Continuous q-Hermite polynomials

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.

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

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

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 $[v^b(k),v^a(k)]$.

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 $V(x,0)$.

$V(x,0)={\max }_{\alpha B_0+\beta S_0=x}\mathbb{𝔼}[1-\exp (-.1\times (\alpha B_T+\beta S_T))]$

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

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

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

$V(x-v^b(1),1)={\max }_{\alpha B_0+\beta S_0=x-v^b(1)}\mathbb{𝔼}[1-\exp (-.1\times (\alpha B_T+\beta S_T+C_T))]$

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

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

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 $[v^b(k),v^a(k)]$ are the indifference price bounds for a claim then by definition $v^b(k)=-v^a(-k)$.
• 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 $v^{sub}(k)\le v(k)\le v^{sup}(k)$. Therefore, in a complete market the indifference price is always equal to the price to hedge the claim.

References

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