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The narrative of a Bitcoin is a bubble is very common. We employ statistical techniques to empirically evaluate such claim. A branch of literature links the existence of a bubble in some financial asset's price to strict local martingales --- a finitely lived asset has a bubble if, and only if, it is a strict local martingale under the equivalent risk-neutral measure. A diffusion process is a strict local martingale if its volatility increases faster than linearly as its level grows. We apply a nonparametric method to estimate the volatility function of Bitcoin daily and high frequency prices, as well as of more traditional financial assets. We then estimate the stochastic volatility model of Andersen and Pitterbarg (2007), whose parameter space has a specific subset under which the asset's price is a strict local martingale. Results suggest the existence of a bubble in Bitcoin prices from early 2013 to mid 2014, but, interestingly, not in late 2017.
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Is Bitcoin a bubble?
Pedro Chaima, M´arcio P. Laurinib,
aDepartment of Economics - FEA-RP USP
bDepartment of Economics - FEA-RP USP
Abstract
The narrative of a Bitcoin is a bubble is very common. We employ statistical
techniques to empirically evaluate such claim. A branch of literature links
the existence of a bubble in some financial asset’s price to strict local mar-
tingales — a finitely lived asset has a bubble if, and only if, it is a strict local
martingale under the equivalent risk-neutral measure. A diffusion process is
a strict local martingale if its volatility increases faster than linearly as its
level grows. We apply a nonparametric method to estimate the volatility
function of Bitcoin daily and high frequency prices, as well as of more tra-
ditional financial assets. We then estimate the stochastic volatility model of
Andersen and Piterbarg (2007), whose parameter space has a specific subset
under which the asset’s price is a strict local martingale. Results suggest
the existence of a bubble in Bitcoin prices from early 2013 to mid 2014, but,
interestingly, not in late 2017.
Keywords: Bitcoin, Cryptocurrencies, Financial Bubbles, Strict Local
Martingales
JEL: G19
IThe authors acknowledge funding from CNPq (303738/2015-4) and FAPESP
(2018/04654-9).
Corresponding author - Av. dos Bandeirantes 3900, 14040-905, Ribeir˜ao Preto, SP,
Brazil. Tel.: +55-16-33290867 - email - mplaurini@gmail.com
Preprint submitted to Physica A November 19, 2018
1. Introduction
The surge in Bitcoin and other cryptocurrencies prices in recent years has
attracted remarkable media and academic attention. A common narrative
is that Bitcoin is undergoing a price bubble (Krugman (2018), Bloomberg
(2017)). We employ statistical techniques to empirically evaluate such claim.
Intuitively, an asset undergoing bubble pricing has market price that ex-
ceeds the price a rational agent would pay for it — its “fundamental value”.
The fundamental price of a financial asset is often thought as the price a
risk-neutral rational agent would pay for it, and calculated by discounting
expected future cash flows from the asset. The existence of a bubble in Bit-
coin’s prices can be justified by several factors. The first is the presence of
exaggerated expectations about the adoption of Bitcoin (and cryptocurren-
cies in general) as a practical mean of payment substitution of traditional
money assets. A second possible motivation is the cheapening of mining
technologies in conjunction with higher mining difficulty. For example, Kris-
toufek (2015) argues mining difficulty measures are positively correlated with
price, and that there is a feedback relation in which higher prices and higher
popular attention lead to more miners. Another possible interpretation is re-
lated to convenience yield explanations of the Dot-com (internet) bubble in
late 1990s and early 2000s (Cochrane (2002)), and also feedback mechanisms
between Internet searches on virtual currencies and the prices of these assets
(e.g., Kristoufek (2013) and Yelowitz and Wilson (2015)).
Camerer (1989) reviews early attempts of identifying financial bubbles
which involves proposing a model for some asset’s fundamental price. Defin-
ing an asset’s fundamental price is problematic since future cash flows are not
observed (Obayashi et al. (2017)). Because in those tests one has to specify a
model for the fundamental value, discrepancy between the market price and
the fundamental price can be either due to the presence of a bubble, or due
to model mispecification. A critical issue with this joint hypothesis is that
the model for the fundamental value cannot be independently validated. The
theory outlined by Jarrow et al. (2007), Jarrow et al. (2010) and Jarrow et al.
(2011a) might prove helpful. It links the presence of a bubble in some asset
price Sto the nonexistence of an equivalent risk-neutral martingale measure.
The key point is that there exists a bubble if and only if Sis not a “true”
martingale, but a strict local martingale under the equivalent risk-neutral
probability measure.
