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Topics in stochastic control with applications to algorithmic trading

Bates, Tom (2016) Topics in stochastic control with applications to algorithmic trading. PhD thesis, The London School of Economics and Political Science (LSE).

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Abstract

This thesis considers three topics in stochastic control theory. Each of these topics is motivated by an application in finance. In each of the stochastic control problems formulated, the optimal strategy is characterised using dynamic programming. Closed form solutions are derived in a number of special cases. The first topic is about the market making problem in which a market maker manages his risk from inventory holdings of a certain asset. The magnitude of this inventory is stochastic with changes occurring due to client trading activity, and can be controlled by making small adjustments to the so-called skew, namely, the quoted price offered to the clients. After formulating the stochastic control problem, closed form solutions are derived for the special cases that arise if the asset price is modelled by a Brownian motion with drift or a geometric Brownian motion. In both cases the impact of skew is additive. The optimal controls are time dependent affine functions of the inventory size and the inventory process under the optimal skew is an Ornstein-Uhlenbeck process. As a result, the asset price is mean reverting around a reference rate. In the second topic the same framework is expanded to include a hedging control that can be used by the market maker to manage the inventory. In particular, the market impact is assumed to be of the Almgren and Chriss type. Explicit solutions are derived in the special case where the asset price follows a Brownian motion with drift. The third topic is about Merton’s portfolio optimisation problem with the additional feature that the risky asset price is modelled in a way that exhibits support and resistance levels. In particular, the risky asset price is modelled using a skew Brownian motion. After formulating the stochastic control problem, closed form solutions are derived.

Item Type: Thesis (PhD)
Additional Information: © 2016 Tom Bates
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Departments > Mathematics
Supervisor: Zervos, Mihail
URI: http://etheses.lse.ac.uk/id/eprint/3476

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