Fragility of arbitrage and bubbles in local martingale diffusion models

Abstract

For any positive diffusion with minimal regularity, there exists a semimartingale with uniformly close paths that is a martingale under an equivalent probability. As a result, in models of asset prices based on such diffusions, arbitrage and bubbles alike disappear under proportional transaction costs or under small model mis-specifications. Thus, local martingale diffusion models of arbitrage and bubbles are not robust to small trading and monitoring frictions.

Appendix:  Proofs

In the sequel, we denote by $C_{0}^{\infty}([s,T]\times\mathbb{R}^{d})$ the space of smooth functions with compact support on $[s,T]\times\mathbb{R}^{d}$, equipped with the topology of uniform convergence. The following lemma is essentially (part of) Theorem 10.1.1 of Stroock and Varadhan [33]. The latter result is stated on the canonical space, and we could not find a convenient reference for the present setting.

Lemma A.1

Let $R^{s}_{\omega}$ be a regular version of the conditional law of X(t),t∈[s,T] (on C[s,T]) with respect to $\mathcal{F}^{X}_{s}$. Then for almost every ω∈Ω, $R^{s}_{\omega}$ solves the martingale problem on [s,T] related to (5.1) with initial condition X(s)(ω).

Proof

Recalling the definition of martingale problems from Chaps. 6 and 10 of Stroock and Varadhan [33], we need to prove that for almost every ω∈Ω and for each $f\in C^{\infty}_{0}([s,T]\times\mathbb{R}^{d})$,

$$f \bigl(t,\pi(t,\cdot) \bigr)-f \bigl(s,X(s) \bigr)-\int_s^t (L_u f) \bigl(u,\pi (u,\cdot) \bigr)\,du,\quad t\in[s,T], $$

is an $(R^{s}_{\omega},(\mathcal{G}_{u})_{u\in[s,T]})$-martingale, where π(u,⋅),u∈[s,T], denote the coordinate mappings on the canonical space C[s,T], $\mathcal{G}_{u}:= \sigma(\pi(r,\cdot ),s\leq r\leq u)$, and the operator L acts as

$$\begin{aligned} (L_uf) (u,p)&:=\frac{\partial}{\partial u}f(u,p)+\frac{1}{2}\sum _{i,j=1}^d \bigl(\sigma\sigma^T \bigr)_{ij}(u,p) \frac{\partial^2}{\partial p_i\partial p_j}f(u,p) \\ &\quad +\sum_{i=1}^d b_i(u,p) \frac{\partial}{ \partial p_i}f(u,p). \end{aligned}$$

Denote by w a generic element of C[s,T]. Then it is clearly sufficient to prove that for almost all ω,

$$\begin{aligned} & \int_{C[s,T]} \biggl(f \bigl(t,\pi(t,w) \bigr)-f \bigl(r,\pi(r,w) \bigr)-\int_r^t (L_u f) \bigl(u,\pi(u,w) \bigr)\,du \biggr) \\ &\quad {}\times g(u_1,\ldots,u_n) R^s_{\omega}(dw)=0 \end{aligned}$$

(A.1)

for suitable countable collections of $f\in C_{0}^{\infty}([s,T]\times \mathbb{R}^{d})$, bounded measurable g, and for all rationals s<r<t≤T and u _i ∈(s,r), i=1,…,n. Equality (A.1) follows from

$$\begin{aligned} &E \biggl[g \bigl(X({u_1}),\ldots,X({u_n}) \bigr) \\ &\quad {}\times \biggl(f \bigl(t,X(t) \bigr)-f \bigl(r,X(r) \bigr)-\int _{r}^t (L_uf) \bigl(u,X(u) \bigr)\,du \biggr) \bigg\vert \mathcal{F}^X_s \biggr]=0 \end{aligned}$$

a.s. This holds by Itô’s formula, the tower law, and

$$E \biggl[ \int_r^t \frac{\partial}{\partial p}f \bigl(u,X(u) \bigr)\sigma \bigl(u,X(u) \bigr)\,dW(u) \bigg\vert \mathcal{F}_r \biggr]=0, $$

which is an elementary property of stochastic integrals since f and its derivatives have compact support and σ is continuous and hence bounded on compact sets. □

