Testing for asset price bubbles: three new approaches

Abstract

Testing for the existence of asset price bubbles is a difficult task. This paper shows that the local martingale theory of bubbles provides three new approaches that can be used to test for the existence of asset price bubbles, two of which are currently unexplored in the literature. The first is based on specifying a stochastic process for the asset's price process. The second exploits option price data, if options trade on the asset suspected to exhibit a price bubble. The third uses a multiple-factor model for the return process of the underlying asset.

Keywords:

• Asset price bubbles
• Local martingale measures
• Option prices
• Multiple-factormodels

JEL Classification:

• G12
• G13
• G14

1. Introduction

This paper proposes three alternative methods, two of which are unexplored in the literature, to test for the existence of asset price bubbles. The traditional approach used to identify asset price bubbles is to estimate a model for an asset's fundamental value and then to compare it to the market price. Although frequently employed, this approach is confounded by a joint hypothesis, which is the model for the asset's fundamental value. The joint hypothesis makes the evidence inconclusive, see Camerer (1989).

Using the local martingale theory of bubbles (see Loewenstein and Willard 2000a,b, Cox and Hobson 2005, Heston et al. 2007, Jarrow et al. 2007, 2010), we propose three alternative methods to test for the existence of asset price bubbles that avoid this joint hypothesis problem. Although the estimation procedures still contain a joint hypothesis, unlike the traditional approach, they allow an independent validation of the second hypothesis, thereby enabling a direct test for asset price bubbles. These alternative approaches are based on investigating properties of the asset's price process or options on the asset's price.

The first approach specifies a stochastic process for the underlying asset's market price that can be used to test to see if the market price contains a price bubble. This approach has been recently proposed in the literature by Jarrow et al. (2011a), but its usage is still in its infancy. The second approach requires the trading in call and put options on the asset conjectured to contain a price bubble. If a bubble exists, a comparison of the asset's put and call prices can identify it. The last and third approach is to use a return factor model to decompose an asset's return into the part due to its fundamental value and the part due to its bubble. Estimation using historical time series and cross-sectional data can differentiate the two components. The last two methods for identifying asset price bubbles are currently unexplored in the literature, and they provide a fruitful area for future research.

An outline for this paper is as follows. Section 2 presents the local martingale theory of bubbles. Section 3 discusses the traditional approach to test for asset price bubbles, while Sections 4–6 present the new methodologies. Section 7 concludes.

2. The local martingale theory of bubbles

This section briefly reviews the local martingale theory of bubbles, see Loewenstein and Willard (2000a,b), Cox and Hobson (2005), Heston et al. (2007), and Jarrow et al. (2007) and Jarrow et al. (2010). For an excellent review of the local martingale theory of bubble's mathematics, see Protter (2013).

We consider a continuous time model on a finite horizon . Let be a filtered complete probability space where the filtration satisfies the ‘usual hypotheses’ (see Protter 2005). The markets are competitive and frictionless. Competitive means that the traders act as price takers, believing their trades have no quantity impact on the market price. Frictionless means that there are no transaction costs and no trading constraints, e.g. short sale constraints, margin requirements, and shares are not infinitely divisible.

Traded in the economy are a risky asset and a money market account that is locally riskless. We denote the time t value of a money market account as (1) where is a non-negative adapted process representing the default-free spot rate of interest.

Let be a càdlàg (left continuous with right limits) semimartingale process^† adapted to which represents the cumulative cash flow process from holding the risky asset. Let be the time T terminal or liquidation value of the asset. We assume that both . The market price of the risky asset is given by the non-negative càdlàg semimartingale representing the price ex-cash flow (since S is càdlàg) where, by the previous assumption, at the liquidation date .

Let G be the cumulative gains process associated with the market price of the risky asset, i.e. (2) Note that all the asset's cash flows are invested in the money market account, and since , .

We assume that the economy satisfies no free lunch with vanishing risk (NFLVR), a technical extension of the standard definition of no arbitrage. This extension excludes both (i) zero investment self-financing trading strategies that have non-negative liquidation values which are strictly positive with positive probability, and (ii) limiting arbitrage opportunities, i.e. the limits of sequences of zero investment self-financing trading strategies that have a small probability of a loss.

Trading strategies are holdings in the risky asset and money market account (for all times and states) that are predictable (depend on only current and past information) and that are admissible (the value of the trading strategy is bounded below), the purpose of which is to exclude doubling strategies from the economy (see Jarrow and Protter 2008 for further discussion).

