Download - How to Detect an Asset Bubble

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SIAM J. FINANCIAL MATH. c© 2011 Society for Industrial and Applied MathematicsVol. 2, pp. 839–865

How to Detect an Asset Bubble∗

Robert Jarrow†, Younes Kchia‡, and Philip Protter§

Abstract. After the 2007 credit crisis, financial bubbles have once again emerged as a topic of current concern.An open problem is to determine in real time whether or not a given asset’s price process exhibits abubble. Due to recent progress in the characterization of asset price bubbles using the arbitrage-freemartingale pricing technology, we are able to propose a new methodology for answering this questionbased on the asset’s price volatility. We limit ourselves to the special case of a risky asset’s pricebeing modeled by a Brownian driven stochastic differential equation. Such models are ubiquitousboth in theory and in practice. Our methods use sophisticated volatility estimation techniquescombined with the method of reproducing kernel Hilbert spaces. We illustrate these techniquesusing several stocks from the alleged Internet dot-com episode of 1998–2001, where price bubbleswere widely thought to have existed. Our results confirm the suspicions of the presence of bubblesin many of the dot-com stocks of 1998–2001.

Key words. asset bubbles, strict local martingales, reproducing kernel Hilbert spaces, dot-com stocks

AMS subject classifications. 60G44, 60G48, 60G07, 60H05, 60H20, 6207

DOI. 10.1137/10079673X

1. Introduction. This paper is interested in the detection of financial bubbles. The ques-tion we address is a timely one. Recently William Dudley, the President of the New YorkFederal Reserve, in an interview with Planet Money [10] stated, “...what I am proposing isthat we try to identify bubbles in real time, try to develop tools to address those bubbles, tryto use those tools when appropriate to limit the size of those bubbles and, therefore, try tolimit the damage when those bubbles burst.”

Asset price bubbles have been recently characterized in frictionless, competitive, and con-tinuous trading economies using the arbitrage-free martingale pricing technology underlyingoption pricing theory (see [23], [24], [2], [11], [18], and [19]). In this classical setting, Jarrow,Protter, and Shimbo [18], [19] show that there are three types of asset price bubbles possible.Two of these price bubbles exist only in infinite horizon economies, the third—called type 3bubbles—exist in finite horizon settings. Consequently, type 3 bubbles are those most rele-vant to actual market experiences. For this type of bubble, saying whether or not a bubbleexists amounts to determining whether the price process under a risk neutral measure is amartingale or a strict local martingale: if it is a strict local martingale, there is a bubble. Thedifference between a martingale and a strict local martingale has been recently investigated

∗Received by the editors May 28, 2010; accepted for publication (in revised form) July 11, 2011; publishedelectronically October 12, 2011.

http://www.siam.org/journals/sifin/2/79673.html†Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853 ([email protected]) and Kama-

kura Corporation, 2222 Kalakaua Ave., Honolulu, HI 96815.‡Centre de Mathematiques Appliquees, Ecole Polytechnique, Paris ([email protected]).§Statistics Department, Columbia University, New York, NY 10027 ([email protected]). This author was

supported in part by NSF grant DMS-0906995.

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mailto:[email protected]




840 ROBERT JARROW, YOUNES KCHIA, AND PHILIP PROTTER

by several authors ([1], [26], [21], for instance). However, the distinction is subtle, and in thecase of a diffusion it amounts to understanding the asymptotic behavior of the asset’s pricevolatility. If the asset’s price volatility is large enough, then a bubble exists.

More formally, we model the asset price process by a standard stochastic differentialequation (SDE) driven by a Brownian motion W :

(1) dSt = σ(St)dWt + μ(St)dt

for all t in [0, T ], in some filtered probability space (Ω,F , P,F), where F = (Ft)t≥0. We makethe standing assumption that the asset price S is nonnegative. The asset’s price volatilityσ(St) is stochastic since it depends on the level of the asset’s price. Assuming no arbitrage inthe sense of no free lunch vanishing risk (NFLVR), there exists a risk neutral measure (see [7])under which this SDE simplifies to

(2) St = S0 +

∫ t

0σ(Ss)dWs.

