arXiv subjects

Arne Lokka

Publications and source records attributed to Arne Lokka.

3 recordsLinked to original sources

arXiv2020

Optimal liquidation for a risk averse investor in a one-sided limit order book driven by a Levy process

In a one-sided limit order book, satisfying some realistic assumptions, where the unaffected price process follows a Levy process, we consider a market agent that wants to liquidate a large position of shares. We assume that the agent has constant absolute risk aversion and aims at maximising the expected utility of the cash position at the end of time. The agent is then faced with the problem of balancing the market risk and the cost of a rapid execution. In particular we are interested in how the agent should go about optimally submitting orders. Since liquidation normally takes place within a short period of time, modelling the risk as a Levy process should provide a realistic model with good statistical fit to observed market data, and thus the model should provide a realistic reflection of the agent's market risk. We reduce the optimisation problem to a deterministic two-dimensional singular problem, to which we are able to derive an explicit solution in terms of the model data. In particular we find an expression for the optimal intervention boundary, which completely characterise the optimal liquidation strategy.

q-fin.TR↗

arXiv2020

Optimal liquidation trajectories for the Almgren-Chriss model with Levy processes

We consider an optimal liquidation problem with infinite horizon in the Almgren-Chriss framework, where the unaffected asset price follows a Levy process. The temporary price impact is described by a general function which satisfies some reasonable conditions. We consider an investor with constant absolute risk aversion, who wants to maximise the expected utility of the cash received from the sale of his assets, and show that this problem can be reduced to a deterministic optimisation problem which we are able to solve explicitly. In order to compare our results with exponential Levy models, which provides a very good statistical fit with observed asset price data for short time horizons, we derive the (linear) Levy process approximation of such models. In particular we derive expressions for the Levy process approximation of the exponential Variance-Gamma Levy process, and study properties of the corresponding optimal liquidation strategy. We then provide a comparison of the liquidation trajectories for reasonable parameters between the Levy process model and the classical Almgren-Chriss model. In particular, we obtain an explicit expression for the connection between the temporary impact function for the Levy model and the temporary impact function for the Brownian motion model (the classical Almgren-Chriss model), for which the optimal liquidation trajectories for the two models coincide.

q-fin.TR↗

arXiv2004

Stochastic partial differential equations driven by Levy space-time white noise

In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Levy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Levy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d\leq 3, then this solution can be represented as a classical random field in L2(μ), where μis the probability law of the Levy process. The starting point of our theory is a chaos expansion in terms of generalized Charlier polynomials. Based on this expansion we define Kondratiev spaces and the Levy Hermite transform.

math.PR↗