2.1. Time transformation

The key tool for translating (2.2) into a constrained SEP is the Dambis–Dubins–Schwarz theorem. However, we need to be careful in defining the time change, since we want to be able to shift pathwise inequalities from ${\mathcal C}_{s_0}[0,T]$ to the Wiener space and back. Moreover, the time change will be a useful tool to precisely define the options we want to consider as well as the set of feasible paths for the insider.

For $\omega\in{\mathcal C}({\mathbb{R}}_+)$ and $n\in{\mathbb{N}}$ , we define the sequence of times

\begin{equation*}\sigma^n_0(\omega)\,:\!=\,0,\quad \sigma^n_{k+1}(\omega)\,:\!=\,\inf\{t>\sigma^n_k(\omega) \,:\, |\omega(t)-\omega(\sigma^n_k)|\geq 2^{-n}\},\qquad k\in{\mathbb{N}}.\end{equation*}

We say that $\omega$ has quadratic variation if the sequence $(V_n(\omega))_{n\in{\mathbb{N}}}$ of functions

\begin{equation*}V_n(\omega)(t)\,:\!=\,\sum_{k=0}^\infty(\omega(\sigma^n_{k+1}\wedge t)-\omega(\sigma^n_k\wedge t))^2,\qquad t\in{\mathbb{R}}_+,\end{equation*}

converges uniformly on compacts to some function in ${\mathcal C}_0({\mathbb{R}}_+)$ , and the limit function has the same intervals of constancy as $\omega$ . We denote this limit function by $\langle \omega\rangle$ . We write ${\Omega^{\mathsf{qv}}}$ for the space of all paths $\omega$ in ${\mathcal C}_{s_0}({\mathbb{R}}_+)$ possessing such quadratic variation and such that either $\langle \omega\rangle$ diverges at infinity, or $\langle \omega \rangle$ is bounded and $\omega$ has a well defined limit at infinity. These conditions are necessary in order for the map ${\text{ntt}}$ given below to be well defined. It is not hard to show that ${\Omega^{\mathsf{qv}}}$ is a measurable subset of ${\mathcal C}({\mathbb{R}}_+)$ .

We define the space of stopped paths as

\begin{equation*}{\mathcal S}\,:\!=\,\left\{(\kern1.4pt f,s) \,:\, f\in{\mathcal C}_{s_0}[0,s], s\in{\mathbb{R}}_+\right\},\end{equation*}

and equip it with the distance $d_{\mathcal S}$ defined for $s<t$ by

\begin{equation*}d_{\mathcal S}((\kern1.4pt f,s),(g,t))=\max\Big\{t-s,\sup_{0\leq u\leq s}|f(u)-g(u)|,\sup_{s\leq u\leq t}|g(u)-f(s)|\Big\},\end{equation*}

which turns ${\mathcal S}$ into a Polish space. The space ${\mathcal S}$ is a convenient way of encoding optionality of a process in our pathwise setup (see e.g. [Reference Dellacherie and Meyer26, Theorem IV.97]); note that optionality is equivalent to predictability, since we consider only continuous paths. More precisely, we set

\begin{equation*} r\,:\,{\mathcal C}_{s_0}({\mathbb{R}}_+)\times {\mathbb{R}}_+ \to {\mathcal S}, \quad (\omega,t)\mapsto (\omega|_{[0,t]},t),\end{equation*}

where $\omega|_{[0,t]}$ denotes the restriction of $\omega$ to [0,t]. Then a process X with $X_0=s_0$ is optional if and only if there is a Borel function $H\,:\,{\mathcal S}\to{\mathbb{R}}$ such that $X=H\circ r$ .

We denote by $\Omega_T^{\mathsf{qv}}$ the set of paths in ${\mathcal C}_{s_0}[0,T]$ which have a continuation in $\Omega^{\mathsf{qv}}$ , and for $\omega\in\Omega_T^{\mathsf{qv}}$ we define the following new clock:

(2.3) \begin{equation}\tau_t(\omega)=\inf\{s\in[0,T] \,:\, \langle\omega\rangle_s > t\}\wedge T,\qquad t\in{\mathbb{R}}_+,\end{equation}

with the usual convention $\inf \emptyset=+\infty$ .

We will work with the normalising time transformation introduced by Vovk [Reference Vovk48], which is written as ${\text{ntt}}_T\,:\,{\Omega^{\mathsf{qv}}}_T\to{\mathcal S}$ , defined by

\begin{equation*}{\text{ntt}}_T(\omega)=((\omega_{\tau_t})_{t \le \langle\omega\rangle_T},\langle\omega\rangle_T).\end{equation*}

That is, ${\text{ntt}}_T(\omega)$ is a version of the path $\omega$ run at a speed such that, for every t, its pathwise quadratic variation at time t is exactly t. It will also be notationally useful at times to ‘forget’ the time component, and consider the function ${\text{ntt}}(\omega)$ , which is equal to $(\omega_{\tau_t})_{t \le \langle\omega\rangle_T} \in {\mathcal C}_{s_0}[0, \langle\omega\rangle_T]$ . Of course, the two quantities are mathematically equivalent. The normalising time transformation is the tool that will allow us to define the class of time-invariant derivatives and the kind of time-invariant additional information which are suitable for developing the SEP approach to robust pricing with insider information.

In this article, we consider payoff functions $F\,:\,{\mathcal C}_{s_0}[0,T]\to{\mathbb{R}}$ which on ${\Omega^{\mathsf{qv}}_{T}}$ satisfy

(2.4) \begin{equation}F=\gamma\circ {\text{ntt}}_T,\end{equation}

for some Borel-measurable $\gamma\,:\,{\mathcal S}\to{\mathbb{R}}$ . This means that the payoff function F is identical for all paths which are time-transformations of each other, that is, which coincide after normalisation of the speed at which they run.

A key additional component in our model will be the information which is held by the insider, and which is not known to the market. We will model this by assuming that the insider knows a set of feasible paths ${\mathcal A}\subseteq{\mathcal C}_{s_0}[0,T]$ . Thanks to Remark 1, we may assume without loss of generality that ${\mathcal A}\subseteq{\Omega^{\mathsf{qv}}_{T}}$ . As with the payoff function, we will assume that the set ${\mathcal A}$ of feasible paths is time-invariant. More precisely, we will consider sets ${\mathcal A}$ given by

(2.5) \begin{align}\mathcal A = {\text{ntt}}_T^{-1}(\Lambda)\end{align}

for some measurable subset $\Lambda\subseteq {\mathcal S}$ , so that $1_{\mathcal A}(\omega)=1_\Lambda \circ {\text{ntt}}_T(\omega)$ . We will call $\Lambda$ the feasibility set. In this way, feasibility of a path $\omega\in{\mathcal C}_{s_0}[0,T]$ is shifted to admissibility of the stopped path $({\text{ntt}}(\omega),\langle\omega\rangle_T)$ . In particular, if a path $\omega \in {\mathcal C}_{s_0}[0,T]$ is feasible, so is any other path which is a time-transformation of $\omega$ .

2.2. Informed robust pricing as constrained SEP

The time transformation introduced above enables us to express the robust pricing problem (2.2) as a constrained optimal stopping problem for Brownian motion.

Proposition 1. Let F and ${\mathcal A}$ satisfy (2.4) and (2.5). The pricing problem for the insider, (2.2), can be formulated as

(2.6) \begin{equation}P^*_\Lambda\,:\!=\,\sup\left\{{\mathbb{E}}[\gamma((W_t)_{t\leq\tau},\tau)] \,:\, \begin{aligned}&(\tilde\Omega,({\mathcal G}_t)_{t\geq 0},{\mathcal G},{\mathbb{P}}) {supporting\ Brownian\ motion}\ W, \\[4pt] & W_0=s_0, \tau \textrm{ a } {\mathcal G} \mbox{-}{stopping\ time\ s.t. }\ W_\tau\sim\mu,\\[4pt] & \, (W_{t\wedge\tau})_{t\geq 0}\ {is\ u.i.},\, {and}\ {\mathbb{E}}[1_\Lambda((W_t)_{t\leq\tau},\tau)]=1\end{aligned}\right\}.\end{equation}

The condition ${\mathbb{E}}[1_\Lambda((W_t)_{t\leq\tau},\tau)]=1$ means that, when moving along a path of W, we can stop only at times such that the stopped path lies in $\Lambda$ . This corresponds to the fact that informed agents only need to take into account the paths in the feasibility set, $\Lambda$ . The condition of uniform integrability on $W_{\cdot\wedge\tau}$ is—in the current setup—equivalent to $\tau$ being minimal; cf. [Reference Monroe40]. This means that, for any other stopping time $\tau^{\prime}$ in the same filtered probability space,

(2.7) \begin{align}\tau^{\prime}\leq \tau\, \text{ and }\, W_{\tau^{\prime}}\sim \mu\, \text{ imply }\, \tau^{\prime}=\tau\, \text{ almost surely (a.s.).}\end{align}

The reformulation in (2.6) shows how the maturity time T, at which we know the marginal distribution $\mu$ of the price process, does not play any role in the pricing problem. This is a consequence of the time-invariance assumption.

