Multifactor Stochastic Volatility Research Articles

Efficient 'Black-Scholes' Approximations of the Heston and Multi-Factor Heston Option Prices for a Correct Spot Price

This paper proposes Black-Scholes closed-form approximations for a correct spot price to estimate prices of European call and put options in the Heston and multi-factor Heston stochastic volatility models. These approximations are inspired by the Romano and Touzi (1997) formula and are obtained by a perturbation solution to the Riccati equations which define the probability density function of the log-price process as the volatility of volatility (vol of vol for short) goes to zero. This approach, which can be extended to affine jump models, shows that the accuracy of these Black-Scholes formulas depends on how fast the Airy function for a specific argument depending on vol of vol converges to the Dirac delta function when the vol of vol goes to zero. Simulations and empirical studies show that performance of these formulas is twofold. They outperform the Heston and multi-factor Heston option pricing formula in terms of computational time while satisfactorily approximating the model option prices so they can be used to develop fast and efficient procedures to estimate the model parameters. Finally, this perturbation approach provides an alternative way to approximate the implied volatility for small values of the vol of vol for models in the larger affine class.

Optimal investment under multi-factor stochastic volatility

We consider a model for multivariate intertemporal portfolio choice in complete and incomplete markets with a multi-factor stochastic covariance matrix of asset returns. The optimal investment strategies are derived in closed form. We estimate the model parameters and illustrate the optimal investment based on two stock indices: S&P500 and DAX. It is also shown that the model satisfies several stylized facts well known in the literature. We analyse the welfare losses due to suboptimal investment strategies and we find that investors who invest myopically, ignore derivative assets, model volatility by one factor and ignore stochastic covariance between asset returns can incur significant welfare losses.

On the source of stochastic volatility: Evidence from CAC40 index options during the subprime crisis

Abstract This paper investigates the performance of time-changed Levy processes with distinct sources of return volatility variation for modeling cross-sectional option prices on the CAC40 index during the subprime crisis. Specifically, we propose a multi-factor stochastic volatility model: one factor captures the diffusion component dynamics and two factors capture positive and negative jump variations. In-sample and out-of-sample tests show that our full-fledged model significantly outperforms nested lower-dimensional specifications. We find that all three sources of return volatility variation, with different persistence, are needed to properly account for market pricing dynamics across moneyness, maturity and volatility level. Besides, the model estimation reveals negative risk premium for both diffusive volatility and downward jump intensity whereas a positive risk premium is found to be attributed to upward jump intensity.

A Multifactor Stochastic Volatility Model of Commodity Prices

A Multifactor Stochastic Volatility Model of Commodity Prices

Pricing VIX Options with Multifactor Stochastic Volatility

By exploiting the flexibility of the Wishart process, we propose an application of this framework to the pricing of Chicago Board Options Exchange (CBOE) volatility index (VIX) options. Our methodology is analytically tractable and yet flexible enough to efficiently price CBOE VIX options. In particular, the dynamics for the CBOE VIX is carried out in a linear affine way and the discounted Laplace transform exhibits an exponentially affine property. The tractable model structure lightens the computational burden and facilitates a fast identification of the parameter estimates. We empirically show that modeling the stochastic co-volatility factor can significantly improve the in-sample fitting results due to the improved modeling of higher conditional moments in the underlying transition probability density.

A Hybrid SLV Model with Multifactor Stochastic Volatility

The need for more satisfactory pricing models has brought the attention of researchers and practitioners on Stochastic Local Volatility models. Despite the growing interest on the topic, however, it seems that no particular attention has been paid to the use of multifactor specifications for the stochastic volatility part. The additional flexibility given by this kind of approach in describing the skew dynamics becomes extremely important when we deal with the pricing of forward volatility sensitive contracts like forward start or cliquet options.This paper tries to fill the gap: we introduce the Wishart Stochastic Local Volatility model that describes the stochastic volatility by means of a Wishart process as proposed. The goal of the new framework is twofold: adding a realistic volatility dynamics to Local Volatility models without the limitations of standard 1-factor Stochastic Volatility approaches, and improving European claims pricing performance of pure Wishart Stochastic Volatility model. The latter is quite relevant since we need to introduce stringent parameters restrictions to satisfy existence and uniqueness conditions for the solution of Wishart SDE. These conditions are required when we want to properly simulate the variance process in order to price more exotic derivatives. However such conditions are not usually met when market calibration is performed and so the resulting constrained parameters set is not able to reproduce accurately the market implied volatility surface. The additional Local Volatility component acts then as compensator.We further present an innovative simulation scheme for the asset path in the pure stochastic volatility specification based on the approach. The simulation scheme is also used to tackle the calibration of the so-called leverage function and price derivatives in the new hybrid framework.

