Derivative Pricing and Hedging in Asynchronous Markets

We consider the problem of how to price and hedge derivatives on underlyings that trade on exchanges with no overlap in opening hours. For a simple two-stock model we derive the dynamics of closing prices, show how they can be simulated efficiently and what value we should put into pricing formulae for a non-observable stock. We then use utility maximisation to show that the price can still be approximated with the 2-stock version of the Black-Scholes PDE and derive expressions for the optimal delta hedges, which are path-dependent but can be written as the ideal delta hedge plus a proxy hedge for the untradable stock