We explore two empirical approaches. In the first one we estimate the
2
volatility function σ2(·) of the diffusion process driving S. A testable condi-
tion (Condition (7) detailed in Section 2) states that Shas a price bubble
(i.e, is a strict local martingale) if its volatility σ2(·) grows faster than linearly
with higher price level St. The second one involves specifying a stochastic
volatility model for the risky asset price Stfor which the parameter space
can be partitioned in two mutually exclusive sets, one where the process ex-
hibits a bubble and one where it does not (this draws on results of Andersen
and Piterbarg (2007)). Because market prices Stare observed, the estimated
model can be independently validated.
We analyze Bitcoin daily1and high-frequency five minute prices between
January 2013 and September 2018. Motivated by major developments in
Bitcoin markets, we split the data in three subsamples. The first goes from
January 1st 2013 to March 31 2014. This period is marked by the Mt. Gox
affair, it covers the first time Bitcoin crossed the $1000 mark (by the end of
2013), then fell around $400. The second subsample ranges from April 1st
2014 to December 31 2016; this is a period of (relatively) low volatility and
average returns in cryptocurrency markets. Volatility functions are estimated
for daily and high-frequency data2. In order to provide comparison context
we repeat the exercises done on Bitcoin for more “traditional” financial assets:
the S&P500 index, the euro-dollar exchange rate, gold dollar prices, and
Brent oil prices.
Results from both nonparametric estimation of volatility functions and
Bayesian estimation of Andersen and Piterbarg (2007) model point to the
presence of a Bitcoin price bubble in the first subsample — price volatility
function is increasing for higher observed prices, and parameter estimates
from Andersen and Piterbarg (2007) model fall in the strict local martingale
range. Interestingly, the 2017-2018 period, although very volatile, is not
characterized as a bubble following our criteria.
Statistical analysis of cryptocurrency markets is a rapidily expanding re-
search field. Literature seems specially keen on market efficiency questions,
and potential price (and volatility) “drivers”. Urquhart (2016) employ tem-
poral dependence tests to Bitcoin returns to evaluate the weak market ef-
ficiency hypothesis. He finds that as the Bitcoin market grows bigger with
1An average accross major Bitcoin exchanges calculated by blockchain.com. Prices are
collected at 22:00 or 21:00. We chose to leave out early periods due to low market liquidity.
2We do not estimate the stochastic volatility model with high-frequency data because
the number of observations makes Bayesian posterior simulation prohibitively expensive.
3
time, it also appears more informatinally efficient. Ciaian et al. (2016) build
on the gold standard price level framework of Barro (1979) to formulate and
test hypotheses regarding the determination of Bitcoin prices; they find sup-
ply and demand forces and investor attractiveness are relevant in determin-
ing Bitcoin prices, but macrofinancial developments do not impact it. Garcia
et al. (2014) employ multivariate autorregressive models to study the rela-
tionship between Bitcoin prices and what they call “digital socio-economic
signals”. They find a feedback relation between Bitcoin (differenced) prices
and volume of word-of-mouth communication, and volume of new adopters3.
There is some literature regarding bubbles in Bitcoin prices. Cheah and
Fry (2015) analyze daily bitcoin prices from 2011 to 2014. They argue bit-
coin’s sharp rise in late 2013, followed by a bust was a bubble. This is the
same period in which price manipulation occurred, as documented by Gan-
dal et al. (2018). MacDonell (2014) fits a log-periodic law model to bitcoin
prices, and argues it ex-ante predicts the late 2013 crash. Corbet et al.
(2018) relate Bitcoin and Ethereum prices to “fundamental drivers” to date
several bubble periods, including late 2013 and the second semester of 2017.
Cheung et al. (2015), Su et al. (2018), Bouri et al. (2018) apply the method-
ology of Phillips et al. (2015), which is based on rolling window Augmented
Dickey-Fuller tests, to datestamp bubles in Bitcoin markets. Several periods
of “explosivity” are highlighted, specially through 2017.
The remainder of this paper is structured as follows. Section 2 reviews the
relation between financial bubbles and strict local martingale, and discusses
key results and empirical exercises in this literature. Section 3 presents and
discusses our results. Section 4 concludes.
2. Asset Price Bubbles and Strict Local Martingales
It is very difficult to say exactly if a bubble is present in the value of
an asset. An informal definition of a bubble is the existence of a systematic
deviation of the market price from the fair price of that asset, corresponding
to the net present value of the future cash flows from that asset. A first diffi-
culty in evaluating a possible bubble is that future cash flows are uncertain,
so this assessment requires forecasting these values for the entire future time
3Authors argue this leads to “price bubble in absence of exogenous stimuli”.
4
horizon4.
