The value of informational arbitrage

Abstract

In the context of a general semimartingale model, we aim at determining how much an investor is willing to pay to learn additional information that allows achieving arbitrage. If such a value exists, we call it the value of informational arbitrage. We are interested in the case where the information yields arbitrage opportunities but not unbounded profits with bounded risk. As in Amendinger et al. (Finance Stoch. 7:29–46, 2003), we rely on an indifference valuation approach and study optimal consumption–investment problems under initial information and arbitrage. We establish some new results on models with additional information and characterise when the value of informational arbitrage is universal.

Appendix: Proofs of the results in Sect. 3

Proof of Lemma 3.3

Let $(\vartheta ,c)\in \mathcal{A}^{\mathbf{H},k}(v)$. For ease of notation, we set $V:=V^{v+k,\vartheta ,c}$, $C:=\int _{0}^{\cdot }c_{u}\,\!\mathrm{d} \kappa _{u}$ and $\widetilde{C}:=\int _{0}^{\cdot }Z^{\mathbf{H}}_{u}\, \!\mathrm{d}C_{u}$. Integration by parts gives for all $t\in [0,T]$ that

$$\begin{aligned} Z^{\mathbf{H}}_{t}V_{t} + \widetilde{C}_{t} &= Z^{\mathbf{H}}_{t} \bigl(v+k+(\vartheta \cdot S)_{t}\bigr) - Z^{\mathbf{H}}_{t}C_{t} + \int _{0}^{t}Z^{\mathbf{H}}_{u} \,\! \mathrm{d}C_{u} \\ &= Z^{\mathbf{H}}_{t}\bigl(v+k+(\vartheta \cdot S)_{t}\bigr) - (C _{-}\cdot Z^{\mathbf{H}})_{t}. \end{aligned}$$

Since $Z^{\mathbf{H}}\in \mathcal{M}_{\text{loc}}(\mathbb{P}, \mathbf{H})$ and $Z^{\mathbf{H}}S\in \mathcal{M}_{\text{loc}}( \mathbb{P},\mathbf{H})$, this implies that $Z^{\mathbf{H}}V + \widetilde{C}$ is a sigma-martingale on $(\Omega ,\mathbf{H}, \mathbb{P})$ (see e.g. Fontana [18, Lemma 4.2]). Being nonnegative, it is also a supermartingale. Therefore, since $V^{v,\vartheta ,c}_{T}\geq 0$ a.s., we get

$$ v+k \geq \mathbb{E}[Z^{\mathbf{H}}_{T}V_{T}+\widetilde{C}_{T}| \mathcal{H}_{0}] \geq \mathbb{E}[kZ^{\mathbf{H}}_{T} + \widetilde{C} _{T}|\mathcal{H}_{0}] $$

so that $\mathbb{E}[\widetilde{C}_{T}|\mathcal{H}_{0}]\leq v+k(1- \mathbb{E}[Z^{\mathbf{H}}_{T}|\mathcal{H}_{0}])$ a.s. Conversely, let $C:=\int _{0}^{\cdot }c_{u}\,\!\mathrm{d}\kappa _{u}$ and suppose that

$$ \mathbb{E}\left [\int _{0}^{T}Z^{\mathbf{H}}_{u}\,\!\mathrm{d}C_{u} \bigg|\mathcal{H}_{0}\right ]\leq v+k(1-\mathbb{E}[Z^{\mathbf{H}}_{T}| \mathcal{H}_{0}]) \qquad \text{a.s.} $$

Consider the process $\widehat{V}=(\widehat{V}_{t})_{t\in [0,T]}$ defined for $t\in [0,T]$ by

$$\begin{aligned} \widehat{V}_{t} &:= v + Z^{\mathbf{H}}_{t}C_{t} - \int _{0}^{t}Z^{ \mathbf{H}}_{u} \,\! \mathrm{d}C_{u} + \mathbb{E}\bigg[ \int _{0}^{T}Z^{\mathbf{H}}_{u} \,\! \mathrm{d}C_{u}\bigg|\mathcal{H}_{t}\bigg] - \mathbb{E}\bigg[ \int _{0}^{T}Z^{\mathbf{H}}_{u} \,\! \mathrm{d}C_{u}\bigg|\mathcal{H}_{0}\bigg] \\ & \phantom{=::} +k(1-\mathbb{E}[Z^{\mathbf{H}}_{T}|\mathcal{H}_{0}]+\mathbb{E}[Z^{ \mathbf{H}}_{T}|\mathcal{H}_{t}]-Z^{\mathbf{H}}_{t}). \end{aligned}$$