Proof. The proof of this essentially follows from [Reference Beiglböck, Cox, Huesmann, Perkowski and Prömel13, Section 4]: let ${\mathbb{Q}}\in{\mathcal M}(\mu)$ , $(\tau_t)_{t\in{\mathbb{R}}_+}$ be the time change defined in (2.3), and $({\mathcal F}^S_t)_{t\in[0,T]}$ the usual augmentation of the filtration generated by $(S_t)_{t\in[0,T]}$ . It is easy to verify that $\langle S\rangle_T$ is a stopping time with respect to the filtration $({\mathcal F}^S_{\tau_t\wedge T})_{t\in{\mathbb{R}}_+}$ . Then, the Dambis–Dubins–Schwarz theorem implies that the process $(X_t)_{t\in{\mathbb{R}}_+}=({\text{ntt}}(S)_{t\wedge\langle S\rangle_T})_{t\in{\mathbb{R}}_+}$ is a stopped Brownian motion under ${\mathbb{Q}}$ in the filtration $({\mathcal F}^S_{\tau_t\wedge T})_{t\in{\mathbb{R}}_+}$ . Moreover, it is uniformly integrable and satisfies ${\text{ntt}}(S)_{\langle S\rangle_T}\sim\mu$ . Vice versa, let W be a Brownian motion on some probability space $(\tilde\Omega,({\mathcal G}_t)_{t\geq 0},{\mathbb{P}})$ , and let $\tau$ be a stopping time such that $W_{.\wedge\tau}$ is uniformly integrable with $W_\tau\sim\mu$ . Then, for $M=(M_t)_{t\in[0,T]}$ defined by

\begin{equation*}M_t\,:\!=\,W_{\frac{t}{T-t}\wedge\tau},\end{equation*}

we have that ${\mathbb{P}}\circ M^{-1}\in{\mathcal M}(\mu)$ . The result follows.

To be able to analyse the optimisation problem (2.6), we introduce another optimisation problem living on a single probability space, the Wiener space $({\mathcal C}_{s_0}({\mathbb{R}}_+),{\mathcal F},{\mathbb{W}})$ . To this end we consider the set

\begin{align*}M=\{\xi\in{\mathcal P}({\mathcal C}_{s_0}({\mathbb{R}}_+)\times{\mathbb{R}}_+) \,:\, \xi(d\omega,dt)=\xi_\omega(dt){\mathbb{W}}(d\omega), \ \xi_\omega &\in {\mathcal P}({\mathbb{R}}_+)\, \\&\textrm{for ${\mathbb{W}}$-almost every $\omega$}\},\end{align*}

where ${\mathcal P}({\mathcal X})$ denotes the set of probability measures on a space ${\mathcal X}$ , and $(\xi_\omega)_{\omega\in{\mathcal C}_{s_0}({\mathbb{R}}_+)}$ is a regular disintegration of $\xi$ with respect to the first coordinate $\omega$ . We equip M with the weak topology induced by the continuous bounded functions on ${\mathcal C}_{s_0}({\mathbb{R}}_+)\times{\mathbb{R}}_+$ . Each $\xi\in M$ can be uniquely characterised by the cumulative distribution function $A^{\xi}(\omega,t)=\xi_\omega[0,t]$ .

Remark 2. It is well known that any randomised stopping time $\xi$ can be identified with a stopping time $\tau_\xi$ on the extended probability space $({\mathcal C}_{s_0}({\mathbb{R}}_+)\times [0,1], {\mathcal F}\otimes \mathcal B,{\mathbb{W}}\otimes\mathcal L)$ , where $\mathcal B$ denotes the Borel $\sigma$ -algebra on [0,1], and $\mathcal L$ the Lebesgue measure on [0,1]. One way of defining $\tau_\xi$ is via

\begin{equation*}\tau_\xi(\omega,u)\,:\!=\,\inf\{t\geq 0\,:\, \xi_\omega([0,t])\geq u\}.\end{equation*}

As a consequence, the optional stopping theorem applies for randomised stopping times. Indeed, any process X on ${\mathcal C}_{s_0}({\mathbb{R}}_+)$ can be lifted to a process $\bar X$ on ${\mathcal C}_{s_0}({\mathbb{R}}_+)\times [0,1]$ by setting $\bar X_t(\omega,u)\,:\!=\, X_t(\omega)$ , and then the classical optional stopping theorem applies for, e.g., uniformly integrable martingales.

Considering the martingale $W_t^2-t$ , it follows from classical results on stopping times (e.g. [Reference Hobson34, Corollary 3.3], [Reference Beiglböck, Cox and Huesmann12, Lemma 3.12], Remark 2 of this paper) that, for $\xi\in\mathsf{RST}$ with $W_\xi=\mu$ , the condition $\int\! t \xi(d\omega, dt)<\infty$ is equivalent to

(2.8) \begin{align}\int\! t \xi(d\omega, dt)=\int\! (x-s_0)^2 \mu(dx) = V,\end{align}

which is assumed to be finite in our setup.

By [Reference Beiglböck, Cox and Huesmann12, Theorem 3.14], $\mathsf{RST}(\mu)$ is non-empty and compact with respect to the topology induced by the continuous and bounded functions on ${\mathcal C}_{s_0}({\mathbb{R}}_+)\times {\mathbb{R}}_+$ . As a direct consequence we get the following result.

Corollary 1. Let $\Lambda\subseteq{\mathcal S}$ be closed. Then the set of feasible randomised stopping times

(2.9) \begin{equation}\mathsf{RST}(\mu; \Lambda)\,:\!=\,\left\{\xi\in\mathsf{RST}(\mu) \,:\, \int_{{\mathcal C}_{s_0}({\mathbb{R}}_+)\times{\mathbb{R}}_+}1_\Lambda\circ r(\omega,t)\xi(d\omega,dt)=1\right\}\end{equation}

is convex and compact with respect to the topology induced by the continuous and bounded functions on ${\mathcal C}_{s_0}({\mathbb{R}}_+)\times {\mathbb{R}}_+$ .

We highlight here the important feature that $\mathsf{RST}(\mu;\ \Lambda)$ might be empty, which can be understood as a robust arbitrage opportunity; see Proposition 3 and Section 4.

Proof. Since $\Lambda$ is assumed to be closed, the function $1_\Lambda\circ r$ is upper semi-continuous. Hence

\begin{equation*}\int_{{\mathcal C}_{s_0}({\mathbb{R}}_+)\times{\mathbb{R}}_+} 1_\Lambda\circ r(\omega,t)\xi(d\omega,dt)=1\end{equation*}

is a closed condition by the portmanteau theorem.

Another important property of the feasible randomised stopping times is that they are precisely the joint distributions on ${\mathcal C}_{s_0}({\mathbb{R}}_+)\times{\mathbb{R}}_+$ of pairs $(W,\tau)$ satisfying the constraints in (2.6). This is a straightforward extension of [Reference Beiglböck, Cox and Huesmann12, Lemma 3.11]. Putting everything together we have derived a formulation of our optimisation problem (2.2) (resp. (2.6)) on the Wiener space as a constrained SEP.

