  In this article, we tackle the problem of a market maker in charge of a book
of options on a single liquid underlying asset. By using an approximation of
the portfolio in terms of its vega, we show that the seemingly high-dimensional
stochastic optimal control problem of an option market maker is in fact
tractable. More precisely, when volatility is modeled using a classical
stochastic volatility model -- e.g. the Heston model -- the problem faced by an
option market maker is characterized by a low-dimensional functional equation
that can be solved numerically using a Euler scheme along with interpolation
techniques, even for large portfolios. In order to illustrate our findings,
numerical examples are provided.
