Multifactor Stochastic Volatility Model Research Articles

Parametric risk-neutral density estimation via finite lognormal-Weibull mixtures

This paper proposes a new parametric risk-neutral density (RND) estimator based on a finite lognormal-Weibull mixture (LWM) density. We establish the consistency and asymptotic normality of the LWM method in a general misspecified parametric framework. Based on the theoretical results, we propose a sequential test procedure to evaluate the goodness-of-fit of the LWM model, which leads to an adaptive choice for the number and type of mixture components. Our simulation results show that, in finite samples with various observation error specifications, the LWM method can approximate complex RNDs generated by state-of-the-art multi-factor stochastic volatility models with a few (typically less than 4) mixtures. Application of the LWM model on index options confirms its reliability in recovering empirical RNDs with a heavy left tail or bimodality, which can be incorrectly identified as bimodality or a heavy left tail by existing (semi)-nonparametric methods if the goodness-of-fit to the observed data is ignored.

Optimal reinsurance-investment with loss aversion under rough Heston model

The paper investigates optimal reinsurance-investment strategies with the assumption that the insurers can purchase proportional reinsurance contracts and invest their wealth in a financial market consisting of one risk-free asset and one risky asset whose price process obeys the rough Heston model. The problem is formulated as a utility maximization problem with a minimum guarantee under an S-shaped utility. Since the rough Heston model is non-Markovian and non-semimartingale, the utility maximization problem cannot be solved by the classical dynamical programming principle and related approaches. This paper uses semi-martingale approximation techniques to approximate the utility maximization problem and proves the rates of convergence for the optimal strategies. The approximate problem is a kind of classical stochastic control problem under multi-factor stochastic volatility models. As the approximate control problem still cannot be solved analytically, a dual-control Monte-Carlo method is developed to solve it. Numerical examples and implementations are provided.

Projecting and Forecasting the Latent Volatility for the Nasdaq OMX Nordic/Baltic Financial Electricity Market Applying Stochastic Volatility Market Characteristics

In this empirical study, multifactor stochastic volatility models for the financial Nordic/Baltic power markets are developed, implemented, and analyzed. Stochastic volatility projections are the primary aim, followed by volatility forecasts and market repercussions. The research provides a functional variant of the conditional distribution (f(x|y)) based on conditional moments and a long-simulated state vector realization (MCMC-GMM) that is evaluated on observed data (a non-linear Kalman Filter) and applicable for step-forward volatility forecasts. For front year and quarter financial electricity contracts, the SV model creates two mean-reverting factors: one persistent and slowly moving component and one choppy, rapidly moving component. According to these factors, static volatility predictions with optimum and generous lags have a Theil covariance percentage of well over 97 percent for the front year contracts and 86 percent for the front quarter contracts. The volatility visibility and its associated static forecasts improve market transparency and will eventually make diversification and risk management easier to implement.

International portfolio choice under multi-factor stochastic volatility

In this article, we develop an identifiable multi-factor stochastic volatility model for international portfolio choice problems in complete and incomplete markets. Allowing for stochastic covariance between financial asset returns and foreign exchange rates, optimal investment strategies are derived in closed form and welfare losses arising from suboptimal investment strategies are analysed. Moreover, we provide a two-step procedure for estimating as well as calibrating the model parameters and use this ansatz to illustrate optimal investment decisions for the S&P 500, the German blue chip index DAX, and the USD/EUR foreign exchange rate. We find, both theoretically and empirically, that the model satisfies various well-known stylized facts of equity and foreign exchange rate markets and that investors who invest myopically or ignore derivative assets can incur substantial welfare losses implying strong evidence for significant welfare benefits from international diversification across different asset classes.

Pricing of the geometric Asian options under a multifactor stochastic volatility model

This paper focuses on the pricing of continuous geometric Asian options (GAOs) under a new multifactor stochastic volatility model. The model considers fast and slow mean reverting factors of volatility, where slow volatility factor is approximated by a quadratic arc. The asymptotic expansion of the price function is assumed, and the first order price approximation is derived using the perturbation techniques for both floating and fixed strike GAOs. Much simplified pricing formulae for the GAOs are obtained in this multifactor stochastic volatility framework. The zeroth order term in the price approximation is the modified Black–Scholes price for the GAOs. This modified price is expressed in terms of the Black–Scholes price for the GAOs. The accuracy of the approximate option pricing formulae is established, and also verified numerically by comparing the model prices with the Monte Carlo simulation prices and the Black–Scholes prices for the GAOs. The model parameter is estimated by capturing the volatility smiles. The sensitivity analysis is also performed to investigate the effect of underlying parameters on the approximated prices.

A Closed Form Solution for Pricing Variance Swaps Under the Rescaled Double Heston Model.

As is well known, multi-factor stochastic volatility models are necessary to capture the market accurately in pricing financial derivatives. However, the multi-factor models usually require too many parameters to be calibrated efficiently and they do not lead to an analytic pricing formula. The double Heston model is one of them. The approach of this paper for this difficulty is to rescale the double Heston model to reduce the number of the model parameters and obtain a closed form analytic solution formula for variance swaps explicitly. We show that the rescaled double Heston model is as effective as the original double Heston model in terms of fitting to the VIX market data in a stable condition and yet the computing time is much less than that under the double Heston model. However, in a turbulent situation after the start of the COVID-19 pandemic in 2020, we acknowledge that even the double Heston model fails to capture the market accurately.

