Arbitrage and utility maximization in market models with an insider

Abstract

We study arbitrage opportunities, market viability and utility maximization in market models with an insider. Assuming that an economic agent possesses an additional information in the form of an $\mathscr {F}_T$-measurable discrete random variable G, we give criteria for the no unbounded profits with bounded risk property to hold, characterize optimal arbitrage strategies, and prove duality results for the utility maximization problem faced by the insider. Examples of markets satisfying NUPBR yet admitting arbitrage opportunities are provided. For the case when G is a continuous random variable, we consider the notion of no asymptotic arbitrage of the first kind (NAA1) and give an explicit construction for unbounded profits if NAA1 fails.

Appendix

Lemma 4.11

Assume that X, Y are two independent exponential random variables with parameters $\alpha , \beta$, respectively. Then the random variable $Z = \frac{\alpha X}{\beta Y}$ has density $1/(1+ z)^2.$

Proof

For $z > 0$. we compute the cumulative distribution of Z

$$\begin{aligned} \mathbb {P}[Z \le z]&= \mathbb {P}\left[ Y \ge \frac{\alpha X}{\beta z}\right] = \int \limits _0^{\infty } {\left( \int \limits _{(\alpha x) / (\beta z)}^{\infty } {\beta {e}^{-\beta y}dy} \right) \alpha {e}^{-\alpha x}{d}x}\\&= \int \limits _0^{\infty } {{e}^{\frac{-\alpha x}{z}} \alpha {e}^{-\alpha x}{d}x}= \frac{z}{1 + z}. \end{aligned}$$

The density of Z is obtained by taking derivative of the cumulative distribution of Z with respect to z. $\square$

Definition 4.12

(Optional projection—Definition 5.2.1 of [23]) Let X be a bounded (or positive) process, and $\mathbb {F}$ a given filtration. The optional projection of X is the unique optional process ${}^{o}X$ which satisfies

$$\begin{aligned} \mathbb {E}[X_{\tau }1_{\{\tau<\infty \}}] ={}^{o}X_{\tau }1_{\{\tau < \infty \}} \end{aligned}$$

almost surely for any $\mathbb {F}$-stopping time $\tau .$

The following result helps us to find the compensator of a process when passing to smaller filtrations.

Lemma 4.13

Let $\mathbb {G}, \mathbb {H}$ be filtrations such that $\mathscr {G}_t \subset \mathscr {H}_t,$ for all $t \in [0,T]$. Let X be a $\mathbb {G}$-adapted process. Suppose that the process $M_t := X_t - \int \nolimits _0^t {\lambda _udu}$ is a $\mathbb {H}$-martingale, where $\lambda \ge 0$. Then the process $M^G_t :=X_t - \int \nolimits _0^t {^{o}\lambda _u du}$ is a $\mathbb {G}$-martingale, where $^{o}\lambda$ is the optional projection of $\lambda$ onto $\mathbb {G}$.

Proof

Since $\lambda _u \ge 0,$ the optional projection $^{o}\lambda$ exists and for fixed u, it holds that $^{o}\lambda _u = \mathbb {E}[\lambda _u| \mathscr {G}_u]$ almost surely. If $0\le s <t$ and H is bounded and $\mathscr {G}_s$-measurable, then, by Fubini’s Theorem

$$\begin{aligned} \mathbb {E}\left[ H\left( M^G_t - M^G_s\right) \right]&= \mathbb {E}[H(X_t - X_s)] - \int \limits _s^t {\mathbb {E}[H\mathbb {E}[\lambda _u|\mathscr {G}_u]]du}\\&= \mathbb {E}[H(X_t - X_s)] - \int \limits _s^t {\mathbb {E}[H\lambda _u]du}\\&= \mathbb {E}[H(M_t - M_s)] = 0. \end{aligned}$$

Hence $M^G$ is a $\mathbb {G}$-martingale.$\square$