It is well known [1], [26] that this process S is a strict local martingale if and only if

(3)

∫ ∞

α

x

σ2(x)dx < ∞

for all α > 0. Kramkov has recently pointed out that this follows quite simply from Feller’stest for explosions [22]. This last condition forms the basis of our bubble testing methodology;i.e., type 3 bubbles exist if and only if this integral is finite. This uses the theory of bubblesas presented, for example, in [2]. The intuition behind the distinction between a martingaleand a strict local martingale (in the case where the local martingale S > 0) derives from thefact that S is always a supermartingale and is a martingale if and only if it has constantexpectation. So for a strict local martingale its expectation decreases with time. Thus onaverage under the risk neutral measure the buy and hold strategy is a losing one. A “typical”path of such a nonnegative continuous local martingale is to shoot up to high values and thenquickly decrease to small values and remain at them; this is also the typical behavior of pricesof assets undergoing speculative bubbles.

We can also replace (1) for S with a more general equation, for example,

dSt = σ(St)dWt + μ(St, Yt)dt,(4)

dYt = s(Yt)dBt + g(Yt)dt,

where B is a Brownian motion which is independent of W . This gives a model for S inthe context of an incomplete market. This is perhaps the simplest model that implies anincomplete market. An alternative incomplete market model, and one that we do not considerhere, introduces a stochastic volatility function as well. In any event, for our purposes, weare inevitably led in both situations (complete or incomplete models) to (2), under any riskneutral measure. If the models are complete, we appeal to the NFLVR framework of Cox andHobson [2], and if the models are incomplete, we can use the no dominance framework of [18],[19], which we consider a better framework for models of bubbles.D

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HOW TO DETECT AN ASSET BUBBLE 841

Many authors have proposed estimators for the volatility function σ(x). Florens-Zmirou[8] proposes a nonparametric estimator based on the local time of the diffusion process. Genon-Catalot and Jacod [9] propose an estimation procedure for parameterized volatility functions.Hoffmann [13] constructs a wavelets-based estimator. In the first part of this paper, we recallFlorens-Zmirou’s results. Since the constraint on the grid step, denoted hn in what follows,required by Florens-Zmirou’s theorem cannot be satisfied due to the limited data available, wepropose another local time-based estimator, using a smooth kernel, where the condition on hnis easier to satisfy. Florens-Zmirou also obtained a limit theorem when σ(x) is bounded aboveand away from zero. We were able to relax this assumption and extend the limit theoremby working in an enlarged filtration. We do not reproduce these results herein since they aretangential to this paper, but they are available to the reader upon request.

The main difficulty in using nonparametric estimators is that one can estimate σ(x) onlyat points visited by the process. We, therefore, cannot know the tails of the volatility functionand determine whether the integral in (3) is finite or infinite. In the second part of this paperwe propose two methods to deal with this “extrapolation problem.”

The first method is based on a comparison theorem. We compare the behavior of para-metric and nonparametric estimators of σ(x). When the two estimators are statisticallysimilar within the observation interval, we extrapolate into the tails using the parametricform’s asymptotic behavior. The second method is based on reproducing kernel Hilbert space(RKHS) theory. In fact, a roughly analogous problem arises in physical chemistry for po-tentials whose asymptotic behavior is known (cf. [14]). In our case, we do not know theasymptotic behavior of the volatility (that is what we are looking for!). To overcome thisproblem, we introduce a parameterized family (Hm) of RKHSs. Different m’s allow us toconstruct interpolating functions with different asymptotic behaviors. We optimize over m ina sense that will be explained below and identify the reproducing kernel Hilbert space, whichallows us to construct an interpolating function that extends the nonparametric estimatorfrom the observation interval to the entire real line.

We devote the last section to illustrating these various estimation methodologies. Wefocus on stocks from the alleged internet dot-com bubble of 1998–2001 (see, for instance, [30],[25]) for which we could find relevant tick data. We selected four stocks: Lastminute.com,eToys, Infospace, and Geocities.1 The data was obtained from WRDS [31]. We used ourmethodology to see whether these stocks exhibited price bubbles. The evidence supports theexistence of price bubbles. In addition, these four stocks allow us to illustrate the strengthsand weaknesses of our testing methodology. We can also develop (and have done so) tests forbubble behavior via an analysis of derivatives (calls and puts). We have not used that heresince the data sets from 1998–2001 are limited, and we could not find a source for large enoughquantities of good data on derivatives during that period. Nevertheless such techniques mightwork well in a more contemporary setting.

We note that the estimation is performed in the real (and not the risk neutral) world.However, Florens-Zmirou shows that the estimators we use do not involve the drift; hence,

1Lastminute.com bid prices were from 14 March 2000 to 30 July 2004; eToys bid prices from 20 May 1999to 26 February 2001; Infospace bids from 15 December 1998 to 19 September 2002; and Geocities bids from 11August 1998 to 28 May 1999.D

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without loss of generality, we assume throughout the remainder of the paper that μ is iden-tically null. Therefore, we consider the SDE in (2) where the function σ(x) is unknown. Wedefine our nonparametric estimator of σ(x) based on discrete time observations St1 , . . . , Stn

on the finite time interval [0, T ]. We assume a regular sampling; that is, ti =inT .

