Conditional Davis pricing

Abstract

We study the set of Davis (marginal utility-based) prices of a financial derivative in the case where the investor has a non-replicable random endowment. We give a new characterisation of the set of all such prices, and provide an example showing that even in the simplest of settings – such as Samuelson’s geometric Brownian motion model –, the interval of Davis prices is often a non-degenerate subinterval of the set of all no-arbitrage prices. This is in stark contrast to the case with a constant or replicable endowment where non-uniqueness of Davis prices is exceptional. We provide formulas for the endpoints of these intervals and illustrate the theory with several examples.

Appendix A: Davis prices and related derivative prices

The purpose of this appendix is to place the notion of Davis pricing within the wider framework of arbitrage- and utility-based pricing concepts. We start by noting that conditional Davis prices are always arbitrage-free in the sense of the following definition from Siorpaes [34, Sect. 2].

Definition A.1

A constant $p = p(\varphi )\in {\mathbb{R}}$ is called an arbitrage-free price of $\varphi$ if for all $\pi \in {\mathcal{A}}$ and $q'\in {\mathbb{R}}$, we have that

$$\begin{aligned} q'(\varphi -p) + \int _{0}^{T} \pi _{u} dS_{u} \ge 0 \quad \text{implies}\quad q'(\varphi -p) + \int _{0}^{T} \pi _{u} dS_{u} = 0, \qquad {\mathbb{P}}\text{-a.s. } \end{aligned}$$

(A.1)

Proposition A.2

Given$B\in {\mathbb{L}}^{\infty }_{++}$, each$B$-conditional Davis price$p$for a contingent claim$\varphi \in {\mathbb{L}}^{\infty }$is also an arbitrage-free price of$\varphi$.

Proof

Let $p$ be a conditional Davis price and suppose to the contrary of (A.1) that we can find $q'\in {\mathbb{R}}$ and $\pi \in {\mathcal{A}}$ such that the nonnegative random variable

$$ A:= q'(\varphi -p) + \int _{0}^{T} \pi _{u} dS_{u} $$

is strictly positive with strictly positive probability, i.e., that we have ${\mathbb{P}}[A\geq 0]=1$ and ${\mathbb{P}}[A>0]>0$. Then for $n\in {\mathbb{N}}$, the inequality (3.1) implies that

$$\begin{aligned} \mathfrak{U}(B) & \geq \mathfrak{U}\big(B+nq'(\varphi -p)\big) \\ &\geq {\mathbb{E}}\bigg[ U\bigg( B+nq'(\varphi -p) + n\int _{0}^{T} \pi _{u}\, dS_{u} \bigg)\bigg] \\ &= {\mathbb{E}}[ U(B +nA)]. \end{aligned}$$

It remains to let $n\to \infty$ and use the monotone convergence theorem to reach a contradiction with the fact that $\mathfrak{U}(B)<\infty$. □

Next, we relate several popular utility-based pricing methods to Davis prices (see e.g. Becherer [1]). As always, $B\in {\mathbb{L}}^{\infty }_{++}$ is the random endowment and $\varphi \in {\mathbb{L}}^{\infty }$ is the claim’s payoff. However, we also allow a dependence on the quantity $q$ of claims held.

Definition A.3

Let $q\ne 0$. A constant $m$ is called a utility-based derivative price for the payoff $\varphi$ at quantity $q$ if

$$\begin{aligned} q \in \mathop{\mathrm{argmax}} \limits_{q' \in {\mathbb{R}}} \mathfrak{U}\big(B + q' ( \varphi -m)\big). \end{aligned}$$

(A.2)

A constant $h=h(q) = h(q;\varphi |B)$ is called a utility indifference (Hicks, reservation) price for $\varphi$ at quantity $q$ if

$$\begin{aligned} \mathfrak{U}\Big(B + q \big(\varphi - h(q)\big) \Big) = \mathfrak{U}(B). \end{aligned}$$

(A.3)

