General Stochastic Volatility Model Research Articles

Sequential Itô–Taylor expansions and characteristic functions of stochastic volatility models

AbstractThis study proposes a new approach to derive the characteristic function of a general stochastic volatility model by sequentially utilizing the Itô–Taylor expansions. In particular, our method applies to non‐affine stochastic volatility models with jumps, for which the corresponding characteristic functions do not have closed‐form expressions. Numerically inverting these characteristic functions can yield accurate probability density functions of stochastic volatility models to serve for various pricing and hedging purposes in quantitative finance. The proposed sequential Itô–Taylor expansion allows us to handle derivatives with medium to long maturities. Numerical experiments illustrate the accuracy and effectiveness of our approach.

A new options pricing method: semi-stochastic kernel regression method with constraints

This paper presents a unified semi-stochastic kernel regression method for pricing options under general stochastic volatility model. The method combines semi-stochastic sampling for initial asset values with Monte Carlo simulations to construct a least-squares based kernel function regression solution. This approach can not only approximates option prices, but also determines the Greeks of option. The least square problem is augmented with weighted derivative constraints, which enables flexible adjustment of approximate errors for both option prices and Greeks. Numerical results show the efficiency of the proposed method for the Vanilla option and some exotic options: Asian option, Lookback option, discretely monitored Barrier option and the Basket option with several assets under the stochastic volatility model.

MARKOVIAN STOCHASTIC VOLATILITY WITH STOCHASTIC CORRELATION — JOINT CALIBRATION AND CONSISTENCY OF SPX/VIX SHORT-MATURITY SMILES

In this paper, we show how to calibrate a general Markovian stochastic volatility model with stochastic correlation to the VIX implied volatility smile and the overall level, slope and curvature of the SPX smile in the [Formula: see text] limit. Explicit formulae are obtained for the asymptotic VIX smile for Heston and SABR-type models with mean reversion, and the Lewis CEV-p-model. We also discuss how the Bass martingale can be used to give an exact fit to a single VIX smile for [Formula: see text]. In the second half of this paper, we derive a more involved integral equation for the correlation function [Formula: see text] to be perfectly consistent with the short-maturity SPX and VIX smiles at all strikes (or all strikes in an interval) as [Formula: see text], and discuss consistency conditions between the wings of the two asymptotic smiles and how to avoid [Formula: see text] for the calibrated [Formula: see text] in practice.

Traders’ heterogeneous beliefs about stock volatility and the implied volatility skew in financial options markets

The paper represents an initial effort to unfold some of the determinants of the implied volatility skew empirically observed in financial (derivative) markets. In particular, in a general stochastic volatility model, we theoretically relate traders’ heterogenous expectations about the underlying stock volatility to the emergence of the implied volatility skew. We also used our model to predict sampled option prices. The analysis provides new characterizations of the behavior of the equilibrium option price as a mixture of Black and Scholes prices, and the associated Black and Scholes implied volatility that hold promise for practical modeling and forecasting.

Optimal contracting with asymmetric belief and complementarity

PurposeIn this study, a dynamic contracting model is developed between a venture capitalist (VC) and an entrepreneur (EN) to explore the influence of asymmetric beliefs regarding output-relevant parameters, agency conflicts and complementarity on the VC's posterior beliefs through the EN's unobservable effort choices to influence the optimal dynamic contract.Design/methodology/approachThe authors construct the contracting model by incorporating the VC's effort, which is ignored in most studies. Using backward induction and a discrete-time approximation approach, the authors solve the continuous-time contract design problem, which evolves into a nonlinear ordinary differential equation (ODE).FindingsThe optimal equity share that the VC provides to the EN decreases over time. In accordance with the empirical evidence, the EN's optimistic beliefs regarding the project's profitability positively affect its equity share. However, the interactions between the optimal equity share, project risk and both partners' degrees of risk aversion are not monotonic. Moreover, the authors find that the optimal equity share increases with the degree of complementarity, which indicates that the EN is willing to cooperate with the VC. This study’s results also show that the optimal equity shares at each time are interdependent if the VC is risk-averse and independent if the VC is risk-neutral.Research limitations/implicationsIn conclusion, the authors highlight two potential directions for future research. First, the authors only considered a single VC, whereas in practice, a risk project may be carried out by multiple VCs, and it is interesting to discuss how the degree of complementarity affects the number of VCs that ENs contract. Second, the authors may introduce jumps and consider more general multivariate stochastic volatility models for output dynamics and analyze the characteristics of the optimal contracts. Third, further research can deal with other forms of discretionary output functions concerning complementarity, such as Cobb–Douglas and constant elasticity of substitution (See Varian, 1992).Social implicationsThe results of this study have several implications. First, it offers a novel approach to designing dynamic contracts that are specific and easy to operate. To improve the complicated venture investment situation and abate conflict between contractual parties, this study plays a good reference role. Second, the synergy effect proposed in this study provides a theoretical explanation for the executive compensation puzzle in economics, in which managers are often “rewarded for luck” (Bertrand and Mullainathan, 2001; Wu et al., 2018). This result indicates a realistic perspective on financing and establishing cooperative relationships, which enhances the efficiency of venture investment. Third, from an empirical standpoint, one can apply this framework to study research and development (R&D) problems.Originality/valueFirst, the authors introduce asymmetric beliefs and Bayesian learning to study the dynamic contract design problem and discuss their effects on equity share. Second, the authors incorporate the VC's effort into the contracting problem, and analyze the synergistic effect of effort complementarity on the optimal dynamic contract.

