# Indifference price

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In [finance](/source/finance), '''indifference pricing''' is a method of pricing [financial securities](/source/financial_securities) with regard to a [utility function](/source/utility_function).  The ''' indifference price''' is also known as the '''[reservation price](/source/reservation_price)''' or '''private valuation'''.  In particular, the indifference price is the price at which an agent would have the same [expected utility](/source/expected_utility) level by exercising a [financial transaction](/source/financial_transaction) as by not doing so (with optimal trading otherwise).  Typically the indifference price is a pricing range (a [bid–ask spread](/source/bid%E2%80%93ask_spread)) for a specific agent; this price range is an example of [good-deal bounds](/source/good-deal_bounds).<ref name="FE">{{cite book|title=Financial Engineering|author=John R. Birge|year=2008|publisher=Elsevier|pages=521–524|isbn=978-0-444-51781-4}}</ref>

==Mathematics==
Given a utility function <math>u</math> and a claim <math>C_T</math> with known payoffs at some terminal time <math>T,</math> let the function <math>V: \mathbb{R} \times \mathbb{R} \to \mathbb{R}</math> be defined by
: <math>V(x,k) = \sup_{X_T \in \mathcal{A}(x)} \mathbb{E}\left[u\left(X_T + k C_T\right)\right]</math>,
where <math>x</math> is the initial endowment, <math>\mathcal{A}(x)</math> is the set of all [self-financing portfolio](/source/self-financing_portfolio)s at time <math>T</math> starting with endowment <math>x</math>, and <math>k</math> is the number of the claim to be purchased (or sold).  Then the indifference bid price <math>v^b(k)</math> for <math>k</math> units of <math>C_T</math> is the solution of <math>V(x - v^b(k),k) = V(x,0)</math> and the indifference ask price <math>v^a(k)</math> is the solution of <math>V(x + v^a(k),-k) = V(x,0)</math>.  The indifference price bound is the range <math>\left[v^b(k),v^a(k)\right]</math>.<ref name=C09>{{cite book|first=Rene|last=Carmona|title=Indifference Pricing: Theory and Applications|publisher=Princeton University Press|year=2009|isbn=978-0-691-13883-1}}</ref>

==Example==
Consider a market with a risk free asset <math>B</math> with <math>B_0 = 100</math> and <math>B_T = 110</math>, and a risky asset <math>S</math> with <math>S_0 = 100</math> and <math>S_T \in \{90, 110, 130\}</math> each with probability <math>1/3</math>.  Let your utility function be given by <math>u(x) = 1 - \exp(-x/10)</math>.  To find either the bid or ask indifference price for a single European call option with strike 110, first calculate <math>V(x,0)</math>.

: <math>V(x,0) = \max_{\alpha B_0 + \beta S_0 = x} \mathbb{E}[1 - \exp(-.1 \times (\alpha B_T + \beta S_T))]</math>
:: <math> = \max_{\beta} \left[1 - \frac{1}{3} \left[\exp\left(-\frac{1.10 x - 20 \beta}{10}\right) + \exp\left(-\frac{1.10 x}{10}\right) + \exp\left(-\frac{1.10 x + 20 \beta}{10}\right)\right]\right]</math>.
Which is maximized when <math>\beta = 0</math>, therefore <math>V(x,0) = 1 - \exp\left(-\frac{1.10 x}{10}\right)</math>.

Now to find the indifference bid price solve for <math>V(x - v^b(1),1)</math>

: <math>V(x - v^b(1),1) = \max_{\alpha B_0 + \beta S_0 = x - v^b(1)} \mathbb{E}[1 - \exp(-.1 \times (\alpha B_T + \beta S_T + C_T))]</math>
:: <math> = \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]</math>
Which is maximized when <math>\beta = -\frac{1}{2}</math>, therefore <math>V(x - v^b(1),1) = 1 - \frac{1}{3} \exp(-1.10 x/10) \exp(1.10 v^b(1)/10)\left[1 + 2 \exp(-1)\right]</math>.

Therefore <math>V(x,0) = V(x - v^b(1),1)</math> when <math>v^b(1) = \frac{10}{1.1} \log\left(\frac{3}{1+2 \exp(-1)}\right) \approx 4.97</math>.

Similarly solve for <math>v^a(1)</math> to find the indifference ask price.

==See also==
*[Willingness to pay](/source/Willingness_to_pay)
*[Willingness to accept](/source/Willingness_to_accept)

==Notes==
* If <math>\left[v^b(k),v^a(k)\right]</math> are the indifference price bounds for a claim then by definition <math>v^b(k) = -v^a(-k)</math>.<ref name=C09 />
* If <math>v(k)</math> is the indifference bid price for a claim and <math>v^{sup}(k),v^{sub}(k)</math> are the [superhedging price](/source/superhedging_price) and subhedging prices respectively then <math>v^{sub}(k) \leq v(k) \leq v^{sup}(k)</math>.  Therefore, in a [complete market](/source/complete_market) the indifference price is always equal to the price to hedge the claim.

==References==
{{Reflist}}

Category:Mathematical finance
Category:Utility
Category:Pricing

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