Optimal Investment Under Multi-Factor Stochastic Volatility

We consider a model for multivariate intertemporal portfolio choice in complete and incomplete markets with multi-factor stochastic covariance matrix of asset returns. The optimal investment strategies are derived in closed form. We estimate the model parameters and illustrate the optimal investment based on two stock indices: S&P500 and DAX. It is also shown that the model satisfies several stylized facts well known in the literature. We analyse the welfare losses due to suboptimal investment strategies and we find that the investors who invest myopically/ignore derivative assets/model volatility by one factor and ignore stochastic covariance between asset returns can incur significant welfare losses.

A Dual-Curve Short Rate Model with Multi-Factor Stochastic Volatility: I. Asymptotic Analysis

We present a stochastic-volatility, short rate term structure model, which extends the classic multi-factor Hull-White model. This model is designed to fit the swaption implied volatility cube and to incorporate the two-curve modeling paradigm. The model exhibits non-Gaussian forward swap rates whose distributions are parameterized across the dimensions of the volatility cube: underlying tenor, option strike and option expiration. To facilitate rapid model calibration, we establish suitable asymptotic expressions for the bond prices. Furthermore, we derive an effective SABR dynamics for each forward swap rate. Finally, we use the mean field approximation to match the effective SABR parameters corresponding to each swaption to the market levels.

Stochastic volatility models for the Brent oil futures market: forecasting and extracting conditional moments

AbstractThis paper builds and implements a multifactor stochastic volatility model for the latent (and observable) volatility from the front month future contracts at the Intercontinental Commodity Exchange (ICE), London, applying Bayesian Markov chain Monte Carlo simulation methodologies for estimation, inference and model adequacy assessment. Stochastic volatility is the main way time‐varying volatility is modelled in financial markets. An appropriate scientific model description, specifying volatility as having its own stochastic process, broadens the applications into derivative pricing purposes, risk assessment and asset allocation and portfolio management. From an estimated optimal and appropriate stochastic volatility model, the paper reports risk and portfolio measures, extracts conditional one‐step‐ahead moments (smoothing), forecasts one‐step‐ahead conditional volatility (filtering), evaluates shocks from conditional variance functions, analyses multistep‐ahead dynamics and calculates conditional persistence measures. (Exotic) option prices can be calculated using the re‐projected conditional volatility. Observed market prices and implied volatilities establish market risk premiums. The analysis adds insight and enables forecasts to be made, building up the methodology for developing valid scientific commodity market models.

Variance swap with mean reversion, multifactor stochastic volatility and jumps

This paper examines variance swap pricing using a model that integrates three major features of financial assets, namely the mean reversion in asset price, multi-factor stochastic volatility (SV) and simultaneous jumps in prices and volatility factors. Closed-form solutions are derived for vanilla variance swaps and gamma swaps while the solutions for corridor variance swaps and conditional variance swaps are expressed in a one-dimensional Fourier integral. The numerical tests confirm that the derived solution is accurate and efficient. Furthermore, empirical studies have shown that multi-factor SV models better capture the implied volatility surface from option data. The empirical results of this paper also show that the additional volatility factor contributes significantly to the price of variance swaps. Hence, the results favor multi-factor SV models for pricing variance swaps consistent with the implied volatility surface.

