Abstract
We derive optimal spreads for market makers based on the Hamilton-Jacobi-Bellman equation with an integral utility function which takes inventory risk, volatility risk and a discount into account. A discount is introduced so that we can obtain an optimal control in T → ∞ limit, i.e. an optimal control that does not depend on the terminal time. We show that a difference between market buy and sell order intensity acts as a drift. In the limit T → ∞, terms proportional to T − t, a difference between current time and the terminal time, is replaced by ω −1 , an inverse discount factor. We then perform Monte Carlo simulation using obtained optimal spreads for various risk aversion parameter and see that there is “optimal” risk aversion parameter.
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