By the first fundamental theorem of asset pricing (see Delbaen and Schachermayer 1998), NFLVR holds if and only if there exists a probability measure Q equivalent to P such that is a local martingale.^‡ A local martingale is an adapted and càdlàg stochastic process where there exists a sequence of stopping times such that and is a martingale for all n. Q is called an equivalent local martingale measure (ELMM).

To define an asset's price bubble, we need to first define an asset's fundamental value. To do this, we need to identify a unique ELMM. If the market is complete, then by the second fundamental theorem of asset pricing (see Jarrow and Protter 2008) the ELMM is unique, and it is determined by the market. If the economy is incomplete, then there exists a continuum of possible ELMMs. To obtain a unique ELMM from this set, we assume that the model considered is embedded in a larger economy that is complete. This larger economy uniquely determines the ELMM. For example, if enough static trading in call options exist, then the ELMM can be uniquely identified via the traded call option prices (see Jacod and Protter 2010). For the remainder of the paper, we assume that a unique ELMM Q has been selected.

We can now define the risky asset's fundamental value, denoted by the following expression: (3) where denotes expectation under the ELMM Q.

The risky asset's fundamental value is equal to the price a trader would pay if after purchase, she had to hold the asset in her portfolio until time T. Indeed, expression (3(3) ) is the expected discounted cash flows received from holding the asset until time T, where the ELMM Q implicitly adjusts for risk.

The asset market price bubble is defined as (4) A bubble reflects the notion that the resale value of the asset is higher than the price paid if it were to be held forever. A bubble is non-negative because a lower bound on its market price is its value obtained from buying the asset and not retrading.^§ The key observation from this definition is that the asset's price exhibits a bubble if and only if if and only if is a strict local martingale (i.e. it is not a martingale). Necessary and sufficient conditions for the existence of an equivalent martingale probability measure are given in Jarrow and Larsson (2012).

3. Traditional rests

The traditional tests for asset price bubbles are in discrete time, infinite horizon models. The analogue of the traditional approach in continuous time, finite horizon models is based on expression (4(4) ), which we repeat here in an expanded form for convenience (5) To identify the bubble using this expression, one needs to specify and estimate a stochastic model for the default-free spot rate, the dividend process, the asset's liquidation value, and the ELMM, i.e. , thereby generating a joint hypothesis.

Testing the null hypothesis that using this expression will suffer from the same difficulties inherent in testing for bubbles in the discrete time, infinite horizon models.^¶ This difficulty is due to the joint hypothesis. Rejecting the null hypothesis under this joint hypothesis does not necessarily imply there are no price bubbles. It could mean only that the model for is misspecified. The difficultly in this joint hypothesis is that the model for cannot be independently validated.

Because of this joint hypothesis, the empirical literature testing for bubbles in discrete time, infinite horizon models using the traditional approach is inconclusive (see Camerer 1989 for a review of this literature). Although use of this traditional test for continuous time, finite horizon models have not been utilized yet, there is no reason to believe the evidence would be different from that obtained in the existing empirical literature. For this reason, the next three sections present alternative approaches to test for the existence of asset price bubbles in continuous time, finite horizon models. These alternative approaches are designed to overcome the limitations of the joint hypothesis in the traditional approach.

4. Stochastic processes

An alternative approach for testing the existence of asset price bubbles is based on specifying a stochastic process for the market price of the risky asset S where the parameters of the asset's price evolution can be partitioned into two mutually exclusive and exhaustive sets, one where the process exhibits a bubble and one where it does not. Then, standard statistics procedures can be used to estimate the parameters using historical time series data to determine whether the asset price process exhibits a bubble or not. Simply stated, this approach tests whether or not the asset price process is a strict local martingale.

To illustrate this approach, assume the risky asset has no cash flows. Let the asset's market price process be given by the following stochastic differential equation driven by a Brownian motion Z. (6) Here, the asset's price volatility is stochastic and depends on the level of the asset's price.

This evolution implies a complete market. Under suitable integrability conditions, there exists a unique equivalent probability measure Q such that (7) where is a Brownian motion under Q. Hence, given the necessary integrability conditions, this evolution is consistent with NFLVR.

The following theorem provides the basis for the first alternative testing approach.