An outline for our paper is as follows. Sections 2 and 3 present the Florens-Zmirou andsmooth kernel volatility estimators on a compact domain. Section 4 extends these estimatorsto the nonnegative real line. Section 5 illustrates our testing methodology for asset pricebubbles, and section 6 concludes.

2. Florens-Zmirou’s estimators. This section reviews Florens-Zmirou’s estimators for oursubsequent usage. Her estimator is based on the local time of a diffusion and is explainedheuristically as follows. The local time is given by

�T (x) = limε→0

1

2ε

∫ T

01{|Ss−x|<ε}d〈S, S〉s,

where d〈S, S〉s = σ2(Ss)ds so that �T (x) = σ2(x)LT (x), and

LT (x) = limε→0

1

2ε

∫ T

01{|Ss−x|<ε}ds.

Hence, the ratio �T (x)LT (x) = σ2(x) yields the volatility at x. These limits and integrals can be

approximated by the following sums:

LnT (x) =

T

2nhn

n∑i=1

1{|Sti−x|<hn},

�nT (x) =T

2nhn

n∑i=1

1{|Sti−x|<hn}n(Sti+1 − Sti)

2,

where hn is a sequence of positive real numbers converging to 0 and satisfying some constraints.This allows us to construct an estimator of σ(x) given by

(5) Sn(x) =

∑ni=1 1{|Sti

−x|<hn}n(Sti+1 − Sti)2∑n

i=1 1{|Sti−x|<hn}.

Indeed, Florens-Zmriou [8] proves the following theorems.Theorem 1. If σ is bounded above and below from zero and has three continuous and

bounded derivatives, and if (hn)n≥1 satisfies nhn → ∞ and nh4n → 0, then Sn(x) is a consis-tent estimator of σ2(x).

The proof of this theorem is based on the expansion of the transition density. The choiceof a sequence hn converging to 0 and satisfying nhn → ∞ and nh4n → 0 allows one to showthat Ln

T (x) and �nT (x) converge in L2(dQ) to LT (x) and σ2(x)LT (x), respectively. Hence Sn(x)is a consistent estimator of σ2(x) for any x that has been visited by the diffusion.

We also have the following limit theorem, useful in obtaining confidence intervals for theestimator Sn(x) of σ(x).D

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Theorem 2. If, in addition to the hypotheses of Theorem 1, one has nh3n → 0, then√Nn

x (Sn(x)σ2(x)

− 1) converges in distribution to√2Z, where Z is a standard normal random

variable and Nnx =

∑ni=1 1{|Sti−x|<hn}.

The aim of the first part of this paper is to construct another estimator based on thelocal time of the diffusion but using a smooth kernel. Theorem 1 will remain true under theconstraint nh2n → ∞. This is important for the purpose of this paper: requiring that hnsatisfy the conditions of Theorem 1 would provide useless estimators (not smooth enough towork with in practice) due to the limited data available to us. When testing our procedure,we will always provide these two estimators, although theoretically we cannot be sure that theestimator of Florens-Zmirou converges due to the requirement that hn = n− 1

4 , which needsmore data than we have.

3. A smooth kernel estimator. We introduce a smooth kernel estimator to relax thecondition on hn to nh2n → ∞. In practice we often do not have enough data, and theconvergence condition nh4n → 0 is too restrictive. The key theorem of this section proves theconvergence in probability of our sequence of smooth kernel estimators Sn(x) to σ2(x).

For this section and without loss of generality, we assume that T = 1 and ti =in . We con-

sider again the discrete observation S(n) = (S0, S1/n, . . . , S1), defined through the SDE (2). Weassume that σ(x) is bounded above and away from zero and with three continuous, boundedderivatives. These assumptions guarantee the existence of a unique strong solution which doesnot explode. We denote by Q the law on the space of continuous functions equipped withthe canonical filtration (Ft)t∈[0,1] and under which the canonical process (St, 0 ≤ t ≤ 1) is asolution to the previous SDE.

Note that, from a statistical point of view, it is more natural to work with weak solutions.The smoothness assumption and the boundedness of σ(x) and its derivatives are requiredto obtain some estimates. We also consider a compact interval D, which represents theobservation interval, i.e., the domain on which the estimation is performed. We emphasizethe fact that we are able to estimate σ(x) only for those points that have been visited by thediffusion.