The process $\widehat{V}$ is well defined as an element of $\mathcal{M}_{\text{loc}}(\mathbb{P},\mathbf{H})$. Due to Assumption 2.1 (and Proposition 2.5, in the case $\mathbf{H}= \mathbf{G}$), there exists $\psi \in L(S,\mathbf{H})$ such that

$$ \widehat{V}_{t} = Z^{\mathbf{H}}_{t}\bigl(v + (\psi \cdot S)_{t} \bigr) \qquad \text{a.s. for all }t\in [0,T]. $$

The process $V^{v+k,\psi ,c}=(V^{v+k,\psi ,c}_{t})_{t\in [0,T]}$ associated to the pair $(\psi ,c)$ satisfies

$$\begin{aligned} Z^{\mathbf{H}}_{t}V^{v+k,\psi ,c}_{t} + \int _{0}^{t}Z^{\mathbf{H}} _{u} \,\! \mathrm{d}C_{u} &= v+k +k(\mathbb{E}[Z^{\mathbf{H}}_{T}|\mathcal{H} _{t}]-\mathbb{E}[Z^{\mathbf{H}}_{T}|\mathcal{H}_{0}]) \\ & \phantom{=:}+ \mathbb{E}\bigg[\int _{0}^{T}Z^{\mathbf{H}}_{u} \,\! \mathrm{d}C_{u}\bigg|\mathcal{H}_{t}\bigg] - \mathbb{E}\bigg[\int _{0} ^{T}Z^{\mathbf{H}}_{u} \,\! \mathrm{d}C_{u}\bigg|\mathcal{H}_{0}\bigg] \end{aligned}$$

a.s. for all $t\in [0,T]$. By construction, we have $Z^{\mathbf{H}} _{t}V_{t}^{v+k,\psi ,c}\geq 0$ a.s. for all $t\in [0,T]$ and $Z^{\mathbf{H}}_{T}V^{v,\psi ,c}_{T}\geq 0$ a.s. Therefore $(\psi ,c)\in \mathcal{A}^{\mathbf{H},k}(v)$, proving that $c\in \mathcal{C}^{\mathbf{H},k}(v)$. □

Proof of Proposition 3.5

Under the given assumptions, the process $c^{\mathbf{H}}=(c^{\mathbf{H}}_{t})_{t\in [0,T]}$ satisfies

$$ \mathbb{E}\bigg[\int _{0}^{T}Z^{\mathbf{H}}_{u}c^{\mathbf{H}}_{u}\,\! \mathrm{d}\kappa _{u}\bigg|\mathcal{H}_{0}\bigg]=v+k(1-\mathbb{E}[Z _{T}^{\mathbf{H}}|\mathcal{H}_{0}]) \qquad \text{a.s.} $$

so that $c^{\mathbf{H}}\in \mathcal{C}^{\mathbf{H},k}(v)$ by Lemma 3.3. Consider an arbitrary consumption process $c\in \mathcal{C}^{\mathbf{H},k}(v)$. By the Fenchel–Legendre duality (see e.g. Karatzas and Shreve [27, Lemma 3.4.3]), the definitions of the stochastic field $I$ and of the process $c^{ \mathbf{H}}$ imply that

$$ U(t,c^{\mathbf{H}}_{t}) - \Lambda ^{\mathbf{H},k}(v)Z^{\mathbf{H}}_{t}c ^{\mathbf{H}}_{t} \geq U(t,c_{t}) - \Lambda ^{\mathbf{H},k}(v)Z^{ \mathbf{H}}_{t}c_{t} \qquad \text{for all }t\in [0,T]. $$

Therefore, it holds that

$$\begin{aligned} \mathbb{E}\bigg[ \int _{0}^{T}U(u,c^{\mathbf{H}}_{u})\,\!\mathrm{d} \kappa _{u} \bigg|\mathcal{H}_{0}\bigg] &\geq \mathbb{E}\bigg[ \int _{0}^{T}U(u,c_{u})\,\!\mathrm{d}\kappa _{u} \bigg|\mathcal{H}_{0}\bigg] \\ & \phantom{=:} + \Lambda ^{\mathbf{H},k}(v) \mathbb{E}\bigg[ \int _{0}^{T} Z^{ \mathbf{H}}_{u}c^{\mathbf{H}}_{u}\,\!\mathrm{d}\kappa _{u} \bigg| \mathcal{H}_{0}\bigg] \\ & \phantom{=:}- \Lambda ^{\mathbf{H},k}(v) \mathbb{E}\bigg[ \int _{0} ^{T}Z^{\mathbf{H}}_{u}c_{u}\,\!\mathrm{d}\kappa _{u} \bigg|\mathcal{H} _{0}\bigg] \\ &\geq \mathbb{E}\bigg[ \int _{0}^{T}U(u,c_{u})\,\!\mathrm{d}\kappa _{u} \bigg|\mathcal{H}_{0}\bigg], \end{aligned}$$

