1 Introduction
Over the last 20 years, a mathematical theory of bubbles for continuous-time models has been developed based on the concept of strict local martingales; see the seminal papers by Loewenstein and Willard [
12], Cox and Hobson [
4], Jarrow et al. [
9,
10] as well as the survey article by Protter [
17] and the references therein. In economic terms, an asset price bubble exists if the fundamental value of the asset deviates from its current price. If the fundamental value is understood to be the expectation of the (discounted) price process
\(S = (S_{t})_{t \geq 0}\) under the equivalent local martingale measure ℙ, then the asset
\(S\) has a ℙ-bubble if
for some fixed time
\(T>0\), i.e., if
\(S\) is a strict local ℙ-martingale, that is, a local ℙ-martingale that fails to be a ℙ-martingale.
While many implications and extensions of this definition have been discussed in the literature (see e.g. Ekström and Tysk [
6], Bayraktar et al. [
1], Biagini et al. [
2], Herdegen and Schweizer [
7]), the strict local martingale definition of bubbles has no direct analogue in discrete-time models. The reason for this is that a nonnegative local martingale
in discrete time with
\(S_{0}\in L^{1}\) is automatically a (true) martingale. Hence, a definition of bubbles based on strict local martingales is void. Also, trying to define a bubble in discrete time as a martingale
that is not uniformly integrable does not lead to a meaningful concept as this would imply that virtually all relevant models such as the standard binomial model (considered on an unbounded time horizon) are bubbles, which seems absurd.
Despite the above negative results, the goal of this paper is to introduce a new definition of bubbles for discrete-time models on an unbounded time horizon – keeping the standard assumption that the discounted stock price is a martingale. This definition has to satisfy at least two conditions.
(I) It should split martingales that are not uniformly integrable into two sufficiently rich classes: those that are bubbles and those that are not. In particular, standard discrete-time models with i.i.d. returns like the binomial model should not be bubbles.
(II) It should be consistent with the strict local martingale definition in continuous time in the sense that a continuous local martingale in continuous time is a strict local martingale if and only if all appropriate discretisations thereof are bubbles in discrete time.
To the best of our knowledge, there has been no attempt in the extant literature to extend the martingale theory of bubbles to discrete time. The only slight exception is Roch [
18] who introduced the notion of asymptotic asset price bubbles using the concept of weakly convergent discrete-time models (“large financial market”). More precisely, he showed that even if the price process is a martingale in a sequence of weakly convergent discrete-time models, it can have properties similar to a bubble in that the fundamental value in the asymptotic market can be lower than the current price in the asymptotic market. In contrast to [
18], our approach is non-asymptotic.
To motivate our definition of a bubble in a discrete-time model, consider a non-sophisticated investor who follows a simple buy-and-hold strategy to invest into an asset with (discounted) price process . The investor buys the asset at time \(k=0\) and hopes that it will rise and rise. When the (discounted) asset price drops for the first time, the investor fears to lose money and sells the asset. Denoting by \(\tau _{1}:=\inf \{j>0 : S_{j}< S_{j-1}\}\) the time of the first drawdown of the asset, the fundamental value under ℙ of \(S\) at time 0 (viewed with regard to the first drawdown) is , where ℙ denotes an equivalent martingale measure. As the process \(S\) is a nonnegative supermartingale, we always have . If , the fundamental value of the asset (viewed with regard to the first drawdown) is lower than its initial price and hence \(S\) might be considered a ℙ-bubble. Indeed, if the market is complete, the predictable representation theorem implies that a sophisticated investor might choose a dynamic trading strategy \(\vartheta \) whose (discounted) value process satisfies and \(V_{\tau}(\vartheta ) = S_{\tau}\).
Of course, the requirement that \(S\) loses mass at the first drawdown is somewhat arbitrary and unrealistic. For this reason, our precise definition of a bubble in discrete time is more general and only requires \(S\) to lose mass at the \(k\)th drawdown for some . While this definition is very simple, it leads to a rich theory.
In Sect.
2, we provide several equivalent probabilistic characterisations for a nonnegative discrete-time martingale to be a bubble. We also provide necessary and sufficient characterisations for a discrete-time martingale with independent increments to have a bubble. In particular, we show that i.i.d. returns models such as the standard binomial model do not have a bubble, which implies that Condition (I) above is satisfied.
In Sect.
