No arbitrage in continuous financial markets

The absence of arbitrage is of fundamental interest in many areas of financial mathematics. Our goal is to provide a systematic discussion for a financial market with one risky asset modeled via its discounted price process \(P = (P_t)_{t \in [0, T]}\), which we assume to be either the stochastic exponential of an Itô process, i.e. to have dynamicsor to be a positive diffusion with Markov switching, i.e. to have dynamicswhere \(\xi = (\xi _t)_{t \in [0, T]}\) is a continuous-time Markov chain and \(W = (W_t)_{t \in [0, T]}\) is a Brownian motion.

For semimartingale markets the classical concepts of no arbitrage are the notions of no free lunch with vanishing risk (NFLVR) as defined by Delbaen and Schachermayer [15, 16] and no feasible free lunch with vanishing risk (NFFLVR) as defined by Sin [52]. The difference between (NFLVR) and (NFFLVR) is captured by the concept of a financial bubble in the sense of Cox and Hobson [9]. For our market it is well-known that (NFLVR) is equivalent to the existence of an equivalent local martingale measure (ELMM), see [16], and that (NFFLVR) is equivalent to the existence of an equivalent martingale measure (EMM), see [6, 52, 55]. The no arbitrage condition used in the stochastic portfolio theory of Fernholz [22] is no relative arbitrage (NRA). In complete markets Fernholz and Karatzas [21] showed that (NRA) is equivalent to the existence of a strict martingale density (SMD). A weaker concept is no unbounded profit with bounded risk (NUPBR), which is known to be equivalent to the existence of a strict local martingale density (SLMD), see [7]. (NUPBR) is considered to be the minimal notion needed for portfolio optimization, see [35].

The first findings of this article are integral tests for the existence and non-existence of SMDs, ELMMs and EMMs. For (1.1) the tests are formulated in terms of Markovian upper and lower bounds on the volatility coefficient \(\sigma \) and for (1.2) the tests depend on \(x \mapsto \sigma (x, j)\) with \(j\) in the state space of the Markov chain \(\xi \). The main novelty of our results is that they apply in the presence of multiple sources of risk. Beside the Markov switching framework, this is for instance the case in diffusion models with a change point, which represents a change of the economical situation caused for instance by a sudden adjustment in the interest rates or a default of a major financial institution. In general, the question whether (NFLVR) and/or (NFFLVR) hold for a model with a change point is difficult, see [24] for some results in this direction. Our integral tests provide explicit criteria, which are easy to verify. For many applications of the Markov switching model (1.2) it is important to know how the change to an ELMM affects the dynamics of the Markov chain \(\xi \). As a second contribution, we study this question form a general perspective for independent sources of risk modeled via martingale problems. In particular, we show that the minimal local martingale measure (MLMM), see [25], preserves the independence and the laws of the sources of risk. To our knowledge, this property has not been reported in the literature. A third contribution of this article are integral tests for the martingale property of certain stochastic exponentials driven by Itô processes or switching diffusions. These characterizations are our key tools to study the absence of arbitrage.

We comment on related literature. For continuous semimartingale models the absence of arbitrage has been studied by Criens [11], Delbaen and Shirakawa [17], Lyasoff [40] and Mijatović and Urusov [43]. Criens, Delbaen and Shirakawa and Mijatović and Urusov proved integral tests for the existence of SMDs, ELMMs and EMMs in diffusion frameworks. Our results can be viewed as generalizations to an Itô process or Markov switching framework. For a model comparable to (1.1), Lyasoff proved that the existence of an ELMM is determined by the equivalence of a probability measure to the Wiener measure. The structure of this characterization is very different from our results. In Sect. 3.3 below we comment in more detail on the results in [11, 17, 40, 43]. The martingale property of stochastic exponentials is under frequent investigation. At this point we mention the articles of Blanchet and Ruf [3], Cheridito et al. [5], Criens [11] and Kallsen and Muhle-Karbe [34]. Criens used arguments based on Lyapunov functions and contradictions to verify the martingale property of certain stochastic exponential in a multi-dimensional diffusion setting. We transfer these techniques to a general Itô process setting. Cheridito et al. and Kallsen and Muhle-Karbe related the martingale property of a stochastic exponential to an explosion probability via a method based on the concept of local uniqueness as defined in [30]. This technique traces back to work of Jacod and Mémin [29] and Kabanov et al. [31, 32]. We use a similar argument for the Markov switching setting. The main difficulties are the proofs of explosion criteria and local uniqueness. Both approaches have a close relation to the work of Blanchet and Ruf, where a tightness criterion for the martingale property of non-negative local martingales has been proven. The connection between Lyapunov functions, explosion and tightness is for instance explained in [12].

