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 (with optimal trading otherwise). Typically the indifference price is a pricing range (a bid–ask spread) for a specific agent; this price range is an example of good-deal bounds.[1]
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 (or sold). 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
.[2]
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
![{\displaystyle =\max _{\beta }\left[1-{\frac {1}{3}}\left[\exp \left(-{\frac {1.10(x-v^{b}(1))-20\beta }{10}}\right)+\exp \left(-{\frac {1.10(x-v^{b}(1))}{10}}\right)+\exp \left(-{\frac {1.10(x-v^{b}(1))+20\beta +20}{10}}\right)\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62dfe4af1dff2cb016c70488d635e894cab425b7)
Which is maximized when
, therefore
.
Therefore
when
.
Similarly solve for
to find the indifference ask price.
- If
are the indifference price bounds for a claim then by definition
.[2]
- If
is the indifference bid price for a claim and
are the superhedging price and subhedging prices respectively then
. Therefore, in a complete market the indifference price is always equal to the price to hedge the claim.