This forecast stage usually involves a great deal of subjectivity, since it
involves all possible scenarios in the future. A second difficulty in this calcu-
lation is the discount rate to be used, also requiring a forecast for the entire
future horizon of the asset. Normally these two steps are evaluated jointly,
through risk premium on the discount rate as protection of the uncertainty
present in the valuation, thus incorporating a risk-return relationship in the
pricing of this asset. Note that this does not completely resolve the great
uncertainty in the assessment of the fair value of this asset. Usually the
existence of a bubble in the price of an asset is linked to the presence of
exacerbated expectations about future cash flows, and bubble rupture may
occur when there is a disappointment about the expected values of these
future yields. Note that there is an additional difficulty in evaluating the
presence of a bubble in the value of Bitcoin, since in this case there are no
direct cash flows associated with this asset, such as payment of dividends in
the case of equity shares. Thus the present value of this asset depends on the
expectations about future capital gains (price appreciation), which makes
this valuation even more difficult. See for example Jarrow et al. (2011a) for
a discussion about this feature.
It is possible to notice that this form of evaluation is based on unobserved
elements (Obayashi et al. (2017)), and thus the definition of the presence of
a bubble is not consensual, since different discount rates or expectations of
future cash flows or price gains can lead to different probabilities for the
probability of a bubble. Camerer (1989) argues this identification problem
makes so that this traditional empirical bubble detection literature is in-
conclusive, but also suggests the issue can be circumvented if bubbles have
common statistical properties that do not depend on the specification of
market fundamentals that drives intrinsic value.
To avoid this possible ambiguity, an alternative is to use the so-called
no-arbitrage valuation, which provides a direct way to check if an asset is
correctly priced. In no-arbitrage theory an asset is correctly priced if the
pricing process is consistent with a stochastic process known as martingale
after adjusting for financing costs and risk, which is mathematically accom-
plished through a change of measure (the equivalent martingale measure).
4See Flood and Hodrick (1986), Smith et al. (1988), Camerer and Weigelt (1991) for
examples of this literature.
5
A martingale is a process whose forecast for all future horizons of time is
given by the current value of the asset, and thus a simple way to test for
the correct pricing is to check the consistency of the price observed with a
martingale process. The presence of a bubble imposes some additional math-
ematical difficulties in the assessment by no-arbitrage, which requires some
additional characterizations. The no-arbitrage theory characterizes the ex-
istence of a bubble through the presence of a process known as strict local
martingale, and thus provides a testable condition for the presence of a bub-
ble that circumvents the difficulties existing in the usual cash flow assessment
procedures.
Let us formalize the idea of a bubble. Because the literature on the subject
is exceedingly vast and diverse, we focus on the strict local martingale theory
of financial bubbles developed by Jarrow et al. (2007), Jarrow et al. (2010),
Jarrow et al. (2011a).
Following Protter (2013), consider finitely lived economy in which are
traded a risky asset and a money market account in the time interval [0, T ].
For simplicity, assume there are no cash flows associated with the asset.
Consider a complete probability space (Ω,Ft, P ), and a filtration F= (Ft)t0.
Let r= (rt)t0denote the instantaneous default-free spot interest rate, then
the money market account value at time tis
Bt= exp Zt
0
rudu.(1)
No arbitrage in the sense of No Free Lunch with Vanishing Risk (NFLVR)
implies that there exists a probability measure Q, equivalent to the physical
probability measure P, such that
St
Bt
(2)
is a local martingale under Q. This is the first theorem of asset pricing, as
stated by Delbaen and Schachermayer (1998).
A local martingale is an adapted c`adl`ag stochastic process Mt, for which
there exists an increasing sequence of stopping times (τn)n0, such that
limn→∞ τn=, and Mmin{t,τn}is a martingale with probability one for all n.
Jarrow (2012) explains that a local martingale is a generalization of a mar-
tingale that extends the martingale’s “fair game” property to a “game” that
has a random termination time, which depends on information generated
when “playing the game”.
6
Let STbe the liquidation value of the asset at time T. Protter (2013)
defines the fundamental price of the asset, S?
t, which pays no dividends, as
the expected value of the liquidation value STunder the equivalent local
martingale measure Q,
S?
t=EQST
BT
1{tT}
FtBt.(3)
The risky asset’s fundamental value is equal to the price one would pay for it,
if after the purchase, one had to hold the asset until time T. Protter (2013)
shows S?is a true martingale under Q.
The asset price bubble βtis defined as the difference between the observed
market price Stand the fundamental value S?
t. Such difference is assumed
nonnegative.
βt=StS?
t0.(4)
Protter (2013) shows that saying some finitely-lived risky asset Sthas non-
zero bubble βton [0, T ] is equivalent to say Stis a strict Q-local martingale.
Jarrow et al. (2011a) discuss the intuition behind the distinction between
a nonnegative martingale and a strict local martingale, S0, follows from
the fact that Sis always a supermartingale and is a true martingale if and
only if it has constant expectation. Then if Sis a strict local martingale
its expectation decreases over time. The typical behavior of a strict local
martingale process is to shoot up to high values, to then drop down and stay
low.