Let us also comment on consecutive problems and extensions of our results: In case the discounted price process \(P\) is a positive Itô process of the typeour results on the martingale property of stochastic exponentials can be used to obtain characterizations for no arbitrage with a similar structure as for the model (1.2). Moreover, in case \(P\) is the stochastic exponential of a diffusion with Markovian switching, i.e.our martingale criteria yield conditions for no arbitrage with a similar structure as for (1.1). It is also interesting to ask about multi-dimensional models. In this case, results in the spirit of [11] can be proven by similar arguments as used in this article. However, the conditions are rather complicated to formulate and space consuming. Therefore, we restrict ourselves to the one-dimensional case.

The article is structured as follows. In Sect. 2 we give conditions for the martingale and strict local martingale property of certain stochastic exponentials. In Sect. 3.1 we study the model (1.1) and in Sect. 3.2 we study the model (1.2). In Sect. 4 we show that the MLMM preserves independence and laws for sources of risk and we explain how the MLMM can be modified to affect the law of an additional source of risk. The proofs are collected in the remaining sections.

Fix a finite time horizon \(0< T < \infty \) and let \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {P})\) be a complete filtered probability space with right-continuous and complete filtration \(\mathbf {F}= (\mathcal {F}_t)_{t \in [0, T]}\). Moreover, fix a state space \(I \triangleq (l, r)\) with \(- \infty \le l < r \le + \infty \).

In the following two sections we provide conditions for the martingale and strict local martingale property of certain stochastic exponentials.

Let \(0< T < \infty \) be a finite time horizon and let \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {P})\) be a complete filtered probability space with right-continuous and complete filtration \(\mathbf {F}= (\mathcal {F}_t)_{t \in [0, T]}\). We consider a financial market consisting of one risky asset with discounted price process \(P = (P_t)_{t \in [0, T]}\), which is assumed to be a positive continuous semimartingale with deterministic initial value.

Recall the following classical terminology: A probability measure \(\mathbb {Q}\) is called an equivalent (local) martingale measure (E(L)MM) if \(\mathbb {Q}\sim \mathbb {P}\) and \(P\) is a (local) \(\mathbb {Q}\)-martingale. A strictly positive local \(\mathbb {P}\)-martingale \(Z = (Z_t)_{t \in [0, T]}\) with \(Z_0 = 1\) is called a strict (local) martingale density (S(L)MD) if \(ZP\) is a (local) \(\mathbb {P}\)-martingale.

In the following we study existence and non-existence of SMDs, ELMMs and EMMs in case \(P\) is either the stochastic exponential of an Itô process or a positive switching diffusion. In case \(P\) is a positive Itô process or the stochastic exponential of a real-valued switching diffusion similar results can be deduced from the martingale criteria in Sect. 2.

In Sect. 3.2.1 we proved conditions for the existence of the minimal (local) martingale measure in a Markov switching framework. We ask the following consecutive questions: In this section we answer these questions from a general perspective under an independence assumption, which holds in our Markov switching framework.

The following section is divided into three parts. In the first part we prove Lyapunov-type conditions for non-explosion of Itô processes, in the second part we prove non-existence conditions for Itô processes and in the third part we deduce Theorems 2.1 and  2.3.

The section is split into two parts: First, we prove existence, non-existence and local uniqueness for switching diffusions and second, we deduce Theorem 2.5.

