A financial market with singular drift and no arbitrage

TL;DR

This paper models a jump diffusion financial market with a singular drift influenced by delays, demonstrating that market arbitrage disappears when delays are present, but emerges as delays vanish, with implications for high-frequency trading.

Contribution

It introduces a jump diffusion market model with a singular drift and delay, analyzing arbitrage conditions and explicit optimal strategies using white noise calculus.

Findings

Maximal value of the model is finite with delay, indicating no arbitrage.

As delay approaches zero, the value becomes infinite, indicating arbitrage.

The model is relevant for understanding high-frequency trading dynamics.

Abstract

We study a financial market where the risky asset is modelled by a geometric It\^o-L\'{e}vy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow & Protter for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas & Shreve (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay \theta>0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as the delay \theta> 0. This implies that there is no arbitrage in the market in that case. However, when \theta goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al and the references therein.