Speculative Bubble - Revision history

Tebaldi at 19:39, 2 April 2023

2023-04-02T19:39:58Z

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	====='''1. DEFINITION OF BUBBLE'''=====		====='''1. DEFINITION OF BUBBLE'''=====

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2023-02-10T09:58:07Z

1. DEFINITION OF BUBBLE

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BIOGRAPHY

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	</math>		</math>



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	<math>f_{t, n}=\phi^n v_t+(1+r)^n B_t </math>		<math>f_{t, n}=\phi^n v_t+(1+r)^n B_t </math>



	And by rewriting Eq. (2.9) the future spot rate at ''t + n'' is given by:		And by rewriting Eq. (2.9) the future spot rate at ''t + n'' is given by:



	<math>s_{t+n} = v_{t+n} +B_{t+n} </math>		<math>s_{t+n} = v_{t+n} +B_{t+n} </math>



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	<math> s_{t+n}=\phi^n v_t+\left(\frac{1+r}{\pi}\right)^n B_t+\varepsilon^*_{t+n} </math>		<math> s_{t+n}=\phi^n v_t+\left(\frac{1+r}{\pi}\right)^n B_t+\varepsilon^*_{t+n} </math>

	Where the term <math>\varepsilon^*{t+n} </math> consists of two moving average processes, <math display="inline">\~~sum~~{i=1}^n \~~emptyset~~^{n-i} \theta_{t+i} </math> and <math display="inline">\sum_{i=1}^n\left(\frac{1+r}{\pi}\right)^{n-i} \varepsilon_{t+i} </math>. By comparing Eq. (2.12) with Eq. (2.14), it is clear that the actual future spot rate is in expectations greater than the forward rate. As noted before, such a difference result from the expectations of the bubble to burst in a future period with probability (following Blanchard’s definition of rational bubbles, Eq. (2.1)). Consequently, the expected growth rate of the bubble <math>(1+r)^n </math> , is lower than the actual rate of growth <math display="inline">(\frac{1+r}{\pi})^n </math>. This is because investors are compensated more and more as the probability of a crash (<math>1 - \pi </math>) increases (and <math>\pi </math> decreases).		Where the term <math>\varepsilon^*_{t+n} </math> consists of two moving average processes, <math display="inline">\sum_{i=1}^n \phi^{n-i} \theta_{t+i} </math> and <math display="inline">\sum_{i=1}^n\left(\frac{1+r}{\pi}\right)^{n-i} \varepsilon_{t+i} </math>. By comparing Eq. (2.12) with Eq. (2.14), it is clear that the actual future spot rate is in expectations greater than the forward rate. As noted before, such a difference result from the expectations of the bubble to burst in a future period with probability (following Blanchard’s definition of rational bubbles, Eq. (2.1)). Consequently, the expected growth rate of the bubble <math>(1+r)^n </math> , is lower than the actual rate of growth <math display="inline">(\frac{1+r}{\pi})^n </math>. This is because investors are compensated more and more as the probability of a crash (<math>1 - \pi </math>) increases (and <math>\pi </math> decreases).

	By subtracting Eq. (2.14) from Eq. (2.12), the following result holds:		By subtracting Eq. (2.14) from Eq. (2.12), the following result holds:

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2023-02-05T20:29:33Z

BIOGRAPHY

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3210664: Bubble (incomplete)

2022-11-29T08:34:42Z

Bubble (incomplete)

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====='''1. DEFINITION OF BUBBLE'''=====

As said in the introduction, bubbles are generally defined as periods where financial assets are traded in high volume, at prices significantly higher than the fundamental value (Shiller, 2006).

Can we detect such mispricing effect? The short answer is yes, there are some econometric tests that can be used to detect for asset price bubbles. However, this answer comes with two major warnings. First, this doesn’t imply that we can predict when a bubble will burst, and secondly, econometric models works beautifully until they don’t. Before delving deeper in reviewing some, among the numerous methods to detect bubbles, it is useful to take a bird view and explore some key features of financial bubbles as described in the scientific literature. This will allow us to grasp some general understanding of the mechanics behind financial bubbles that will be useful later when considering the various methods available in the literature to test for bubbles.

Although several specifications of asset price bubbles have been put forward in the literature, a formal, widely accepted definition of asset price bubble is the one pioneered by Blanchard (1979). After his research, a large and growing number of papers used his specification of ''rational bubbles'' to describe asset price misalignment. Blanchard and Watson (1982) further enhance the introductory work of Blanchard (1979), pointing out that under the assumption of rational expectation, asset prices do not have to reflect solely their fundamental value. In other words, there can be rational deviations of the price from its fundamentals, i.e. rational bubbles[1].

To show this, let us consider the single period identity that links asset prices, future cashflows, and returns. For an asset that has a constant positive expected return, ''R'' , its price, ''P'' , is a linear function of future expected cashflows. Considering the case of a stock, whereby denotes the real (ex-dividend) price today (i.e. at the end of period ''t''), <math>D_{t+1}</math> is the real dividend that is paid to the owner of the stock between ''t'' and ''t+1'' , and indicates the positive rate of return from owning such asset between ''t'' and ''t+1'' , we have:
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[1] In economics, the rational expectations theory implies that all investors not only understand the structure of the economic model but also use information in an efficient manner as they do not make systematic mistakes when formulating expectations.

<math display="inline"> (1+R_{t+1} )\equiv \tfrac {P_{t+1}+D_{t+1}} {P_t}</math>

Under the assumption of rational expectations, risk neutrality, and further assuming a constant expected return, , the price of a dividend paying stock in a two-period setting equals the present value of all its expected cashflows, namely dividends paid by the stock and its resale value. Algebraically:

<math>P_t = \tfrac{1} {1+R} E_t(P_{t+1} + D_{t+1}), t > 0 </math>

Eq. (1.1) is the starting point of most empirical asset pricing tests. However, the literature proposes many different alternatives to Eq. (1.1). As an example, since dividends may not always capture the fundamental value attached to a stock, Diba and Grossman (1988), as well as Phillips et al. (2015), enriched Eq. (1.1) with the process {Ut}, representing the unobservable fundamental component[1]. Other specifications of (1.1) involve lessening the assumptions made above, for instance the one regarding the constant expected rate of return. Nonetheless, as argued by Phillips et al. (2011), changes to Eq. (1.1) would not change the behaviour and the properties of the bubble but will complicate the analysis of the rational bubble solution. Therefore, without loss of generality, I will disregard the amendments proposed in the literature, at least in this section, and proceed with the standard set of assumptions outlined above.

It is worth noticing that Eq. (1.1) is a first-order stochastic difference equation. This implies that the overall set of solutions can be found by solving it forward. By repeatedly substituting subsequent prices, and by using the law of iterated expectations to eliminate future dated expectation, the following result holds[3]:

<math>P_t = \sum_{i=1}^\infty (\tfrac{1}{1+R})^i E_t(D_{t+1}) + \lim_{i \to \infty} (\tfrac{1}{1+R})^i E_t(P_{t+1)}) </math>

And by denoting:

----[2] The unobservable fundamental component is assumed to be stationary.

[3] This procedure consists of replacing with in Eq. (1.1), then and so on. By exploiting the Law of Iterated Expectations, i.e. , it is possible to derive Eq. (1.2).