A financial market with singular drift and no arbitrage
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A financial market with singular drift and no arbitrage Nacira Agram1 · Bernt Øksendal2 Received: 10 February 2020 / Accepted: 30 October 2020 © The Author(s) 2020
Abstract We study a financial market where the risky asset is modelled by a geometric Itô-Lévy 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 and Protter (J Bank Finan 29:2803–2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (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 θ > 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 θ > 0. This implies that there is no arbitrage in the market in that case. However, when θ 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. (Math Finan Econom 10(3):223–262, 2016) and the references therein.
This research was carried out with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20.
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Nacira Agram [email protected] Bernt Øksendal [email protected]
1
Department of Mathematics, Linnaeus University, SE-351 95 Växjö, Sweden
2
Department of Mathematics, University of Oslo, Box 1053 Blindern, NO-0316 Oslo, Norway
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Mathematics and Financial Economics
Keywords Jump diffusion · Financial market with a local time drift term · Arbitrage · Optimal portfolio · Delayed information · Donsker delta function · White noise calculus Mathematics Subject Classification 60H05 · 60H40 · 93E20 · 91G80 · 91B70
1 Introduction It is well-known that in the classical Black-Scholes market, there is no arbitrage. However, if we include a singular term in the drift of the risky asset, it wa
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