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

TL;DR

This paper investigates the asymptotic behavior of utility indifference prices for path-dependent options in a discretized Bachelier model, revealing a limit that involves volatility control as trading frequency increases.

Contribution

It introduces a probabilistic approach to derive the scaling limit of utility indifference prices in a discretized setting, connecting it to a volatility control problem.

Findings

Derives a non-trivial scaling limit for utility indifference prices as trading frequency increases.

Transforms the pricing problem into an optimal drift control problem via duality.

Shows the limiting problem involves a volatility control formulation.

Abstract

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.