Up to a sign convention, these prices are further divided into utility indifference buy and sell prices (see e.g. [4, Chap. 2]). Finally, a constant $c = c(q) = c(q; \varphi |B)$ is called a certainty equivalent for $\varphi$ at quantity $q$ if

$$ \mathfrak{U}(B + q \varphi ) = \mathfrak{U}\big(B+q c(q)\big). $$

While the three concepts introduced above differ from each other in general, we show that under appropriate conditions, their limits as $q\to 0\pm$ coincide with the endpoints of the interval $P(\varphi |B)$ of conditional Davis prices.³ To streamline the presentation, we define $p_{\min }$ and $p_{\max }$ by

$$ [p_{\min }, p_{\max }] = P(\varphi |B). $$

Similarly, let $m_{\min }(q) \leq m_{\max }(q)$ be the endpoints of the interval of utility-based derivative prices at quantity $q$ defined in (A.2) above.

In Proposition A.4 below, we assume that both $\varphi$ and $B$ are positive, bounded and bounded away from zero. This entails virtually no loss of generality, but makes the value function $u(x,q)$ defined in (4.1) strictly increasing in each argument. On the other hand, the assumption that $-B$ is minimally superreplicable is a major one. We leave the question of the validity of Proposition A.4 without this for future research.

Proposition A.4

Suppose that Assumption4.5holds, $B, \varphi \in {\mathbb{L}}^{\infty }_{++}$and$-B$is minimally superreplicable. Then

$$\begin{aligned} \lim _{q\searrow 0} c(q) = \lim _{q\searrow 0} h(q) = \lim _{q \searrow 0} m_{\max }(q) = \lim _{q\searrow 0} m_{\min }(q) = p_{ \max } \end{aligned}$$

(A.4)

and

$$\begin{aligned} \lim _{q\nearrow 0} c(q) = \lim _{q\nearrow 0} h(q) = \lim _{q \nearrow 0} m_{\max }(q) = \lim _{q\nearrow 0} m_{\min }(q) = p_{ \min }. \end{aligned}$$

(A.5)

Proof

We prove only (A.4) as the proof of (A.5) is completely analogous. Since the value function $u$ from (4.1) is strictly increasing in each argument, $h(q)$ and $c(q)$ are well defined and unique for $q\ne 0$. Moreover, we have for all $q>0$ the bounds

$$\begin{aligned} 0< \operatorname{essinf} \varphi &\leq \, c(q)\, \leq \operatorname{esssup} \varphi < \infty , \\ 0 < \operatorname{essinf} \varphi &\leq \, h(q)\, \leq \operatorname{esssup} \varphi < \infty . \end{aligned}$$

(A.6)

Hence $\lim _{q\searrow 0} q c(q)=0$ and we can let $q\searrow 0$ in

$$\begin{aligned} \frac{u(0,q) - u(0,0)}{q} &= \frac{ \mathfrak{U}(B+q\varphi )- \mathfrak{U}(B)}{q} \\ &=\frac{ \mathfrak{U}(B+q c(q) )- \mathfrak{U}(B)}{q} \\ &=\frac{ \mathfrak{U}(B+q c(q) )- \mathfrak{U}(B)}{q c(q)}c(q), \end{aligned}$$

and use Corollary 5.8 (where the constant $y_{B}>0$ is defined) to obtain

$$ \partial _{q+} u(0,0) = \lim _{q\searrow 0} y_{B} c(q). $$

It remains to use Proposition 4.1 to conclude that $\lim _{q\searrow 0} c(q)$ is indeed the right endpoint of the interval of $B$-conditional Davis prices.