Step 1 Let \(g \in A\) and setDue to the definition of the martingale problem \((A, L, T)\), the process \(M^g\) is a local martingale with localizing sequence \((\rho _n (\xi ))_{n \in \mathbb {N}}\). Thus, the quadratic variation process \([M^g, W]\) is well-defined. Our first step is to show that a.s. \([M^g, W] = 0\). We explain that \(WM^g\) is a local martingale for the completed right-continuous version of the natural filtration of \(\xi \) and \(W\). Let \(0 \le s < t \le T\), \(G \in \sigma (W_r, r \in [0, s]) \triangleq \mathcal {W}_s\) and \(F \in \sigma (\xi _r, r \in [0, s]) \triangleq \mathcal {E}_s\). The independence assumption yields thatBy a monotone class argument, we havefor all \(B \in \mathcal {W}_s \vee \mathcal {E}_s\). Due to the downwards theorem (see [48, Theorem II.51.1]), the process \(W M^g_{\cdot \wedge \rho _m(\xi )}\) is a martingale for the completed right-continuous version \(\mathbf {G}\triangleq (\mathcal {G}_t)_{t \in [0, T]}\) of \((\mathcal {W}_t \vee \mathcal {E}_t)_{t \in [0, T]}\). Consequently, because \(\rho _m (\xi )\nearrow \infty \) as \(m \rightarrow \infty \), \(W M^g\) is a local \(\mathbf {G}\)-martingale. By the tower rule, also \(W\) and \(M^g\) are local \(\mathbf {G}\)-martingales. Integration by parts implies thatwhere the stochastic integrals are defined as local \(\mathbf {G}\)-martingales. Here, we use that \([W, M^g]\) can be defined independently of the filtration. We deduce that the process \([W, M^g]\) is a continuous local \(\mathbf {G}\)-martingale of finite variation and hence a.s. \([W, M^g] = 0\).

Step 2 In this step we identify the laws of \(B\) and \(\xi \) under \(\mathbb {Q}\). Clearly, \(B\) is a \(\mathbb {Q}\)-Brownian motion due to Girsanov’s theorem. Next, we show that on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {Q})\) the process \(\xi \) is a solution process for the martingale problem \((A, L, j_{0}, T)\). By Step 1 and Girsanov’s theorem, the processis a local \(\mathbb {Q}\)-martingale. The equivalence \(\mathbb {Q}\sim \mathbb {P}\) implies that \(\mathbb {Q}(\xi _0 = j_0) = 1\) and that \(M^g_{\cdot \wedge \rho _n(\xi )}\) is \(\mathbb {Q}\)-a.s. bounded. Thus, the claim follows.

Step 3 We prove \(\mathbb {Q}\)-independence of \(B\) and \(\xi \) borrowing an idea from [20, Theorem 4.10.1]. We define \(C^2_b(\mathbb {R})\) to be the set of all bounded twice continuously differentiable functions \(\mathbb {R} \rightarrow \mathbb {R}\) with bounded first and second derivative. Suppose that \(f \in C^2_b(\mathbb {R})\) with \(\inf _{x \in \mathbb {R}} f(x) > 0\) and defineBy Itô’s formula, we haveThus, \(K^f\) is a \(\mathbb {Q}\)-martingale, as it is a bounded local \(\mathbb {Q}\)-martingale. Recall that the quadratic variation process is not affected by an equivalent change of measure. By Step 1, \(\mathbb {Q}\)-a.s. \([B, M^g] = 0\). Due to integration by parts, we obtain thatwhich implies that \(K^f M^g_{\cdot \wedge \rho _m(\xi )}\) is a \(\mathbb {Q}\)-martingale, as it is a bounded local \(\mathbb {Q}\)-martingale.

