New paper out!

Post date: Sep 22, 2015 7:34:21 AM

Just notifying the new paper with Toufik Mansour, Lorenzo Sindoni, and Simone Severini:

On moments of the integrated exponential Brownian motion

We present new exact expressions for a class of moments for the geometric Brownian motion, in terms of determinants, obtained using a recurrence relation and combinatorial arguments for the case of a Ito’s Wiener process. We then apply the obtained exact formulas to computing averages of the solution of the logistic stochastic differential equation via a series expansion, and compare the results to the solution obtained via Monte Carlo.

http://arxiv.org/pdf/1509.05980.pdf

a prelude of the one with Lorenzo Sindoni, Fabio Caccioli and Cozmin Ududec:

Optimal dynamical leverage with market impact

When dealing with financial instruments which are modelled by some geometric Brownian motion it is important to consider time-average growth rates (or use something like a log-utility function, or the closely related Kelly criterion) when evaluating decisions, rather than simple expectations or ensemble averages at a single time.

In the case of the Ornstein-Ulhenbeck process this is simplified by the independence of the expectation value at some time from the past history of the process. This feature is lost in the presence of a time-dependent drift term, and thus a more careful analysis is required.

In this paper we study the optimal leverage in the case of a specific functional form which takes into account the effect of market impact. This is closer to realistic processes involved in financial markets, in particular when one considers trades on the order of the market size.

We show analytically that when market impact is included, one has to appropriately increase risk exposure by including a term which which describes how far the investor is from the asymptotic classic equilibrium. We further test numerically the validity of our analytical result.

(upcoming)