A class of non-zero-sum stochastic differential games between two mean–variance insurers under stochastic volatility

This paper studies the open-loop equilibrium strategies for a class of non-zero-sum reinsurance–investment stochastic differential games between two insurers with a state-dependent mean expectation in the incomplete market. Both insurers are able to purchase proportional reinsurance contracts and invest their wealth in a risk-free asset and a risky asset whose price is modeled by a general stochastic volatility model. The surplus processes of two insurers are driven by two standard Brownian motions. The objective for each insurer is to find the equilibrium investment and reinsurance strategies to balance the expected return and variance of relative terminal wealth. Incorporating the forward backward stochastic differential equations (FBSDEs), we derive the sufficient conditions and obtain the general solutions of equilibrium controls for two insurers. Furthermore, we apply our theoretical results to two special stochastic volatility models (Hull–White model and Heston model). Numerical examples are also provided to illustrate our results.

The characteristic function of Gaussian stochastic volatility models: an analytic expression

Stochastic volatility models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylised facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample paths. This is made possible by virtue of the flexibility introduced in the choice of the covariance function of the Gaussian process. The price to pay is that in general, such models are no longer Markovian nor semimartingales, which limits their practical use. We derive, in two different ways, an explicit analytic expression for the joint characteristic function of the log-price and its integrated variance in general Gaussian stochastic volatility models. That analytic expression can be approximated by closed-form matrix expressions. This opens the door to fast approximation of the joint density and pricing of derivatives on both the stock and its realised variance by using Fourier inversion techniques. In the context of rough volatility modelling, our results apply to the (rough) fractional Stein–Stein model and provide the first analytic formulas for option pricing known to date, generalising that of Stein–Stein, Schöbel–Zhu and a special case of Heston.

Effective Markovian projection: application to CMS spread options and mid-curve swaptions

Pricing of interest rate derivatives, such as CMS spread or mid-curve options, depends on the modelling of the underlying single rates. For flexibility and realism, these rates are often described in the framework of stochastic volatility models. In this paper we allow rates to be modelled within a class of general stochastic volatility models, which includes the SABR, ZABR, free SABR and Heston models. We provide a versatile technique called Effective Markovian Projection, which allows a tractable model to be found that mimics the distribution of the more complex models used to price multi-rate derivatives. Three different numerical approaches are outlined and applied to relevant examples from practice. Finally, a new method that involves moment-matching of Johnson distributions is applied to facilitate closed-form pricing formulas.

CTMC integral equation method for American options under stochastic local volatility models

In this paper, a continuous-time Markov chain (CTMC) approach is proposed to solve the problem of American option pricing under stochastic local volatility (SLV) models. The early exercise premium (EEP) representation of the value function, which contains the corresponding European option term and the EEP term, is in general not available in closed-form. We propose to use the CTMC to approximate the underlying asset, and derive explicit closed-form expressions for both the European option term and the EEP term, so that the integral equation characterizing the early exercise surface can be explicitly expressed through characteristics of the CTMC. The integral equations are then solved by the iteration method and the early exercise surface can be computed, and semi-explicit expressions for the values and Greeks of American options are derived. We denote the new method as the CTMC integral equation method, and establish both the theoretical convergence and the precise convergence order. Numerical examples are given for the classical Black-Scholes model and the general stochastic (local) volatility models, such as the stochastic alpha beta rho (SABR) model, the Heston model, the 4/2 model and the α−hypergeometric models. They illustrate the high accuracy and efficiency of the method.

Small‐time, large‐time, and asymptotics for the Rough Heston model

AbstractWe characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small‐time, large‐time, and (i.e., ) limits. We show that the short‐maturity smile scales in qualitatively the same way as a general rough stochastic volatility model , and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of and so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log‐moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher‐order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher‐order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large‐maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price tends weakly to a nonsymmetric random variable as (i.e., ) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with , and we show that tends weakly to a nonsymmetric random variable as , which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log‐moneyness as , and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the process converges on pathspace to a random tempered distribution which has the same law as the hyper‐rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).

Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

Rough volatility and CGMY jumps with a finite history and the Rough Heston model – small-time asymptotics in the regime

A small-time Edgeworth expansion is established for near-the-money European options under a general rough stochastic volatility (RSV) model driven by a Riemann-Liouville (RL) process plus an additional generalised tempered stable Lévy process with when (this relaxes the more complicated and restrictive condition which appeared in an earlier version of the article.), in the regime where log-moneyness as for z fixed, conditioned on a finite volatility history. This can be viewed as a more practical variant of Theorem 3.1 in Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] (Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] does not allow for jumps or a finite history and uses the somewhat opaque Muravlev representation for fractional Brownian motion), or if we turn off the rough stochastic volatility, the expansion is a variant of the main result in Mijatovic and Tankov [A new look at short-term implied volatility in asset price models with jumps. Math. Finance, 2016, 26(1), 149–183] and Theorem 3.2 in Figueroa-López et al. [Third-order short-time expansions for close-to-the-money option prices under the CGMY model. J. Appl. Math. Finance, 2017, 24, 547–574]. The regime is directly applicable to FX options where options are typically quoted in terms of delta (.10,.25 and .50) not absolute strikes, and we also compute a new prediction formula for the Riemann-Liouville process, which allows us to express the history term for the Edgeworth expansion in a more useable form in terms of the volatility process itself. We later relax the assumption of bounded volatility, and we also compute a formal small-time expansion for implied volatility in the Rough Heston model in the same regime (without jumps) which includes an additional at-the-money, convexity and fourth order correction term, and we outline how one can go to even higher order in the three separate cases , and . (We thank Masaaki Fukasawa and Hongzhong Zhang for fruitful discussions, Adam Hesse for help with Matlab computations and Vian Dinh for help with IT issues.)

Efficient Simulation of Stochastic Differential Equations Based on Markov Chain Approximations With Applications

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms \textit{both biased and unbiased} simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory. Note: this is an earlier version of the work Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations.

Data cloning estimation for asymmetric stochastic volatility models

The paper proposes the use of data cloning (DC) to the estimation of general univariate asymmetric stochastic volatility (ASV) models with flexible distributions for the standardized returns. These models are able to capture the asymmetric volatility, the leptokurtosis and the skewness of the distribution of returns. Data cloning is a general technique to compute maximum likelihood estimators, along with their asymptotic variances, by means of a Markov chain Monte Carlo (MCMC) methodology. The main aim of this paper is to illustrate how easily general univariate ASV models can be estimated and consequently studied via data cloning. Changes of specifications, priors and sampling error distributions are done with minor modifications of the code. Using an intensive simulation study, the finite sample properties of the estimators of the parameters are evaluated and compared to those of a benchmark estimator that is also user-friendly. The results show that the proposed estimator is computationally efficient, and can be an effective alternative to the existing estimation methods applied to ASV models. Finally, we use data cloning to estimate the parameters of general ASV models and forecast the one-step-ahead volatility of S&P 500 and FTSE-100 daily returns.

Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility

This paper investigates the open-loop equilibrium reinsurance-investment (RI) strategy under general stochastic volatility (SV) models. We resolve difficulties arising from the unbounded volatility process and the non-negativity constraint on the reinsurance strategy. The resolution enables us to derive the existence and uniqueness result for the time-consistent mean variance RI policy under both situations of constant and state-dependent risk aversions. We apply the general framework to popular SV models including the Heston, the 3/2 and the Hull–White models. Closed-form solutions are obtained for the aforementioned models under constant risk aversion, and the non-leveraged models under state-dependent risk aversion.

SMALL-TIME ASYMPTOTICS IN GEOMETRIC ASIAN OPTIONS FOR A STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL

The aim of this paper is to study the small time to maturity of the behavior of the geometric Asian option price and implied volatility under a general stochastic volatility model with Lévy process. The volatility process does not need to be a diffusion or a Markov process, but the future average volatility in the model is a nonadapted process. An anticipating Itô formula for Lévy process and the decomposition of the price (Hull–White formula) are obtained using the Malliavin calculus techniques. The decomposition formula is applied to find the small-time limit of the geometric Asian option price and the implied volatility for the model in at-the-money and out-of-the-money cases.

Pricing Bermudan Variance Swaptions Using Multinomial Trees

In a recent study, Zhao et al. (2017) presented a tree methodology to evaluate the expected generalized realized variance in a general stochastic volatility model; it provided an efficient way of calculating the fair value of the strike for variance swaps. In this article, the authors expand the methodology to price nonlinear derivatives written on realized variance. They introduce a new option contract, a Bermudan variance swaption, defined as an option on variance swap with early exercise dates. Within the same framework they also show how to value forward-start variance swaps, VIX futures, and VIX options. Numerical tests show that the methodology is efficient and accurate.

Affine Stochastic Volatility Models: Supplementary Material

Affine jump diffusion models in general and affine stochastic volatility models in particular are important modeling tools in finance. Their popularity resides in their exibility coupled with their analytical tractability, especially with respect to characteristic functions and polynomial moments. Within a generic affine jump diffusion model and a generic stochastic volatility model, nested in the former as a special case, this paper collects explicit expressions for various characteristic functions and polynomial moments.

Short-Term At-the-Money Asymptotics under Stochastic Volatility Models

A small-time Edgeworth expansion of the density of an asset price is given under a general stochastic volatility model, from which asymptotic expansions of put option prices and at-the-money implied volatilities follow. A limit theorem for at-the-money implied volatility skew and curvature is also given as a corollary. The rough Bergomi model is treated as an example.