Proposition 2. In the setting described above,

\begin{equation*}P^*_\Lambda=\sup\left\{\int_{{\mathcal C}_{s_0}({\mathbb{R}}_+)\times{\mathbb{R}}_+}\gamma\circ r(\omega,t)\xi(d\omega,dt) \,:\, \xi\in\mathsf{RST}(\mu;\,\Lambda)\right\}. {(\mathsf{conSEP})}\end{equation*}

We will say that ( $\mathsf{conSEP}$ ) is well posed if

\begin{equation*}\textstyle{\int_{{\mathcal C}_{s_0}({\mathbb{R}}_+)\times{\mathbb{R}}_+}\gamma\circ r d\xi}\end{equation*}

exists with values in $[{-}\infty,\infty)$ for all $\xi \in\mathsf{RST}(\mu;\ \Lambda)$ , and is finite for one such $\xi$ . In particular, ( $\mathsf{conSEP}$ ) is not well posed if $\mathsf{RST}(\mu;\ \Lambda)=\emptyset$ , which has a pleasing financial interpretation (cf. Proposition 3). The (unconstrained) SEP corresponds to the case $\Lambda={\mathcal S}$ , when all paths are feasible, and hence the above supremum is taken over $\mathsf{RST}(\mu)$ .

From an analytical point of view, the formulation ( $\mathsf{conSEP}$ ) is extremely useful since we are now dealing with a linear optimisation problem over a convex and compact set on a single probability space. A direct consequence is the following result.

Theorem 1. Let $\gamma\,:\,{\mathcal S}\to{\mathbb{R}}$ be upper semi-continuous and bounded from above in the sense that, for some constants $a, b, c\in{\mathbb{R}}_+$ ,

(2.10) \begin{align}\gamma((\omega(s))_{s\leq t},t)\leq a+bt+c\sup_{s\leq t}\omega(s)^2,\qquad (\omega,s)\in{\mathcal C}_{s_0}({\mathbb{R}}_+)\times{\mathbb{R}}_+. \end{align}

Assume that $\Lambda\subseteq{\mathcal S}$ is closed and that $\mathsf{RST}(\mu;\ \Lambda)$ is non-empty. Then the optimisation problem ( $\mathsf{conSEP}$ ) admits a maximiser.

Proof. We claim that without loss of generality we can assume that $\gamma$ is bounded from above. Indeed, by the pathwise version of Doob’s inequality (see [Reference Acciaio, Beiglböck, Penkner, Schachermayer and Temme2]),

\begin{equation*} \sup_{s\leq t} \omega(s)^2\leq M_t + 4\omega(t)^2\end{equation*}

for some martingale $M_t$ starting in zero. Hence the condition (2.10) implies that

\begin{equation*}\tilde\gamma\circ r(\omega,t)\,:\!=\,\gamma\circ r(\omega,t) - a^{\prime} - b^{\prime}t -c^{\prime} (M_t + \omega(t)^2)\end{equation*}

is bounded from above, and the term $\int a^{\prime} - b^{\prime}t -c^{\prime} (M_t + \omega(t)^2) d\xi$ is independent of $\xi\in \mathsf{RST}(\mu)$ by (2.8) and the assumed second moment of $\mu$ . Therefore, we can assume $\gamma$ to be bounded from above.

Finally, since $\mathsf{RST}(\mu;\ \Lambda)$ is compact and r is continuous, and by the portmanteau theorem the map $\xi\mapsto \int \gamma\circ r d\xi$ is upper semi-continuous, we deduce the result.

3.1. Duality

In this section, we first show a duality result for the problem $P_\Lambda^*$ defined in (2.6), that is, for ( $\mathsf{conSEP}$ ), and then from it deduce a duality result for the original robust pricing problem $P_{\mathcal A}$ defined in (2.2). The latter is the analogue of the super-replication duality in the present robust setting with additional information/beliefs. As in the classical (non-robust) case, this in turn leads to a dichotomy between existence of martingale measures and existence of arbitrage opportunities, the so-called fundamental theorem of asset pricing, which we prove at the end of this section.

A martingale $\phi$ is called ${\mathcal S}$ -continuous if there exists a continuous $H\,:\,{\mathcal S}\to{\mathbb{R}}$ such that $\phi=H\circ r$ . Note that a martingale which is ${\mathcal S}$ -continuous has continuous paths, but the other implication is in general not true.

Theorem 2. Let $\gamma \,:\,{\mathcal S} \to {\mathbb{R}}$ be upper semi-continuous and bounded from above in the sense of (2.10), and let $\Lambda\subseteq{\mathcal S}$ be closed. Set

(3.1) \begin{align}D^*_\Lambda\,:\!=\,\inf\left\{ \int\psi d\mu \,:\, \begin{array}{l}\psi \in {\mathcal C}({\mathbb{R}}), \exists\ { an\ {\mathcal S}\hbox{-}continuous\ martingale\ \phi, \phi_0=0\ s.t.}\\[5pt] \phi_t(\omega)+\psi(\omega(t))\geq \gamma\circ r(\omega,t)\ { for\ all }\ (\omega,t)\in r^{-1}(\Lambda)\end{array}\! \right\},\end{align}

where $\phi, \psi$ satisfy

(3.2) \begin{equation}|\phi_t(\omega)| \leq a + bt+c \omega(t)^2,\;\; |\psi(y)| \leq a+ b y^2,\quad \forall\ \omega\in{\mathcal C}_{s_0}({\mathbb{R}}^+),\ {for\ some\ a,b,c > 0}.\end{equation}

Then we have

\begin{equation*} P_\Lambda^*=D_\Lambda^* .\end{equation*}

Proof. Put

(3.3) \begin{equation}\overline\gamma(\kern1.4pt f,t)=\begin{cases} \gamma(\kern1.4pt f,t), & \quad (\kern1.4pt f,t) \in \Lambda,\\[5pt] -\infty, & \quad (\kern1.4pt f,t) \in{\mathcal S} \setminus \Lambda.\end{cases}\end{equation}

Since we assume that $\gamma$ is upper semi-continuous and bounded from above in the sense of (2.10), it is easy to see that these properties are inherited by $\overline\gamma$ . Moreover, the well-posedness assumption of ( $\mathsf{conSEP}$ ) for $\gamma$ implies that ( $\mathsf{conSEP}$ ) is still well posed for $\overline\gamma$ and $\Lambda={\mathcal S}$ . Hence, the result follows from [Reference Beiglböck, Cox and Huesmann12, Theorem 4.2].

This duality result is already of interest in its own right. However, to identify it as a super-replication result we need to recover the hedging strategies corresponding to the martingale. For this we need some kind of pathwise martingale representation theorem. In fact Theorem 6.2 of [Reference Vovk48] can be interpreted as such. To this end, we need to introduce some more notation.

We will need the concept of simple strategy, by which we mean a process $H\,:\,\Omega_T^{\mathsf{qv}}\times{\mathbb R}^+\to{\mathbb R}$ of the form

\begin{equation*}H_t(\omega)=\sum_{n\geq 0}K_n(\omega)1_{(\tau_n(\omega),\tau_{n+1}(\omega)]}(t),\qquad (\omega,t)\in \Omega_T^{\mathsf{qv}}\times{\mathbb R}^+,\end{equation*}

where $0=\tau_0(\omega)<\tau_1(\omega)<\ldots$ are ${\mathcal F}^S$ -stopping times such that for every $\omega$ one has $\lim_{n\to\infty}\tau_n(\omega)=\infty$ , and $K_n\,:\,\Omega_T^{\mathsf{qv}}\to{\mathbb R}$ are ${\mathcal F}^S_{\tau_n}$ -measurable bounded functions for $n\in{\mathbb N}$ . For such a strategy, we can define the corresponding pathwise stochastic integral as

\begin{equation*}(H\cdot S)_t(\omega)=\sum_{n\geq 0}K_n(\omega)(S_{\tau_{n+1}(\omega)\wedge t}-S_{\tau_{n}(\omega)\wedge t})(\omega).\end{equation*}

Then, following exactly the line of reasoning as for the proof of [Reference Beiglböck, Cox, Huesmann, Perkowski and Prömel13, Theorem 3.1], one can get the following result. We recall that $F = \gamma \circ {\text{ntt}}_T$ and ${\mathcal A} = {\text{ntt}}^{-1}_T (\Lambda)$ , and that the robust pricing problem for the insider was defined in (2.2).