To deal with the indifference price $h$, we use its definition together with the concavity of the value function $u$ to conclude that for $q>0$ and $\lambda \in (0,1)$, we have

$$\begin{aligned} u\big( - \lambda q\, h (\lambda q), \lambda q \big) &= u(0,0) \\ &= \lambda u\big(-q\, h(q),q\big)+ (1-\lambda ) u(0,0) \\ &\leq u\big( - \lambda q\, h(q), \lambda q\big). \end{aligned}$$

Since $u$ is strictly increasing in its second argument, we have $h(\lambda q) \geq h(q)$. Therefore $q \mapsto h(q)$ is a nonincreasing function and by (A.6), its limit $h(0+) := \lim _{q\searrow 0} h(q)$ exists in $(0,\infty )$. We use that fact to pass to the limit $q\searrow 0$ in both sides of the equality

$$ 0 = \frac{ u ( - q\, h(q),q ) - u(0,0)}{q}. $$

So the directional derivative of $u$ at $(0,0)$ in the direction $(-h(0+),1)$ is 0, and

$$ \inf _{(y,r) \in \partial u(0,0)} \big( r - h(0+) y \big) = 0. $$

Corollary 5.8 implies that we have $(y,r) \in \partial u(0,0)$ only if $y = y_{B}$, which then yields $h(0+) = \frac{1}{y_{B}} \partial _{q+} u(0,0)$. Proposition 4.1 completes the argument.

Finally, we treat utility-based derivative prices. Let $(q_{n})_{n\in \mathbb{N}}$ be a sequence of positive numbers decreasing to 0 and define the sequence $(m_{n})_{n\in \mathbb{N}}$ by $m_{n} := m_{\max }(q_{n})$. Because the lower endpoints $m_{\min }(q_{n})$ can be treated similarly, we omit the details for that case. By (A.2), the concave function

$$ q' \mapsto u( - m_{n} q', q') $$

admits a maximum at $q'=q_{n}$. Therefore the partial derivatives of $u$ at $(-m_{n} q_{n}, q_{n})$ are nonpositive in the directions $(-m_{n},1)$ and $(m_{n}, -1)$. The proof of Proposition 4.1 produces a pair $(y_{n}, r_{n}) \in \partial u( - m_{n} q_{n}, q_{n})$ such that $r_{n} = m_{n} y_{n}$.

Because of (A.6), we have $0< m_{n} \leq \operatorname{esssup} \varphi$ so that $(- m_{n} q_{n}, q_{n}) \to (0,0)$. The graph of the supergradient correspondence of a concave function is closed (see e.g. Hiriart-Urruty and Lemaréchal [18, Proposition 6.2.1]) and locally bounded (see e.g. [18, Proposition 6.2.2]). Therefore, by passing to a subsequence if necessary, we may conclude that $(y_{n}, r_{n}) \to (y^{*},r^{*})$ for some $(y^{*},r^{*}) \in \partial u(0,0)$. By Corollary 5.8, the value $y^{*}$ must equal $y_{B}>0$, and so $y_{n}>0$ for large enough $n$. This implies that $m_{n} \to m$, where $m=r^{*}/y_{B}$, and Proposition 4.1 guarantees that $m$ is a Davis price for the payoff $\varphi$.

On the other hand, for any conditional Davis price $p \in P(\varphi |B)$ and for any $(y,r_{B})\in \partial u(0,0)$ such that $p = r/y_{B}$, the monotonicity of the supergradient correspondence, i.e.,

$$ \langle (- m_{n} q_{n}, q_{n}) - (0,0), (y_{n},r_{n}) - (y,r_{B}) \rangle \geq 0, $$

see e.g. [18, Proposition 6.1.1], implies that

$$ (r_{n} - r) - m_{n}( y_{n}-y_{B}) \geq 0 \qquad \text{for all}\ n\in { \mathbb{N}}. $$

Since $y_{n} \to y_{B}$, $r_{n} \to r^{*}$ and $(m_{n})_{n\in \mathbb{N}}$ is a bounded sequence, we conclude that $p = r/y_{B} \leq r^{*}/y_{B} = m$. Consequently, $m = p_{\max }$.

It remains to show that $m_{\max }(q) \to p_{\max }$ as $q\to 0$. If it did not, there would exist a sequence $(q_{n})_{n\in \mathbb{N}}$ with $q_{n} \searrow 0$ such that $m_{\max }(q_{n}) \to p_{0}\ne p_{\max }$. The same would be true for any subsequence of $(q_{n})_{n\in \mathbb{N}}$, which contradicts the conclusion of the previous paragraph. □