The idea underlying the smooth kernel estimator is to replace the kernel K(x) = 121{|x|≤1}

by a smooth kernel φ, which is a C6 positive function with compact support and such that∫R+

φ = 1. We are interested in some Lp (p is stated later) convergence of the quantities

V xn =

1

nhn

n−1∑i=0

φ

(S i

n− x

hn

)n(S i+1

n− S i

n

)2,(6)

Lxn =

1

nhn

n−1∑i=0

φ

(S i

n− x

hn

)(7)

to σ2(x)Lx and Lx, respectively, where hn satisfies nh2n → ∞.

Our convergence theorem requires the use of various lemmas. The first lemma involvesthe convergence of Lx

n, and it follows from Proposition 3 of Hoffmann [13, p. 468].

Lemma 1. Assume that φ given in (7) is taken to be C3. For each γ ≥ 2, there exists aDow

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constant C such that

supx∈D

E(|Lxn − Lx|γ) ≤ C

(h

γ2n +

(1

nh2n

)γ).

Hence the Lp convergence of Lxn to Lx is guaranteed, for all p ≥ 2 and all x ∈ D, and

the Lp convergence for all p > 0 follows by the Cauchy–Schwarz method. Our next lemmainvolves the L1 convergence of V x

n .Lemma 2. For each x ∈ D, V x

n converges in L1 to σ2(x)Lx.In order to prove this lemma, we write V x

n − σ2(x)Lx = An(x) +Bn(x), where

An(x) =1

hn

∫ 1

0φ

(Xs − x

hn

)σ2(Xs)ds − σ2(x)Lx,

Bn(x) = V xn − 1

hn

∫ 1

0φ

(Xs − x

hn

)σ2(Xs)ds.

We study each of those two terms separately. Let x be fixed in D, where D is the domain overwhich the estimation is performed. Since x is fixed, we omit it from now on and prove thatAn and Bn converge in L1 to 0.

Lemma 3 (study of An). For each γ ≥ 2, there exists c > 0 such that E|An|γ ≤ chγ/2n .

Proof. Let l = (lx1 ) be the local time of the diffusion at time t = 1. First, we use theoccupation time formula,

An =1

hn

∫ 1

0φ

(Xs − x

hn

)σ2(Xs)ds− σ2(x)Lx

=1

hn

∫R+

φ

(y − x

hn

)lydy − 1

hn

∫R+

φ

(y − x

hn

)lxdy =

1

hn

∫R+

φ

(y − x

hn

)(ly − lx)dy.

Applying Jensen’s lemma to the integral, a straightforward change of variables and takingexpectations give

E|An|γ ≤ 1

hγnE

(∫R+

|lzhn+x − lx|γφγ(z)hγndz

).

The following inequalities follow from an application of Fubini’s theorem and the Holderproperty of the local time paths of a continuous local martingale. (This is a well-known andclassic result, given, for example, by Revuz and Yor [28, p. 227].)

E|An|γ ≤∫R+

φγ(z)E(|lzhn+x − lx|γ)dz ≤ hγ2n

∫R+

|z| γ2 φγ(z)dz.

Since φ has compact support, we can take c =∫R+ |z|γ/2φγ(z)dz. Then E|An|γ ≤ ch

γ/2n , and

the lemma is proved.We now focus on the study of Bn, which we write −Bn = Cn +Dn, where

Cn =1

hn

n−1∑i=0

∫ i+1n

in

(φ

(Ss − x

hn

)σ2(Ss)− φ

(S i

n− x

hn

)σ2(S i

n)

)ds,

Dn =1

hn

n−1∑i=0

∫ i+1n

in

φ

(S i

n− x

hn

)(σ2(S i

n)− n(S i+1

n− S i

n)2)ds.

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We need the following lemma borrowed from Genon-Catalot and Jacod, which we recall in theone dimensional setting. We refer to [9] for a proof. We define Xn

i =√nσ(Si/n)(W(i+1)/n −

Wi/n) and Y ni =

√n∫ (i+1)/ni/n σ(Ss)dWs.

Lemma 4 (Genon-Catalot and Jacod). Let g ∈ C2. Assume there exists γ > 0 such that forall x, |g(x)| + |g′

(x)|+ |g′′(x)| ≤ γ(1 + |x|γ). Then there exists a constant C such that

E((

g(Xni )− g(Y n

i ))2|F i

n

)≤ C

n.