Stochastic Skew and Target Volatility Options

AbstractTarget volatility options (TVO) are a new class of derivatives whose payoff depends on some measure of volatility. These options allow investors to take a joint exposure to the evolution of the underlying asset, as well as to its realized volatility. In equity options markets the slope of the skew is largely independent of the volatility level. A single‐factor Heston based volatility model can generate steep skew or flat skew at a given volatility level but cannot generate both for a given parameterization. Since the payoff corresponding to TVO is a function of the joint evolution of the underlying asset and its realized variance, the consideration of stochastic skew is a relevant question for the valuation of TVO. In this sense, this article studies the effect of considering a multifactor stochastic volatility specification in the valuation of the TVO as well as forward‐start TVO. © 2015 Wiley Periodicals, Inc. Jrl Fut Mark 36:174–193, 2016

From the Samuelson Volatility Effect to a Samuelson Correlation Effect: Evidence from Crude Oil Calendar Spread Options

We introduce a multi-factor stochastic volatility model based on the CIR/Heston stochastic volatility process. In order to capture the Samuelson effect displayed by commodity futures contracts, we add expiry-dependent exponential damping factors to their volatility coefficients. The pricing of single underlying European options on futures contracts is straightforward and can incorporate the volatility smile or skew observed in the market. We calculate the joint characteristic function of two futures contracts in the model in analytic form and use the one-dimensional Fourier inversion method of Caldana and Fusai (2013) to price calendar spread options. The model leads to stochastic correlation between the returns of two futures contracts. We illustrate the distribution of this correlation in an example. We then propose analytical expressions to obtain the copula and copula density directly from the joint characteristic function of a pair of futures. These expressions are convenient to analyze the term-structure of dependence between the two futures produced by the model. In an empirical application we calibrate the proposed model to volatility surfaces of vanilla options on WTI. In this application we provide evidence that the model is able to produce the desired stylized facts in terms of volatility and dependence. In a separate appendix, we give guidance for the implementation of the proposed model and the Fourier inversion results by means of one and two-dimensional FFT methods.

Seasonal Stochastic Volatility and Correlation Together with the Samuelson Effect in Commodity Futures Markets

We introduce a multi-factor stochastic volatility model based on the CIR/Heston volatility process that incorporates seasonality and the Samuelson effect. First, we give conditions on the seasonal term under which the corresponding volatility factor is well-defined. These conditions appear to be rather mild. Second, we calculate the joint characteristic function of two futures prices for different maturities in the proposed model. This characteristic function is analytic. Finally, we provide numerical illustrations in terms of implied volatility and correlation produced by the proposed model with five different specifications of the seasonality pattern. The model is found to be able to produce volatility smiles at the same time as a volatility term-structure that exhibits the Samuelson effect with a seasonal component. Correlation, instantaneous or implied from calendar spread option prices via a Gaussian copula, is also found to be seasonal.

It's all about volatility of volatility: Evidence from a two-factor stochastic volatility model

The persistent nature of equity volatility is investigated by means of a multi-factor stochastic volatility model with time varying parameters. The parameters are estimated by means of a sequential matching procedure which adopts as an auxiliary model a time-varying generalization of the HAR model for the realized volatility series. It emerges that during the recent financial crisis the relative weight of the daily component dominates over the monthly term. The estimates of the two factor stochastic volatility model suggest that the change in the dynamic structure of the realized volatility during the financial crisis is due to the increase in the volatility of the persistent volatility term. A set of Monte Carlo simulations highlights the robustness of the methodology adopted in tracking the dynamics of the parameters. • We study the dynamics of the log-realized volatility with a time-varying HAR. • Parameters are estimated with an on-line method. • The parameters of a two-factor stochastic volatility model are recursively estimated. • The vol-of-vol parameter of the slow factor increases during the financial crisis. • The increase of vol-of-vol generates higher persistence and higher variability.

From the Samuelson Volatility Effect to a Samuelson Correlation Effect: Evidence from Crude Oil Calendar Spread Options

We introduce a multi-factor stochastic volatility model with expiry dependent damping factors. This futures-based model is able to capture the Samuelson effect and the volatility smile displayed by commodity futures and option contracts. We calculate the joint characteristic function of two futures contracts in the model in analytic form and use it to price calendar spread options. We then propose analytical expressions to obtain the copula and copula density directly from the joint characteristic function of a pair of futures. In an empirical application we perform a joint calibration to market prices of vanilla and calendar spread options on WTI. We provide evidence that the model is able to produce the desired stylized facts in terms of implied volatility and at the same time to fit market prices of calendar spread options. In particular, we observe a phenomenon we call the Samuelson correlation effect.