Proof of Theorem 5.2

To ease notation, consider d=1; the same argument carries over to the general case. Recall the concept of conditional full support: A strictly positive adapted process S defined on $(\varOmega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},P)$ with continuous paths has conditional full support (CFS) if

$$\operatorname{supp} P [S\vert_{[t,T]}\in\cdot \vert\mathcal {F}_t ]=C_{S_t}^+[t,T]\quad \mbox{a.s. for all }t \in[0,T], $$

where $\operatorname{supp} \mu$ denotes the support of a measure μ, and C _v [c,d] (resp. $C_{v}^{+}[c,d]$) is the set of continuous functions (resp. strictly positive continuous functions) f on [c,d] with f(c)=v.

Theorem 1.2 of Guasoni et al. [13] states that for a process satisfying CFS, there exist Q∼P and a Q-martingale $\tilde{S}$ (with respect to $(\mathcal{F}^{X}_{t})$) that is ε-close to S. Thus, $M(t):=\tilde{S}(t)E[\frac{dQ}{dP}\vert{\mathcal{F}^{X}_{t}}]$ is a P-martingale (with respect to $(\mathcal{F}^{X}_{t})$), and if $(\mathcal {F}^{X}_{t})$ is the natural filtration of some Brownian motion, then M as well as the (positive) P-martingale $(E[\frac{dQ}{dP}\vert {\mathcal{F}^{X}_{t}}])$ must have continuous paths, and so must $\tilde{S}$.

It remains to prove the CFS property for S with respect to $(\mathcal {F}^{X}_{t})$. Let P ^s,y denote the law of a solution of (5.1) starting from $\bar{X}(s)=y$ on C[s,T]. Replacing $\mathcal{F}^{X}_{t}$, t≥s, by $\mathcal{F}^{\bar{X}}_{t}$, t≥s, in Lemma A.1, the same argument gives that P ^s,y solves the martingale problem related to (5.1) with initial condition $\bar{X}(s)=y$ (we do not even need to work with conditional expectations in this case, $\mathcal{F}_{s}^{\bar{X}}$ being trivial). We can now take a regular version of P ^s,y as provided by Theorem 10.1.1 of Stroock and Varadhan [33]. Let us now notice that uniqueness in law (see Assumption 5.1) and Lemma A.1 necessarily entail $P^{s,X(s)(\omega)}=R_{\omega}^{s}$ a.s.

Thus, it suffices to show that $\operatorname{supp} R^{s}_{\cdot}=\operatorname{supp} P^{s,X(s)}=C_{X(s)}[s,T]$ a.s. for each 0≤s<T. To achieve this, we prove that for all $y\in\mathbb{R}$, η∈(0,1), and g∈C _y [s,T],

$$P^{s,y} \bigl[\bigl\{ f\in C_y[s,T]:\Vert f-g \Vert_{\infty}< \eta\bigr\} \bigr]>0. $$

Define K:=∥g∥_∞+1, $\tilde{b}(t,x):=b(t,x)1_{\{ \vert x\vert\leq K\}}$ and $\tilde{\sigma}(t,x):=\sigma(t,x)$ for (t,x)∈[0,T]×[−K,K]. Note that by continuity there exists h>0 such that

$$\vert\tilde{\sigma}(t,x)\vert> h\quad\text{for all }(t,x)\in [0,T] \times[-K,K] . $$