Theorem 3. Let $\gamma$ be upper semi-continuous and bounded from above in the sense of (2.10), and let $\Lambda\subseteq{\mathcal S}$ . Set

\begin{equation*}D_{\mathcal A}\,:\!=\,\inf\left\{\int \psi(y)\, d\mu(y)\,:\, \begin{array}{l} \psi \in {\mathcal C}({\mathbb{R}}), \exists\ { simple\ strategies}\ (H^n)_n\ {s.t.} \\[5pt] \liminf_n(H^n\cdot S)_T(\omega)+\psi(\omega(T))\geq F(\omega)\ {for\ all\ \omega\in{\mathcal A}} \end{array}\! \right\}, \end{equation*}

where $|\psi(y)| \leq a+ b y^2$ and $(H^n\cdot S)_t\geq -a-bt$ for some $a,b>0$ and all $t \in [0,T]$ . Then we have

\begin{equation*}P_{\mathcal A}=D_{\mathcal A}.\end{equation*}

Theorem 3 is the analogue of the classical super-replication duality theorem, in the present robust insider setting. Moreover, like its classical counterpart, it additionally implies a version of the first fundamental theorem of asset pricing. In the following we will use Theorem 3 with different payoff functions. To stress the dependence on the cost function we will sometimes write $P_{\mathcal A}(F)$ , $D_{\mathcal A}(F)$ .

The property (iii) means that one cannot make arbitrary profits by starting with zero capital. Indeed, if (3.4) holds for some $\hat{\varepsilon}>0$ , then it does so for any ${\varepsilon}>0$ .

Proof. The equivalence between (i) and (ii) follows from the arguments around Proposition 1.

$(i)\Rightarrow(iii)$ : Note that (i) implies $D_{\mathcal A}(\tilde F_0)=P_{\mathcal A} (\tilde F_0) =0$ for any derivative $\tilde F_0$ such that $\tilde F_0=0$ on ${\mathcal A}$ , by Theorem 3. Pick

\begin{equation*} F_0=\begin{cases} 0 & \text{ on } {\mathcal A},\\-\infty & \text{ else.} \end{cases}\end{equation*}

Suppose, for contradiction, that there exist ${\varepsilon}$ , $(H^n)_n$ , and $\psi$ such that (3.4) is satisfied. Then the pair $((H^n)_n,\psi-{\varepsilon})$ is admissible for the dual problem $D_{\mathcal A}(F_0)$ . However, this implies $ D_{\mathcal A}(F_0)\leq-{\varepsilon}$ for $F_0$ , which gives the desired contradiction.

$(iii)\Rightarrow(i)$ : By Theorem 3, if there is no measure ${\mathbb Q}\in{\mathcal M}(\mu)$ such that ${\mathbb Q}({\mathcal A})=1$ , then $D_{\mathcal A}=P_{\mathcal A} = -\infty$ for all derivatives F. In particular, for

\begin{equation*} F_{\varepsilon}=\begin{cases} {\varepsilon} & \text{ on } {\mathcal A},\\-\infty & \text{ else,} \end{cases}\end{equation*}

there exist $(H^n)_n$ and $\psi$ , with $\int \psi d\mu = 0$ , such that (3.4) holds.

3.2. Constrained monotonicity principle

In this section, we provide a modified version of the monotonicity principle of [Reference Beiglböck, Cox and Huesmann12] giving necessary geometric conditions on the support set of an optimiser to ( $\mathsf{conSEP}$ ).

To this end, we denote the concatenation of two paths $(\kern1.4pt f,s),(g,t)\in{\mathcal S}$ by $f\oplus g$ ; i.e.,

\begin{equation*} f\oplus g(u)=\begin{cases} f(u), & u \leq s,\\[4pt] f(s)+g(u-s)-g(0), & s\leq u\leq s+t. \end{cases}\end{equation*}

For $(\kern1.4pt f,s)\in{\mathcal S}$ we define the process $\gamma^{(\kern1.4pt f,s)\oplus}(\omega,t)\,:\!=\, \gamma(\kern1.4pt f\oplus \omega|_{[0,t]},s+t).$

Definition 2. A pair $((\kern1.4pt f,s),(g,t))\in{\mathcal S}\times{\mathcal S}$ is called a feasible stop-go pair, written $((\kern1.4pt f,s),(g,t))\in\mathsf{SG}_\Lambda$ , if $f(s)=g(t)$ ; $(\kern1.4pt f,s)\in\Lambda$ ; the set of $({\mathcal F}_t^W)_{t\geq 0}$ stopping times $\sigma$ satisfying $0<{\mathbb{E}}[\sigma]<\infty$ and $1_\Lambda\circ r(\kern1.4pt f\oplus W,s+\sigma)=1$ a.s. is non-empty; and every such stopping time satisfies

(3.5) \begin{align} {\mathbb{E}}[\gamma^{(\kern1.4pt f,s)\oplus}((W_u)_{u\leq\sigma},\sigma)] + \gamma(g,t) < \gamma(\kern1.4pt f,s) + {\mathbb{E}}[\gamma^{(g,t)\oplus}((W_u)_{u\leq\sigma},\sigma)] \end{align}

and $1_\Lambda\circ r(g\oplus W,t+\sigma)=1$ a.s., where both sides of (3.5) are well defined and the left-hand side is finite. Here, the probability space is assumed to be rich enough to support a Brownian motion W, and $({\mathcal F}^W_t)_{t\geq 0}$ denotes the natural filtration generated by W.

The interpretation is that on average it is better to stop a path at time s with history f, and to run the paths that would have carried on from (f, s) from a previously stopped history (g, t) (to let (g, t) go), as long as this results in a feasible stopping rule. Note that since $f(s) = g(t)$ , the law of the stopped process is not changed. We remark here that—as a consequence of only considering $({\mathcal F}^W_t)_{t\geq 0}$ stopping times—the definition of feasible stop-go pairs is independent of the probability space on which $\sigma$ lives, as long as it is rich enough to support the Brownian motion W. In a similar manner to [Reference Beiglböck, Cox and Huesmann12, Section 5], one could introduce an even stronger notion of feasible stop-go pairs by only considering one particular candidate stopping time. In this article, we do not need this generality.

For a set $\Gamma\subset {\mathcal S}$ we denote by $\Gamma^<$ the set of all stopped paths which have a proper extension in $\Gamma$ :

\begin{equation*}\Gamma^<\,:\!=\,\{(\kern1pt f,s)\in{\mathcal S} \,:\, \exists (g,t)\in\Gamma,\, s<t,\, g|_{[0,s]}=f\}.\end{equation*}

Definition 3. A set $\Gamma\subset \Lambda$ is called feasible $\gamma$ -monotone if

\begin{equation*}\mathsf{SG}_\Lambda\cap(\Gamma^<\times\Gamma)=\emptyset .\end{equation*}

A set $\Gamma\subset {\mathcal S}$ should be viewed as a possible stopping set, i.e. a set of paths $(\omega|_{[0,\tau]},\tau)$ for an admissible stopping strategy $\tau$ in ( $\mathsf{conSEP}$ ). If such a set $\Gamma$ is feasible $\gamma$ -monotone then there is no way of changing the stopping rule in a pathwise fashion, as in (3.5), to produce a feasible stopping rule with higher payoff.

Theorem 4. (Constrained monotonicity principle.) Let $\gamma\,:\,{\mathcal S}\to{\mathbb{R}}$ be Borel. Assume that ( $\mathsf{conSEP}$ ) is well posed and that $\xi\in\mathsf{RST}(\mu;\ \Lambda)$ is an optimiser. Then there exists a feasible $\gamma$ -monotone set $\Gamma\subset{\mathcal S}$ such that

\begin{equation*}\xi(r^{-1}(\Gamma))=1.\end{equation*}

Proof. Taking $\overline\gamma$ as in (3.3), the result follows from [Reference Beiglböck, Cox and Huesmann12, Theorem 5.7].

The constrained monotonicity principle will be an important tool for characterising solutions to ( $\mathsf{conSEP}$ ), and in particular will allow us to deduce geometric features of optimisers. We will illustrate this in the subsequent sections.

4.1. Information as barrier in a certain phase space

We now consider the case where the additional information is of the kind of $\Lambda_1$ in (4.1) and translates into having a barrier in a certain phase space. We will see how in this situation the no-arbitrage condition (cf. Proposition 3, Theorem 5) imposes an order between such a barrier and the barrier characterising the unique optimal stopping for the uninformed agent in such a phase space. These results are notable since the ordering of barriers is a much weaker condition than the convex order condition, significantly strengthening the results of Theorem 5.