where the last inequality follows from the fact that in view of Lemma 3.3,

$$ \mathbb{E}\bigg[ \int _{0}^{T}Z^{\mathbf{H}}_{u}c^{\mathbf{H}}_{u}\, \!\mathrm{d}\kappa _{u} \bigg|\mathcal{H}_{0}\bigg] = v + k (1- \mathbb{E}[Z^{\mathbf{H}}_{T}|\mathcal{H}_{0}] ) \geq \mathbb{E}\bigg[ \int _{0}^{T}Z^{\mathbf{H}}_{u}c_{u}\,\!\mathrm{d}\kappa _{u} \bigg| \mathcal{H}_{0}\bigg] \qquad \text{a.s.} $$

The result then follows by the arbitrariness of $c\in \mathcal{C}^{ \mathbf{H},k}(v)$. □

Proof of Corollary 3.8

In view of Proposition 3.5, to compute $u^{\mathbf{H},k}(v)$, it suffices to find the $\mathcal{H}_{0}$-measurable random variable $\Lambda ^{\mathbf{H},k}(v)$ satisfying (3.3).

(i) If $U(\omega ,t,x)=\log x$, then $I(\omega ,t,y) = 1/y$ for all $y \in (0,+\infty )$. Therefore (3.3) can be explicitly solved and we find

$$ \Lambda ^{\mathbf{H},k}(v) = \frac{\mathbb{E}[\kappa _{T}|\mathcal{H} _{0}]}{v+k(1-\mathbb{E}[Z^{\mathbf{H}}_{T}|\mathcal{H}_{0}])}. $$

By Proposition 3.5, the solution $c^{\mathbf{H}}=(c^{ \mathbf{H}}_{t})_{t\in [0,T]}$ is therefore given by

$$ c^{\mathbf{H}}_{t} = \frac{1}{\Lambda ^{\mathbf{H},k}(v)Z^{\mathbf{H}} _{t}} = \frac{v+k(1-\mathbb{E}[Z^{\mathbf{H}}_{T}|\mathcal{H}_{0}])}{Z ^{\mathbf{H}}_{t}\,\mathbb{E}[\kappa _{T}|\mathcal{H}_{0}]} \qquad \text{for all }t\in [0,T]. $$

Under the integrability assumption in the corollary, the optimal expected utility $u^{\mathbf{H},k}(v)$ in (3.4) can be obtained by means of a straightforward computation.

(ii) If $U(\omega ,t,x)=x^{p}/p$, then $I(\omega ,t,y) = y^{1/(p-1)}$ for all $y \in (0,+\infty )$. By Proposition 3.5, the $\mathcal{H}_{0}$-measurable random variable $\Lambda ^{\mathbf{H},k}(v)$ must solve

$$ \mathbb{E}\bigg[ \int _{0}^{T}(Z^{\mathbf{H}}_{u})^{\frac{p}{p-1}} \big( \Lambda ^{\mathbf{H},k}(v) \big)^{\frac{1}{p-1}} \,\! \mathrm{d}\kappa _{u} \bigg| \mathcal{H}_{0} \bigg] = v+k(1-\mathbb{E}[Z ^{\mathbf{H}}_{T}|\mathcal{H}_{0}]). $$

Therefore if $\mathbb{E}[\int _{0}^{T}(Z^{\mathbf{H}}_{u})^{p/(p-1)}\, \!\mathrm{d}\kappa _{u}|\mathcal{H}_{0}]<+\infty$ a.s., it holds that

$$ \Lambda ^{\mathbf{H},k}(v) = \big(v+k(1-\mathbb{E}[Z^{\mathbf{H}}_{T}| \mathcal{H}_{0}])\big)^{p-1} \mathbb{E}\bigg[ \int _{0}^{T} (Z^{ \mathbf{H}}_{u})^{\frac{p}{p-1}}\,\!\mathrm{d}\kappa _{u} \bigg| \mathcal{H}_{0} \bigg]^{1-p}. $$

By Proposition 3.5, the optimal consumption process $c ^{\mathbf{H}}=(c^{\mathbf{H}}_{t})_{t\in [0,T]}$ is given by $c_{t} ^{\mathbf{H}} = (\Lambda ^{\mathbf{H},k}(v) Z^{\mathbf{H}}_{t})^{1/(p-1)}$ for all $t\in [0,T]$. If

$$ \mathbb{E}\bigg[\int _{0}^{T}(Z^{\mathbf{H}}_{u})^{p/(p-1)}\,\! \mathrm{d}\kappa _{u}\bigg|\mathcal{H}_{0}\bigg]^{1-p} \in L^{1}( \mathbb{P}), $$

the optimal expected utility $u^{\mathbf{H},k}(v)$ is finite and can be explicitly computed as in (3.5).

