Martingale Property of Exponential Semimartingales: A Note on Explicit Conditions and Applications to Financial Models

Local martingales are the core object of stochastic integration. Thus they provide a natural access to time evolutionary stochastic modeling, which is a cornerstone of mathematical finance. The fundamental theorem of asset pricing states that the absence of arbitrage is essentially equivalent to the local martingale property of discounted asset prices under some equivalent probability measure. One important benefit of the true martingale property of discounted asset price processes is their use for density processes of a change of measure. In financial terms this corresponds to a change of numeraire. Since the seminal work of Geman et al. (1995) this concept became indispensable for both computational and modeling aspects. Often a change of numeraire facilitates option pricing by reducing complexity of computations. Moreover, it is a building stone of the construction of Libor market models introduced by Brace et al. (1997) and Miltersen et al. (1997). More fundamentally, a change of measure connects historical and risk-neutral probability measures. On the other hand if the discounted asset price process is a strict local martingale, i.e. a local martingale which is not a true martingale, this is sometimes interpreted as financial bubble. However, the definition and existence of financial bubbles critically depends on the specific notion of the market price, arbitrage and admissible strategies, see for example Cox and Hobson (2005) and Jarrow et al. (2010). In a typical modeling situation it is enjoyable to work with true martingales.

Usually price processes are non-negative and therefore are modeled as exponentials of semimartingales, which form a wide and flexible class of positive processes. One can characterize the local martingales in this class by a drift condition. It is, however, more involved to identify conditions for their true martingale property. In order to formulate the problem more precisely denote by $X$ an ${\mathbb{R}^{d}}$-valued semimartingale and by $\lambda$ an ${\mathbb{R}^{d}}$-valued predictable process which is integrable with respect to $X$. Then $\lambda\cdot X:=\sum_{i\leq d}\int_{0}^{\cdot}\lambda^{i}\operatorname{d}\!X^{i}$ denotes the real-valued stochastic integral process of $\lambda$ with respect to $X$. Moreover, let $V$ be a predictable process with finite variation. We pose the following question: Under which conditions on the characteristics of $X$ is a real-valued semimartingale $Z$ of the form

a (uniformly integrable) martingale?

If $e^{\lambda\cdot X}$ is a special semimartingale, there exists a unique predictable process of finite variation $V$ such that $Z$ is a local martingale. In this case, $V$ is called the exponential compensator of $\lambda\cdot X$, see Section 2 for details. Various criteria for the more delicate true martingale property of $Z$ have been proposed. The seminal paper by Novikov (1972) treats the continuous semimartingale case. Sufficient conditions for general semimartingales are provided for example in Lepingle and Mémin (1978), Kallsen and Shiryaev (2002), Jacod (1979), Cheridito et al. (2005) and Protter and Shimbo (2008), see also a recent paper by Larsson and Ruf (2014) for further generalizations of Novikov-Kazamaki type conditions based on convergence results for local supermartingales. Moreover, we refer to Section 1 and Section 3 of Kallsen and Shiryaev (2002) for an exhaustive literature overview. In the special case when $X$ is a process with independent increments and absolutely continuous characteristics and $\lambda$ deterministic, Eberlein et al. (2005) show that if $Z$ is a local martingale, it is also a true martingale. Deterministic conditions ensuring the martingale property of an exponential of an affine process are given in Kallsen and Muhle-Karbe (2010). The conditions for more general semimartingales are not as explicit.

Our contribution is to give explicit conditions for the martingale property of an exponential quasi-left continuous semimartingale in terms of its characteristics. In Section 2 we introduce the notation and describe the general semimartingale setting following Jacod and Shiryaev (2003). Section 3 contains the main results. The advantage of the explicit conditions is their convenience for applications. We illustrate this by investigating the true martingale property of asset prices in semimartingale stochastic volatility models in Section 4.1. Finally, in Section 4.2 we prove the well-definedness of the backward construction of Lévy Libor models. More precisely, we show that the candidate density processes for the measure changes are indeed true martingales which has not been rigorously proved earlier. Moreover, we present a natural extension to the semimartingale Libor model.