Assume the risky asset price S= (St)t0is driven by a standard stochastic
differential equation
dSt=µ(St)dt +σ(St)dW S
t.(5)
NFLVR implies that there exists an equivalent local martingale measure un-
der which the SDE (5) simplifies to
St=S0+Zt
0
σ(Su)dW S
u.(6)
It is shown by Kotani (2006) and Mijatovi´c and Urusov (2012) that Sis
a strict local martingale if, and only if
Z
δ
St
σ2(St)dSt<,(7)
7
for all δ > 0. Jarrow (2012) notes condition (7) states that an asset is a strict
local martingale if and only if its volatility, as function of its price, increases
faster than linearly. As Jarrow et al. (2011b) explain, this theory tells us
that if the graph of the volatility versus the asset price tends to infinity at a
faster rate than does the graph of f(x) = x, then there is a bubble.
The integral (7) can be empirically evaluated by estimating the diffusion
coefficient, or volatility function, σ2(·). Whether Sis a true Q-martingale
or a strict Q-local martingale will depend on the behavior of the estimated
volatility ˆσ2(·) as St→ ∞.
The volatility function σ2(·) can be estimated by kernel-based nonpara-
metric estimators, such as the one introduced by Florens-Zmirou (1993) and
Jiang and Knight (1997). This approach is appealing because little structure
is imposed on the dynamics of S, but is hindered by the fact that one can
only compute ˆσ(St) for observed values of S.
Jarrow et al. (2011a) introduce an alternative method of estimation for the
volatility function based on a parameterized family of Reproducing Kernel
Hilbert Spaces, which allows for the extrapolation of ˆσ2(x) for values xnot
visited by the sample path of process S. Jarrow et al. (2011b) employ such
method to analyze LinkedIn stock prices through May 2011, they infer the
presence of a bubble. The same method is applied by Jarrow (2016) to gold
prices, no bubble is detected.
Now consider the dynamics of Sfollow the stochastic volatility model
studied by Andersen and Piterbarg (2007).
dSt=λStpVtdW S
t,
dVt=κ(θVt) + V p
tdW V
t,(8)
where λ, κ, θ, p > 0, and WS
t, W V
tis a two dimensional Brownian motion
with correlation coefficient ρ.
Results of Andersen and Piterbarg (2007) and Protter (2013) describe how
the parameter space of model (8) can be divided in two disjoint sets, such
that a specific parameter range characterizes Sas a strict local martingale;
thus allowing for a testable implication of the martingale hypothesis.
Theorem 1. For model (8): if ρ0,Sis a true martingale; if ρ > 0and
p1/2or p > 3/2,Sis a true martingale; if ρ > 0and 1/2<p<3/2, then
Sis a strict local martingale. For the case p= 3/2,Sis a true martingale
if ρ1/2λ1, and Sis a strict local martingale if ρ > λ1/2.
8
Proof. See Proposition 2.5 of Andersen and Piterbarg (2007).
Baldeaux et al. (2018) investigate the presence of bubbles in money mar-
kets by estimating models similar to (8), and comparing estimated parameter
values to the conditions in Theorem 1. They find the fitted models can pro-
duce bubbles.
3. Empirical Evidence
3.1. Descriptive Statistics
Selected descriptive statistics of Bitcoin log-returns are displayed in Ta-
ble 3.1. Motivated by major developments in Bitcoin markets and existing
empirical evicence, we split the data in three subsamples. The first goes
from January 1st 2013 to March 31 2014. This period is marked by the Mt.
Gox affair, it covers the first time Bitcoin crossed the $1000 mark (by the
end of 2013), then fell around $400; this episode is identified by Cheah and
Fry (2015) as a bubble. The second subsample ranges from April 1st 2014
to December 31 2016; this is a period of low volatility and average returns
(relatively, see second line of Table 3.1) in cryptocurrency markets. The
thirds starts on January 1st 2017 and goes until the end of our sample, on
September 1st 2018 — it captures the remarkable attention cryptocurrencies
received during 2017 and early 2018, and is dated as a bubble by Corbet
et al. (2018). End dates for subsamples were motivated by the conditional
volatility level of Bitcoin returns when contrasted to unconditional average
volatility (see Chaim and Laurini (2019)).
In order to provide some context, we repeat the exercises performed on
Bitcoin for a few, more traditional, financial assets. We analyze the S&P 500
index, the euro-dollar exchange rate, Brent oil prices, and dollar gold prices.
Descriptive statistics for log-returns of those assets are also shown in Table
1.