(iii) We first show that (3.7) admits an a.s. unique solution for every $v>v^{\mathbf{H}}_{k}$. Arguing similarly as in Muraviev [32, Theorem 3.2], define the $\mathcal{H}_{0}$-measurable function $g:\Omega \times (0,+\infty ) \rightarrow \mathbb{R}_{+}$ by

$$ g(\lambda ) := \frac{1}{\alpha }\mathbb{E}\bigg[\int _{0}^{T}Z^{ \mathbf{H}}_{u}\bigg(\log \frac{\alpha }{\lambda Z^{\mathbf{H}}_{u}} \bigg)^{+} \,\! \mathrm{d}\kappa _{u}\bigg|\mathcal{H}_{0}\bigg] \qquad \text{for }\lambda \in (0,+\infty ). $$

Note that $g$ is well defined since

$$ g(\lambda ) = \frac{1}{\alpha }\mathbb{E}\bigg[\int _{0}^{T}Z^{ \mathbf{H}}_{u}\log \bigg(\frac{\alpha }{\lambda Z^{\mathbf{H}}_{u}} \bigg)\mathbf{1}_{\{Z^{\mathbf{H}}_{u}\leq \alpha /\lambda \}} \,\! \mathrm{d}\kappa _{u}\bigg|\mathcal{H}_{0}\bigg] \leq \frac{\mathbb{E}[ \kappa _{T}|\mathcal{H}_{0}]}{\lambda } < +\infty \text{ a.s.} $$

Clearly, $g$ is a decreasing function. Furthermore, dominated convergence implies that $g$ is continuous. Again by dominated convergence, we get $\lim _{\lambda \rightarrow +\infty }g( \lambda )=0$ a.s., and a straightforward application of Fatou’s lemma yields that $\lim _{\lambda \downarrow 0}g(\lambda )=+\infty$ a.s. Moreover, for all $0<\lambda '<\lambda <+\infty$, it holds that $g(\lambda ')>g( \lambda )$ a.s. on $\{g(\lambda )>0\}$. Indeed, if the $\mathcal{H}_{0}$-measurable set $G_{\lambda ,\lambda '}:=\{g( \lambda )=g(\lambda '),g(\lambda )>0\}$ has strictly positive probability, then it holds on that set that

$$ \mathbb{E}\bigg[\int _{0}^{T}Z^{\mathbf{H}}_{u}\bigg(\Big(\log \frac{ \alpha }{\lambda ' Z^{\mathbf{H}}_{u}}\Big)^{+}-\Big(\log \frac{ \alpha }{\lambda Z^{\mathbf{H}}_{u}}\Big)^{+}\bigg) \,\! \mathrm{d}\kappa _{u}\bigg|\mathcal{H}_{0}\bigg] = 0. $$

However, since $\log (\alpha /(\lambda 'Z^{\mathbf{H}}_{u}))>\log ( \alpha /(\lambda Z^{\mathbf{H}}_{u}))$ for all $u\in [0,T]$, this contradicts the assumption that $g(\lambda )>0$. In view of these observations,

$$ v+k\bigl(1-\mathbb{E}[Z^{\mathbf{H}}_{T}|\mathcal{G}_{0}](\omega ) \bigr)\in \{g(\lambda )(\omega ):\lambda \in (0,+\infty )\} $$

for a.a. $\omega \in \Omega$. Therefore, by Benes̆ [10, Lemma 1], (3.7) admits a unique strictly positive $\mathcal{H}_{0}$-measurable solution $\Lambda ^{\mathbf{H},k}(v)$ for every $v>v^{\mathbf{H}}_{k}$. If $U( \omega ,t,x)=-e^{-\alpha x}$, then $I(\omega ,t,y)=(1/\alpha )( \log (\alpha /y))^{+}$ for all $y\in (0,+\infty )$. By Proposition 3.5, the optimal consumption process $c^{\mathbf{H}}=(c ^{\mathbf{H}}_{t})_{t\in [0,T]}$ is given by

$$ c^{\mathbf{H}}_{t}=\frac{1}{\alpha }\bigg(\log \frac{\alpha }{ \Lambda ^{\mathbf{H},k}(v)Z^{\mathbf{H}}_{t}}\bigg)^{+} $$

for all $t\in [0,T]$, thus proving (3.6). □