In this section we introduce the notation and summarize the basic notions and facts from the semimartingale theory in order to keep the paper self-contained. Our main reference is Jacod and Shiryaev (2003), whose notation we use throughout the paper. Other standard references for stochastic calculus and semimartingales are e.g. Jacod (1979), Métivier (1982) and Protter (2004).

Let $(\Omega,{\mathcal{F}},({\mathcal{F}}_{t})_{t\geq 0},{\mathbb{P}})$ denote a stochastic basis, i.e. a filtered probability space with right-continuous filtration. For a class of processes $\mathcal{C}$, we say that a process $X$ is in the localized class $\mathcal{C}_{\textup{loc}}$ if there exits a sequence of stopping times $(\tau_{n})_{n\in\mathbb{N}}$ such that a.s. $\tau_{n}\uparrow\infty$ as $n\to\infty$ and $X^{\tau_{n}}\in\mathcal{C}$. Denote by $\mathcal{M}$ the class of càdlàg uniformly integrable martingales. The processes in the localized class ${\mathcal{M}}_{{\mathrm{loc}}}$ are called local martingales. We denote by ${\mathcal{V}}^{+}$ (resp. ${\mathcal{V}}$) the set of all real-valued càdlàg processes starting from zero that have non-decreasing paths (resp. paths with finite variation over each finite interval $[0,t]$). Let ${\mathcal{A}}^{+}$ denote the set of all processes $A\in{\mathcal{V}}^{+}$ that are integrable, i.e. such that ${\mathbb{E}}[A_{\infty}]<\infty$, where $A_{\infty}(\omega):=\lim_{t\to\infty}A_{t}(\omega)\in\overline{{\mathbb{R}}}_{+}$ for every $\omega\in\Omega$. Moreover, let ${\mathcal{A}}$ denote the set of all $A\in{\mathcal{V}}$ that have integrable variation, i.e. $\operatorname{Var}\,(A)\in{\mathcal{A}}^{+}$, where for every $t\geq 0$ and every $\omega\in\Omega$, $\operatorname{Var}\,(A)_{t}(\omega)$ is defined as the total variation of the function $s\mapsto A_{s}(\omega)$ on $[0,t]$. A process $X$ is called a semimartingale if it has a decomposition of the form

where $X_{0}$ is finite-valued and ${\mathcal{F}}_{0}$-measurable, $M\in{\mathcal{M}}_{{\mathrm{loc}}}$ with $M_{0}=0$ and $A\in{\mathcal{V}}$. If $A$ in decomposition (2.1) is predictable, $X$ is called a special semimartingale and the decomposition is unique. A semimartingale is called quasi-left continuous if a.s. $\Delta X_{\tau}=0$ on the set $\{\tau<\infty\}$ for all predictable times $\tau$.

Let $X$ be an ${\mathbb{R}^{d}}$-valued semimartingale, i.e. each component of $X$ satisfies (2.1). Denoting by $\varepsilon_{a}$ the Dirac measure at point $a$, the random measure of jumps $\mu^{X}$ of $X$ is an integer-valued random measure of the form

There is a version of the predictable compensator of $\mu^{X}$, denoted by $\nu$, such that the $\mathbb{R}^{d}$-valued semimartingale $X$ is quasi-left continuous if and only if $\nu(\omega,\{t\}\times\mathbb{R}^{d})=0$ for all $\omega\in\Omega$, c.f. Jacod and Shiryaev (2003), Corollary II.1.19.

In general, $\nu$ satisfies

The semimartingale $X$ admits a canonical representation

where $h:{\mathbb{R}^{d}}\to{\mathbb{R}^{d}}$ is a truncation function, i.e. a function that is bounded and behaves like $h(x)=x$ around 0, $B(h)$ is a predictable ${\mathbb{R}^{d}}$-valued process with components in ${\mathcal{V}}$, and $X^{c}$ is the continuous martingale part of $X$.

