by Charlie Woodman, Research Assistant, Cambridge Centre for Finance and Cambridge Endowment for Research in Finance
One of the most frequently asked questions in the financial news media this year has been whether the stock market is in a ‘bubble’? Since the onset of the pandemic in early 2020, the S&P500 experienced a -35% return between the middle of February and the middle March before bouncing back to record levels by September. One year since the low in February, the index is now sitting around $3900, representing an annual return of more than 75%. This has led many to question, and in some cases proclaim, that US and international stock markets are in a bubble. But what exactly is a bubble, and how can we discern one?
A bubble is defined as the difference between price and fundamental value, where fundamental value is defined as the discounted sum of conditional expected future payoffs (Jarrow et al, 2010). Broadly speaking, there are two types of bubble: rational and irrational. During an irrational bubble, price deviates from fundamental value as a consequence of behavioural biases and limits to arbitrage, which prevents new information from being fully incorporated into prices. Irrational bubbles represent an arbitrage opportunity and a failure of market efficiency. Testing for an irrational bubble requires a model for fundamental value, which is unobservable, or on the model implied dynamics of a bubble component that has irrational origins. Thus, tests for irrational bubbles are plagued by the joint hypothesis problem. Moreover, theories of irrational bubble formation are most often designed to fit a pre-conceived notion that a bubble is a period of dramatic price ascent followed by a spectacular collapse. In other words, the starting point becomes one of an idealised price trajectory, rather than a deviation of price from fundamental value. This is unsatisfactory because there are many possible explanations for a rapid increase in prices that do not require behavioural explanations, chief among which is a decrease in the discount rate, which may be driven by changes in risk preferences or investors rationally updating their expectations about future returns. Bubbles can also occur during periods of sustained price decline, so long as fundamental value is falling at a greater rate. This leads to significant sample bias when designing and testing models using a handful of examples that are chosen on the basis of their convenient price characteristics.
The rational bubble literature can be decomposed into that which models prices in discrete-time and that which models prices in continuous-time. In the discrete-time setting, a rational bubble is born form a failure of the transversality condition in the rational asset pricing equation for stocks. Put simply, the bubble is expected to grow at a rate equal to the discount rate. The implied price behaviour is explosive, that is the expected price tomorrow is equal to the price today multiplied by a factor greater than 1. Phillips, Wu and Yu (2011) and Phillips, Shi and Yu (2015) tested this implication by applying recursive econometric tests for explosiveness to the NASDAQ and the S&P500. The results of these tests offer support for the dotcom bubble thesis and also identify several other shorter periods of explosiveness before and after. However, the discrete-time rational bubble framework does not permit the reformation of bubbles after they have burst and by implication, a bubble can only exist if it existed since the start of trading. Furthermore, discrete-time bubbles can only exist for infinitely lived assets. To that extent, the empirical literature is removed from the theory that motivates it, such that it is difficult to say whether the results are attributable to bubbles or other possible sources of explosive behaviour. For example, explosiveness could also be a consequence of a convex decrease in discount rates, which is not synonymous with the definition of a rational bubble.
In a continuous-time framework, bubbles can arise on finitely lived assets, exhibit finite duration and reform after they have burst. Moreover, continuous-time bubbles can be detected without need of a model for fundamental value and where a model is needed, it can be independently verified without incurring the joint hypothesis problem. Continuous-time bubbles are characterised as periods in which the discounted price process is a strict local martingale, which we understand as a local martingale that is not a martingale. An assets price process is a strict local martingale if volatility is a convex function of price. Therefore, continuous-time bubbles differ from discrete-time bubbles insofar as the emphasis shifts from explosiveness in price to explosiveness in volatility. The price paths produced by continuous-time bubbles are often similar to the typical J-shaped runups associated with historically recognised bubble periods, but the model also embeds a much more diverse set of potential price trajectories that are not captured by discrete-time alternatives. This is because strict local martingales are strictly continuous-time phenomena, reflecting the additional flexibility of continuous-time models in finance. Note also that both discrete and continuous-time rational bubbles are consistent with the notion of no-arbitrage, but neither can exist unless the market is incomplete (Jarrow et al, 2010).
Existing testing procedures for continuous-time bubbles can be classified into two groups: parametric and non-parametric. Parametric procedures postulate a model for price, estimate the parameters that govern the process and check if the estimates fall within the interval that renders price a strict local martingale. Non-parametric testing procedures estimate the distribution of volatility over the observable price interval and then extrapolate into the tail to check that volatility is a convex function of price. However, existing parametric approaches are slow to respond to changing market conditions and non-parametric methods often produce false positive signals in real-time applications.
Research on the theoretical modelling and empirical detection of asset price bubbles remains a fertile area of research. For example, recent research by Fusari et al (2020) uses option price data to detect bubbles. The premise is that call option prices can contain bubbles but put option prices cannot, such that one can estimate the magnitude of the bubble in the underlying asset by calculating the difference between observed and model implied prices of call options using a model that accurately prices puts. Another recent approach from Bashchenko and Marchal (2020) applies machine learning techniques, specifically Recurrent Neural Networks (RNNs) with Long-Short Term Memory (LSTM) cell architecture to estimate a parametric model that classifies assets as true martingales or strict local martingales at each time interval. The model is trained on simulated price data and achieves an out-of-sample accuracy in excess of 83%. Finally, recent advances in the theoretical literature continue to provide new insights and new testable hypotheses that will contribute to an improved understanding of bubbles, and the implications of these phenomena for investors, policy makers and the wider economy.
Bashchenko, O. and Marchal, A. (2020) “Deep learning for asset bubbles detection.” (Available at SSRN 3531154)
Fusari, N., Jarrow, R. and Lamichhane, S. (2020) “Testing for asset price bubbles using options data.” (Available at SSRN 3670999)
Jarrow, R.A., Protter, P. and Shimbo, K. (2010) “Asset price bubbles in incomplete markets.” Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 20(2): 145-185
Phillips, P.C., Wu, Y. and Yu, J. (2011) “Explosive behavior in the 1990s Nasdaq: when did exuberance escalate asset values?” International Economic Review, 52(1): 201-226
Phillips, P.C., Shi, S. and Yu, J. (2015) “Testing for multiple bubbles: historical episodes of exuberance and collapse in the S&P 500.” International Economic Review, 56(4): 1043-1078