3, we look at the special case that
\(S\) is a Markov martingale. We provide characterisations for the presence or absence of bubbles, depending on the
probability of going down, and the
relative recovery when going down. Loosely speaking, it turns out that
\(S\) is a bubble if and only if
\(b(x)\) converges to 0 fast enough as
\(x \to \infty \). To make this precise, however, is quite involved. While we are able to give sufficient conditions in the general case, we provide necessary and sufficient conditions in the case of complete markets.
In Sect.
4, we continue our study of Markov martingales by looking more closely at the underlying Markov kernel. We show that the existence of bubbles for
\(S\) is directly linked to the existence of a non-trivial nonnegative solution to a linear Volterra integral equation of the second kind involving the Markov kernel. Among other things, this allows us to give some additional sufficient conditions for the existence of bubbles that cannot be covered with the results from Sect.
3.
Finally, in Sect.
5, we discuss how our definition of a bubble in discrete time relates to the strict local martingale definition in continuous time. We show that when discretising a positive continuous strict local martingale along sequences of stopping times in a certain somewhat canonical class, one obtains a bubble in discrete time. Conversely, we show that a positive continuous local martingale is a strict local martingale if for all localising sequences in the same class, the corresponding discretised martingales are bubbles. This shows that Condition (II) above is also satisfied.
1 To prove these discretisation results, we rely on the deep change of measure techniques first employed by Delbaen and Schachermayer [
5] and further developed by Pal and Protter [
15], Kardaras et al. [
11] and Perkowski and Ruf [
16] that allow turning the inverse of a nonnegative strict local martingale into a true martingale under a locally dominating probability measure. Some technical proofs of this section are shifted to the
Appendix.
2 Definition and characterisation of bubbles
In this section, we introduce our definition of a bubble in discrete time and provide equivalent probabilistic characterisations of this concept.
Some comments on the above definition are in order.
We proceed to give a first simple example of a bubble in a complete market model.
The following result provides two equivalent characterisations of bubbles. The first shows that a nonnegative martingale \(S \) is a bubble if and only if there exists a deterministic time such that \(S\) loses mass at the first drawdown after \(k\). The second provides a limit characterisation. The latter characterisation is particularly useful for checking whether or not a martingale \(S\) is a bubble.
The characterisation (c) in Theorem
2.4 is generally the most useful to decide whether or not
\(S\) is a bubble. The following corollary strengthens this characterisation. It shows directly that bounded martingales fail to be bubbles. (Of course, this follows directly from the fact that a bounded martingale is uniformly integrable.)
While characterisations (b) and (c) in Corollary
2.5 are an improvement of Theorem
2.4 (c), they still depend on
\(S_{\infty}\), of which we generally do not have a good knowledge. The following corollary provides a mild condition on
\(S\) under which the bubble behaviour of
\(S\) can be characterised without involving
\(S_{\infty}\).
The following result gives simple necessary and sufficient conditions for a nonnegative martingale with independent increments to be a bubble.
We illustrate the above theorem by two examples. The first gives a bubble in a time-dependent binomial-type model, where the downward jumps get more and more severe.
The second example shows that a martingale with i.i.d. returns is never a bubble. In particular, a standard binomial model is never a bubble, which agrees with our intuition.
3 Characterisation of bubble measures for Markov chains
Throughout this section, we suppose that
is a positive martingale which is a Markov process with transition kernel
\(K:(0,\infty )\times \mathcal{B}{(0,\infty )} \to [0,\infty )\) and starting from
\(S_{0}=x>0\). Our goal is to determine under which conditions on the kernel
\(K(x, \,\mathrm{d}y)\) the measure
is a bubble measure. To this end, we define the functions
\(a, b: (0, \infty ) \to [0, 1)\) by
(3.1)
(3.2)
Hence
\(a(x)\) denotes the
probability of a downward jump and
\(b(x)\) the
relative recovery in case of a downward jump.
First, we show we show that \(S\) cannot be a bubble unless the relative recovery function \(b\) converges to zero at infinity.
We proceed to formulate a mild condition on the function \(a\) which allows us to characterise the bubble behaviour of \(S\) without involving \(S_{\infty}\).
We next aim to give a sufficient condition for
\(S\) to be a bubble. To this end, we need to slightly relax the definition of the function
\(b\) from (
3.2). For
\(\varepsilon > 0\), define the function
\(b_{\varepsilon }: (0, \infty ) \to (0, 1]\) by
We illustrate the above result by two examples. The first is a “smooth” version of Example
2.3; the second is an example of a “discrete diffusion” for the log price.
