Utility Indifference Pricing

Ordinary option pricing assumes you can hedge the risk away completely, so the fair price is whatever it costs to build that perfect replicating hedge, no personal risk appetite required. Many real risks — an illiquid position, a real-estate holding, an employee stock grant that can't be sold — can't be hedged away, so there is no single "correct" price that everyone would agree on. Utility indifference pricing sidesteps this by pricing the risk relative to one specific investor's own risk tolerance: the price is whatever amount of guaranteed cash would make that investor's expected utility exactly the same whether they hold the risky payoff or hold the cash instead.

Concretely, an investor with utility function $U$ and initial wealth $w$ compares two worlds: holding the risky payoff $X$, giving expected utility $E[U(w + X)]$, versus being paid a fixed amount $p$ instead and holding no risk, giving utility $U(w + p)$. The indifference price $p$ solves $U(w+p) = E[U(w+X)]$ — the payment that makes the investor exactly indifferent between the two outcomes. Because $U$ is a personal risk-aversion function, this price differs from one investor to the next: a more risk-averse investor demands a bigger discount to hold the same unhedgeable risk, so their indifference price for buying it is lower than a less risk-averse investor's.