- Mathematics
- Example
- See also
- Notes
- References
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]
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 (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]
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.
See also
- Willingness to pay
- Willingness to accept
Notes
- 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.
References
2. ^1 {{cite book|first=Rene|last=Carmona|title=Indifference Pricing: Theory and Applications|publisher=Princeton University Press|year=2009|isbn=978-0-691-13883-1}}
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