When replication is impossible, the hedger cannot eliminate risk — but they can minimise it in a well-defined sense. This lesson develops two approaches: mean-variance hedging (minimise the variance of the hedging error) and utility indifference pricing (set the price at which the seller is indifferent between selling the derivative and not, given their risk preferences). Both approaches yield prices and hedges that differ from the delta-hedge in a model-complete world, and the difference is precisely the jump-risk premium.
Learning objectives. By the end of this note you should be able to:
- state the mean-variance hedging problem and write down the optimal hedge ratio in a jump-diffusion model;
- compute the utility indifference price for a simple payoff under exponential utility;
- compare the mean-variance hedge to the delta-hedge and quantify the residual risk.
In this lesson
- 2.1 Mean-variance hedging
- 2.2 Utility indifference pricing
- 2.3 Residual risk
Before you start
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