A risk reserve model for hedging in incomplete markets

This paper presents a new approach to the pricing and hedging problem for contingent claims in incomplete markets. We assume that traders wish to maximize the expected final payoff of the hedging portfolio and the claims, and we avoid the use of utility functions. Instead, we model how traders are punished when taking excessive risks in practice. To do so, we introduce an extra reserve bank account, which earns a smaller rate of return than a standard deposit bank account. The reserve account should always contain a minimal amount of money, which depends on the risk that the trader's portfolio is exposed to. We focus on a specific example which uses option price sensitivities (the ‘Greeks’) to specify the risk. The resulting optimization problem can then be solved in a rather explicit form, and we show how the solution naturally leads to bid-ask spreads, prices which depend on the trader's current position and implied volatility smiles.