Heston Stochastic Volatility Model Research Articles

On the RND under Heston's Stochastic Volatility Model

We consider Heston's (1993) stochastic volatility model for valuation of European options to which (semi) closed form solutions are available and are given in terms of characteristic functions. We prove that the class of scale-parameter distributions with mean being the forward spot price satisfies Heston's solution. Thus, we show that any member of this class could be used for the direct risk-neutral valuation of the option price under Heston's SV model. In fact, we also show that any RND with mean being the forward spot price that satisfies Hestons' option valuation solution, must be a member of a scale-family of distributions in that mean. As particular examples, we show that one-parameter versions of the {\it Log-Normal, Inverse-Gaussian, Gamma, Weibull} and the {\it Inverse-Weibull} distributions are all members of this class and thus provide explicit risk-neutral densities (RND) for Heston's pricing model. We demonstrate, via exact calculations and Monte-Carlo simulations, the applicability and suitability of these explicit RNDs using already published Index data with a calibrated Heston model (S&P500, Bakshi, Cao and Chen (1997), and ODAX, Mrazek and Pospisil (2017)), as well as current option market data (AMD).

Recovering the Variance Premium

This paper generalizes Ross (2015) recovery theory to accommodate growth, including the Black-Scholes and stochastic volatility option models. The new theory recovers information about equity risk premia and variance risk premia from options prices. In the Heston (1993) stochastic volatility model, the theory predicts an exact (negative) value for the variance risk premium as a function of the equity premium. Recovery theory also predicts that the stochastic discount factor is the reciprocal return on a model-free portfolio of index options. This paper tests the theory on returns from one-month VIX and three-month VIX3M option portfolios from 2007-2018. Recovery theory links the equity pre- mium to the values of both conditional and unconditional variance premia. It also predicts how VIX3M option variance is a biased predictor of future one-month VIX variance. Empirically, recovery theory simultaneously matches the average S&P 500 equity premium, the average variance premium, and observed biases in the variance expectations hypothesis. Autocorrelation properties of VIX indices imply a 12% annual equity premium.

PRICING VARIANCE SWAPS UNDER DOUBLE HESTON STOCHASTIC VOLATILITY MODEL WITH STOCHASTIC INTEREST RATE

In this paper, we discuss the problem of pricing discretely sampled variance swaps under a hybrid stochastic model. Our modeling framework is a combination with a double Heston stochastic volatility model and a Cox–Ingersoll–Ross stochastic interest rate process. Due to the application of the T-forward measure with the stochastic interest process, we can only obtain an efficient semi-closed form of pricing formula for variance swaps instead of a closed-form solution based on the derivation of characteristic functions. The practicality of this hybrid model is demonstrated by numerical simulations.

The Generalized Gamma Distribution as a Useful RND Under Heston’s Stochastic Volatility Model

Following Boukai (2021) we present the Generalized Gamma (GG) distribution as a possible RND for modeling European options prices under Heston's (1993) stochastic volatility (SV) model. This distribution is seen as especially useful in situations in which the spot's price follows a negatively skewed distribution and hence, Black-Scholes based (i.e. the log-normal distribution) modeling is largely inapt. We apply the GG distribution as RND to modeling current market option data on three large market-index ETFs, namely the SPY, IWM and QQQ as well as on the TLT (an ETF that tracks an index of long term US Treasury bonds). The current option chain of each of the three market-index ETFs shows of a pronounced skew of their volatility `smile' which indicates a likely distortion in the Black-Scholes modeling of such option data. Reflective of entirely different market expectations, this distortion appears not to exist in the TLT option data. We provide a thorough modeling of the available option data we have on each ETF (with the October 15, 2021 expiration) based on the GG distribution and compared it to the option pricing and RND modeling obtained directly from a well-calibrated Heston's (1993) SV model (both theoretically and empirically, using Monte-Carlo simulations of the spot's price). All three market-index ETFs exhibit negatively skewed distributions which are well-matched with those derived under the GG distribution as RND. The inadequacy of the Black-Scholes modeling in such instances which involve negatively skewed distribution is further illustrated by its impact on the hedging factor, delta, and the immediate implications to the retail trader. In contrast, for the TLT ETF, which exhibits no such distortion to the volatility `smile', the three pricing models (i.e. Heston's, Black-Scholes and Generalized Gamma) appear to yield similar results.

Expansion method for pricing foreign exchange options under stochastic volatility and interest rates

Some expansion methods have been proposed for pricing options approximately in analytical form. One of these is the smart expansion method based on the Malliavin calculus, which is used to price options in the Heston stochastic volatility model with deterministic interest rates. In this paper, we apply the method to the Heston–Hull–White model, which admits stochastic interest rates to enhance the model, and we obtain the expansion formula for pricing options in the model up to second order. Then numerical studies are performed to compare our approximation formula with the Monte Carlo simulation. Our formula shows numerically comparable results with another method using the approximation of the characteristic function, and can also be applied for parameter configurations where the latter method is not useful. The control variate is also used to improve the accuracy for high volatility-of-volatility cases.

