An analytically tractable multi-asset stochastic volatility model

Abstract

A multi-asset stochastic volatility model is presented. This is based on the Heston stochastic volatility model. It has the advantage of being sufficiently flexible to interpret the individual asset dynamics as well as the interdependencies among them, while still being analytically tractable. A formula for the joint transition probability density function of the stochastic process implicitly defined by the model is deduced. This formula is expressed as a one dimensional integral of an explicitly known function and allows us to derive elementary formulae for the conditional mean and variance of the asset price log-returns. A calibration procedure is proposed and tested on simulated data.

Asset allocation models in a generalized Heston framework using a gradient like vector optimization algorithm

“…. , n, t > t are the expected value and the variance of the variable x i,t conditioned to the observation made at t (see [20] for further details):…”

“…The main result is the formulation of two long-term portfolio optimization problems and their solution through the use of new computational methods. These optimization problems are formulated assuming that the returns of the risky assets in the portfolio are modeled with a generalization of the Heston stochastic volatility model (GH model for short) [20].…”

“…The GH model is a stochastic volatility model that describes the behavior of n assets having stochastic variances prescribed by a mean reverting process. This model is obtained by the Heston stochastic volatility model adding to the stochastic differential equation satisfied by each asset price an extra term (see [20] for further details). A semi-explicit formula for the transition probability density function associated with the stochastic process is derived in [20] and this formula is used to calibrate the GH model parameters with the maximum likelihood approach.…”

“…The paper is organized as follows. In Section 2 we derive the stochastic process that prescribes the dynamic of the portfolio when the asset prices are described by the generalization of the Heston model introduced in [20]. In Section 3 we derive a formula for the probability density function associated with the stochastic variable that describes the portfolio dynamics.…”

“…In fact, it is easy to see that the function p f is only a function of τ = t − t instead of being a function of t and t separately. Using the fact that τ = t − t and (28), equation (20) can be rewritten as an equation for p b , that is:…”