Hedging and pricing illiquid options with market impacts

Abstract

In this paper, we consider hedging and pricing of illiquid options on an untradable underlying asset, where an alternative asset is used as a hedging instrument. Particularly, we consider the situation where the trade price of the hedging instrument is subject to market impacts caused by the hedger and the liquidity costs paid as a spread from the mid price. Pricing illiquid options, which often appears in trading of structured products, is a critical issue in practice because of its difficulties in hedging mainly due to untradability of the underlying asset as well as the liquidity costs and market impacts of the hedging instrument. First, by setting the problem under a discrete time model, where the optimal hedging strategy is defined by the local risk-minimization, we present algorithms to obtain the option price along with the hedging strategy by an asymptotic expansion. Moreover, we provide numerical examples. This model enables the estimation of the effect of both the market impacts and the liquidity costs on option prices, which is important in practice.