Denote by $C$ the predictable ${\mathbb{R}}^{d}\otimes\mathbb{R}^{d}$-valued covariation process defined as $C^{ij}:=\langle X^{i,c},X^{j,c}\rangle$. Then the triplet $(B(h),C,\nu)$ is called the triplet of predictable characteristics of $X$ (or simply the characteristics of $X$). It can be shown (see Proposition II.2.9 in Jacod and Shiryaev (2003)) that there exists a predictable process $A\in{\mathcal{A}}_{{\mathrm{loc}}}^{+}$ such that

where $b(h)$ is a $d$-dimensional predictable process, $c$ is a predictable process taking values in the set of symmetric non-negative definite $d\times d$-matrices and $F$ is a transition kernel from $(\Omega\times{\mathbb{R}}_{+},{\mathcal{P}})$ into $({\mathbb{R}^{d}},{\mathcal{B}}({\mathbb{R}^{d}}))$. Here ${\mathcal{P}}$ denotes the predictable $\sigma$-field on $\Omega\times{\mathbb{R}}_{+}$. We call $(b(h),c,F;A)$ the triplet of differential (or local) characteristics of $X$. If $X$ admits the choice $A_{t}=t$ above, we say that $X$ has absolutely continuous characteristics (or shortly AC) and call $X$ an Itô semimartingale.

An important subclass of semimartingales is the class of Itô semimartingales with independent increments. These processes are known as time-inhomogeneous Lévy processes or as Processes with Independent Increments and Absolutely Continuous characteristics (PIIAC), see e.g. Section 2 in Eberlein et al. (2005). The differential characteristics $(b(h),c,F)$ of a PIIAC $X$, for every truncation function $h$, are deterministic and satisfy the following integrability assumption: For every $T>0$

where $\|\cdot\|$ denotes any norm on the set of $d\times d$-matrices. For every $t>0$, the law of $X_{t}$ is characterized by a Lévy-Khintchine type formula for its characteristic function, see again Section 2 in Eberlein et al. (2005). This property makes the class of PIIAC particularly suitable for applications. The following definition and results on exponentials of semimartingales are given in Definition 2.12, Lemma 2.13 and Lemma 2.15 in Kallsen and Shiryaev (2002).

Let $X$ be an ${\mathbb{R}^{d}}$-valued semimartingale with differential characteristics $(b(h),c,F;A)$ and $\lambda\in L(X)$, where $L(X)$ denotes the set of predictable processes integrable with respect to $X$, c.f. Jacod and Shiryaev (2003), page 207. Moreover, assume that $\lambda\cdot X$ is exponentially special. Following Jacod and Shiryaev (2003), Section III.7.7a we define the Laplace cumulant process

where

and the modified Laplace cumulant process $K^{X}(\lambda):=\ln({\mathcal{E}}(\widetilde{K}^{X}(\lambda)))$, where ${\mathcal{E}}$ denotes the stochastic exponential, and

The following results are proved in Proposition III.7.14 and Theorem III.7.4 in Jacod and Shiryaev (2003):

In the following section we give sufficient conditions for the martingale property of exponential semimartingales.

Integrability conditions ensuring the (UI) martingale property of a non-negative or positive local martingale were studied from many perspectives and in various levels of generality. It started with the classical conditions of Novikov (1972) which applies to continuous exponential local martingales. A natural generalization included jumps was given in the seminal paper of Lepingle and Mémin (1978). Various related conditions are given by Kallsen and Shiryaev (2002). A profound overview on Novikov-type conditions as well as boundedness conditions is given in the monograph of Jacod (1979). In this section we collect conditions for exponential semimartingales and express them in terms of semimartingale characteristics. Thanks to these expression we usually call these type of conditions predictable conditions. Let us start with a Novikov-type integrability condition which is based on the main result of Lepingle and Mémin (1978). We follow its statement given by Jacod (1979) as Corollary 8.44.

