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 (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]

MathematicsEdit

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]

ExampleEdit

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 alsoEdit

NotesEdit

  • 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.

ReferencesEdit

  1. ^ John R. Birge (2008). Financial Engineering. Elsevier. pp. 521–524. ISBN 978-0-444-51781-4.
  2. ^ a b Carmona, Rene (2009). Indifference Pricing: Theory and Applications. Princeton University Press. ISBN 978-0-691-13883-1.