Let \(\zeta \) be a stopping time such that \(\zeta \le T\) and setBecause \(K^f M^g_{\cdot \wedge \rho _m(\xi )}, K^f\) and \(M^f_{\cdot \wedge \rho _m(\xi )}\) are \(\mathbb {Q}\)-martingales (see also Step 2), the optional stopping theorem implies that for all stopping times \(\psi \le T\)Consequently, by [47, Proposition II.1.4], \(M^g_{\cdot \wedge \rho _m(\xi )}\) is a \(\widetilde{\mathbb {Q}}\)-martingale. Because \(\widetilde{\mathbb {Q}} \sim \mathbb {Q}\), this implies that on \((\Omega , \mathcal {F}, \mathbf {F}, \widetilde{\mathbb {Q}})\) the process \(\xi \) is a solution process for the martingale problem \((A, L, j_0, T)\). The uniqueness assumption for the martingale problem \((A, L, j_0, T)\) implies thatfor allwhere \(G_1, \dots , G_n \in \mathcal {B}(J)\) and \(0 \le t_1< \dots < t_n \le T\). We fix \(\Gamma \) such that \(\mathbb {Q}(\Gamma ) > 0\) and defineUsing the definition of \(\widetilde{\mathbb {Q}}\), (7.2), the fact that \(K^f\) is a \(\mathbb {Q}\)-martingale and the optional stopping theorem, we obtainBecause \(\zeta \) was arbitrary, we conclude that \(K^f\) is a \(\widehat{\mathbb {Q}}\)-martingale. Furthermore, \(\widehat{\mathbb {Q}}(B_0 = 0) = 1\) follows from the fact that \(B\) is a \(\mathbb {Q}\)-Brownian motion. Finally, due to [20, Proposition 4.3.3], the process \(B\) is a \(\widehat{\mathbb {Q}}\)-Brownian motion. We conclude thatfor all \(F_1, \dots , F_k \in \mathcal {B}(\mathbb {R})\) and \(0 \le s_1< \dots < s_k \le T\). By the definition of \(\widehat{\mathbb {Q}}\), we have proven thatwhich implies that the \(\sigma \)-fields \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (B_t, t \in [0, T])\) are \(\mathbb {Q}\)-independent. The proof is complete. \(\square \)

Because \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (W_t, t \in [0, T])\) are assumed to be \(\mathbb {P}\)-independent, it follows as in the proof of Theorem 4.3 that a.s. \([Z, W] = 0\). Thus, Girsanov’s theorem implies that \(W\) is a \(\mathbb {Q}\)-Brownian motion.

Take \(0 \le s_1< \dots< s_m \le T, 0 \le t_1< \dots < t_n \le T\),\((G_k)_{k \le m} \subset \mathcal {B}(J)\) and \((F_k)_{k \le n} \subset \mathcal {B}(\mathbb {R})\), and setThe \(\mathbb {P}\)-independence of \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (W_t, t \in [0, T])\) and the uniqueness of the Wiener measure yield thatWe conclude that \(\sigma (\xi _t, t \in [0, T])\) and \(\sigma (W_t, t \in [0, T])\) are \(\mathbb {Q}\)-independent.

For \(g \in A^*\) we setThe processes \(K^f\) and \(K^{fg}\) are local \(\mathbb {P}\)-martingales. We setIntegration by parts implies thatUsing again integration by parts and the identity \(L^* g = \frac{1}{f} (L (fg) - g Lf)\) yieldsWe conclude that \(Z M^g\) is a local \(\mathbb {P}\)-martingale and it follows from [30, Proposition III.3.8] that \(M^g\) is a local \(\mathbb {Q}\)-martingale. Due to the equivalence \(\mathbb {Q}\sim \mathbb {P}\), we conclude that on \((\Omega , \mathcal {F}, \mathbf {F}, \mathbb {Q})\) the process \(\xi \) is a solution process to the martingale problem \((A^*, L^*, j_0, T)\). This completes the proof.\(\square \)

Let \((X_t)_{t \ge 0}\) be the coordinate process on \(D (\mathbb {R}_+, J)\) and denoteDefine by \(\mu \triangleq \mathbb {P}\circ \xi ^{-1}\) a Borel probability measure on \(D (\mathbb {R}_+, J)\). We have to show thatIt follows from [27, Lemma 2.9] that \(M^f\) is a local \(\mu \)-martingale with localizing sequence \((\rho _n)_{n \in \mathbb {N}}\). For all \(n \in \mathbb {N}\), define a Borel probability measure \(\mu _n\) on \(D (\mathbb {R}_+, J)\) via the Radon–Nikodym derivativeThe following lemma is proven after the proof of Theorem 4.9 is complete.

Recalling that \(\{\rho _n > T\} \in \mathcal {D}^o_{T \wedge \rho _n}\), Lemma 9.1 implies thatThis completes the proof. \(\square \)