Given a utility function ${\displaystyle u}$  and a claim ${\displaystyle C_{T}}$  with known payoffs at some terminal time ${\displaystyle T,}$  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 ${\displaystyle x}$  is the initial endowment, ${\displaystyle {\mathcal {A}}(x)}$  is the set of all self-financing portfolios at time ${\displaystyle T}$  starting with endowment ${\displaystyle x}$ , and ${\displaystyle k}$  is the number of the claim to be purchased (or sold). Then the indifference bid price ${\displaystyle v^{b}(k)}$  for ${\displaystyle k}$  units of ${\displaystyle C_{T}}$  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]

Consider a market with a risk free asset ${\displaystyle B}$  with ${\displaystyle B_{0}=100}$  and ${\displaystyle B_{T}=110}$ , and a risky asset ${\displaystyle S}$  with ${\displaystyle S_{0}=100}$  and ${\displaystyle S_{T}\in \{90,110,130\}}$  each with probability ${\displaystyle 1/3}$ . 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 ${\displaystyle \beta =0}$ , 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.

• Willingness to pay
• Willingness to accept

• 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.