On arbitrages arising with honest times

Abstract

In the context of a general continuous financial market model, we study whether the additional information associated with an honest time τ gives rise to arbitrage profits. By relying on the theory of progressive enlargement of filtrations, we explicitly show that no kind of arbitrage profit can ever be realised strictly before τ, whereas classical arbitrage opportunities can be realised exactly at τ as well as after τ. Moreover, arbitrages of the first kind can only be obtained by starting to trade as soon as τ occurs. We carefully study the behavior of local martingale deflators and consider no-arbitrage-type conditions weaker than no free lunch with vanishing risk.

Appendix

1.1 Proof of Lemma 3.5

Due to the representation (3.4), for any $\mathbb {F}$-stopping time σ and any local martingale deflator L in $\mathbb {G}$ on the time interval [0,σ∧τ], we can write

$$ \begin{aligned} E\left[L_{\sigma\wedge\tau}\right] &= E\left[L_{\sigma} \mathbf {1}_{\left\{\sigma<\tau\right\}}\right] +E\left[L_{\tau} \mathbf {1}_{\left\{\tau\leq\sigma\right\}}\right] \\ &= E\left[\frac{1}{N_{\sigma}}\exp\left(-\int_0^{\sigma} \frac {k_s}{N^*_s}\,dN^*_s\right) \mathbf {1}_{\left\{\sigma<\tau\right\}}\right] \\ &\quad\ \ {} +E\left[\frac{1}{N_{\tau}}\exp\left(-\int_0^{\tau} \frac {k_s}{N^*_s}\,dN^*_s\right)\left(1+k_{\tau}+\eta\right)\mathbf {1}_{\left\{\tau \leq\sigma\right\}}\right]. \end{aligned} $$

(A.1)

Let us first focus on the term on the second line of (A.1). Recall that τ<ν P-a.s. (see the proof of Proposition 3.1) and that $Z=\left(Z_{t}\right)_{t\geq0}$ is the $\mathbb {F}$-optional projection of $(\mathbf {1}_{\left\{\tau>t\right\}})_{t\geq0}$ and $Z_{\sigma}/N_{\sigma}=1/N^{*}_{\sigma}$ on the set {σ<ν} (see Lemma 2.10). Then we can write

$$ \begin{aligned} &E\left[\frac{1}{N_{\sigma}}\exp\left(-\int_0^{\sigma} \frac {k_s}{N^*_s}\,dN^*_s\right) \mathbf {1}_{\left\{\sigma<\tau\right\}}\right] \\ &\quad= E\left[\frac{1}{N_{\sigma}}\exp\left(-\int_0^{\sigma} \frac {k_s}{N^*_s}\,dN^*_s\right) \mathbf {1}_{\left\{\sigma<\tau\right\}}\mathbf {1}_{\left\{\sigma<\nu\right\}}\right ] \\ &\quad = E\left[\frac{1}{N_{\sigma}}\exp\left(-\int_0^{\sigma} \frac {k_s}{N^*_s}\,dN^*_s\right) Z_{\sigma} \mathbf {1}_{\left\{\sigma<\nu\right\}}\right] \\ &\quad= E\left[\frac{1}{N^*_{\sigma}}\exp\left(-\int_0^{\sigma} \frac {k_s}{N^*_s}\,dN^*_s\right) \mathbf {1}_{\left\{\sigma<\nu\right\}}\right] \\ &\quad = E\left[\exp\left(-\int_0^{\sigma} \frac{1+k_s}{N^*_s}\,dN^*_s\right) \mathbf {1}_{\left\{\sigma<\nu\right\}}\right]. \end{aligned} $$

(A.2)