While Proposition
3.1 and Theorem
3.5 give useful general sufficient conditions for the absence or presence of a bubble, respectively, these conditions are not necessary. In the complete Markov case, we can give a necessary and sufficient characterisation of bubbles under mild assumptions on the functions
\(a\) and
\(b\).
The above process corresponds to a binomial-type model where the probability of downward jumps is bounded away from 0 and 1 and the relative recovery in case of a downward jump decreases for large values.
4 A fixed point equation associated to a Markovian bubble
In this section, we continue our study of Markov martingales, taking a more analytic perspective. We assume throughout that is a positive Markov martingale with kernel \(K:(0,\infty )\times \mathcal{B}{(0, \infty )}\to [0,\infty )\), starting from \(S_{0}=x>0\). The key object of this section is the default function of \(S\).
It follows from (
2.2) (with
\(k = 0\)) that
, so that
\(M_{S}\) measures the loss of mass at the first drawdown of
\(S\). It is clear that
is a bubble measure for
\(S\) if
\(M_{S}(x) > 0\). The following two results show that
\(M\) essentially fully characterises the bubble behaviour of
\(S\) under
for all
\(x > 0\).
In the remainder of this section, we seek to characterise the function \(M_{S}\) in an analytic way and provide conditions for it to be non-zero.
First, we show that
\(M_{S}\) solves a fixed point equation, more precisely a homogeneous
Volterra integral equation of the second kind; cf. Brunner [
3, Chap. 1.2] for a textbook treatment.
Note that (
4.1) is non-standard in that the domain is non-compact. Therefore, we cannot apply standard existence and uniqueness results for Volterra integral equations, cf. [
3, Chap. 8]. In fact, existence is anyway not an issue since the zero function always solves (
4.1). Since the bubble case corresponds to (
4.1) having a non-zero (nonnegative) solution, we are actually interested in
non-uniqueness, i.e., the case that (
4.1) has multiple nonnegative solutions. By homogeneity of (
4.1), we then always have infinitely many solutions, and so it is clear that we need an additional condition to pin down the default function
\(M_{S}\).
It follows from the definition of
\(M_{S}\) that
\(M_{S}(x) \leq x\) for all
\(x > 0\). So we consider nonnegative solutions to (
4.1) that are dominated by the identity. To this end, denote by ℐ all Borel-measurable functions
\(M: (0, \infty ) \to [0, \infty )\) satisfying
\(M(x) \leq x\) for all
\(x > 0\). Using that
$$\begin{aligned} 0 \leq \int _{[x,\infty )} M(y)K(x,\mathrm{d}y) \leq \int _{(0, \infty )} M(y)K(x,\mathrm{d}y) \leq \int _{(0,\infty )} y K(x, \mathrm{d}y) = x \end{aligned}$$
for all
\(M \in \mathcal{I}\) and
\(x > 0\), we can define the map
\(\mathcal{K}: \mathcal{I}\to \mathcal{I}\) by
$$ \mathcal{K}(M)(x) = \int _{[x,\infty )} M(y)K(x,\mathrm{d}y), \qquad x > 0. $$
Then the nonnegative solutions to (
4.1) dominated by the identity are precisely given by fixed points of
\(\mathcal{K}\).
While the map
\(\mathcal{K}\) is in general not a contraction (and therefore (
4.1) may have multiple solutions on ℐ), it is
monotone, and this property will prove crucial for our subsequent analysis.
Due to monotonicity of
\(\mathcal{K}\), it is very useful to consider
subsolutions and
supersolutions to (
4.1) on ℐ.
The following result shows that we can construct from each sub- or supersolution a solution to (
4.1) by Picard iteration. To this end, for
, define
\(\mathcal{K}^{n}(M)\) recursively by
\(\mathcal{K}^{0}(M) := M\) and
\(\mathcal{K}^{n}(M) := \mathcal{K}(\mathcal{K}^{n-1}(M))\) for
\(n \geq 1\).
We note the following important corollary.
It follows from Lemma
4.4 and Corollary
4.8 that the default function
\(M_{S}\) is dominated by
\(\mathcal{K}^{\infty}(\operatorname{id})\). Under a mild assumption on the kernel
\(K\), we can assert that
\(M_{S}\) coincides with
\(\mathcal{K}^{\infty}(\operatorname{id})\). Thus in this case, we can characterise the default function
\(M_{s}\) as the maximal solution to (
4.1) dominated by the identity.