Extend $\tilde{\sigma}$ to $[0,T]\times\mathbb{R}$ so that $|\tilde {\sigma}(t,x)|\geq h$ for all (t,x) and $\tilde{\sigma}$ remains continuous. Since $\tilde{b},\tilde{\sigma}$ are bounded and $\tilde{\sigma}$ is bounded away from 0 and continuous, the martingale problem

$$d\tilde{X}(t)=\tilde{b} \bigl(t,\tilde{X}(t) \bigr)\,dt+\tilde{\sigma } \bigl(t, \tilde{X}(t) \bigr)\,dW(t),\quad \tilde{X}(s)=y, $$

admits unique solution measures $\tilde{P}^{s,y}$ on the space C _y [s,T] for all 0≤s<T; see Stroock and Varadhan [31] and Chap. 7 of Stroock and Varadhan [33].

Take τ(w):=inf{t>s:|w(t)|≥K}∧T, where w is the generic element of the canonical space C _y [s,T]. Let us denote by π(t,⋅) the coordinate mapping on C _y [s,T] corresponding to s≤t≤T. Obviously, τ is a $(\mathcal{G}_{t})_{t\in[s,T]}$-stopping time (recall that $\mathcal{G}$ is the filtration generated by the coordinate mappings). The last part of Theorem 6.1.2 of Stroock and Varadhan [33] implies that both measure-concatenations $\tilde{P}^{s,y}\otimes_{\tau(\cdot)}{P}^{\tau(\cdot),\pi(\tau (\cdot ),\cdot)}$ and P ^s,y⊗ _τ(⋅) P ^{τ(⋅),π(τ(⋅),⋅)} solve the martingale problem (5.1) on [s,T] (see Theorem 6.1.2 of Stroock and Varadhan [33] for unexplained notation), and hence these measures are equal on $\mathcal{G}_{T}$. A fortiori, $P^{s,y}\vert\mathcal{G}_{\tau}=\tilde {P}^{s,y}\vert\mathcal{G}_{\tau}$. The event {f∈C _y [s,T]:∥f−g∥_∞<η} is in $\mathcal{G}_{\tau}$; consequently,

$$\begin{aligned} {P}^{s,y} \bigl[\bigl\{ f\in C_{y}[s,T]:\Vert f-g \Vert_{\infty}< \eta\bigr\} \bigr]= \tilde{P}^{s,y} \bigl[\bigl\{ f \in C_{y}[s,T]:\Vert f-g\Vert_{\infty}<\eta\bigr\} \bigr]. \end{aligned}$$

The functions $\tilde{\sigma}$ and $\tilde{b}$ satisfy the conditions of Sect. 3 in Stroock and Varadhan [32]; hence, Lemma 3.1 of Stroock and Varadhan [32] (or Exercise 6.7.5 of Stroock and Varadhan [33]) implies that $\tilde{P}^{s,y}[\{ f\in C_{y}[s,T]:\Vert f-g\Vert_{\infty}<\eta\}]>0$, concluding the proof. □

Proof of Theorem 6.2

Let us define the multifunction

$$D(t,x):= \biggl\{ u\in\mathbb{R}^{(N-d)\times N}: \det \begin{bmatrix} \sigma(t,x)\\ u \end{bmatrix} >0 \biggr\} $$

for $(t,x)\in[0,T]\times\mathbb{R}^{d}$. Clearly, D takes convex nonempty values in a finite-dimensional space, and its graph is $\mathcal{B}([0,T]\times \mathbb{R}^{d})\otimes\mathcal{B}(\mathbb{R}^{(N-d)\times N})$-measurable. It is also lower semicontinuous. Indeed, by Proposition 2.1 of Michael [24] it suffices to prove that if u∈D(t,x) and ϵ>0, then there is δ>0 such that for all t′,x′ with |t−t′|+|x−x′|<δ and there is u′∈D(t′,x′) with |u−u′|<ϵ. But this is obvious by the continuity of σ and of the determinant function.