4.1.1. The Root phase space

We recall that the Root solution of the (unconstrained) SEP for the distribution $\mu$ is given by

\begin{equation*}\tau_{Root}(\mu)=\inf\{t\geq 0 \,:\, (t,W_t)\in\mathcal{R}\},\end{equation*}

where $\mathcal{R}$ is a closed barrier; that is, $(t,x) \in \mathcal{R}$ implies $(s,x) \in \mathcal{R}$ for $s>t$ ; see [Reference Root44]. This was one of the first known solutions to the SEP; it is optimal when $\gamma(\kern1.4pt f,t)=h(t)$ for a strictly convex function h. The Root solution is illustrated in Figure 1. To avoid trivialities, we assume that our barriers are regular (see [Reference Cox, Obłój and Touzi22]); that is, they are closed and $\{x\,:\, (0,x) \not\in \mathcal{R}\}$ is an open interval containing the origin. Any barrier which is not regular can be replaced by a regular barrier without changing the hitting time. Any regular barrier can be described by its lower semi-continuous barrier function $\mathcal R(x)=\inf\{t\,:\,(t,x)\in\mathcal R\}.$

Figure 1:

The Root solution to the SEP.

For the informed agent we assume that

\begin{equation*}\Lambda=\{r(\omega,t) \,:\, t\leq\overline{\tau}\},\end{equation*}

where the stopping time $\overline{\tau}$ is the hitting time of a regular barrier ${\mathcal B}$ in the phase space (t,W), i.e. a Root-type barrier:

(4.3) \begin{equation}\overline{\tau}=\inf\{t\geq 0 \,:\, (t,W_t)\in{\mathcal B}\}.\end{equation}

As in Theorem 7 below, we are able to determine whether $\mathsf{RST}(\mu;\ \Lambda)$ is empty, and hence whether there is an arbitrage for the informed agent, through properties of the barriers.

Theorem 6. Let the set $\Lambda$ be given by (4.5), with $\overline{\tau}$ of the form in (4.3). Then the set $\mathsf{RST}(\mu;\ \Lambda)$ is non-empty if and only if

(4.4) \begin{equation} {\mathcal B} \subseteq {\mathcal R}, \end{equation}

which yields $\tau_{Root}(\mu)\leq\overline{\tau}$ . In particular, if (4.4) holds, the stopping rule $\tau_{Root}(\mu)$ is admissible for the informed agent, in the sense that $((W_t)_{t\leq\tau_{Root}(\mu)},\tau_{Root}(\mu))\in\Lambda$ a.s.

Proof. We first observe that if (4.4) holds, then we immediately have $\tau_{Root}(\mu)\leq\overline{\tau}$ , and since $\tau_{Root}(\mu)\in \mathsf{RST}(\mu;\ \Lambda)$ , it follows that $\mathsf{RST}(\mu;\ \Lambda) \neq \emptyset$ . To show the reverse implication, suppose, for contradiction, that $\mathsf{RST}(\mu;\ \Lambda)$ is non-empty and ${\mathcal B} \not\subseteq {\mathcal R}$ . This means that there exist pairs $(t,x)\in{\mathcal B}\setminus{\mathcal R}$ . Among those pairs, we consider a fixed $(\hat t, \hat x)$ such that there are no $(t,\hat x)\in{\mathcal B}\setminus{\mathcal R}$ with $t<\hat t$ , as in Figure 2.

Figure 2:

Proof of Theorem 6.

Now consider $\tau^{\prime}\in\mathsf{RST}(\mu;\ \Lambda)$ . Denote the local time of Brownian motion in z by $L^z$ . Since the Root embedding maximises ${\mathbb{E}}\left[L^x_{\tau\wedge t}\right]$ among all stopping times $\tau$ which are minimal embeddings of $\mu$ (cf. (2.7)), simultaneously for all $(t,x)\in{\mathbb{R}}_+\times{\mathbb{R}}$ (e.g. by [Reference Gassiat and Oberhauser31, Theorem 3]), in particular

\begin{equation*}{\mathbb{E}}\left[L^{\hat x}_{\tau^{\prime}\wedge \hat t}\right]\leq {\mathbb{E}}\left[L^{\hat x}_{\tau_{Root}(\mu)\wedge \hat t}\right].\end{equation*}

On the other hand, the path stopped at $\tau^{\prime}$ cannot accumulate any more local time at $\hat x$ after $\hat t$ , i.e. ${\mathbb{E}}\left[L^{\hat x}_{\tau^{\prime}\wedge \hat t}\right] = {\mathbb{E}}\left[L^{\hat x}_{\tau^{\prime}\wedge t}\right]$ for all $t \ge \hat{t}$ , while the Root stopping rule will do so (i.e.,

\begin{equation*}{\mathbb{E}}\left[L^{\hat x}_{\tau_{Root}(\mu)\wedge \hat t}\right] <{\mathbb{E}}\left[L^{\hat x}_{\tau_{Root}(\mu)\wedge t}\right]\end{equation*}

when $t>\hat{t}$ ), because the barrier is assumed to be regular. Therefore,

\begin{equation*}{\mathbb{E}}\left[L^{\hat x}_{\tau^{\prime}}\right]={\mathbb{E}}\left[L^{\hat x}_{\tau^{\prime}\wedge \hat t}\right]\leq {\mathbb{E}}\left[L^{\hat x}_{\tau_{Root}(\mu)\wedge \hat t}\right]<{\mathbb{E}}\left[L^{\hat x}_{\tau_{Root}(\mu)}\right].\end{equation*}

This gives the desired contradiction, since, for any $x\in{\mathbb{R}}$ and any stopping time $\tau\in\mathsf{RST}(\mu)$ ,

\begin{equation*}{\mathbb{E}}\left[L^x_\tau\right]={\mathbb{E}}\left[|W_\tau-x|\right]-|x|=-u_\mu(x)-|x|,\end{equation*}

where $u_\mu$ is the potential function associated to $\mu$ , i.e., $u_\mu(x)=-\int|y-x|\mu(dy)$ .

4.1.2. The Azéma–Yor phase space

We let $\overline{\omega}_t\,:\!=\,\sup_{0\leq s\leq t} \omega_s$ , and define the process $\overline{W}$ analogously. We start by recalling the Azéma–Yor solution of the (unconstrained) SEP. The barycentre function $b_\mu$ of a probability measure $\mu$ is defined by

\begin{equation*}b_{\mu}(x) \,:\!=\, \frac{\int_{[x,\infty)} y \, \mu(\textrm{d} y)}{\mu([x,\infty))}.\end{equation*}

Denote the inverse of $b_\mu$ by $\beta_\mu$ . The solution to the SEP by Azéma and Yor (see [Reference Azéma and Yor8]) is given by

\begin{equation*}\tau_{AY}(\mu)=\inf\{t\geq 0 \,:\, W_t\leq \beta_\mu(\overline{W}_t)\}.\end{equation*}

This is arguably the most renowned solution to the SEP, for which many properties are known, among which is that it maximises stochastically the maximum of the stopped Brownian motion. See the survey article [Reference Obłój42] for further details. We illustrate the Azéma–Yor solution in Figure 3.

Figure 3:

The Azéma–Yor construction.

For the informed agent, we assume that

(4.5) \begin{equation}\Lambda=\{r(\omega,t) \,:\, t\leq\overline{\tau}\},\end{equation}

where the stopping time $\overline{\tau}$ is the hitting time of a barrier in the phase space $(\overline{W},W)$ :

\begin{equation*} \overline{\tau}=\inf\{t\geq 0 \,:\, (\overline{W}_t,W_t)\in{\mathcal H}\},\end{equation*}

where ${\mathcal H}$ is a Borel set ${\mathcal H}\subseteq \{(x,y)\in{\mathbb{R}}_+\times{\mathbb{R}} \,:\, y\leq x\}$ induced by some increasing left-continuous Borel function $h\,:\,{\mathbb{R}}_+\to{\mathbb{R}}$ via

\begin{equation*}{\mathcal H} = \{(x,y) \,:\, y \leq h(x) \},\end{equation*}

so that $(x,y)\in{\mathcal H}$ and $z>x$ imply $(z,y)\in{\mathcal H}$ . Note that this gives

(4.6) \begin{equation}\overline{\tau}=\inf\{t\geq 0 \,:\, W_t\leq h(\overline{W}_t)\};\end{equation}

thus the set $\Lambda$ in (4.5) corresponds to the following set of feasible paths for the informed agent:

(4.7) \begin{equation}\mathcal{A}=\{\omega\in{\mathcal C}_{s_0}[0,T] \,:\, \omega_t>h(\overline{\omega}_t)\; \forall t\in[0,T)\};\end{equation}

that is, the paths that satisfy the drawdown constraint $\omega>h(\overline{\omega})$ during the period [0, T).