If g is an even function,∣∣E(g(Xn

i )− g(Y ni )|Fi/n)

∣∣ ≤ Cn .

We can now study the convergence of Dn.

Lemma 5 (study of Dn). There exists C > 0 such that E|Dn| ≤ CnE(

1nhn

∑n−1i=0 φ(

Si/n−x

hn))

and Dn converges to 0 in L1.

Proof. Recall that Dn = 1hn

∑n−1i=0 φ(

Si/n−x

hn)(σ2(Si/n)

n − (∫ (i+1)/ni/n σ(Ss)dWs

)2). Write

g(x) = x2. The following inequalities are straightforward:

E|Dn| ≤ 1

hn

n−1∑i=0

E

⎛⎝φ

(S i

n− x

hn

)∣∣∣∣∣∣E(σ2(S i

n)

n

∣∣∣∣∣F in

)−E

⎛⎝(∫ i+1

n

in

σ(Ss)dWs

)2 ∣∣∣∣∣F in

⎞⎠∣∣∣∣∣∣⎞⎠

≤ 1

hn

n−1∑i=0

E

(φ

(S i

n− x

hn

)∣∣E(g(Xni )− g(Y n

i )|F in)∣∣) 1

n.

Since g is assumed to be an even function, Lemma 4 ensures the existence of a constant Csuch that ∣∣E(g(Xn

i )− g(Y ni )|F i

n

)∣∣ ≤ C

n.

Hence, E|Dn| ≤ CnE(

1nhn

∑n−1i=0 φ(

Si/n−x

hn)). Using Lemma 1, the sum converges to the local

time of the diffusion in x and E|Dn| → 0.In order to study Cn, we introduce the function f(y) = φ(y−x

hn)σ2(y), which is C3 by

assumption. We use a third order Taylor expansion and get that for all s in [ in ,i+1n [ there

exists Ξs,i/n such that

f(Ss) = f(S in) + (Ss − S i

n)f

′(S i

n) +

(Ss − S in)2

2f

′′(S i

n) +

(Ss − S in)3

6f (3)(Ξs, i

n).

We plug this into Cn and obtain Cn = C1n + C2

n + C3n + C4

n, where

C1n =

1

hn

n−1∑i=0

∫ i+1n

in

((Ss − S i

n)φ

(S i

n− x

hn

)(σ2)

′(S i

n)

+1

2(Ss − S i

n)2

(2

hnφ

′(S i

n− x

hn

)(σ2)

′(S i

n) + φ

(S i

n− x

hn

)(σ2)

′′(S i

n)

)

+1

6(Ss − S i

n)3

(3

h2nφ

′′(Ξs, i

n− x

hn

)(σ2)

′(Ξs, i

n) +

3

hnφ

′(Ξs, i

n− x

hn

)(σ2)

′′(Ξs, i

n)

+ φ

(Ξs, i

n− x

hn

)(σ2)(3)(Ξs, i

n)

))dsD

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and

C2n =

1

h2n

n−1∑i=0

φ′(S i

n− x

hn

)σ2(S i

n)

∫ i+1n

in

(Ss − S in)ds,

C3n =

1

h3n

n−1∑i=0

φ′′(S i

n− x

hn

)σ2(S i

n)

∫ i+1n

in

(Ss − S in)2

2ds,

C4n =

1

h4n

n−1∑i=0

∫ i+1n

in

φ(3)

(Ξs, i

n− x

hn

)σ2(Ξs, i

n)(Ss − S i

n)3

6ds.

We prove in the following lemma that C1n converges in L1 to 0 when nh2n tends to infinity.

The idea is to bound C1n by first bounding σ, φ, and their three first derivatives and then

using the Burkholder–Davis–Gundy inequalities (hereafter referred to simply as BDG) toobtain estimates of powers of Ss − Si/n.

Lemma 6 (study of C1). Assume that nh2n → ∞. Then C1n converges in L1 to 0.

Proof. Since φ and σ and their derivatives are bounded, there exist nonnegative constants(ci)1≤i≤6 such that

E|C1n| ≤ E

(n−1∑i=0

c1hn

∫ i+1n

in

|Ss − S in|ds +

(c2hn

+c3h2n

)∫ i+1n

in

|Ss − S in|2ds

+

(c4hn

+c5h2n

+c6h3n

)∫ i+1n

in

|Ss − S in|3ds

).

