Martino Grasselli

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.

Optimal Investment Strategies in the presence of a Minimum Guarantee.

In a continuous-time framework, we consider the problem of a De…ned Contribution Pension Fund in the presence of a minimum guarantee. The problem of the fund manager is to invest the initial wealth and the (stochastic) contribution ‡ow into the …nancial market, in order to maximize the expected util- ity function of the terminal wealth under the constraint that the terminal wealth must exceed the minimum guarantee. We assume that the stochastic interest rates follow the a¢ne dynamics, including the CIR (Cox, Ingersoll and Ross 1985) model and the Vasiµ model. The optimal investment strategies are obtained by assum- ing the completeness of …nancial markets and a CRRA utility function. Explicit formulae for the optimal investment strategies are included for dierent examples of guarantees and contributions.

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.

Solvable Affine Term Structure Models

An Affine Term Structure Model (ATSM) is said to be solvable if the pricing problem has an explicit solution, i.e., the corresponding Riccati ordinary differential equations have a regular globally integrable flow. We identify the parametric restrictions which are necessary and sufficient for an ATSM with continuous paths, to be solvable in a state space , where , the domain of positive factors, has the geometry of a symmetric cone. This class of state spaces includes as special cases those introduced by Duffie and Kan (1996), and Wishart term structure processes discussed by Gourieroux and Sufana (2003). For all solvable models we provide the procedure to find the explicit solution of the Riccati ODE.

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.

The Explicit Laplace Transform for the Wishart Process

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODEs and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.

General Closed-Form Basket Option Pricing Bounds

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms; hence, they do not suffer from the curse of dimensionality and can be applied also to high-dimensional problems where most existing methods fail. In particular, we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate.

THE 4/2 STOCHASTIC VOLATILITY MODEL: A UNIFIED APPROACH FOR THE HESTON AND THE 3/2 MODEL

We introduce a new stochastic volatility model that includes, as special instances, the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). Our model exhibits important features: first, instantaneous volatility can be uniformly bounded away from zero, and second, our model is mathematically and computationally tractable, thereby enabling an efficient pricing procedure. This called for using the Lie symmetries theory for partial differential equations; doing so allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation scheme for the model, which is useful for numerical applications.

An Affine Multi-Currency Model with Stochastic Volatility and Stochastic Interest Rates

We introduce a tractable multi-currency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the FX market and the evolution of yield curves. The pricing of vanilla options on FX rates can be performed effciently through the FFT methodology thanks to the affinity of the model Our framework is also able to describe many non trivial links between FX rates and interest rates: a second calibration exercise highlights the ability of the model to fit simultaneously FX implied volatilities while being coherent with interest rate products.

Pricing Currency Derivatives Under the Benchmark Approach

This paper considers the realistic modelling of derivative contracts on exchange rates. We propose a stochastic volatility model that recovers not only the typically observed implied volatility smiles and skews for short dated vanilla foreign exchange options but allows one also to price payoffs in foreign currencies, lower than possible under classical risk neutral pricing, in particular, for long dated derivatives. The main reason for this important feature is the strict supermartingale property of benchmarked savings accounts under the real world probability measure, which the calibrated parameters identify under the proposed model. Using a real dataset on vanilla option quotes, we calibrate our model on a triangle of currencies and find that the risk neutral approach fails for the calibrated model, while the benchmark approach still works.