rate process and can be combined with the Heston stochastic volatility model to form the so-called Heston-Hull-White model. The Heston-Hull-White model allows for correlation between the equity and interest rate processes, a component that is important when pricing long-dated contingent claims. In this paper, we apply the Heston-Hull-White model to price Guaranteed Minimum Maturity Benefits (GMMBs) and Guaranteed Minimum Death Benefits (GMDBs) offered in the life insurance industry in South Africa. We propose a further extension by including stochastic mortality rates based on either a continuous-time Cox-Ingersoll-Ross short rate process or a discrete-time AR(1)-ARCH(1) model. Our findings suggest that stochastic interest rates are the dominating factor when reserving for GMMB and GMDB products. Furthermore, a delta-hedging strategy can help reduce the variability of embedded derivative liabilities.

Pricing American options with stochastic volatility and small nonlinear price impact: A PDE approach

In this paper, transaction costs as small nonlinear price impact are introduced into American option pricing under the Heston stochastic volatility model, forming a gap between option prices of the holder and writer. Through the use of a dynamic hedging strategy together with a known option independent of transaction costs, we derive two different nonlinear pricing partial differential equation systems for the holder and writer, respectively. A numerical algorithm is designed to solve the systems so that American option prices as well as the optimal exercise boundary can be simultaneously obtained. Examples are presented to illustrate the effect of transaction costs on both option prices and optimal exercise prices.

Asymptotic analysis of the Heston model and its statistical and financial applications

AbstractTransform inversion algorithm is a popular method in the areas of Statistics and Financial Engineering. However, it is often difficult to guarantee its accuracy due to the lack of error control, which depends heavily on the asymptotic feature of the transform function. This article characterizes the asymptotic behavior of the Heston stochastic volatility model, by which one can derive computable error bounds and achieve any pre‐specified error tolerance in the transform inversion algorithm. Moreover, a universal method is proposed to specify a closed‐form upper bound of the discretization error. Numerical examples indicate that the resulting inversion algorithm with our computable error bounds is reliably accurate and efficient, for parameter estimation and option pricing under the Heston model.

Amending the heston stochastic volatility model to forecast local motor vehicle crash rates: A case study of Washington, D.C.

Modelling crash rates in an urban area requires a swathe of data regarding historical and prevailing traffic volumes and crash events and characteristics. Provided that the traffic volume of urban networks is largely defined by typical work and school commute patterns, crash rates can be determined with a reasonable degree of accuracy. However, this process becomes more complicated for an area that is frequently subject to peaks and troughs in traffic volume and crash events owing to exogenous events – for example, extreme weather – rather than typical commute patterns. One such area that is particularly exposed to exogenous events is Washington, D.C., which has seen a large rise in crash events between 2009 and 2020. In this study, we adopt a forecasting model that embeds heterogeneity and temporal instability in its estimates in order to improve upon forecasting models currently used in transportation and road safety research. Specifically, we introduce a stochastic volatility model that aims to capture the nuances associated with crash rates in Washington, D.C. We determine that this model can outperform conventional forecasting models, but it does not perform well in light of the unique travel patterns exhibited throughout the COVID-19 pandemic. Nevertheless, its adaptability to the idiosyncrasies of Washington, D.C. crash rates demonstrates its ability to accurately simulate localised crash rates processes, which can be further adapted in public policy contexts to form road safety targets.