It follows clearly that

E|C1n| ≤

n−1∑i=0

c1hn

∫ i+1n

in

E

⎛⎝ sup

s∈[ in, i+1

n]

|Ss − S in|⎞⎠ ds+

(c2hn

+c3h2n

)∫ i+1n

in

E

⎛⎝ sup

s∈[ in, i+1

n]

|Ss − S in|2⎞⎠ ds

+

(c4hn

+c5h2n

+c6h3n

)∫ i+1n

in

E

⎛⎝ sup

s∈[ in, i+1

n]

|Ss − S in|3⎞⎠ ds.

We now apply BDG inequalities for continuous local martingales. For each 1 ≤ p ≤ 3 thereexist nonnegative constants Cp such that

E

⎛⎝ sup

s∈[ in, i+1

n]

|Ss − S in|p⎞⎠ ≤ E

⎛⎝(∫ i+1

n

in

σ2(Ss)ds

) p2

⎞⎠ ≤ Cp

np2

.

Integrating, summing, and taking the expectation, we finally obtain

E|C1n| ≤

c1hn

C1√n+

(c2hn

+c3h2n

)C2

n+

(c4hn

+c5h2n

+c6h3n

)C3

n√n

≤ c1C1√nh2n

+C2

nh2n(c2hn + c3) +

C3

nh2n

(c4hn√

n+

c5√n+

c6√nh2n

).

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Since nh2n → ∞ and hn → 0, it follows clearly that C1n converges in L1 to 0.

We turn now to the study of C2n.

Lemma 7 (study of C2n). If nh

2n → ∞, then C2

n converges in L1 to 0.Proof. First,

E|C2n| ≤

1

h2n||σ2||∞E

(n−1∑i=0

|φ′ |(S i

n− x

hn

)∣∣∣∣∣E(∫ i+1

n

in

|Ss − S in|ds∣∣F i

n

)∣∣∣∣∣).

It follows from an application of a BDG inequality that there exists a constant M1 such that

E|C2n| ≤

M1√nhn

E

(1

nhn

n−1∑i=0

∣∣∣∣∣φ′(S i

n− x

hn

)∣∣∣∣∣).

Now using the kernel 1a |φ

′ |, where a =∫ |φ′ |(x)dx, the quantity 1

nhn

∑n−1i=0 |φ′ |(Si/n−x

hn) con-

verges in L1 to aLx. Hence E|C2n| converges to zero as soon as nh2n → ∞, which proves the

lemma.We can provide another proof of this result following the proof in [13, p. 477]. We obtain

that for each γ ≥ 2 there exists a constant C such that

E|C2n|γ ≤ C

1

nγh− 3

2n .

It follows that E|C2n| ≤

√E(|C2

n|2) ≤√C

nh2nh5/4n , which converges to zero. The remaining terms

to estimate are C3n and C4

n. Note that in the expansion of Cn, C3n is the most important term.

Lemma 8 (study of C3n and C4

n). If nh2n → ∞, then C3

n and C4n converge to zero in L1.

Proof. It is straightforward to obtain the estimate

E|Cn3 | ≤

M

h3nE

(n−1∑i=0

|φ′′ |(S i

n− x

hn

)∫ i+1n

in

1

2E((Ss − S i

n)2 | F i

n

)).

Now a BDG inequality implies the existence of a constant C such that E((Ss−Si/n)2 | Fi/n) ≤

C(s− in). Hence

E|Cn3 | ≤

MC

2nh2nE

(1

nhn

n−1∑i=0

|φ′′ |(S i

n− x

hn

)),

which converges to zero when nh2n → ∞. The same techniques as in this lemma and theprevious one can be applied to prove the convergence of C4

n in L1 to 0.We have a stronger result than that stated in the lemma above. Under the assumption

that nh2n → ∞, both C3n and C4

n converge to zero in Lγ for all γ > 2. Define

C3n =

1

h3n

n−1∑i=0

φ′′(S i

n− x

hn

)∫ i+1n

in

(Ss − S in)2

2ds.

Hoffman’s result in [13] guarantees that C3n converges in Lγ to 0 as soon as nh2n → ∞, for all

γ > 2. Since σ and its derivatives are assumed to be bounded, it is not hard to see that wehave the same result for C3

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Putting all these lemmas together proves that Cn converges in L1 to 0, and thus Bn

converges in L1 to 0. We have then proved that Lxn converges to Lx in Lp for all p > 0 and

that V xn − σ2(x)Lx converges in L1 to 0, which ends the proof of Lemma 2. The following

theorem is now straightforward.

Theorem 3. If nh2n → ∞, then Sxn = V x

nLxnconverges in probability to σ2(x) and provides a

consistent estimator of σ2(x).

