A Scaling Limit for Utility Indifference Prices in the Discretized Bachelier Model

We consider the discretized Bachelier model where hedging is done on an equidistant set of times. Exponential utility indifference prices are studied for path-dependent European options and we compute their non-trivial scaling limit for a large number of trading times $n$ and when risk aversion is scaled like $n\ell$ for some constant $\ell>0$. Our analysis is purely probabilistic. We first use a duality argument to transform the problem into an optimal drift control problem with a penalty term. We further use martingale techniques and strong invariance principles and get that the limiting problem takes the form of a volatility control problem.