Continuous q-Hermite polynomials: Difference between revisions

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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 $V : ℝ \times ℝ \to ℝ$ be defined by

V (x, k) = \sup_{X_{T} \in 𝒜 (x)} 𝔼 [u (X_{T} + k C_{T})]

,

where $x$ is the initial endowment, $𝒜 (x)$ 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 $[v^{b} (k), v^{a} (k)]$ .^[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_{α B_{0} + β S_{0} = x} 𝔼 [1 - \exp (- . 1 \times (α B_{T} + β S_{T}))]

= \max_{β} [1 - \frac{1}{3} [\exp (- \frac{1.10 x - 20 β}{10}) + \exp (- \frac{1.10 x}{10}) + \exp (- \frac{1.10 x + 20 β}{10})]]

.

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

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

V (x - v^{b} (1), 1) = \max_{α B_{0} + β S_{0} = x - v^{b} (1)} 𝔼 [1 - \exp (- . 1 \times (α B_{T} + β S_{T} + C_{T}))]

= \max_{β} [1 - \frac{1}{3} [\exp (- \frac{1.10 (x - v^{b} (1)) - 20 β}{10}) + \exp (- \frac{1.10 (x - v^{b} (1))}{10}) + \exp (- \frac{1.10 (x - v^{b} (1)) + 20 β + 20}{10})]]

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

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

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

References

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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]].<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>
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+==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>.  If we 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]]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.
+==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]] and subhedging prices respectively then <math>v^{sub}(k) \leq v(k) \leq v^{sup}(k)</math>.  Therefore, in a [[complete market]] the indifference price is always equal to the price to hedge the claim.
+==References==
+{{Reflist}}
+[[Category:Mathematical finance]]
+[[Category:Utility]]
+[[Category:Pricing]]