Fragility of arbitrage and bubbles in local martingale diffusion models

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.

1. By the dual characterization of superhedging prices [2, 12], this definition is equivalent to (ii) in Definition 3.1.

2. These processes are H ^↑(t)=μ ⁺([0,t]) and H ^↑(t)=μ ⁻([0,t]), where μ=μ ⁺−μ ⁻ is the usual Hahn decomposition of the signed measure defined by μ([0,t])=H(t).

3. To be precise, with a bid price less than or equal to S/(1+ε) and an ask price greater than or equal to S(1+ε). Definition 4.2 considers the buying price S(T)(1+ε) at time T and the selling price S(0)/(1+ε) at time zero, to ensure that the superreplication is robust to the initial cash/stock allocation and to settlement either in cash or in stock. To wit, in the worst case, the strategy ends in cash (i.e., H(T)=0), and delivering one share requires S(T)(1+ε). Likewise, in the worst case, the strategy begins in cash (H(0)=0), but the initial capital is all in stock, and its cash value is S(0)/(1+ε).

4. Regrettably, in (c) of the statement of Theorem 3.1’’’, there is a misprint: there should be \(\mathcal{D}(Y)\) instead of \(\mathcal{K}(Y)\), which is clear from the ensuing remarks. See p. 372 of Michael [24] for the definition of \(\mathcal{D}(Y)\). This family contains finite-dimensional convex sets; hence, we may indeed apply Theorem 3.1’’’ in the current setting.

Authors and Affiliations

1. Department of Mathematics and Statistics, Boston University, 111 Cummington St., Boston, MA, 02215, United States

   Paolo Guasoni

2. School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

   Paolo Guasoni

3. MTA Alfréd Rényi Mathematical Institute, Reáltanoda utca 13-15, 1053, Budapest, Hungary

   Miklós Rásonyi

4. School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh, EH9 3JZ, United Kingdom

   Miklós Rásonyi

Corresponding author

Correspondence to Paolo Guasoni.

We thank Dörte Kreher, Johannes Muhle-Karbe, Vilmos Prokaj, and Walter Schachermayer for their helpful comments. We are especially indebted to two anonymous referees, the Associate Editor, and the Editor, Martin Schweizer, for their patient and tireless effort in improving the paper’s technical and conceptual contribution. Any remaining errors are our own.

Paolo Guasoni is partially supported by the ERC (279582), NSF (DMS-1109047), SFI (07/SK/M1189, 08/SRC/FMC1389), and the European Commission (RG-248896).

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 \(f\in C^{\infty}_{0}([s,T]\times\mathbb{R}^{d})\),

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

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

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

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

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

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],

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

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

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,

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

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).^{Footnote 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

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

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.

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