Camerer (1989) argues intuitively that since bubbles will cause some ex-
tremely large positive price changes as they grow (especially during the last
stages of bubble growth), and even larger negative price changes when they
burst, the distribution of returns should have negative skewness and large
kurtosis if a bubble exists. Notice from Table 1 that Bitcoin returns are
leptokurtic, with kurtosis well above what one would expect of a Normal
distribution, and have negative skewness. One can take this as some initial
positive evidence towards the presence of price bubbles.
9
Figure 1: Daily Bitcoin dollar prices and returns
0
5000
10000
15000
20000
2013−01
2013−07
2014−01
2014−07
2015−01
2015−07
2016−01
2016−07
2017−01
2017−07
2018−01
2018−07
a) Bitcoin daily closing prices
−50.0%
−25.0%
0.0%
25.0%
2013−01
2013−07
2014−01
2014−07
2015−01
2015−07
2016−01
2016−07
2017−01
2017−07
2018−01
2018−07
b) Bitcoin daily log−returns
Note: figure plots Bitcoin daily prices and returns from January 2nd 2013 to September
1st 2018. Dashed vertical lines represent subsample endpoints.
10
Table 1: Descriptive statistics of daily log-returns
Mean Std.Error Min. Median Max. Skew. Kurt.
BTC 2013-01 to 2014-03 0.775 7.105 -47.831 0.521 35.879 -1.049 11.083
BTC 2014-04 to 2016-12 0.053 3.214 -26.862 0.096 21.168 -0.377 10.698
BTC 2017-01 to 2018-09 0.326 4.771 -22.571 0.522 24.661 -0.255 3.267
BTC full sample 0.302 4.769 -47.831 0.246 35.879 -0.764 15.162
S&P 500 0.048 0.771 -4.184 0.057 3.829 -0.591 3.268
EUR-USD -0.009 0.533 -2.672 -0.008 3.064 0.042 2.099
Gold -0.023 0.903 -8.913 -0.025 4.788 -0.685 8.239
Oil -0.027 1.972 -8.083 0.000 9.896 0.370 2.994
Note: table reports descriptive statistics of daily log-returns of Bitcoin, including three
subsamples, the S&P500 index, the EUR-USD exchange rate, dollar gold prices, and
Brent oil prices, for the sample 2013-01 - 2018-09. First column displays mean. Second
column displays standard errors. Third and fifth columns show the smallest and largest
observations, respectively. Fourth column displays medians. Sixth column show skewness
coefficients. Seventh column displays raw kurtosis coefficients.
Table 2 shows descriptive statistics for high-frequency log-returns data
of the Bitcoin series. These returns are constructed using a 5-minute price
aggregation, in this case using the last five-minute value of the tick-by-tyck
operations in Bitcoin Bitstamp exchange. The use of high-frequency data
is especially interesting for this asset since it is traded continuously, unlike
traditional assets with fixed periods of operation. The use of high-frequency
data allows us to analyze whether the bubble detection processes are sensitive
to the analyzed frequency of the data. We used a 5-minute aggregation to
have a regular frequency of data, and also to mitigate possible contaminations
by microstructure noise that affect high-frequency estimates. We can see that
important characteristics remain present at this frequency of observation. In
particular it is possible to note the existence of extreme values of returns,
asymmetry patterns and especially high kurtosis, which may be associated
with the presence of bubbles, as discussed above.
3.2. Volatility Function
Recall from our discussion in Section 2 that condition (7) implies a con-
tinuous stochastic process S= (St)t0is a strict local martingale if, and only
if, its volatility increases fast enough as its level grows. That is, if the integral
11
Table 2: Descriptive statistics of five minutes Bitcoin log-returns
Mean Std.Error Min. Median Max. Skew. Kurt.
BTC 2013-01 to 2014-03 0.003 0.713 -22.019 0.000 55.675 4.853 451.039
BTC 2014-04 to 2016-12 0.000 0.261 -15.069 0.000 9.820 -0.944 95.758
BTC 2017-01 to 2018-09 0.001 0.357 -20.264 0.001 7.450 -1.264 88.963
BTC full sample 0.001 0.429 -22.019 0.000 55.675 4.769 798.267
Note: Table reports descriptive statistics of high frequency five minute returns of
Bitcoin, including three subsamples. First column displays mean. Second column
displays standard errors. Third and fifth columns show the smallest and largest
observations, respectively. Fourth column displays medians. Sixth column show skewness
coefficients. Seventh column displays raw kurtosis coefficients.
bellow is finite. Z
δ
St
σ2(St)dSt<,δ > 0.
We present estimates the volatility function σ2(·) based on the nonpara-
metric estimator of Florens-Zmirou (1993) and Jiang and Knight (1997).