Series Expansions and Direct Inversion for the Heston Model

Efficient sampling for the conditional time integrated variance process in the Heston stochastic volatility model is key to the simulation of the stock price based on its exact distribution. We construct a new series expansion for this integral in terms of double infinite weighted sums of particular independent random variables through a change of measure and the decomposition of squared Bessel bridges. When approximated by series truncations, this representation has exponentially decaying truncation errors. We propose feasible strategies to largely reduce the implementation of the new series to simulations of simple random variables that are independent of any model parameters. We further develop direct inversion algorithms to generate samples for such random variables based on Chebyshev polynomial approximations for their inverse distribution functions. These approximations can be used under any market conditions. Thus, we establish a strong, efficient, and almost exact sampling scheme for the Heston model.

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.

High-Order Expansion Based Saddlepoint Approximation Errors of the Heston Stochastic Volatility Model

High-Order Expansion Based Saddlepoint Approximation Errors of the Heston Stochastic Volatility Model

Fast Hybrid Schemes for Fractional Riccati Equations (Rough Is Not So Tough)

We solve a family of fractional Riccati equations with constant (possibly complex) coefficients. These equations arise, for example, in fractional Heston stochastic volatility models, which have received great attention in the recent financial literature because of their ability to reproduce a rough volatility behavior. We first consider the case of a zero initial value corresponding to the characteristic function of the log-price. Then we investigate the case of a general starting value associated to a transform also involving the volatility process. The solution to the fractional Riccati equation takes the form of power series, whose convergence domain is typically finite. This naturally suggests a hybrid numerical algorithm to explicitly obtain the solution also beyond the convergence domain of the power series. Numerical tests show that the hybrid algorithm is extremely fast and stable. When applied to option pricing, our method largely outperforms the only available alternative, based on the Adams method.

Pricing variance swaps under hybrid CEV and stochastic volatility

In this paper, we consider the problem of pricing a variance swap whose underlying asset price dynamics is modeled under a hybrid framework of constant elasticity of variance and stochastic volatility (SVCEV). Applying the multi-scale asymptotic analysis approach, we obtain a semi-closed form approximation of the fair continuous variance strike. We conduct numerical experiments by applying this approximation formula to calculate the square root of the fair continuous variance strike with different values of parameters. The market data of S&P 500 options are used to obtain calibrations of the SVCEV model, and then the estimated parameters are further used to compute the values of the square root of fair continuous variance strike. In addition, we also analyze and compare the performance of the CEV model, the SVCEV model and the Heston stochastic volatility model.

The valuation of variance swaps under stochastic volatility, stochastic interest rate and full correlation structure

This paper considers the case of pricing discretely-sampled variance swaps under the class of equity-interest rate hybridization. Our modeling framework consists of the equity which follows the dynamics of the Heston stochastic volatility model, and the stochastic interest rate is driven by the Cox-Ingersoll-Ross (CIR) process with full correlation structure imposed among the state variables. This full correlation structure possesses the limitation to have fully analytical pricing formula for hybrid models of variance swaps, due to the non-affinity property embedded in the model itself. We address this issue by obtaining an efficient semi-closed form pricing formula of variance swaps for an approximation of the hybrid model via the derivation of characteristic functions. Subsequently, we implement numerical experiments to evaluate the accuracy of our pricing formula. Our findings confirm that the impact of the correlation between the underlying and the interest rate is significant for pricing discretely-sampled variance swaps.

A novel term structure stochastic model with adaptive correlation for trend analysis

AbstractThe prediction of underlying price continues to draw extensive attention in academic research. Based on a review of the advantages and disadvantages of different volatility models, we find that the Heston stochastic volatility model with an adaptive correlation coefficient is most suitable for analysing the Hong Kong options market. We subsequently propose a model‐free implied volatility term structure formulated using options with different strikes and different maturities. The implied volatility is calculated by integrating the option price and strike price from the current time to the expiry date. Discrete points of term structure data are used to fit a term structure curve. Finally, we use the model‐free implied volatility term structure as the long‐run mean level of the Heston model to fully exploit the information content contained in the implied volatility term structure. We simulate the distribution of the underlying asset price based on the Heston model and constant elasticity of variance (CEV) model. The adaptive correlation Heston model provides superior results in terms of one‐day‐ahead prediction performance and the 79‐day distribution of the underlying asset price compared with the CEV model.

Variance and Interest Rate Risk in Unit-Linked Insurance Policies

One of the risks derived from selling long-term policies that any insurance company has arises from interest rates. In this paper, we consider a general class of stochastic volatility models written in forward variance form. We also deal with stochastic interest rates to obtain the risk-free price for unit-linked life insurance contracts, as well as providing a perfect hedging strategy by completing the market. We conclude with a simulation experiment, where we price unit-linked policies using Norwegian mortality rates. In addition, we compare prices for the classical Black-Scholes model against the Heston stochastic volatility model with a Vasicek interest rate model.

