Mathematical models of bubbles

Abstract

In this paper we review recent attempts to model mathematically the price evolution of risk assets when they are undergoing bubble pricing. We consider both continuous processes and processes with jumps, and use the framework that, under a risk-neutral measure, the price process will be a strict local martingale (and not a ‘true’ martingale) when bubble pricing is present. Finally, we mention briefly the issue of causes for bubbles, and one approach to modelling them mathematically.

Keywords:

• Financial bubbles
• Bubble lifetimes
• Strict local martingales
• Generalized gamma distributions

JEL Classification:

• G100
• G120
• C810

1. Introduction

Over the last decade an increasingly popular approach to the modelling of financial bubbles involving positive risky asset prices is via stochastic differential equations where the solution, under a risk-neutral measure, is no longer a martingale but instead is a local martingale which is not a martingale, known as a strict local martingale. In this short paper we review recent progress in the modelling of bubbles in this manner, both for continuous path processes and for processes with jumps. We include the now classic models of Delbaen and Shirakawa (2002) as discussed in the companion piece of Jarrow (2015), and go on to include stochastic volatility models and strict local martingales with jumps. We close with a comment concerning the causes for bubbles, using the work of Dandapani (2016).

2. The basic idea for the modeling of financial bubbles

In this section we establish the mathematical framework for the modelling of a stock price process that can give rise to a financial bubble. We begin with a complete probability space and a filtration satisfying the ‘usual hypotheses’. (See Protter 2005 for all otherwise undefined terms.) We let be at least progressively measurable, and it denotes the instantaneous default-free spot interest rate, and (1) is then the time t value of a money market account. We work on a time interval where T is a finite fixed time. Working with one chosen and fixed stock, we let be the dividend process, and we assume it is a semimartingale. We let be nonnegative and denote the semimartingale price process of the stock. Since S has càdlàg paths, it represents the price process ex cash flow. By ex cash flow, we mean that the price at time t is after all dividends have been paid, including the time t dividend. We also assume that S is such that there is an absence of arbitrage opportunities, in the sense of Delbaen and Schachermayer (1998). Let be the time τ terminal payoff or liquidation value of the asset. We assume that .

Next, let W be the wealth process associated with the market price of the risky asset plus accumulated cash flows: (2) Note that all cash flows are invested in the money market account. We assume the market is incomplete, so that there are an infinite number of risk-neutral measures. For a given risk-neutral measure Q, if we take conditional expectations in equation (2(2) ) and rearrange the terms, this translates into (3) The superscript ⋆ will be used systematically to denote fundamental values.

Definition 1.

We define by the difference between the market price and the fundamental price. In a well functioning market, this difference is 0. The process β is called a bubble.

Note that the possibility of buy and hold strategies and the absence of arbitrage keeps β nonnegative.

The next theorem (see Jarrow et al. 2010; Protter 2013 for proofs and more details) allows us not to consider the fundamental price of our stock given in equation (3(3) ), which is vague and ultimately unknowable, with a precise mathematical criterion provided by Theorem 1:

Theorem 1.

A risky asset price process S is undergoing bubble pricing on the compact time interval if and only if under the chosen risk-neutral measure the bubble process β is not a martingale but is a strict local martingale. This is equivalent to the price process S being a strict local martingale since is always a martingale and not a martingale.

With Definition 1, which involves and hence implicitly the risk-neutral measure, and despite the recent work Herdegen and Schweizer (2015) where as many objects as possible are defined without reference to a risk-neutral measure, there is nevertheless the issue of which risk-neutral measure should one use. Fortunately, in simple cases that are nevertheless sophisticated enough to be useful, we can finesse this issue. For example, if we model the dynamic evolution of the stock price as a solution of a stochastic differential equation of the form (4) where b is controlled as not to be too big, and Y represents a stochastic process reflecting relevant market forces, then under any one of the infinite choice of risk-neutral measures Q, we have that the drift disappears via a Girsanov-type transformation, to get (5) and therefore it does not matter which risk-neutral measure we use! (see, e.g. Jarrow et al. 2011; Protter 2013 for details of this procedure.) From now on we work under a fixed risk-neutral measure, whose choice is arbitrary.

The theory developed in Delbaen and Shirakawa(2002); Mijatović and Urusov (2012) that says if satisfies the two conditions (6) (7) then X>0 and also X is a strict local martingale. If on the other hand, the integral in the inequality (7(7) ) equals ∞, then X is a martingale. This gives a way to check to see if a price process, assumed to evolve following an equation of type (4(4) ), is a strict local martingale or a true martingale. These techniques are discussed by Jarrow in his contribution to this volume (Jarrow 2015). For recent (more technical) results in this area, see, e.g. Obayashi et al. (2016).

3. Stochastic volatility and strict local martingales

The model given in equation (4(4) ), reduced to equation (5(5) ) under a risk-neutral measure, is largely sufficient for most applications, as can be seen in the largely empirical work of Obayashi et al. (2016). But one might nevertheless find it limiting, since it does not include a criterion for (for example) stochastic volatility models, such as Heston-type models. As it turns out, in Sin (1998), and later and more generally, Lions and Musiela (2007), and also Andersen and Piterbarg (2007), both in 2007, gave necessary and sufficient conditions for a simple stochastic volatility model to be a martingale or a strict local martingale. Their motivations are unrelated to bubbles (except implicitly since they deal with the lack of constancy of the expectation of the price process in the strict local martingale case), but nevertheless today they have great applicability to bubbles. For example, Dandapani (2016) has recently applied the ideas of Lions and Musiela (2007) to showing how bubbles can form via the addition of new information to the underlying filtration of market observable events.