Consider the continuous stochastic process S= (St)0tT, let {St=
St1, St2, . . . , Stn}be nequispaced observations of process Sat times {t1=
n, t2= 2∆n, . . . , tn=nn}, with ∆n=T/n. When the sampling path of
the diffusion process Stvisits x,
ˆσ2(x) = Pn1
i=1 nK Sinx
hnS(1+i)∆nSin2
Pn
i=1 T K Sinx
hn,(9)
where hn>0 is a bandwidth parameter, and K(·) is a kernel function. We
employ a Gaussian kernel, and set the bandwidth parameter specified by
Kullback-Leibler cross-validation (Hurvich et al. (1998)).
Inference on whether Sis a true martingale or a strict local martingale
comes from the properties of σ2(St) as St→ ∞. That is, if volatility grows
faster than linearly as level increases, then the underlying process is a strict
local martingale according to condition (7). A limitation of this nonpara-
metric estimation approach is that we can only estimate σ2(x) for values of
xvisited by the sample path of St.
Figure 2 displays nonparametric estimates of Bitcoin daily volatility func-
tions ˆσ2(St) (left panels), and the ratio St/ˆσ2(St), integrand of condition (7),
12
for the three considered subsamples and the whole sample. Figure 3 shows
these estimated quantities for high frequency Bitcoin prices, for each subsam-
ple. Because there are more observations for each price level, high frequency
estimates are more precise. We repeat this exercise for the S&P500 index, the
USD-EUR exchange rate, dollar gold prices, and Brent oil prices in Figure
4. We experimented with log-prices instead of prices but results were similar
and uninformative, thus we do not report here.
Top panels (a) present estimates for the first Bitcoin subsample (2013-
01-01 to 2014-03-31). Daily volatility function in Figure 2 seems to increase
with level, but confidence bands are broad, particularly for prices above $900.
Turning to high frequency estimates in panel (a) of Figure 3 we see a steep
slope of the variance function for higher price levels. This suggests condition
(8) holds and Bitcoin follows a strict local martingale during this period.
Mid panels (b) in Figure 2 display daily estimates for the second sub-
sample (2014-04-01 to 2016-12-31). If we compare the magnitudes of vertical
axes, it is clear that Bitcoin is much less volatile during this second subsam-
ple, confirming descriptive statistics in Table 1. Volatility function estimates
from both daily and high frequency data stabilize for higher Bitcoin prices.
Integrand St/ˆσ2(St) in right panel (b) is more similar to those from Oil and
Gold (Figure 4) then to other Bitcoin subsamples. The results obtained for
the traditional assets indicate the absence of bubble evidence in the period
under analysis. This indicates Bitcoin do not display a bubble during this
period.
Estimates for the third subsample (2017-01-01 to 2018-09-01) are dis-
played in mid panels (c). In Figure 2 we see daily estimates are monoton-
ically increasing but confidence bands are very broad. Interestingly, high-
frequency estimates in Figure 3 show volatility increases steeply until around
the $13,000 mark, remains stable until $15,000, and clearly falls for higher
price levels. We take this evidence as indication the late 2017 period was not
a bubble according to condition (7).
3.3. Stochastic Volatility Model
We now estimate the stochastic volatility model of Andersen and Piter-
barg (2007), discussed in Section 2.
dSt=λStpVtdW S
t,
dVt=κ(θVt) + V p
tdW V
t,
13
Figure 2: Variance function estimates - daily Bitcoin prices
Note: figure displays nonparametric estimations of variance functions σ2(St) (left panels)
and relation St2(St) (right panels) for the three considered subsamples and the full
sample for daily Bitcoin prices. Prices are collected at 21:00 or 22:00. We employed the
Florens-Zmirou (1993) estimator with a Gaussian kernel. Shaded area represents a 95%
confidence band, obtained from 10000 bootstrap simulations.
14
Figure 3: Variance function estimates - high-frequency Bitcoin prices
Note: figure displays nonparametric estimations of variance functions σ2(St) (left panels)
and relation St2(St) (right panels) for the three considered subsamples and the full
sample for high-frequency Bitcoin prices, collected every 10 minutes. We employed the
Florens-Zmirou (1993) estimator with a Gaussian kernel. Shaded area represents a 95%
confidence band, obtained from 10000 bootstrap simulations.
15
Figure 4: Variance function estimates - S&P500, EUR-USD, Gold, and Oil
Note: figure displays nonparametric estimations of variance functions σ2(St) (left panels)
and relation St2(St) (right panels) for four selected daily financial asset: the S&P500
index, the EUR-USD exchange rate, dollar gold prices, and Brent oil prices. We
employed the Florens-Zmirou (1993) estimator with a Gaussian kernel. Shaded area
represents a 95% confidence band, obtained from 10000 bootstrap simulations.