Forecasting Stochastic Volatility Characteristics for the Financial Fossil Oil Market Densities

This paper builds and implements multifactor stochastic volatility models for the international oil/energy markets (Brent oil and WTI oil) for the period 2011–2021. The main objective is to make step ahead volatility predictions for the front month contracts followed by an implication discussion for the market (differences) and observed data dependence important for market participants, implying predictability. The paper estimates multifactor stochastic volatility models for both contracts giving access to a long-simulated realization of the state vector with associated contract movements. The realization establishes a functional form of the conditional distributions, which are evaluated on observed data giving the conditional mean function for the volatility factors at the data points (nonlinear Kalman filter). For both Brent and WTI oil contracts, the first factor is a slow-moving persistent factor while the second factor is a fast-moving immediate mean reverting factor. The negative correlation between the mean and volatility suggests higher volatilities from negative price movements. The results indicate that holding volatility as an asset of its own is insurance against market crashes as well as being an excellent diversification instrument. Furthermore, the volatility data dependence is strong, indicating predictability. Hence, using the Kalman filter from a realization of an optimal multifactor SV model visualizes the latent step ahead volatility paths, and the data dependence gives access to accurate static forecasts. The results extend market transparency and make it easier to implement risk management including derivative trading (including swaps).

Calibration to FX triangles of the 4/2 model under the benchmark approach

We calibrate a novel multifactor stochastic volatility model that includes as special cases the Heston-based model of De Col et al. (J Bank Finance 37(10):3799–3818, 2013) and the 3/2-based model of Baldeaux et al. (J Bank Finance 53:34–48, 2015). Using a dataset on vanilla option quotes in a triangle of currencies, we find that the risk neutral approach typically fails for the calibrated model, in line with the results of Baldeaux et al. (2015).

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.

Variance swaps with double exponential Ornstein-Uhlenbeck stochastic volatility

Recent empirical studies find that with a high sampling frequency, stochastic volatility models can be differentiated from each other and the log-normal stochastic volatility model can capture the volatility of real market better than the Heston model. Unless certain restriction such as fast mean reversion or zero mean reversion is assumed, however, the log-normal stochastic volatility model does not allow an analytic solution for the characteristic function. When we consider derivative products such as variance swaps, the restriction may cause a significant gap between theoretical price and actual price. On the other hand, there are a good amount of studies showing that a multi-factor stochastic volatility model can capture the market more accurately than a single-factor stochastic volatility model. So, in this paper, we propose a two-factor log normal stochastic volatility model in which log volatility is given by an Ornstein-Uhlenbeck process which is a mean-reverting, temporally homogeneous, Markov Gaussian process. We derive an exact semi-analytic solution as well as an approximate analytic solution for the fair strike price of discretely sampled variance swap by using the first and second moments of the moment generating function. We check the validity of our solution through Monte Carlo simulation and show how to estimate the model parameters by implementing calibration to real market data and compare our two-factor model with the one-factor version and also the Heston and double Heston models. Furthermore, the sample discretization risk is examined for the four different models.

Quantization meets Fourier: a new technology for pricing options

In this paper we introduce a novel pricing methodology for a broad class of models for which the characteristic function of the log-asset price can be efficiently computed. The method is based on a new quantization procedure, crucially exploiting for the first time the Fourier transform of the asset process, which fully characterizes the distribution of the log-asset. As opposed to previous quantizations based on Euler (or more sophisticated) discretization schemes, our method reveals to be fast and accurate, to the point that it is possible to calibrate the models on real data. Moreover, our approach allows to price options in multi factor stochastic volatility models including jumps. As a motivating example, we calibrate a Tempered Stable model on market data. This represents the first application of quantization to a pure jump process.

Quadratic approximation of the slow factor of volatility in a multifactor stochastic volatility model

A new multifactor stochastic volatility model is proposed in which the slow volatility factor is approximated by a quadratic arc. The perturbation technique is used to obtain the approximate expression for the European option price. The notion of a modified Black‐Scholes price is introduced. A simplified expression for the European option price, perturbed around the modified Black‐Scholes price, is obtained. An expression of modified price is also obtained in terms of the Black‐Scholes price. The effect of this modification on pricing is explained, the accuracy of the approximate option pricing formula established, and its computational cost discussed.

Monte Carlo calibration to implied volatility surface under volatility models

The calibration problem of implied volatility surface under complex financial models can be formulated as a nonlinear high-dimensional optimization problem. To resolve this problem for genuine volatility models, we develop a sequential methodology termed two-stage Monte Carlo calibration method. It consists of the first stage-dimension separation for splitting parametric set into two subsets, and the second stage-standard error reduction for efficient evaluation of option prices. The first stage dimension separation aims to reduce dimensionality of the optimization problem by estimating some volatility model parameters a priori under the historical probability measure such that the total number of model parameters under an option pricing measure is significantly reduced. The second stage standard error reduction aims simultaneously to reduce variance of option payoffs by the martingale control variate algorithm, and to increase the total number of Monte Carlo simulation by the hardware graphics processing unit (GPU) for parallel computing. This two-stage Monte Carlo calibration method is capable of solving a variety of complex volatility models, including hybrid models and multifactor stochastic volatility models. Essentially, it provides a general framework to analyze backward information from the historical spot prices and the forward information from option prices.

A multifactor stochastic volatility model of commodity prices

We propose a novel representation of commodity spot prices in which the cost-of-carry and the spot price volatility are both driven by an arbitrary number of risk factors, nesting many existing specifications. The model exhibits unspanned stochastic volatility, provides simple closed-form expressions of commodity futures, and yields analytic formulas of European options on futures. We estimate the model using oil futures and options data, and find that the pricing of traded contracts is accurate for a wide range of maturities and strike prices. The results suggest that at least three risk factors in the spot price volatility are needed to accurately fit the volatility surface of options on oil futures, highlighting the importance of using general multifactor models in pricing commodity contingent claims.

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.

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

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.

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.