Indifference price


In finance, indifference pricing is a method of pricing financial securities with regard to a utility function. The indifference price is also known as the reservation price or private valuation. In particular, the indifference price is the price at which an agent would have the same expected utility level by exercising a financial transaction as by not doing so. Typically the indifference price is a pricing range for a specific agent; this price range is an example of good-deal bounds.

Mathematics

Given a utility function and a claim with known payoffs at some terminal time let the function be defined by
where is the initial endowment, is the set of all self-financing portfolios at time starting with endowment, and is the number of the claim to be purchased. Then the indifference bid price for units of is the solution of and the indifference ask price is the solution of. The indifference price bound is the range.

Example

Consider a market with a risk free asset with and, and a risky asset with and each with probability. Let your utility function be given by. To find either the bid or ask indifference price for a single European call option with strike 110, first calculate.
Which is maximized when, therefore.
Now to find the indifference bid price solve for
Which is maximized when, therefore.
Therefore when.
Similarly solve for to find the indifference ask price.