Theorem 1

(Test for a Local Martingale)

The process is a strict local martingale if and only if (8) for some .

A proof of this theorem can be found in Delbaen and Shirakawa (2002) and Mijatovic and Urusov (2012).

Hence, a bubble exists for this asset price process if and only if expression (8(8) ) is finite. The finiteness of the integral in expression (8(8) ) can be empirically tested. Testing this expression involves a joint hypothesis with . The advantage of a joint hypothesis with , versus , is that the hypothesis for can be independently empirically tested since historical time series for the spot rate of interest and the asset's price are observable. In contrast, it is not possible to independently test the hypothesis for .

A statistical methodology for testing stock price bubbles based on example (8(8) ) was recently developed by Jarrow et al. (2011a). The methodology consists of two steps. The first step is to estimate the volatility as a function of the asset's price level. In this step, only a compact set of prices can be observed. The second step is to extrapolate the shape of the volatility function to the nonnegative real line, to check the finiteness of the integral in expression (8(8) ). Various extrapolation methods can be employed. We illustrate this procedure using two recent examples: one where the methodology documents the existence of a bubble and one where it does not.

The first example is for LinkedIn's IPO. This analysis is taken from Jarrow et al. (2011b). To perform the bubble test, we obtained minute by minute stock price tick data for LinkedIn for the 4 business days 5/19/2011 to 5/24/2011 following its IPO. This gave 1535 price observations, which are presented in figure 1.

Figure 1. Times series of LinkedIn stock prices.

Using this time series data, we estimated the volatility function as shown in figure 2. This volatility function is similar to that of the polynomial , confirming the finiteness of the integral in condition (8(8) ), proving that LinkedIn's stock price exhibited a price bubble after its IPO.

Figure 2. Estimated volatility function for LinkedIn.

The second example is for gold prices for the period August 25, 2011 to September 1, 2011, taken from Jarrow et al. (2011c). Again, we obtained gold price tick data giving 73 695 data points depicted in figure 3. The estimated local volatility function for gold prices with its 95% confidence interval is contained in figure 4. As seen in figure 4, the volatility function is decreasing as gold prices tend to ∞, implying that the integral in condition (8(8) ) is infinite. There is no gold price bubble over this time period.

Figure 3. Time series of gold spot prices.

Figure 4. Non-parametric gold price volatility estimate with 95% confidence intervals.

This approach based on a joint hypothesis with for the empirical validation of asset price bubbles is just in its infancy. Extensions and implementations of this approach represent a fruitful area for additional research.

5. Option prices

Another approach to test for the existence of asset price bubbles requires the trading of call and put options on the asset. If these options trade, then they can be used to determine if the underlying asset price exhibits a bubble or not. For this section, we assume that there are no cash flows on the risky asset and the economy satisfies no dominance. An asset is dominated if there exists a trading strategy whose cash flows and liquidation value are (i) always greater than or equal to the cash flows and liquidation value of the risky asset and strictly greater with positive probability, and (ii) the cost of constructing the trading strategy is less than the market price of the asset. An economy satisfies no dominance (ND) if no traded asset is dominated.

We assume that European call and put options trade on the risky asset with strike price K and maturity date . The call's payoff at expiration is , and the put's payoff at expiration is . Let be the call's and put's time market prices, respectively. Since the put's payoff is bounded and the call's value is unbounded, one can prove the following result (see Jarrow et al. 2010).

Theorem 2

(Put and Call Valuation)

Assume there are no cash flows on the risky asset and the economy satisfies ND. Then, (9) (10)

The theorem shows that for the put option, the standard risk neutral valuation methodology applies, even when the underlying asset exhibits a price bubble. In contrast, however, the standard risk neutral valuation methodology does not apply to the call option. The call's market price deviates from its risk neutral valuation by the difference between the current level of the price bubble and the expected level of the asset's price bubble at the option's maturity. This difference can be used to design a test for the existence of an asset price bubble.

To apply this approach, needed are historical time series of market prices for interest rates, the underlying asset price, and option prices, i.e. . First, one hypothesizes a put option price model, i.e. one specifies . This is standard practice in the empirical literature evaluating option price models. Then, one validates or rejects the model by testing to see if the model price for the put, expression (9(9) ), equals the market price. If accepted, then using the market prices for the call options, one compares the market prices to expression (10) to determine if . If so, then the underlying stock exhibits a bubble.