We now give a result which shows that, when the agent’s information is given by $\mathcal{A}$ as in (4.7), we can provide a simple necessary and sufficient condition for the existence of consistent models for the informed agent; cf. (1) of Theorem 5. If there is no ambiguity we write $\beta_\mu=\beta$ in the following.

Theorem 7. Let the set $\Lambda$ be given by (4.5), with $\overline{\tau}$ of the form in (4.6). Then the set $\mathsf{RST}(\mu;\ \Lambda)$ is non-empty if and only if

(4.8) \begin{equation} h(x)\leq\beta(x) \quad \textrm{for all $x\in{\mathbb{R}}_+$}, \end{equation}

which yields $\tau_{AY}(\mu)\leq\overline{\tau}$ . In particular, if (4.8) holds, the stopping rule $\tau_{AY}(\mu)$ is admissible for the informed agent, in the sense that $((W_t)_{t\leq\tau_{AY}(\mu)},\tau_{AY}(\mu))\in\Lambda$ a.s.

Proof. We first observe that if (4.8) holds, then we immediately have $\tau_{AY}(\mu) \le \overline{\tau}$ , and since $\tau_{AY}(\mu) \in \mathsf{RST}(\mu;\ \Lambda)$ , it follows that $\mathsf{RST}(\mu;\ \Lambda) \neq \emptyset$ .

For the reverse implication, we suppose that there exists $\hat x\in{\mathbb{R}}_+$ such that $h(\hat x)>\beta(\hat x)$ , as in Figure 4. Then we fix $\tau^{\prime}\in \mathsf{RST}(\mu;\ \Lambda)$ and argue as follows. Define a measure

\begin{equation*} \eta(A)\,:\!=\, {\mathbb{P}}(W_{\tau^{\prime}} \in A, \overline{W}_{\tau^{\prime}} \ge \hat{x})\end{equation*}

and note that, by the martingale property, $\int y \,\eta(\textrm{d} y) =\hat{x} \cdot \eta({\mathbb{R}})$ . Moreover, $\eta(A \cap [\hat{x},\infty)) = \mu(A\cap [\hat{x},\infty))$ , and $\eta(A) \le \mu(A)$ for all Borel sets A.

Figure 4:

Proof of Theorem 7

Define functions $\Phi_{\eta}, \Phi_\mu\,:\,({-}\infty,\hat{x}] \to {\mathbb{R}}$ by

\begin{equation*} \Phi_\eta(x) = \int_{[x,\infty)} y \, \eta(\textrm{d} y) - \hat{x} \cdot \eta([x,\infty)) = \int_{[x,\infty)} (y-\hat{x}) \, \eta(\textrm{d} y),\end{equation*}

and similarly for $\mu$ . Then $\Phi_\mu$ and $\Phi_\eta$ are both increasing on $({-}\infty, \hat{x}]$ , $\Phi_\mu(\hat{x}) =\Phi_\eta(\hat{x})$ , and $\Phi_\mu(x) - \Phi_\eta(x)$ is increasing in x for $x \in ({-}\infty,\hat{x}]$ since $\mu(\textrm{d} y) \ge \eta(\textrm{d}y)$ . Hence we deduce that $\Phi_\mu(x) \le \Phi_\eta(x)$ for $x \le\hat{x}$ .

Now we observe that $\eta(({-}\infty,h(\hat{x}))) = 0$ , so $\Phi_\eta(h(\hat{x})) = 0$ . On the other hand, by the definition of the barycentre function,

\begin{equation*} \beta(\hat{x}) \,:\!=\, \sup\left\{y <\hat{x}\,:\, \Phi_\mu(y) \le 0\right\}.\end{equation*}

It follows from $\Phi_\mu(x) \le \Phi_\eta(x)$ that $h(\hat{x}) \le\beta(\hat{x})$ , contradicting our original assumption.

Let us consider the drawdown constraint in (4.2) which corresponds to $\Lambda=[\![ 0,\overline{\tau}]\!]$ , with $\overline{\tau}$ given as in (4.6) for $h(x)=x-c$ . In this case Theorem 7 implies that $x-c\leq\beta(x)=b_\mu^{-1}(x)$ must hold for there to exist a feasible solution for the informed agent. This condition can of course be rephrased in terms of barycentre functions, since h is the inverse of the barycentre function associated to $\overline \mu\sim W_{\overline{\tau}}$ (by [Reference Azéma and Yor8]). Therefore, the existence of a consistent solution for the insider is equivalent to $b_{\overline \mu}^{-1}\leq b_\mu^{-1}$ , that is, $b_\mu\leq b_{\overline \mu}$ .

Theorem 7 tells us when the pricing problem for the insider (cf. (2.2) or (2.6)) has a feasible solution, and hence an optimiser under the conditions of Theorem 1, but does not tell us anything about the specific optimiser. On the other hand, the constrained monotonicity principle, Theorem 4, allows us to characterise the geometry of optimisers in various settings. We illustrate this in the special situation where the agent wishes to find the stopping times $\tau$ solving ( $\mathsf{conSEP}$ ) in the case $\gamma(\kern1.4pt f,s)=-s^2$ corresponding to the payoff $F(S)=-\langle S\rangle_T^2.$ We are interested in characterising solutions to

(4.9) \begin{align} \min_{\tau\in\mathsf{RST}(\mu;\ \Lambda)} {\mathbb{E}}[\tau^2],\end{align}

where

(4.10) \begin{equation}\Lambda=\{r(\omega,t)\,:\,t\leq \overline\tau\},\;\; \text{with}\;\; \overline\tau=\inf\{t\geq 0 \,:\, W_t\leq h(\overline W_t)\},\end{equation}

for a step function

\begin{equation*}h(x)=\sum_{i=1}^n a_i 1_{[m_{i-1},m_i)}(x)\end{equation*}

with $m_0=s_0 <m_1<\ldots<m_n$ and $a_1\leq a_2\leq\ldots\leq a_n$ . We set $\tilde\Lambda=\{r(\omega,t)\,:\, t <\overline\tau\}$ , and note that $(\kern1.4pt f,s)\in\tilde\Lambda$ and $m_{i-1}\leq \overline{f}_s <m_i$ imply that $f(s)>a_i$ .

We recall that in the unconstrained case, i.e. $h(x)=-\infty$ , the solution is the Root solution $\tau_\mathsf{Root}$ , the first hitting time of a barrier in space–time (see also Section 4.1.1).

In the current setup, the situation is similar.

Theorem 8. Assume that $\mathsf{RST}(\mu;\ \Lambda)\neq \emptyset$ and that the optimisation problem (4.9) is well posed. Then for any optimiser $\hat\tau$ there exists a sequence of barriers $(\mathcal R_i)_{i=1}^n$ such that

\begin{equation*}\hat\tau= \inf\{t\geq 0\,:\, (t,W_t)\in \mathcal R_{\ell(\overline{W}_t)}\},\end{equation*}

where $\ell(m)=\sum_{i=1}^n i1_{[m_{i-1},m_i)}(m).$

Moreover, for each $j\leq i$ it holds that

\begin{equation*} \left(\mathcal R_j\cap [0,\infty)\times (a_i,\infty)\right) \subset \left(\mathcal R_i\cap [0,\infty)\times (a_i,\infty) \right).\end{equation*}

Proof. To avoid too many minus signs, we redefine $\gamma(\kern1.4pt f,s)=s^2$ , and for this proof we consider the minimisation variant of ( $\mathsf{conSEP}$ ).

By Theorem 1 we can find a minimiser, say $\hat\tau$ , to the optimisation problem (4.9). By Theorem 4 we can pick a feasible $\gamma$ -monotone set $\Gamma$ such that $\hat\tau(r^{-1}(\Gamma))=1$ and $\mathsf{SG}_\Lambda\cap (\Gamma^<\times \Gamma)=\emptyset$ .