Remark 4. After finding and proving this theorem, we learned to our chagrin, from JeanJacod, that he had not only already considered this exact problem more than 10 years ago,but also established similar results which are both more general and more effective; see [16]and [17]. In particular he is able to take hn = 1√

n, and he also obtains a rate of convergence

and an associated central limit theorem. We have decided nevertheless to retain our estimatorand its proof presented here, since it is the one we used to process the data and since it seems toworks well for our purposes. But we wish to signal for future related work that there are morepowerful (if perhaps slightly more complicated) similar estimators available. These remarksalso apply for parts of section 4.

4. Unbounded volatility function estimators. The previous two estimators for the volatil-ity function σ(x) are over a compact domain representing the observation interval. In thissection, for the SDE (2), we relax this boundedness assumption on the volatility function σ(x).Herein, we now assume that σ > 0 on I = ]0,∞[, is identically null elsewhere, and satisfies1σ2 ∈ L1

loc(I).

This is the Engelbert–Schmidt condition (see, e.g., [4], [5], [6], or [20]) under which theSDE has a unique weak solution S that does not explode to ∞. We let P be the law ofthe solution on the canonical space Ω = C([0, T ],R) equipped with the canonical filtration(Ft)t∈[0,T ] and the canonical process S = (St)t∈[0,T ]. We also assume that σ is C3 boundedand with bounded derivatives on every compact set. We add in passing that these hypothesesimply the existence of a strong solution as well. Let τ0(S) be the first time S hits zero. Thefollowing theorem provides straightforward but useful extensions of Theorem 1 and Lemma 2.

Theorem 5. Suppose that σ(x) has three continuous derivatives. Assume that nh4n → 0and nhn → ∞. Then conditional on {τ0(S) > T}, Sn(x) given in (5) converges in probabilityto σ2(x). The same holds for our smooth kernel estimator under the constraint nh2n → ∞.

Proof. Let Tq = inf {t, St ≥ q} and τp = inf {t, St ≤ 1p}. Then limp→∞ τp = τ0(S) and

limq→∞ Tq = ∞ since S does not explode to ∞. We can take σp,q to be a function boundedabove and below away from zero with three bounded derivatives such that σp,q(x) = σ(x) forall 1

p ≤ x ≤ q. Let (Sp,qt )t∈[0,T ] be the unique strong solution to the SDE dSp,q

t = σp,q(Sp,qt )dWt.

Introduce now Sp,qn (x), the estimator computed on the basis of (Sp,q

t )t∈[0,T ] as in (5) or usingour smooth kernel estimator. Then, under the suitable constraints on the sequence (hn)n≥1,Sp,qn (x) converges in probability to σ2

p,q(x). Moreover, Sp,qn (x) = Sn(x) if T < Tq ∧ τp. Then

obviously Sn(x) converges in probability to σ2(x), in restriction to the set {T < τ0(S)}.We can extend Theorem 2 by working in the filtration Gt = σ(Ss, s ≤ t) ∨ σ(τp), where

τp = inf {t, St ∈ ]1p , p[}, and more interestingly in G′t = σ(Ss, s ≤ t) ∨ U , where U = 1{τp>T}

and T is the time horizon. Consider, for instance, the initial enlargement Gt = Ft ∨ σ(τp),where Ft = σ(Ss, s ≤ t) and τp is the first exit time of S from ]1p , p[. τp is an F stoppingtime. Consider the filtered probability space (Ω,G, Q,G), where G = (Gt)t≥0. Under someD

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technical assumptions, and if the sequence hn satisfies hn → 0, nhn → ∞, and nh3n → 0,

then we can prove that 1{τp>t}√

Nna (

Sn(a)σ2(a)

− 1) converges in distribution to√2Z, where Z is

a standard normal random variable and Nna =

∑ni=1 1{|Si/n−a|<hn}. The exact statement of

the theorem and its proof can be provided to the interested reader upon request. We do notprovide these here because the proof’s technicalities use results from the initial enlargement offiltrations [15] (alternatively, see [27]) and subsequently require derivations of many estimates.We focus herein on the methodology for bubble detection and the illustrative tests using dot-com company stocks.

Remark 6 (in practice). Note that this limit theorem can be applied if during the time in-terval [0, T ] the process does not hit 0. The limit theorem also provides us with a confidenceinterval for the volatility estimator.