Stochastic Volatility Models for the European Electricity Markets: Forecasting and Extracting Conditional Moments for Option Pricing and Implied Market Risk Premiums

This paper builds and implements a multifactor stochastic volatility model for the latent (and observable) volatility from the quarter and year forward contracts at the NASDAQ OMX Commodity Exchanges, applying Bayesian Markov chain Monte Carlo simulation methodologies for estimation, inference, and model adequacy assessment. Stochastic volatility is the main way time-varying volatility is modelled in financial markets. An appropriate scientific model description, specifying volatility as having its own stochastic process, broadens the applications into derivative pricing purposes, risk assessment and asset allocation and portfolio management. From an estimated optimal and appropriate stochastic volatility model, the paper reports risk and portfolio measures, extracts conditional one-step-ahead moments (smoothing), forecast one-step-ahead conditional volatility (filtering), evaluates shocks from conditional variance functions, analyses multi-step-ahead dynamics, and calculates conditional persistence measures. (Exotic) option prices can be calculated using the re-projected conditional volatility. Observed market prices and implied volatilities establish market risk premiums. The analysis adds insight and enables forecasts to be made, building up the methodology for developing valid scientific commodity market models.

Stochastic Skew and Target Volatility Options

Target volatility options (TVO) are a new class of derivatives whose payoff depends on some measure of volatility. These options allow investors to take a joint exposure to the evolution of the underlying asset, as well as to its realized volatility. For instance, a target volatility call can be viewed as a European call whose notional amount depends on the ratio of the target volatility (a fixed quantity representing the investor's expectation of the future realized volatility) and the realized volatility of the underlying asset over the life of the option. In equity options markets the slope of the skew is largely independent of the volatility level. A single-factor Heston based volatility model can generate steep skew or flat skew at a given volatility level but cannot generate both for a given parameterization. Since the payoff corresponding to TVO is a function of the joint evolution of the underlying asset and its realized variance, the consideration of stochastic skew is a relevant question for the valuation of TVO. In this sense, this article studies the effect of considering a multifactor stochastic volatility specification in the valuation of the TVO. To this end, we consider the two-factor Heston-based model of Christoffersen et al. (2009) in order to investigate TVO, as well as forward-start TVO, that is, TVO where the strike is determined at a later date.

Modeling credit spreads under multifactor stochastic volatility

The empirical tests of traditional structural models of credit risk tend to indicate that such models have been unsuccessful in the modeling of credit spreads. To address these negative findings some authors introduce single-factor stochastic volatility specifications and/or jumps.In the yield curve literature it is widely accepted that one-factor is not sufficient to capture the time variation and cross-sectional variation in the term structure. This article introduces a two-factor stochastic volatility specification within the structural model of credit risk. One of the factors determines the correlation between short-term firms’ assets returns and variance, whereas the other factor determines the correlation between long-term returns and variance. The numerical tests reveal how the introduction of two volatility factors can generate a wide range of combinations associated with short-term and long-term patters corresponding to credit spreads. In this sense, multi-factor stochastic volatility specifications provide more flexibility than single-factor models to capture a wide range of shapes associated with the term structure of credit spreads consistent with the empirical evidence.

Efficient Monte Carlo Simulation for Pricing Variance Derivatives under Multi-Factor Stochastic Volatility Models

This paper studied the pricing of variance swap derivatives under the multi-factor stochastic volatility models by Monte Carlo simulation. Control variate technique was well used to reduce the variance of the simulation effectively. How to choose the high efficient control variate was also contained. Then the numerical results show the high efficiency of the speed up method. The pricing structure in the paper is also applicable for the valuation of other types of variance swaps and other financial derivatives under multi-factor models.

A Hybrid Data Cloning Maximum Likelihood Estimator for Stochastic Volatility Models

Abstract: In this article, we analyze a maximum likelihood estimator using Data Cloning for Stochastic Volatility models. This estimator is constructed using a hybrid methodology based on Integrated Nested Laplace Approximations to calculate analytically the auxiliary Bayesian estimators with great accuracy and computational efficiency, without requiring the use of simulation methods such as Markov Chain Monte Carlo. We analyze the performance of this estimator compared to methods based on Monte Carlo simulations (Simulated Maximum Likelihood, MCMC Maximum Likelihood) and approximate maximum likelihood estimators using Laplace Approximations. The results indicate that this data cloning methodology achieves superior results over methods based on MCMC, comparable to results obtained by the Simulated Maximum Likelihood estimator. The methodology is extended to models with leverage effects, continuous time formulations, multifactor and multivariate stochastic volatility.