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]

Mathematics

Given a utility function $u$ and a claim $C_{T}$ with known payoffs at some terminal time $T$. If we let the function ${\displaystyle V:\mathbb {R} \times \mathbb {R} \to \mathbb {R} }$ be defined by

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

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 ${\displaystyle v^{b}(k)}$ for $k$ units of $C_{T}$ is the solution of ${\displaystyle V(x-v^{b}(k),k)=V(x,0)}$ and the indifference ask price ${\displaystyle v^{a}(k)}$ is the solution of ${\displaystyle V(x+v^{a}(k),-k)=V(x,0)}$. The indifference price bound is the range ${\displaystyle \left[v^{b}(k),v^{a}(k)\right]}$.^[2]

Example

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

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

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

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

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

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

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

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

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

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

Notes

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

References

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