Stochastic Volatility Models Predictive Relevance for Equity Markets

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.

A smooth estimator for MC/QMC methods in finance

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

We evaluate the properties of mean reversion and mean aversion in asset prices and returns as commonly characterized in the finance literature. The study is undertaken within a class of wellknown dynamic stochastic general equilibrium models and shows that the mean reversion/ aversion distinction is largely artificial. We then propose an alternative measure, the 'Average Crossing Time' that both unifies these concepts and provides an alternative characterization. Ceteris paribus, mean reverting processes have a relatively shorter average crossing time as compared to mean averting processes.

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.

This paper aims to provide a framework for modeling and estimating the implied volatilities of stock prices using stochastic processes. The CIR (Cox-Ingersoll-Ross) model is proposed to capture the mean reversion characteristic as shown in the movement of the daily implied volatilities of the S$\&$P 500 Index and Nikkei 225 Index. The maximum likelihood procedure is applied to estimate the parameters appearing in the model, where both analytical and approximation techniques are sought to handle the difficulties arising from the corresponding optimization problem. The procedure is validated with varying sampling methods by setting different time horizons and observation intervals. Results from numerical experiments show that the stochastic volatility model proposed in this paper functions well in both the US and Japan markets. As one of the applications, our approach is tested to be effective in detecting jumps in likelihood ratios, hence useful for forecasting stock market shocks and crisis.

This article reconsiders the relationship between stock return volatility and trading volume. Based on the multi-factor stochastic volatility model for stock return, we suggest several specifications for the trading volume. This approach enables the unobservable information arrival to follow the ARMA process. We apply the model to the data of Philippine Stock Exchange Composite Index and find that two factors are adequate to describe the movements of stock return volatility and variance of trading volume. We also find that the weights for the factors of the return and volume models are different from each other. The empirical results show (i) a negative correlation between stock return volatility and variance of trading volume, and (ii) a lack of effect of information arrivals on the level of trading volume. These findings are contrary to the results for the equity markets of advanced countries.

Increasing attention has been focused on the analysis of the realized volatility, which can be treated as a proxy for the true volatility. In this paper, we study the potential use of the realized volatility as a proxy in a stochastic volatility model estimation. We estimate the leveraged stochastic volatility model using the realized volatility computed from five popular methods across six sampling-frequency transaction data (from 1-min to 60- min) based on the trust region method. Availability of the realized volatility allows us to estimate the model parameters via the MLE and thus avoids computational challenge in the high dimensional integration. Six stock indices are considered in the empirical investigation. We discover some consistent findings and interesting patterns from the empirical results. In general, the significant leverage effect is consistently detected at each sampling frequency and the volatility persistence becomes weaker at the lower sampling frequency.

A new nonlinear stochastic volatility model with regime switching stochastic mean reversion and its applications to option pricing

This 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.

In this study, we extend the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254] by incorporating a slow varying factor of volatility. The resulting model can be viewed as a multifactor extension of the Heston model with two additional factors driving the volatility levels. An asymptotic analysis consisting of singular and regular perturbation expansions is developed to obtain an approximation to European option prices. We also find explicit expressions for some essential functions that are available only in integral formulas in the work of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254]. This finding basically leads to considerable reduction in computational time for numerical calculation as well as calibration problems. An accuracy result of the asymptotic approximation is also provided. For numerical illustration, the multifactor Heston model is calibrated to index options on the market, and we find that the resulting implied volatility surfaces fit the market data better than those produced by the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254], particularly for long‐maturity call options.

The VIX Future in Bergomi Models: Analytic Expansions and Joint Calibration with S&P 500 Skew

LIBOR Market Model with Stochastic Volatility

We survey the theory and empirical evidence on GARCH option valuation models. Our treatment includes the range of functional forms available for the volatility dynamic, multifactor models, nonnormal shock distributions as well as style of pricing kernels typically used. Various strategies for empirical implementation are laid out and we also discuss the links between GARCH and stochastic volatility models. In the appendix we provide Matlab computer code for option pricing via Monte Carlo simulation for nonaffine models as well as Fourier inversion for affine models.

This paper demonstrates the pricing and hedging efficiency of a three-factor stochastic mean reversion Gaussian model of commodity prices using oil and copper futures and forward contracts. The model is estimated using NYMEX WTI (light sweet crude oil) and LME Copper futures prices and is shown to fit the data well. Furthermore, it shows how to hedge based on a three-factor model and confirms that using three different futures contracts to hedge long-term contracts outperforms the traditional parallel hedge based on a single futures position using time series data and simulation. It also finds that the three-factor model outperforms the two-factor version with respect to the replication of actual term structures and that stochastic mean reversion models outperform constant mean reversion models in Out of Sample hedges.