The following corollary shows that if we can find a non-trivial subsolution to (
4.1), then
\(S\) is a bubble.
A typical candidate for a subsolution in Corollary
4.10 is given by the call function
\(M(x) = (x -L)^{+}\) for some
\(L > 0\). This is illustrated by the following example. Note that this example cannot be addressed with the results from Sect.
3.
While Theorem
4.9 provides a characterisation of the default function
\(M\), it does not provide a criterion to decide whether (
4.1) has a non-trivial, i.e., a non-zero nonnegative solution dominated by the identity. Moreover, it does not provide a criterion to decide whether a given candidate solution
\(M\) to (
4.1) is indeed maximal. Under a stronger assumption on the kernel
\(K\), we can provide a sufficient criterion for the existence of non-trivial solutions to (
4.1) dominated by the identity. Moreover, we obtain a local uniqueness result in this case. To this end, recall the definitions of the functions
\(a\) and
\(b\) from (
3.1) and (
3.2), respectively. Moreover, denote by
\(\Vert \cdot \Vert _{\sup}\) the supremum norm.
We proceed to illustrate Theorem
4.12 by an example.
Combining Theorem
4.12 with Proposition
4.3, we get the following existence results for bubbles. Note that this result covers cases that cannot be treated with the theory of Sect.
3.
5 Relation to the strict local martingale definition of asset price bubbles in continuous-time models
In this final section, we discuss how our definition of bubbles in discrete time relates to the strict local martingale definition of bubbles in continuous time. To approach this question, one first has to discretise a positive continuous local martingale \(X = (X_{t})_{t \geq 0}\) in continuous time in such a way that it becomes a discrete-time martingale. Of course, there are many ways to do this, and we choose a somewhat canonical construction. More precisely, we consider localising sequences of stopping times with \(\tau _{n}\to \infty \) ℙ-a.s. such that for each \(n\), both \(\tau _{n}\) and the stopped process \(X^{\tau _{n}}\) are uniformly bounded. We then define the discrete-time process by \(S_{n} : = X_{\tau _{n}}\). Then \(S\) is a martingale by the stopping theorem and satisfies \(S_{\infty }= X_{\infty}\) , which implies that \(S\) is uniformly integrable if and only if \(X\) is uniformly integrable.
The simplest way to get localising sequences as above is to choose two increasing sequences of positive real numbers
and
converging to infinity and to define the sequence
of stopping times by
\(\tau ^{a,b}_{0}:=0\) and
(5.1)
Then
is a localising sequence of stopping times for
\(X\) with
\(\tau ^{a, b}_{n} \leq a_{n}\) and
\(\sup _{t \geq 0} X^{\tau ^{a, b}_{n}}_{t} \leq b_{n}\), by continuity of
\(X\).
In the special case that
\(X\) is a Markov process, we should like to stop in such a way that the discrete-time process
\(S\) is again a Markov process. In this case, the simplest way to get localising sequences as above is to choose two constants
\(\alpha , \beta > 0\) and to define the sequence of stopping times
by
\(\tau ^{\alpha ,\beta}_{0}:=0\) and
(5.2)
In this case, it is still true that
is a localising sequence of stopping times for
\(X\) and that
\(\tau ^{\alpha , \beta}_{n}\) and
\(X^{\tau ^{\alpha , \beta}_{n}}\) are uniformly bounded.
Our first goal in this section is to show that if
\(X\) is a continuous positive strict local martingale, then the discrete-time process
\(S\) is a bubble for either choice of stopping times above. The proof of this result relies on the following deep characterisation of strict local martingales in continuous time; cf. Meyer [
13], Delbaen and Schachermayer [
5] and Kardaras et al. [
11]. Let
\(X = (X_{t})_{t \geq 0}\) be a positive càdlàg local ℙ-martingale with
\(X_{0} = x\). Then under some technical assumptions on the probability space and the underlying filtration, there exists a probability measure ℚ with
for all
\(t\geq 0\) such that
\(Y:=1/X\) is a nonnegative true ℚ-martingale, and for all bounded stopping times
\(\tau \) and all
\(A\in \mathcal{F}_{\tau}\),
Especially, we have the identity
i.e.,
\(X\) is a strict local martingale on
\([0,t]\) if and only if
.
With this, we have the following two results.
We only establish the proof of Proposition
5.1. The proof of Proposition
5.2 is similar and left to the reader.
We proceed to illustrate Proposition
5.2 by an example.
We finish this section by providing a converse to Proposition
5.1.
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