Let us now compute more explicitly the second term on the right-hand side of (A.1). Recall that $E[\eta|\mathcal {G}_{\tau-}]=0$ (see Jeanblanc and Song [23], Theorem 6.2). Recall also that since all $\mathbb {F}$-local martingales are continuous (by Assumption 2.8 together with the continuity of S), Corollary 2.4 of Nikeghbali and Yor [38] (see also Mansuy and Yor [33], Exercise 1.8) implies that the dual $\mathbb {F}$-predictable projection of the process (1 _{τ≤t}) _t≥0 is given by $(\log N^{*}_{t} )_{t\geq0}$. Moreover, the measure $dN^{*}_{s}$ is supported by the set $\{ s\geq0:Z_{s}=1\}=\{s\geq0:N_{s}=N^{*}_{s}\}$. By first taking the $\mathcal {G}_{\tau -}$-conditional expectation and then recalling that $\{\tau>\sigma\}\in \mathcal {G}_{\tau-}$ and that k _τ is $\mathcal {G}_{\tau-}$-measurable (see Jacod and Shiryaev [22], Proposition I.2.4), we can write

$$ \begin{aligned} &E\left[\frac{1}{N_{\tau}}\exp\left(-\int_0^{\tau} \frac{k_s}{N^*_s}\, dN^*_s\right)\left(1+k_{\tau}+\eta\right)\mathbf {1}_{\left\{\tau\leq\sigma \right\}}\right] \\ &\quad= E\left[\frac{1}{N_{\tau}}\exp\left(-\int_0^{\tau} \frac {k_s}{N^*_s}\,dN^*_s\right)\left(1+k_{\tau}\right) \mathbf {1}_{\left\{\tau\leq\sigma\right\}}\right] \\ &\quad= E\left[\int_0^{\sigma} \frac{1}{N_s}\exp\left(-\int_0^s \frac {k_u}{N^*_u}\,dN^*_u\right)\left(1+k_s\right)\frac{1}{N^*_s}\, dN^*_s\right] \\ &\quad= E\left[\int_0^{\sigma} \frac{1}{N^*_s}\exp\left(-\int_0^s \frac {k_u}{N^*_u}\,dN^*_u\right)\left(1+k_s\right)\frac{1}{N^*_s}\, dN^*_s\right] \\ &\quad= E\left[\int_0^{\sigma} \exp\left(-\int_0^s \frac{1+k_u}{N^*_u}\, dN^*_u\right)\left(1+k_s\right)\frac{1}{N^*_s}\,dN^*_s\right] \\ &\quad= E\left[1-\exp\left(-\int_0^{\sigma} \frac{1+k_s}{N^*_s}\, dN^*_s\right)\right]. \end{aligned} $$

(A.3)

Equation (3.5) then follows by combining (A.2) and (A.3), using the fact that since τ<ν P-a.s., we have σ>τ on the set {σ≥ν}, and noting that the process N ^∗ is constant after τ. In order to show that $\int_{0}^{\tau} \frac{1+k_{s}}{N^{*}_{s}}\,dN^{*}_{s}>0$ P-a.s., it suffices to use the fact that 1+k _τ >0 P-a.s. (see Remark 3.4) and note that

$$E\left[\frac{1+k_{\tau}}{N^*_{\tau}} \mathbf {1}_{\{\int_0^{\tau} \frac {1+k_s}{N^*_s}dN^*_s=0\}}\right] = E\left[\int_0^{\infty}\frac{1+k_s}{N^*_s}\mathbf {1}_{\{\int_0^s \frac {1+k_u}{N^*_u}dN^*_u=0\}}\frac{1}{N^*_s}\,dN^*_s\right] = 0, $$

which implies that $\int_{0}^{\tau} \frac{1+k_{s}}{N^{*}_{s}}\,dN^{*}_{s}>0$ P-a.s. The last assertion follows from the fact that the nonnegative $\mathbb {G}$-local martingale L ^σ∧τ is a uniformly integrable $\mathbb {G}$-martingale if and only if $E[L^{\sigma\wedge\tau}_{\infty }]=E[L_{\sigma\wedge\tau}]=1$. Due to (3.5), the latter holds if and only if P(ν≤σ)=0.  □