Now we may apply Theorem 3.1’’’ of Michael [24], which gives us a continuous $\nu:[0,T]\times\mathbb{R}^{d}\to\mathbb {R}^{(N-d)\times N}$ such that ν(t,x)∈D(t,x) for all (t,x).⁴ Let us define $\hat{\sigma}:[0,T]\times\mathbb{R}^{d}\to\mathbb {R}^{N\times N}$ such that its first d rows equal σ and the remaining rows equal ν. Let us also consider $\hat{b}:(t,x)\mapsto(b(x),\mathbf{0})^{T}$ (where 0 denotes an (N−d)-dimensional row vector of zeros). By abuse of notation, we write from now on $\hat{b}(t,y):=\hat{b}(t,\bar{y}^{d})$ for all $y\in\mathbb{R}^{N}$, where $\bar{y}^{d}$ denotes the vector formed by the first d coordinates of y. We define $\hat{\sigma}$ similarly and hence extend $\hat{b},\hat{\sigma}$ to $[0,T]\times\mathbb{R}^{N}$ (note that the last N−d coordinates are dummy variables only).

Define $\bar{X}^{i}(t):=X^{i}(t)$ for i=1,…,d and set

$$\bar{X}^i(t):=\int_0^t \hat{ \sigma}_i \bigl(s,X(s) \bigr)\,dW(s) $$

for i=d+1,…,N, where $\hat{\sigma}_{i}$ denotes the ith row of $\hat{\sigma}$. By construction, $\bar{X}$ satisfies

$$d\bar{X}(t)=\hat{b} \bigl(t,\bar{X}(t) \bigr)\,dt+\hat{\sigma} \bigl(t,\bar {X}(t) \bigr)\,dW(t),\quad\bar{X}(0)=\hat{x}_0, $$

where the first d components of $\hat{x}_{0}$ coincide with those of x ₀, and the remaining components are zero. By Assumption 6.1, X is unique in law. Theorem 3.1 of [4] or the main result in [3] implies that also the law of the pair (X,W) is unique. Since $\bar{X}$ is a functional of (X,W), its law is unique as well; so $\hat{b},\hat{\sigma}$ clearly satisfy the conditions of Theorem 5.2, and hence for $\bar{S}^{i}(t)=\exp(\bar{X}^{i}(t))$, i=1,…,N, we get an N-dimensional process $\check{S}$ that is ε-close to $\bar{S}$. Notice that the first d coordinates of $\bar{S}$ are precisely S. Denoting by $\tilde{S}$ the d-dimensional process consisting of the first d coordinates of $\check{S}$, we clearly have that $\tilde{S}$ is ε-close to S.

From Theorem 5.2 we have the existence of a probability R∼P such that $\check{S}$ is a (true) R-martingale (with respect to $(\mathcal{F}^{\bar{X}}_{t})$); in particular, $R\in \mathcal{M}(\tilde{S})$. Note that the construction in Guasoni et al. [13] ensures that $\check{S}(T)\in L^{2}(R)$. A fortiori, $\tilde{S}$ is an L ²(R)-martingale, and hence $\sup_{t\in[0,T]}|\tilde{S}_{t}|\in L^{2}(R)$. Now recall that the set of $Q'\in\mathcal{M}(\tilde{S})$ such that dQ′/dR is bounded is dense in $\mathcal{M}(\tilde{S})$ with respect to the total-variation norm (see Kabanov and Stricker [20]). Clearly, $\tilde{S}$ satisfies $\sup_{t\in[0,T]}|\tilde{S}_{t}|\in L^{2}(Q')$ as well; so it is a true Q′-martingale for all such Q′. This finishes the proof. □

Remark A.2

If in the statements of Theorems 5.2 and 6.2, the solutions X are strong and $\mathcal{F}=\mathcal{F}^{W}$, then the proof of Lemma A.1 works with the filtration $\mathcal{F}^{W}$ in lieu of $\mathcal{F}^{X}$. Hence, the proofs of Theorems 5.2 and 6.2 yield $\tilde{S}$ that are $\mathcal{F}^{W}$-martingales and, in particular, continuous.