16
where λ, κ, θ, p > 0, and WS
t, W V
tis a two dimensional Brownian motion
with correlation coefficient ρ. As described in Theorem 1, the parameter
space of this model is divided in two disjoint sets: one under which Sis a
true martingale, and another under which Sis a strict local martingale.
Due to the presence of the latent variable Vtwe estimate model (8) by
Bayesian posterior simulation methods, specifically, the Hamiltonian Monte
Carlo (HMC) algorithm introduced by Duane et al. (1987)5. HMC augments
the usual random walk posterior sampling scheme of the Metropolis-Hastings
(MH) algorithm with Hamiltonian Dynamics, such that new proposed values
can be further apart than they would be under a random walk exploration
scheme, but have a high acceptance probability nevertheless. HMC is then
more efficient than MH in the sense that it takes fewer algorithm iterations in
order to achieve chain convergence. This is specially relevant in our stochas-
tic volatility context because Bayesian statistics treats latent variables as it
treats unknown parameters, what can be computationally expensive. We de-
fine daily returns as the first difference of the log-price. Employing natural
returns here leads to serious convergence problems in the sampling algorithm
and unreliable results.
Preliminary estimation attempts proved parameter λdifficult to identify
from data. We choose to set λ= 1. This affect the strict local martingale
criterion of Theorem 1 only if p= 3/2. This is not unprecedented. Baldeaux
et al. (2018) also attempt to empirically identify strict local martingales by
estimating restricted versions of model (8), all with λ= 1.
Table 3 reports posterior descriptive statistics for Bitcoin. We estimate
model (8) for each subsample, and for the full sample. In Table (4) we
show estimates for the four other “traditional” financial assets: the S&P
500 index, the EUR-USD exchange rate, dollar gold prices, and Brent oil
prices. Remember we are mostly interested in comparing posterior estimates
with the criteria derived by Andersen and Piterbarg (2007), in order to judge
whether there are evidences of strict local martingale bubbles in those assets.
Parameter κgives the speed of mean reversion of the autorregressive pro-
cess driving volatility Vt. From Table 3 we see that for all Bitcoin subsamples,
parameter κhas posterior mean between 0.50 and 0.66, but 95% credibility
intervals are quite broad, specially for the first and third subsamples. Un-
conditional mean volatility level is given by parameter θ. Confirming what
5See Betancourt (2017) for a thorough explanation of the algorithm.
17
we see from standard deviations in Table 3.1, θis estimated largest for the
first subsample (0.0715), and smallest for the second subsample (0.0089).
Parameter pis the exponent of volatility level Vt, which scales the Brow-
nian innovation dW V
tin model (8). According to the criteria of Andersen
and Piterbarg (2007), a value of pbetween 0.5 and 1.5 characterizes Sas a
strict local martingale (remember if p= 0.5 then model (8) reduces to the
well known Heston (1993) model). Posterior estimates of pfall wholly in the
strict local martingale range, for all Bitcoin subsamples.
Evidence from Bayesian estimation of model (8) points to Bitcoin prices
behaving like strict local martingales during our first subsample: the 95%
credibility interval of parameter pfalls in the (0.5,1.5) range, and correlation
ρis estimated positive with high probability. Similar to the results obtained
in the non-parametric estimation of volatility functions, we did not find ro-
bust evidence for the presence of bubbles in the other sub-samples of the
Bitcoin series, as well as no evidence of bubbles in the traditional assets used
as reference.
It is important to note that the stochastic volatility model used represents
a good fit for the behavior observed in the Bitcoin series. Figure 5 shows the
model fit (estimated for the whole sample) for conditional volatility, com-
pared to Bitcoin’s absolute returns series, a proxy for true process volatility.
We can observe that the model adequately captures shifts in conditional
volatility observed in the Bitcoin series.
4. Concluding Remarks
In this paper we explored the Bitcoin bubble narrative through the strict
local martingale theory of financial bubbles (e.g. Jarrow et al. (2007), Jarrow
et al. (2010), Jarrow et al. (2011b)). A certain financial asset displays bubble
pricing if its discounted price is not a “true” martingale under the equivalent
neutral probability measure, but a strict local martingale.
Condition (7) stats an asset price is a strict local martingale if its volatil-
ity increases with level faster than linearly. We estimated the volatility func-
tion of Bitcoin daily and high-frequency prices for three subsamples, and
of other selected financial assets (S&P500, EUR-USD, gold, and Oil) using
the nonparametric estimator of Florens-Zmirou (1993). Comparing volatil-
ity function estimates between Bitcoin and traditional financial assets we see
Bitcoin is much more volatile and displays distinct level-variance dynamics.
