In this paper, we are investigating the market making problem under a stochastic control framework by first deriving the results obtained in seminal work by [AS] for stock market making, which was then expanded by [ElAA3] into an options market making setting. We will focus on the derivation of analytic expressions for optimal controls using dynamic programming principle and their approximation in the case of risk-averse options market makers. We will use the same simplifying assumptions to prove these results, where we consider the market maker as an agent who seeks to maximise the expected utility of terminal wealth. Later we prove derivations in [ElAA3] for approximation of variance of the utility of terminal wealth for a market maker who also seeks to manage the inventory risk. Monte Carlo simulations are used to verify the performance of three different market makers: risk-neutral, risk-averse and zero intelligence as a baseline. In the simulation, we assume the stock price to follow the Heston model for which we briefly investigate some important results concerning option valuation. The aim is to show that stochastic control is a viable method of solving the market making problem under simplifying assumptions.
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[ELAA2] S. El Aoud and F. Abergel, A stochastic control approach for options market making, Market Microstructure and Liquidity, (2015), https://hal.archivesouvertes.fr/hal-01061852v2
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