Proof: Note that the characteristics of $X^{T}$, for any $T>0$, are given by $(B^{T},C^{T},\nu^{T})$, where $\nu^{T}({\mathrm{d}}t,{\mathrm{d}}x):={\mathbf{1}}_{[0,T]\times\mathbb{R}^{d}}\nu({\mathrm{d}}t,{\mathrm{d}}x)$. Now, since local martingales whose localizing sequence is deterministic are martingales, the first claim follows immediately from the second. Note that (A1) implies that $X$ is exponentially special and hence that $M$ is a local martingale. In view of Theorem 2.19 in Kallsen and Shiryaev (2002), the second claim follows from Jacod (1979), Corollary 8.44. $\Box\hskip-1.42262pt$

As an immediate corollary we derive the follows sufficient conditions for the case where $X$ is given as a stochastic integral.

Proof: The characteristics of $\lambda\cdot X$ are given by Proposition IX.5.3 in Jacod and Shiryaev (2003). Now the claim follows by an application of Proposition 3.1. $\Box\hskip-1.42262pt$

For applications the following boundedness condition turns out to be useful, see for instance Corollary 4.1, Proposition 4.3 and 4.2 in the sections below.

Proof: Again, the first part is an immediate consequence of the second. Note that (C1) implies that $\lambda\cdot X$ is exponentially special. Hence, we can deduce the claim from Theorem 2.19 in Kallsen and Shiryaev (2002) together with Lemma 8.8 and Theorem 8.25 in Jacod (1979). $\Box\hskip-1.42262pt$

Let us shortly turn to the subclass of semimartingales with independent increments (SII processes), for which the situation is slightly different than in the more general case. For exponential SII processes the local martingale property is equivalent to the true martingale property. From a mathematical finance perspective this interesting fact for instance implies that exponential SII models cannot include bubbles which are modeled as strict local martingales.

Let us formalize this observation and add some simple deterministic conditions for the martingale property. The main implication $(ii)\Rightarrow(i)$ is essentially thanks to Kallsen and Muhle-Karbe (2010), Proposition 3.12. Note that the assertion does not require quasi-left continuity.

Proof: The implication $(i)$ $\Rightarrow$ $(ii)$ is trivial and equivalence $(ii)$ $\Leftrightarrow$ $(iii)$ holds by definition. The equivalences of $(iii),(iv)$ and $(v)$ are due to Remark 2.2 and Jacod and Shiryaev (2003), Proposition IX.5.3, which shows that $\lambda\cdot X$ has deterministic characteristics with

It is left to show the implication $(iii)$ $\Rightarrow$ $(i)$. In view of (2.6) and since $\lambda\cdot X$ has deterministic characteristics, $K^{X}(\lambda)$ is a deterministic process of finite variation and hence also has deterministic characteristics. Define $f(x,y):=x-y$, then $Y:=\lambda\cdot X-K^{X}(\lambda)=f(\lambda\cdot X,K^{X}(\lambda))$. It follows from Goll and Kallsen (2000), Corollary 5.6 applied to $f(\lambda\cdot X,K^{X}(\lambda))$ that $Y$ has also deterministic characteristics. From the relationship of ordinary and stochastic exponentials given in Jacod and Shiryaev (2003), Theorem II.8.10, we obtain that $M=e^{Y}=\mathcal{E}(\overline{Y}),$ where $\overline{Y}$ is a semimartingale with $\Delta\overline{Y}=(e^{\Delta Y}-1)>-1$. We deduce from Jacod and Shiryaev (2003), Equation II.8.14 that $\overline{Y}$ inherits the property of deterministic characteristics from $Y$. Due to Remark 2.2(b), the condition $e^{\langle\lambda,x\rangle}{\mathbf{1}}_{\{\langle\lambda,x\rangle>1\}}*\nu\in\mathcal{V}$ yields that $\lambda\cdot X$ is exponentially special. Thus, since $K^{X}(\lambda)$ is the exponential compensator of $\lambda\cdot X$, c.f. Proposition 2.3 (i), $M$ is a local martingale. The claim now follows from Proposition 3.12 in Kallsen and Muhle-Karbe (2010). $\Box\hskip-1.42262pt$

In this section we present two applications of the results from Section 3 to financial modeling. A detailed overview concerning applications of general semimartingales in finance is for instance provided by the monographs of Shiryaev (1999), Cont and Tankov (2003), Musiela and Rutkowski (2005) and Jeanblanc et al. (2009).