An iterative splitting method for pricing European options under the Heston model☆

In this paper, we propose an iterative splitting method to solve the partial differential equation (PDE) in option pricing problems. We focus on the Heston stochastic volatility model and the derived two-dimensional PDE. We take the European option as an example and conduct numerical experiments using different boundary conditions. The iterative splitting method transforms the two-dimensional equation into two quasi one-dimensional equations with the variable on the other dimension fixed, which helps to lower the computational cost. Numerical results show that the iterative splitting method together with an artificial boundary condition (ABC) based on the method by Li and Huang (2019) gives the most accurate option price and Greeks compared to the classic finite difference method with the commonly-used boundary conditions in Heston (1993).

Small-Time smile for the multifactor volatility heston model

AbstractWe extend the existing small-time asymptotics for implied volatilities under the Heston stochastic volatility model to the multifactor volatility Heston model, which is also known as the Wishart multidimensional stochastic volatility model (WMSV). More explicitly, we show that the approaches taken in Forde and Jacquier (2009) and Forde, Jacqiuer and Lee (2012) are applicable to the WMSV model under mild conditions, and obtain explicit small-time expansions of implied volatilities.

The mean‐reverting 4/2 stochastic volatility model: Properties and financial applications

AbstractThis article defines and studies a stochastic process that combines two important stylized facts of financial data: reversion to the mean, and a flexible generalized stochastic volatility process: the 4/2 process. This work is motivated by the modeling of at least two financial asset classes: commodities and volatility indexes. We provide analytical expressions for the conditional characteristic functions and closed‐form approximations to relevant cases, in particular a mean‐reverting Heston stochastic volatility model. Our results describe feasible changes of measure with the final aim of pricing financial derivatives. The empirical analysis and the estimation methodology confirm the need of such a model in several examples from the targeted asset classes. Applications to option pricing corroborate the substantial impact on the implied volatility surfaces of the new parameters.

Numerical simulation and applications of the convection–diffusion–reaction equation with the radial basis function in a finite-difference mode

This paper develops two local mesh-free methods for designing stencil weights and spatial discretization, respectively, for parabolic partial differential equations (PDEs) of convection–diffusion–reaction type. These are known as the radial-basis-function generated finite-difference method and the Hermite finite-difference method. The convergence and stability of these schemes are investigated numerically using some examples in two and three dimensions with regularly and irregularly shaped domains. Then we consider the numerical pricing of European and American options under the Heston stochastic volatility model. The European option leads to the solution of a two-dimensional parabolic PDE, and the price of the American option is given by a linear complementarity problem with a two-dimensional parabolic PDE of convection–diffusion–reaction type. Then we use the operator splitting method to perform time-stepping after space discretization. The resulting linear systems of equations are well conditioned and sparse, and by numerical experiments we show that our numerical technique is fast and stable with respect to the change in the shape parameter of the radial basis function. Finally, numerical results are provided to illustrate the quality of approximation and to show how well our approach converges with the results presented in the literature.

Trading strategy with stochastic volatility in a limit order book market

In this paper, we employ the Heston stochastic volatility model to describe the stock’s volatility and apply the model to derive and analyze trading strategies for dealers in a security market with price discovery. The problem is formulated as a stochastic optimal control problem, and the controlled state process is the dealer’s mark-to-market wealth. Dealers in the security market can optimally determine their ask and bid quotes on the underlying stocks continuously over time. Their objective is to maximize an expected profit from transactions with a penalty proportional to the variance of cumulative inventory cost. We provide an approximate, analytically tractable solution to the stochastic control problem. Numerical experiments are given to illustrate the effects of various parameters on the performances of trading strategies.

Variance swaps, volatility swaps, hedging and bounds under multi-factor Heston stochastic volatility model

In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is the multi-factor Heston stochastic volatility model. We obtain pricing formulas for the weighted variance swap and approximate expressions for the weighted volatility swap. The bounds of the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to compute in real time and can be applied efficiently for practical applications. We study the problem of hedging volatility swap with variance swap. We also determined the optimal amount of the underlying asset that has to be held for minimizing the hedging error by taking positions in options and weighted variance swap. From the numerical analysis, a couple of important features of the usefulness of the multi-factor Heston stochastic volatility model are discussed.

Classical Option Pricing and Some Steps Further

This paper considers the asset price p as relations C=pV between the value C and the volume V of the executed transactions and studies the consequences of this definition for the option pricing equations. We show that the classical BSM model implicitly assumes that value C and volume V of transactions follow identical Brownian processes. Violation of this identity leads to 2-dimensional BSM-like equation with two constant volatilities. We show that agents expectations those approve execution of transactions can further increase the dimension of the BSM model. We study the case when agents expectations may depend on the option price data and show that such assumption can lead to the nonlinear BSM-like equations. We reconsider the Heston stochastic volatility model for the price determined by the value and the volume and derive 3-dimensional BSM-like model with stochastic value volatility and constant volume volatility. Variety of the BSM-like equations states the problem of reasonable balance between the accuracy and the complexity of the option pricing equations.