We claim that

(4.11) \begin{align}\mathsf{SG}_\Lambda \supset \{((\kern1.4pt f,s),(g,t))\in\tilde\Lambda\times\Lambda\,:\,f(s)=g(t), s>t, \overline{f}_s \geq \overline{g}_t \}.\end{align}

Indeed, pick $(\kern1.4pt f,s),(g,t)\in\Lambda$ with $s>t$ and $f(s)=g(t)$ . It holds for any $(k,u)\in {\mathcal S}$ by convexity of $s\mapsto s^2$ (and since we are considering minimisation instead of maximisation) that

\begin{equation*} \gamma(\kern1.4pt f,s)+\gamma(g\oplus k, t+u)<\gamma(\kern1.4pt f\oplus k,s+u)+\gamma(g,t),\end{equation*}

so that (3.5) follows by two observations. First, $(\kern1.4pt f,s)\in \tilde\Lambda$ implies the existence of at least one Brownian stopping time $\sigma$ with $0<{\mathbb{E}}[\sigma]<\infty$ such that $1_\Lambda(\kern1.4pt f\oplus W,s+\sigma)=1$ ; e.g. if $m_{i-1}\leq \overline{f}_s<m_i$ , then the first hitting time of Brownian motion of $\{\frac12 (\kern1.4pt f(s)-a_i), \frac12 (m_i-f(s)\}$ is such a stopping time. Second, any stopping time $\sigma$ with $1_\Lambda(\kern1.4pt f\oplus W,s+\sigma)=1$ necessarily satisfies $1_\Lambda(g\oplus W,t+\sigma)=1$ , since $f(s)=g(t)$ , $\overline{f}_s\geq \overline{g}_t$ , and the function h defining $\Lambda$ is increasing.

Put $\Gamma_i=\Gamma\cap \{(\kern1.4pt f,s)\in{\mathcal S}\,:\, m_{i-1}\leq \overline{f}_s<m_i\}$ and set

\begin{align*} \mathcal R_i^\mathsf{op}\,:\!=\,\{(s,x)\,:\, \exists (g,t)\in\Gamma_i, g(t)=x, s>t\},\\ \mathcal R_i^\mathsf{cl}\,:\!=\,\{(s,x)\,:\, \exists (g,t)\in\Gamma_i, g(t)=x, s\geq t\}. \end{align*}

Pick $(g,t)\in\Gamma_i$ . Then we claim that

\begin{equation*} \inf\{s\in[0,t]\,:\, (s,g(s))\in \mathcal R_i^\mathsf{cl}\}\leq t \leq \inf\{s\in[0,t]\,:\, (s,g(s))\in \mathcal R_i^\mathsf{op}\}.\end{equation*}

Since the first inequality holds by construction, suppose for contradiction that $\inf\{s\in[0,t]\,:\, (s,g(s))\in \mathcal R_i^\mathsf{op}\}<t$ . In this case, there is $s<t$ such that $(g|_{[0,s]},s)\,=\!:\,(\kern1.4pt f,s)\in\Gamma_i^<$ , $(s,f(s))\in\mathcal R_i^\mathsf{op}$ , and since $s<t$ it holds that $f(s)>a_i$ . Then there exists $(k,u)\in \Gamma_i$ such that $u<s$ and $k(u)=f(s)>a_i$ so that $(k,u)\in\tilde \Lambda$ . However, by (4.11), this means that $((\kern1.4pt f,s),(k,u)) \in \mathsf{SG}_\Lambda\cap (\Gamma^<\times \Gamma)$ , which cannot be the case.

Pick $\omega$ such that $(W(\omega)_{0\leq t\leq\hat\tau(\omega)},\hat\tau(\omega))\in\Gamma_i$ for some $1\leq i\leq n$ . It then follows that

\begin{align*} \tau_i^{\mathsf{cl}}(\omega)\,:\!=\,\inf\{t\geq 0\,:\, (t, W_t(\omega))\in \mathcal R_i^\mathsf{cl}\}\leq \hat\tau(\omega) \leq \inf\{t\geq 0\,:\, (t, W_t(\omega))\in \mathcal R_i^\mathsf{op}\}\,=\!:\,\tau_i^\mathsf{op}(\omega).\end{align*}

Then we can conclude the existence of the barriers $(\mathcal R_i)_{i=1}^n$ from the observation that conditionally on the event $\{m_{i-1}\leq \overline{W}_{\hat\tau}<m_i \}$ , it holds that $\tau_i^\mathsf{cl}=\tau_i^\mathsf{op}$ a.s. by the strong Markov property and the fact that Brownian motion almost surely immediately returns to its starting point.

To show the final claim, note that (4.11) implies that at each $x\notin\{m_1,\ldots,m_n\}$ the condition $(g,t)\in\Gamma\cap\Lambda$ with $g(t)=x$ implies $g(t)\in\tilde\Lambda$ . Just as in the first part of the proof, it then follows that there is no $(\kern1.4pt f,s)\in\Gamma^<$ with $f(s)=x, \overline{f}_s\geq \overline{g}_t$ and $s>t$ . This gives the result.

Example 1. Consider the case of $\mu=\frac13(\delta_{s_0-1} + \delta_{s_0} + \delta_{s_0+1})$ , when $s_0>1$ . Let h be given by the inverse barycentre function of $\mu$ , i.e. $h=b_\mu^{-1}$ , which equals

\begin{equation*}h(x)= (s_0-1) \cdot 1_{[s_0-1,s_0+1/2)}(x) + s_0 \cdot 1_{[s_0+1/2,s_0+1)} + (s_0+1) \cdot 1_{[s_0+1,\infty)}.\end{equation*}

In particular, in the Root-type optimisation problem (4.9) constrained by h as in (4.10), any path which reaches level $s_0+1/2$ will not be stopped at $s_0-1$ .

The unconstrained Root solution instead is given by the hitting time of

\begin{equation*}\mathcal R=\{(t,s_0-1)\,:\,t\in [0,\infty)\}\cup \{(t,s_0)\,:\,t\in [a,\infty)\} \cup \{(t,s_0+1)\,:\,t\in [0,\infty)\}, \end{equation*}

for some $a>0$ . In particular, there are paths getting arbitrary close to one but which are stopped at $s_0-1$ , so that the constrained Root solution is different from the unconstrained one.

Also note that this is not related to the special case of $\mu$ being atomic. Indeed, keep the same h. Consider $\tilde\mu$ to be the uniform measure on $[s_0-1,s_0+1]$ whose inverse barycentre function is given by $b_{\tilde\mu}^{-1}(m)=2m-s_0-1$ , so that $\mathsf{RST}(\tilde\mu;\ \Lambda)\neq \emptyset$ by Theorem 7. By the same reasoning as before, it is immediate to see that the Root solution is different from the constrained Root solution.

Remark 4. Let us consider (4.9) for the case of a general increasing function h yielding a corresponding set of feasible paths $\Lambda$ as in (4.10). Assume $\mathsf{RST}(\mu;\ \Lambda)\neq \emptyset.$ Approximate h from below by step functions $h^n$ with corresponding sets of feasible paths $\Lambda^n$ and the property that $h^n\leq h^{n+1}$ . Since then $\Lambda^n \supseteq \Lambda^{n+1}\supseteq\Lambda$ , it follows that $\mathsf{RST}(\mu;\ \Lambda^n)\neq\emptyset$ . For each n, pick by Theorem 1 an optimiser to (the corresponding version of) (4.9), say $\hat\tau^n$ . Since $\mathsf{RST}(\mu;\ \Lambda^n)\supseteq \mathsf{RST}(\mu;\ \Lambda)$ , it follows that for all n

\begin{equation*}{\mathbb{E}}[(\hat\tau^n)^2]\leq \inf_{\tau\in\mathsf{RST}(\mu;\ \Lambda)}{\mathbb{E}}[\tau^2].\end{equation*}

Since $\mathsf{RST}(\mu;\ \Lambda^1)$ is compact and $\hat\tau^n\in\mathsf{RST}(\mu;\ \Lambda^1)$ for all n, there is a converging subsequence and any limit point $\hat\tau$ must lie in $\mathsf{RST}(\mu;\ \Lambda).$ Moreover, any limit point $\hat\tau$ must be an optimiser by monotonicity, since

\begin{equation*}{\mathbb{E}}\left[(\hat\tau^n)^2\right] \leq{\mathbb{E}}\left[(\hat\tau^{n+1})^2\right]\leq\inf_{\tau\in\mathsf{RST}(\mu;\ \Lambda)}{\mathbb{E}}[\tau^2].\end{equation*}

Since, by Theorem 8, each $\hat\tau^n$ is given as the hitting time of barriers in space–time indexed by the running maximum, it is then plausible to conjecture that this remains true for $\hat\tau$ as well. To make this argument rigorous seems to be outside the scope of this article; however, we note that (4.11) still holds in the limit.