In contrast to the traditional approach based on a model for , the advantage of using option prices is that any model hypothesized for can be first independently tested on observable put prices for its validity. This option-based method to test for asset price bubbles in currently unexplored in the literature, and as such, is a fruitful area for future research.

6. Return factor models

This section discusses how to use a return factor model as the basis for an alternative approach to test for the existence of asset price bubbles. The method is based on decomposing an asset's return into two components: that due to the fundamental value and that due to the price bubble. An empirical procedure can be designed to identify this difference. The return multiple-factor model utilized depends only on the market structure from Section 2, i.e. risky assets and a money market account trading in a continuous time, finite horizon, frictionless and competitive market that satisfies no-arbitrage (NFLVR).

To state the relevant theorem, we need some notation. Fix a time period , for a given , over which returns are to be computed. Define to be the percentage of the market price of the asset represented by the bubble .

Let be the default-free spot rate over . This is obtained as the return over on a default-free zero coupon bond maturing at time . For simplicity, we assume that the risky asset has no cash flows. Define its return over as We also need to define the return on the asset's price bubble over : Given this notation, we can now state the following theorem, proven in Jarrow (2015).

Theorem 3

(The multiple-factor model with bubbles)

(11) where is the return on the market price of the jth risk-factor over all j, is the return on the bubble embedded in the jth risk-factor over is the percentage of the jth risk-factor's time t market price that its bubble represents, are measurable processes, Φ is a finite collection of risk-factors that span the return of the risky asset, and is an error term with .

This theorem states that under no-arbitrage (NFLVR), the excess return on the risky asset can be decomposed into the excess returns due to its systematic risk , the standard multiple-factor model, plus two modifications due to the possible existence of asset price bubbles. The first modification is that the expected return on the risky asset is increased by the expected return on the bubble component within the asset's market price . This modification implies that positive alphas estimated in standard multiple-factor return models could be due to asset price bubbles and not mispricings. The second modification is that asset price bubbles induce additional systematic bubble risk-factors that may require risk premium . These additional bubble-induced systematic risk-factors reflect the intuitive belief that asset price bubbles are not diversifiable.

This theorem can be the basis for an alternative approach to test for the existence of asset price bubbles. To implement this test, one needs to obtain proxies for the risk-factors, both the market prices and the bubble components . Second, historical time series and cross-sectional data for asset prices and risk-factors are needed to fit expression (11(10) ) using linear regression, perhaps assuming that , are constants. In this estimation, one can also assume that is a constant, representing the intercept in the linear regression. Then, the cross-sectional data can be partitioned into two subsets, one which is believed to contain price bubbles and another set which does not. For example, media stocks could be hypothesized to exhibit price bubbles while bank stocks hypothesized to reflect only their fundamental value. Then, one can test the null hypotheses that for media stocks and for bank stocks.

The joint hypothesis in this test is the structure imposed on the risk-factors and factor loadings , . As with the other two new approaches for identifying asset price bubbles, this structure can be tested independently and then an additional test for asset price bubbles performed, thus avoiding the problems in the traditional tests with the joint hypothesis. This suggested method for identifying the existence of price bubbles is only illustrative. Other, more creative tests, can surely be formulated. The use of this multiple-factor return model to test for the existence of asset price bubbles is also an unexplored research area.

7. Conclusion

Based on recent insights from the local martingale theory of bubbles, this paper discusses three new and unexplored approaches that can be used to test for the existence of asset price bubbles. These three alternatives avoid the joint hypothesis embedded in the traditional methodology for identifying asset price bubbles. These three approaches can be used separately, or in conjunction, to provide new evidence relating the existence of price bubbles. They can be used in real time to exploit bubbles, if they are shown to exist. It is the hope of this author that this brief review spurs empirical investigations based on these methods.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

^† A stochastic process X is a semimartingale if it has a decomposition where (i) , (ii) A is adapted, càdlàg, and of finite variation on compacts, and (iii) M is a local martingale (hence càdlàg) (see Protter 2005, Chapter 2). The definition of a local martingale is provided later in the paper.

^‡ In general, is a martingale, but since it is non-negative (bounded below), it is a local martingale.

^§ Technically, since the discounted cumulative gains process is a nonnegative local martingale, it is a Q supermartingale, implying Equation (4(4) ).

^¶ Besides discrete versus continuous time, the key difference between these two models is that X replaces .