Indifference price

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In finance, indifference pricing is a method of pricing financial securities with regard to a utility function. Also known as the reservation price or private valuation. Particularly the indifference price is the price that an agent would have the same expected utility level between exercising a financial transaction and not (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 $u$ and a claim $C_{T}$ with known payoffs at some terminal time $T$ . If we let the function $V:{\mathbb {R}}\times {\mathbb {R}}\to {\mathbb {R}}$ be defined by

$V(x,k)=\sup _{{X_{T}\in {\mathcal {A}}(x)}}{\mathbb {E}}\left[u\left(X_{T}+kC_{T}\right)\right]$ ,

where $x$ is the initial endowment, ${\mathcal {A}}(x)$ is the set of all self-financing portfolios at time $T$ starting with endowment $x$ , and $k$ is the number of the claim to be purchased (or sold). Then the indifference bid price $v^{b}(k)$ for $k$ units of $C_{T}$ is the solution of $V(x-v^{b}(k),k)=V(x,0)$ and the indifference ask price $v^{a}(k)$ is the solution of $V(x+v^{a}(k),-k)=V(x,0)$ . The indifference price bound is the range $\left[v^{b}(k),v^{a}(k)\right]$ .^[2]

Example

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

$V(x,0)=\max _{{\alpha B_{0}+\beta S_{0}=x}}{\mathbb {E}}[1-\exp(-.1\times (\alpha B_{T}+\beta S_{T}))]$

$=\max _{{\beta }}\left[1-{\frac {1}{3}}\left[\exp \left(-{\frac {1.10x-20\beta }{10}}\right)+\exp \left(-{\frac {1.10x}{10}}\right)+\exp \left(-{\frac {1.10x+20\beta }{10}}\right)\right]\right]$ .

Which is maximized when $\beta =0$ , therefore $V(x,0)=1-\exp \left(-{\frac {1.10x}{10}}\right)$ .

Now to find the indifference bid price solve for $V(x-v^{b}(1),1)$

$V(x-v^{b}(1),1)=\max _{{\alpha B_{0}+\beta S_{0}=x-v^{b}(1)}}{\mathbb {E}}[1-\exp(-.1\times (\alpha B_{T}+\beta S_{T}+C_{T}))]$

$=\max _{{\beta }}\left[1-{\frac {1}{3}}\left[\exp \left(-{\frac {1.10(x-v^{b}(1))-20\beta }{10}}\right)+\exp \left(-{\frac {1.10(x-v^{b}(1))}{10}}\right)+\exp \left(-{\frac {1.10(x-v^{b}(1))+20\beta +20}{10}}\right)\right]\right]$

Which is maximized when $\beta =-{\frac {1}{2}}$ , therefore $V(x-v^{b}(1),1)=1-{\frac {1}{3}}\exp(-1.10x/10)\exp(1.10v^{b}(1)/10)\left[1+2\exp(-1)\right]$ .

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

Similarly solve for $v^{a}(1)$ to find the indifference ask price.

Notes

If $\left[v^{b}(k),v^{a}(k)\right]$ are the indifference price bounds for a claim then by definition $v^{b}(k)=-v^{a}(-k)$ .^[2]
If $v(k)$ is the indifference bid price for a claim and $v^{{sup}}(k),v^{{sub}}(k)$ are the superhedging price and subhedging prices respectively then $v^{{sub}}(k)\leq v(k)\leq v^{{sup}}(k)$ . Therefore, in a complete market the indifference price is always equal to the price to hedge the claim.

References

↑ John R. Birge (2008). Financial Engineering. Elsevier. pp. 521–524. ISBN 978-0-444-51781-4.
↑ 2.0 2.1 Carmona, Rene (2009). Indifference Pricing: Theory and Applications. Princeton University Press. ISBN 978-0-691-13883-1.

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