Pricing contingent claims on illiquid securities in continuous time

We complement arbitrage restrictions with rationality arguments to obtain tight ‘good deal’ bounds on the set of arbitrage-free valuations in incomplete markets with a complete submarket. We show how to select the stochastic discount factors which induce the lowest and highest option prices under risk-neutral no-good-deal valuation. We find that rationality measured by coherent risk measures restrict the volatility of the stochastic discount factor independent of the model parameters and available products, making coherent restrictions robust against model misspecification and giving them an interpretation that is consistent over different calibrations. Good deal bounds are derived in a general pricing model with stochastic volatility and a stochastic interest rate, and the implications of using good deal bounds in a simulation pricing framework are discussed: Most notably, without the completeness assumption the classical Heston stochastic volatility leads to the possibility of exploding discount processes. To mitigate this issue we propose a modified stochastic volatility model. If risk preferences are incoherent, we find that good deal bounds do not necessarily lead to sensible solutions without imposing additional constraints on the market.