José Da Fonseca

A multifactor volatility Heston model

We model the volatility of a single risky asset using a multifactor (matrix) Wishart affine process, recently introduced in finance by Gourieroux and Sufana. As in standard Duffie and Kan affine models the pricing problem can be solved through the Fast Fourier Transform of Carr and Madan. A numerical illustration shows that this specification provides a separate fit of the long-term and short-term implied volatility surface and, differently from previous diffusive stochastic volatility models, it is possible to identify a specific factor accounting for the stochastic leverage effect, a well-known stylized fact of the FX option markets analysed by Carr and Wu.

Riding on the Smiles

Using a data set of vanilla options on the major indexes we investigate the calibration properties of several multifactor stochastic volatility models by adopting the Fast Fourier Transform as the pricing methodology. We study the impact of the penalizing function on the calibration performance and how it affects the calibrated parameters.We consider single asset as well as multiple-asset models, with particular attention to the single asset Wishart Multidimensional Stochastic Volatility model introduced in Da Fonseca et al. (2008b) and the Wishart Affine Stochastic Correlation model proposed by Da Fonseca et al. (2007b), which provides a natural framework for pricing basket options while keeping the stylized smile-skew effects on single name vanillas.For all models we give some option price approximations that are very useful to speed up the pricing process. What is more, these approximations allow us to compare different models by aggregating conveniently the parameters and they highlight the ability of the Wishart-based models in controlling separately the smile and the skew effects. This is extremely important in a risk management perspective of a book of derivatives that includes exotic as well as basket options.

Estimating the Wishart Affine Stochastic Correlation Model Using the Empirical Characteristic Function

This paper provides the first estimation strategy for the Wishart Affine Stochastic Correlation (WASC) model. We provide elements showing that the use of empirical characteristic function-based estimates is advisable as this function is exponential affine in the WASC case. We use a GMM estimation strategy with a continuum of moment conditions based on the characteristic function. We present the estimation results obtained using a dataset of equity indexes. The WASC model captures most of the known stylized facts associated with financial markets, including leverage and asymmetric correlation effects.

Hawkes Process: Fast Calibration, Application to Trade Clustering and Diffusive Limit

This study provides explicit formulas for the moments and the autocorrelation function of the number of jumps over a given interval for a self‐excited Hawkes process. These computations are possible thanks to the affine property of this process. Using these quantities an implementation of the method of moments for parameter estimation that leads to an fast optimization algorithm is developed. The estimation strategy is applied to trade arrival times for major stocks that show a clustering behavior, a feature the Hawkes process can effectively handle. As the calibration is fast, the estimation is rolled to determine the stability of the estimated parameters. Lastly, the analytical results enable the computation of the diffusive limit in a simple model for the price evolution based on the Hawkes process. It determines the connection between the parameters driving the high‐frequency activity to the daily volatility.

Deformation of implied volatility surfaces: an empirical analysis*

The evolution of market prices of options is often represented in terms of the implied volatility surface, which randomly fluctuates through time. Using time series of transaction prices for SP500 index options, we study the dynamics of the implied volatility surface and illustrate how its deformation through time may be captured by a small number of orthogonal factors. These factors are identified and their dynamics is shown to be well approximated by uncorrelated Ornstein Uh-lenbeck processes. Our analysis is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data on SP500 options. Our results provide an empirical basis for factor models of the implied volatility surface.

Pricing Range Notes within Wishart Affine Models

We provide analytic pricing formulas for Fixed and Floating Range Accrual Notes within the multifactor Wishart affine framework which extends significantly the standard affine model. Using estimates for three short rate models, two of which are based on the Wishart process whilst the third one belongs to the standard affine framework, we price these structured products using the FFT methodology. Thanks to the Wishart tractability the hedge ratios are also easily computed. As the models are estimated on the same dataset, our results illustrate how the fit discrepancies (meaning differences in the likelihood functions) between models translate in terms of derivatives pricing errors, and we show that the models can produce different price evolutions for the Range Accrual Notes. The differences can be substantial and underline the importance of model risk both from a static and a dynamic perspective. These results are confirmed by an analysis performed at the hedge ratios level.

Analytic pricing of volatility-equity options within Wishart-based stochastic volatility models

This paper provides the pricing for a new class of derivatives with different affine stochastic volatility models. These products are characterized by payoffs depending on both stock and its volatility. We provide closed-form solutions for different products and two multivariate Wishart-based stochastic volatility models. The methodology is independent of the dimension of the problem. Overall, our results highlight the great flexibility and tractability of Wishart-based stochastic volatility models to develop multivariate extensions of the Heston model.

Correlation and Lead-Lag Relationships in a Hawkes Microstructure Model

The aim of this paper is to develop a multi-asset model based on the Hawkes process describing the evolution of assets at high frequency and to study the lead-lag relationship as well as the correlation between the stocks within this framework. Thanks to its strong analytical tractability several statistical quantities are explicitly computed and give some insight on the impact of the model parameters on these quantities. Furthermore, we compute the covariance matrix associated with the diffusive limit of the model so that the relation between the parameters driving the asset at high and low frequencies is explicit. We illustrate our results using index futures and stocks quoted in the Eurex market. The model can capture the existing lead-lag relationship between the assets.

A joint analysis of market indexes in credit default swap, volatility and stock markets

ABSTRACT This paper analyses the joint dynamics of the CDS, volatility and stock markets using both VAR and Markov regime-switching VAR models with market index data. It shows that the joint behaviour of the three markets is better characterized by the Markov model with two regimes corresponding to low- and high-volatile market conditions. The relationship between changes in the market indexes under a regime is consistent with theory and persistent; the information transmission process of shocks to the markets is similar for the two regimes with a more important role for CDS shock; and the volatility in the money market is an important determinant of regime-switching. The findings have practical implications, particularly for hedging strategies with market indexes under different market conditions.

On Moment Non-Explosions for Wishart-Based Stochastic Volatility Models

This paper provides a result on moment non-explosions for a stock following a Wishart multidimensional stochastic volatility dynamics or a Wishart affine stochastic correlation dynamics when the parameter values satisfy certain constraints. By reformulating the stock dynamics in terms of the volatility path along with standard results on matrix Lyapunov and Riccati equations, a non-explosion result of the moment of order greater than one can be obtained. It extends to these frameworks a property well known for the Heston model.