Remark 5. If in (4.9) we consider a maximisation problem instead of a minimisation problem, the corresponding version of (4.11) turns into

\begin{equation*} \mathsf{SG}_\Lambda \supset \{((\kern1.4pt f,s),(g,t))\in\tilde\Lambda\times\Lambda\,:\,f(s)=g(t), s<t, \overline{f}_s \leq \overline{g}_t \}.\end{equation*}

Following the line of reasoning of Theorem 8, one can show that the optimal stopping time will be the hitting of a sequence of inverse barriers indexed by the running maximum, i.e. the corresponding version of constrained Root solutions. Using similar ideas one can identify the optimal solutions and worst-case scenarios in various other setups.

4.2. Option pricing in the presence of insider information: variance options

In this section, we consider the impact on the insider’s pricing bounds which comes from additional information. Specifically, we suppose that the information is on the drawdown, in a similar manner to the previous discussion, for example, as in (4.10), and we look to find bounds on the prices of options on variance: that is, we consider the motivating example from the introduction, where we think of a trader who believes that the CEO of the company is attempting to satisfy a drawdown constraint, and wishes to understand the impact on pricing bounds of variance options on the same company.

To understand the structure of the derivatives, we consider an asset which follows a model of the form $\textrm{d} S_t = S_t \sigma_t \, \textrm{d} W_t$ , where $S_t$ is the discounted asset price, and $W_t$ a Brownian motion. The process $\sigma_t$ is the volatility, and $\int_0^t \sigma^2_r \, \textrm{d} r$ is known as the integrated variance. A variance option is then a contract which pays the holder $G(\!\kern-0.5pt\int_0^t \sigma^2_r \, \textrm{d} r)$ . The most common example is the variance call, where $G(v) = (v-K)_+$ . Note that the integrated variance process can be determined as $\langle {\textrm{ln}} S\rangle_t$ , the quadratic variation of the logarithm of the asset price. For further details, we refer the reader to [Reference Carr and Lee21, Reference Cox and Wang24, Reference Cox and Wang25, Reference Lee38].

The standard method for pricing such options is to time-change the process $S_t$ by a time change $\tau_t$ such that $X_t \,:\!=\, S_{\tau_t}$ is a geometric Brownian motion. With this time change, $(X_t,t)= (S_{\tau_t},\langle {\textrm{ln}} S\rangle_{\tau_t})$ ; that is, the time-scale in the transformed picture corresponds to the integrated variance process. In particular, the problem of finding a model $S_t$ which minimises ${\mathbb{E}}[G(\langle{\textrm{ln}} S\rangle_T)]$ subject to $S_T \sim \mu$ is equivalent to finding a stopping time $\tau$ for X to minimise ${\mathbb{E}}[G(\tau)]$ subject to $X_\tau \sim \mu$ .

We would therefore like to compare the minimal (model-independent) price of the variance option for the insider, to that for the uninformed agent. To keep things simple, we consider an option which pays the holder the square root of the arithmetic variance, $V_t\, :\!=\, \int_0^t S_r^2 \sigma_r^2 \, \textrm{d} r$ , which corresponds to choosing a time change $\tau_t$ so that $X_t = S_{\tau_t}$ is a Brownian motion. This places us trivially in the setup of the rest of this paper.

Our problem of interest now may be posed as follows: consider an agent who has inside information on the future evolution of the asset, specifically, who knows that the price will never drop below $h(\overline{S}_t)$ , where h is an increasing function. The agent plans to exploit this information by trading in derivatives written on the asset, and does not have strong modelling beliefs, so wishes to profit from their information under any potential model. Suppose variance options with payoff $\sqrt{V_T}$ are liquidly traded. To profit, they plan to sell the derivative and set up a model-independent superhedging strategy. They want to know at what price level they are guaranteed to make a profit. If the agent also knows the feasibility set $\Lambda$ given by (4.10), then their problem becomes to find

(4.12) \begin{align} &\sup\{{\mathbb{E}}_{\mathbb{P}}[\sqrt{\tau}] \,:\, W_\tau\sim\mu,\, W_{.\wedge\tau}\ \textrm{is u.i.},\, \tau \le \overline{\tau} \text{ a.s.} \}. \end{align}

By Theorem 3, if we can identify the solution to this problem, then there exists a corresponding superhedging strategy. However, it follows from Theorem 8 that the solution must be a nested sequence of barriers, which depend on the running maximum. To see how these barriers, and more specifically the price bound, may depend on the information set, we consider the problem numerically under some additional structural examples.

4.2.1. Numerical results

In this section we illustrate the previous example with some numerical evidence. In particular, we are interested in illustrating how the insider’s price changes as the information set changes.

Our basic setup is as follows: we suppose that the insider’s information set $\Lambda$ is determined by (4.10), where the function h is of the form $h(x) = a_1 \boldsymbol{1}_{[m_0,\infty)}$ ; that is, there is a single step in the constraint, which comes in at the point where the maximum first exceeds the level $m_0$ . In the examples, we will consider the case where the information set is determined by the value of $m_0$ . Moreover, we will assume that the measure to be embedded consists of 4 atoms, at points $\{x_0, x_1, x_2, x_3\}$ , and we have $a_1 = x_1$ . It follows that the main issue to be determined is the value of the barrier at the level $x_1$ when the level $m_0$ has not yet been reached, and the barrier at $x_2$ both before and after reaching $m_0$ . From Theorem 8, we know that the barriers at $x_2$ are ordered—that is, the earliest time at which we stop at $x_2$ before reaching $m_0$ is later than the earliest time we stop at $x_2$ after reaching $m_0$ . Since the embedding constraint has two degrees of freedom (there are four atoms of mass, but two values are fixed by the requirement that the probabilities sum to one and the requirement that the embedded mass have mean equal to $s_0$ ), this means that we can compute the optimal barrier by optimising over the single remaining degree of freedom.

We implement a simple numerical algorithm, inspired by the partial-differential-equation characterisation of [Reference Cox and Wang25], which finds the potential of the stopped process. By optimising over the potential functions of the measure embedded before and after reaching $m_0$ , we are able to compute the critical times at which the barriers must start. Here, the potential $u_\lambda(x)$ associated with a measure on ${\mathbb{R}}$ is defined to be $u_\lambda(x) = \int | y-x| \, \lambda(dy)$ . The numerical implementation was performed in Python. A Jupyter notebook containing the code used to produce the figures in this paper can be downloaded from https://github.com/amgc500/SEPInsider. In Figure 5 we plot the price of the variance option as a function of $m_0$ . Moreover, we can see how the law of the quadratic variance in the extremal model varies as we change $m_0$ : this is shown in Figure 6 for several values of $m_0$ , while the values of the barrier at $x_1$ and $x_2$ before and after hitting $m_0$ are shown in Figure 7.

Figure 5:

Effect of changing $m_0$ on barrier. For values below about 17.9, there is no feasible model, and therefore there is an arbitrage if the option can be sold at any finite price. Since the upper range of the price distribution, $x_3 = 25$ , $m_0 = 25$ , corresponds to having no additional information, the maximal price in the plot (attained when $m_0 = 25$ ) is accordingly the model-independent upper bound on the price in the absence of any additional information.

Figure 6:

The cumulative distribution function of the realised quadratic variation of S in the extremal model, for four different values of $m_0$ .

Figure 7:

The values determining the barrier for different values of $m_0$ . In the figure, the values of the barriers at $x_1, x_2$ before hitting $m_0$ and at $x_2$ after hitting $m_0$ are shown. All other barrier values are either 0 or $\infty$ .

Under the restriction to a small number of atomic masses, the optimal models are relatively easy to find numerically in simple examples such as these. However, Theorem 8 provides only necessary conditions for a given barrier to be optimal. An open, and interesting, question is whether it is possible to provide sufficient conditions, and moreover, whether a numerical scheme to compute the corresponding bounds can be implemented. Doing this appears to us to be a challenging problem, and we leave it as an open question for future work.