Besides the work of Sin, Lions and Musiela, and that of Andersen and Piterbarg, the recent work of Jacod and this author (Jacod and Protter 2015) contains a result we describe next.

We begin with the general stochastic volatility framework as follows: (8) where the stochastic process ν is optional, but as of yet is unspecified.

Definition 2.

We let denote the class of functions such that if and that there exists a unique strong solution to equation (8(7) ) then the solution is necessarily nonnegative, with 0 an absorbing point.

Theorem 2.

Assuming the solution Y of equation (8(7) ) is a strict local martingale in the following two situations, where τ is a stopping time and is an -measurable variable such that

• (i) The process ν is constant in time on the interval and for each y the function verifies equation (6(6) )

• (ii) The process ν takes its values in some set Γ on the interval and verifies equation (6(6) ) when .

What the above theorem describes is that a stock price process follows a stochastic volatility equation, since its price evolution is affected by the state of the economy, or some aspect of the economy, which is represented by the process ν in equation (8(7) ). But when the stock price process is in the throes of bubble pricing, the influence of the process ν wanes to such an extent that the price depends only on itself and the driving Brownian motion, at least under the risk-neutral measure, and at least for a positive (but possibly brief) time interval. In this case, we can apply the results of Delbaen and Shirakawa (2002) or better Mijatović and Urusov (2012) to show that in this short-time interval the expectation decreases, and hence the process is a strict local martingale, and a viable model for a price bubble. The proof is more complicated than this description, but nevertheless that is the main idea.

Example 1.

We let Y be as in equation (8(7) ). That is, and assume s is such that the unique solution Y of equation (8(7) ) has the property that for t<T. We choose in such a way that for a given constant α, we have , where verifies equations (6(6) ) and (7(7) ). That is, when the values of Y get sufficiently large, the function becomes a function of y alone, and that function satisfies the Delbaen–Shirakawa conditions (Delbaen and Shirakawa 2002). This is inspired by the behaviour of bubbles: When a stock is shooting up during bubble birth, it tends to lead a life of its own, uninfluenced to a large extent by the behaviour of the economy in general, at least for the short-time during bubble birth. In this case, we have that Y is a strict local martingale. This example is established rigorously in Jacod and Protter (2015).

4. Bubbles for models with jumps

Our discussion in Sections 2 and 3 is restricted to models with continuous paths. However, many scholars feel that in some cases, processes with jumps are better models of reality, even in highly liquid cases; see, for example, the recent book of Aït Sahalia and Jacod (2014), and all of the references therein. Also, many stocks do not trade that often, and if one were to plot their price processes on a tick by tick basis, one would find a graph that is piecewise constant and changes only by jumps. To include both of these situations, it is useful to include models of strict local martingales with jumps. Such models have been proposed before in 2007 by Chybiryakov (2007), but the examples are rather complicated. Instead, we can construct more intuitive examples by shrinking the filtration to such a point that jumps are introduced, and then projecting the models of equations (4(4) ), and (5(5) ) in the risk-neutral framework, onto the smaller filtration.

The next theorem does just this; it is taken from Protter (2015).

Theorem 3.

Let B be a standard Brownian motion on its canonical space and let X be a solution of equation (5(5) ) satisfying equation (6(6) ). Let contain left isolated points such that and let be the filtration generated by the first passage times of for every point in Λ, with satisfying the usual hypotheses. Let M be the projection of X onto . That is, and for a fixed t one has . We take the càdlàg version of M. Then M is a strict local martingale that jumps at every time for β a left isolated point in Λ, and for the first passage time of at β.

These ideas have not been applied to the models of Lions and Musiela (2007), or to the situation presented in Theorem 2, but they could be.

Recently another promising approach to strict local martingales with jumps has been proposed by Keller-Ressel (2015). These models enrich the class of processes we can consider as models for price evolutions with bubbles. The approach presented in Keller-Ressel (2015) is

Example 2.

An example due to Keller-Ressel (2015) of a strict local martingale with jumps is based on the theory of strong Markov processes. We let be a Poisson random measure on , with compensator We next let (9) Then Keller–Ressel shows that is a strict local martingale. It jumps because X is a Markov process that jumps by construction.

5. How bubbles form

These models for bubbles, and their concomitant ability to be tested, as explained in the companion paper of Jarrow (2015) and in the recent paper Obayashi et al. (2016), describe the price evolutions of bubbles, but do not lend insight to how they may be born. The exciting recent work of Dandapani (2016) however does do that, at least for one special case. She considers the addition of new information using an expansion of filtrations technique, doing it in such a way as to avoid introducing arbitrage opportunities. This is challenging, as is pointed out (for example) in the recent paper of Fontana et al. (2014). Dandapani shows how the addition of information, in a Heston-type stochastic volatility model, can change a martingale into a strict local martingale, and as such, change a well-priced risky asset into a bubble priced risky asset. We feel this is a promising direction for the future.

Acknowledgements

The author wishes to thank his friend and former colleague Bob Jarrow for his careful reading of this paper, and also for suggesting that I add examples, resulting in Examples 1 and 2.

Disclosure statement

No potential conflict of interest was reported by the authors.