Marina Marena

Pricing Discretely Monitored Asian Options by Maturity Randomization

We present a new methodology based on maturity randomization to price discretely monitored arithmetic Asian options when the underlying asset evolves according to a generic Levy process. Our randomization technique considers the option expiry to be a random variable distributed according to a geometric distribution of a parameter independent of the underlying process. This allows one to transform the pricing backward procedure into a set of independent integral equations. Numerical procedures for a fast and accurate solution of the pricing problem are provided.

Z-Transform and preconditioning techniques for option pricing

In the present paper, we convert the usual n-step backward recursion that arises in option pricing into a set of independent integral equations by using a z-transform approach. In order to solve these equations, we consider different quadrature procedures that transform the integral equation into a linear system that we solve by iterative algorithms and we study the benefits of suitable preconditioning techniques. We show the relevance of our procedure in pricing options (such as plain vanilla, lookback, single and double barrier options) when the underlying evolves according to an exponential Lévy process.

Portfolio Value at Risk Bounds

This paper develops value at risk (VAR) measures for portfolios of correlated financial assets, without assuming normal returns. The approach can cope with any distribution for marginal returns, the fat–tailed ones included. We provide VAR bounds which hold independently of the joint distribution of returns and their dependence structure. The lower bound can be interpreted as a worst–case scenario VAR. We show that it not only requires little information, but is also easy to compute. In this sense, we suggest it as a practical device for portfolio managers. An application to portfolios of stock–market indices is provided. Comparisons with the VAR values under the normality assumption on returns are discussed.

Option pricing, maturity randomization and distributed computing

We price discretely monitored options when the underlying evolves according to different exponential Levy processes. By geometric randomization of the option maturity, we transform the n-steps backward recursion that arises in option pricing into an integral equation. The option price is then obtained solving n independent integral equations by a suitable quadrature method. Since the integral equations are mutually independent, we can exploit the potentiality of a grid computing architecture. The primary performance disadvantage of grids is the slow communication speeds between nodes. However, our algorithm is well-suited for grid computing since the integral equations can be solved in parallel, without the need to communicate intermediate results between processors. Moreover, numerical experiments on a cluster architecture show the good scalability properties of our algorithm.

Dependence calibration and portfolio fit with factor-based subordinators

The paper explores the properties of a class of multivariate Lévy processes used for asset returns. We focus on describing both linear and non-linear dependence in an economic sensible and empirically appropriate way. The processes are subordinated Brownian motions. The subordinator has a common and an idiosyncratic component, to reflect the properties of trade, which it represents. A calibration to a portfolio of 10 US stock indices returns over the period 2009–2013 shows that the hyperbolic specification has a very good fit to marginal distributions, to the overall correlation matrix and to the return distribution of both long-only and long-short random portfolios, which also incorporate non-linear dependence. Their tail behaviour is also well captured by the variance gamma specification. The main message is not only the goodness of fit, but also the flexibility in capturing dependence and the ease of calibration on large sets of returns.

Dependence Calibration and Portfolio Fit with FactorBased Time Changes

The paper explores the fit properties of a class of multivariate Levy processes, which are characterized as time-changed correlated Brownian motions. The time-change has a common and an idiosyncratic component, to re ect the properties of trade, which it represents. The resulting process may provide Variance-Gamma, Normal-Inverse- Gaussian or Generalized-Hyperbolic margins. A non-pairwise calibration to a portfolio of ten US daily stock returns over the period 2009-2013 shows that fit of the Hyperbolic specification is very good, in terms of marginal distributions and overall correlation matrix. It succeeds in explaining the return distribution of both long-only and long- short random portfolios better than competing models do. Their tail behavior is well captured also by the Variance-Gamma specification.

Neighborhood turnpike theorem for continuous-time optimization models

A neighborhood turnpike theorem is proved for continuous-time, infinite-horizon optimization models with positive discounting. Our approach is a primal one and no differentiability assumption is made. The basic hypothesis is a condition of uniform concavity at the point defining the undiscounted steady state. The main novelty here is that we formulate the theorem by taking the undiscounted steady state as the turnpike.

Asian Options with Jumps

We derive a closed-form formula for the fair value of call and put options written on the arithmetic average of security prices driven by jump diffusion processes displaying (possibly periodical) trend, time varying volatility, and mean reversion. The model allows one for jointly fitting quoted futures curve and the time structure of spot price volatility. Our result extends the no-jump case put forward in [Fusai, G., Marena, M., Roncoroni, A. 2008. Analytical Pricing of Discretely Monitored Asian-Style Options: Theory and Application to Commodity Markets. Journal of Banking and Finance 32 (10), 2033-2045]. A few tests based on commodity price data assess the importance of introducing a jump component on the resulting option prices.

Option pricing, maturity randomization and grid computing

By geometric randomization of the option maturity, we transform the n-steps backward recursion that arises in option pricing into an integral equation. The option price is then obtained solving n independent integral equations. This is accomplished by a quadrature procedure that transforms each integral equation in a linear system. Since the solution of each linear system is independent one of the other, we can exploit the potentiality of the grid architecture AGA1. We compare different quadrature methods of the integral equation with Monte Carlo simulation. Therefore we price options (such as plain vanilla, single and double barrier options) when the underlying evolves according to different exponential Levy processes.

MULTIVARIATE FACTOR-BASED PROCESSES WITH SATO MARGINS

We introduce a class of multivariate factor-based processes with the dependence structure of Levy ρα-models and Sato marginal distributions. We focus on variance gamma and normal inverse Gaussian marginal specifications for their analytical tractability and fit properties. We explore if Sato models, whose margins incorporate more realistic moments term structures, preserve the correlation flexibility in fitting option data. Since ρα-models incorporate nonlinear dependence, we also investigate the impact of Sato margins on nonlinear dependence and its evolution over time. Further, the relevance of nonlinear dependence in multivariate derivative pricing is examined.