18
Table 3: Posterior Distributions of Andersen and Piterbarg (2007) model - Bitcoin
Mean Std q2.5% Median q97.5%
κ0.5020 0.2298 0.0634 0.5045 0.9285
First subsample θ0.0715 0.0562 0.0081 0.0603 0.2082
2013-01-01:2014-03-31 14.9161 1.6122 11.8690 14.8706 18.1869
ρ0.2709 0.1829 -0.0859 0.2643 0.6291
p0.7618 0.0406 0.6830 0.7618 0.8422
κ0.6596 0.2080 0.1982 0.6825 0.9772
Second Subsample θ0.0089 0.0093 0.0003 0.0066 0.0304
2014-04-01:2016-12-31 14.3569 1.5782 11.3895 14.3161 17.5359
ρ0.1131 0.1845 -0.2520 0.1144 0.4669
p0.7503 0.0338 0.6835 0.7505 0.8159
κ0.5483 0.2265 0.0899 0.5573 0.9485
Third Subsample θ0.0345 0.0330 0.0013 0.0279 0.1064
2017-01-01:2018-09-01 10.6551 1.6122 7.5858 10.6211 13.9126
ρ0.0234 0.2350 -0.4482 0.0237 0.4753
p0.8689 0.0458 0.7820 0.8682 0.9608
κ0.6537 0.2162 0.1686 0.6803 0.9770
Full Sample θ0.0040 0.0036 0.0001 0.0028 0.0111
2013-01-01:2018-09-01 10.6557 1.5564 7.7779 10.6001 13.8711
ρ-0.0289 0.1738 -0.3718 0.0128 0.2356
p0.8179 0.0331 0.7527 0.8181 0.8825
Note: table reports descriptive statistics of our estimation of Andersen and Piterbarg
(2007) model for Bitcoin. First column displays posterior means, second column
standard errors, and columns three through display selected quantiles. Estimation was
carried out by Hamiltonian Monte Carlo, with one chain of 100000 replications.
19
Table 4: Posterior Distributions of Andersen and Piterbarg (2007) model - S&P 500,
EUR-USD, Gold, and Oil
Mean Std q2.5% Median q97.5%
κ0.6537 0.2162 0.1686 0.6803 0.9770
θ0.0040 0.0036 0.0001 0.0028 0.0111
S&P 500 10.6557 1.5564 7.7779 10.6001 13.8711
ρ-0.0289 0.1738 -0.3718 0.0128 0.2356
p0.8179 0.0331 0.7527 0.8181 0.8825
κ0.5941 0.2065 0.2065 0.6030 0.9548
θ0.0032 0.0017 0.0017 0.0032 0.0058
EUR-USD 6.2593 1.3470 1.3470 6.1781 9.0996
ρ0.0224 0.0442 -0.0553 0.0174 0.1218
p0.9391 0.0392 0.8603 0.9396 1.0146
κ0.6237 0.2020 0.1969 0.6362 0.9647
θ0.0051 0.0027 0.0005 0.0052 0.0096
Gold 7.1386 1.4592 4.5268 7.0577 10.2148
ρ0.0305 0.0726 -0.0941 0.0206 0.2033
p0.9406 0.0408 0.8599 0.9411 1.0194
κ0.5848 0.2063 0.1559 0.5928 0.9510
θ0.0101 0.0060 0.0012 0.0099 0.0208
Oil 7.5960 1.3407 5.1814 7.5270 10.4190
ρ0.0101 0.0692 -0.1275 0.0083 0.1543
p0.9350 0.8550 0.9353 0.9353 1.0127
Note: table reports descriptive statistics of our estimation of Andersen and Piterbarg
(2007) model for S&P500, EUR-USD, Gold, and Oil for the sample period 2013-01 until
2018-09. First column displays posterior means, second column standard errors, and
columns three through display selected quantiles. Estimation was carried out by
Hamiltonian Monte Carlo, with one chain of 100000 replications.
20
Figure 5: Fitted Volaility - Andersen and Piterbarg (2007) model
Abs Returns x Fitted Volatility
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
01/2013 07/2013 01/2014 07/2014 01/2015 07/2015 01/2016 07/2016 01/2017 07/2017 01/2018 06/2018
Absolut Returns
Fitted Volatility
Note: figure plots daily Bitcoin returns and mean posterior conditional volatility (for the
model estimated with the full sample). Estimation was carried out by the HMC
algorithm with 100000 iterations.
Andersen and Piterbarg (2007) show the parameter space of the stochastic
volatility model (8) can be divided in two disjoint sets, one under which
the asset follows a martingale, and another under which the asset follows a
strict local martingale. We employed a Hamiltonian Monte Carlo simulation
scheme to estimate model (8) for each subsample of Bitcoin daily returns,
and for those other four selected financial assets. Results point to a bubble
in Bitcoin during our first subsample, which goes from January 2013 to April
2014.
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