5. Detecting bubbles. This section illustrates the use of the previous volatility functionestimators for detecting asset price bubbles during the alleged dot-com Internet stock pricebubble episode of 1998–2001 (see [25]). As mentioned in the introduction, asset price bubbleshave been characterized in frictionless, competitive, and continuous trading models using thearbitrage-free martingale pricing technology (see [23], [24], [2], [11], [18], [19]). As shown inJarrow, Protter, and Shimbo [18] and [19], for finite horizon economies a bubble exists if theprice process under a risk neutral measure is a strict local martingale and not a martingale.The difference between a martingale and a strict local martingale has been recently investi-gated by several authors. The following theorem has been proved using different techniques.A proof based on explosion time techniques is provided in [1]. Separating time techniquesare used to prove a more general result in [26]. Kotani gives a PDE-based proof in [21]. Asmentioned previously, Kramkov has recently pointed out that this follows quite simply fromFeller’s test for explosions [22].

Theorem 7. S is a martingale (has no price bubbles) if and only if∫∞ε

xσ2(x)

ds = ∞ for

each ε > 0.This theorem forms the basis for our bubbles testing methodology. Unfortunately, we

are immediately faced with an extrapolation problem. To see this problem, we note that thevolatility function estimators presented in the previous sections provide estimates for σ(x)only on a finite interval—those x that have been visited by the process. Given the availablestock price data, we cannot observe the tails of the volatility function necessary to check forthe divergence of the integral in Theorem 7. To check for divergence, we must extrapolatefrom the observed domain of σ(x) to the entire nonnegative real line.

We propose two extrapolation methods to overcome this problem. The first method isto use a parametric estimator, as in [9], and a comparison theorem to conclude when theparametric and nonparametric volatility estimators are similar. If similar, we extrapolate intothe tail using the parametric form’s asymptotic behavior. The second method is to use RKHStheory to extrapolate the volatility function in the best possible way. We will quantify whatwe mean by “best” below.

5.1. Method 1: Parametric estimation. The appeal of using a parametric form for thevolatility function σ(x) is that we know the tails once the parameters have been estimated.For this estimation we choose a class of volatility functions large enough to include many ofthe forms used in practice (for example, power functions σ(x) = σxα, where σ and α areD

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the unknown parameters that we estimate). For this example of σ(x) = xα we have thatthe process S is a strict local martingale that is always strictly positive if α > 1 and is amartingale if 1

2 ≤ α < 1 and which, however, can assume the value 0. If α = 1, we are in thecase of geometric Brownian motion. We then also use our nonparametric estimators. If theseestimators are comparable, we have a conclusive test for divergence of the volatility integral.If they are not comparable, then the test is inconclusive.

5.1.1. The comparison theorem. This section states and proves the comparison theorem.

Theorem 8 (comparison theorem). Assume that dSt = σ(t, St)dWt and that there exist twofunctions Σ and σ such that, for all t and x, σ(x) ≤ σ(t, x) ≤ Σ(x) and such that σ, Σ, andσ are continuous, locally Holder continuous with exponent 1

2 . Then the following hold:

(i) If for all c > 0,∫∞c

xΣ2(x)

dx = ∞, then S is a martingale.

(ii) If there exists c > 0 such that∫∞c

xσ2(x)

dx < ∞, then S is a strict local martingale.

In order to prove this theorem, we need the following lemma (see [3]).

Lemma 9. Let g be a concave function and αi, i = 1, 2, be two continuous functions, locallyHolder continuous with exponent 1

2 such that, for all (x, t), α1(x, t) ≤ α2(x, t). Let T > 0 befixed. We consider dX

α1,2

t = α1,2(t,Xα1,2

t )dWt and u1,2(x, t) = E(x,t)(g(Xα1,2

T )). Then for allx ∈ R

+ and t ∈ [0, T ], u1(x, t) ≥ u2(x, t).

Proof of Theorem 8. (i) Since g(x) = x is concave, we can apply the previous lemma andget that, for all (x, t), u(x, t) ≥ uΣ(x, t). If

∫∞c

xΣ2(x)

dx = ∞, then by Theorem 7, uΣ(x, t) = x

for all (x, t). Thus, for all (x, t), u(x, t) ≥ x. But, we know that S is a positive local martingaleand hence a supermartingale by Jensen’s lemma; thus, for all (x, t), u(x, t) ≤ x. This provesthat E(ST |St = x) = x and S is a martingale.

(ii) Let T > 0 be fixed and u(x, t) = E(SσT |Sσ

t = x). Let c > 0 such that∫∞c x/σ2(x)ds <

∞. We know that Sσ is a strict local martingale by Theorem 7 and u(x, t) ≤ x, and thereexists a t such that u(x, t) < x. Again since g(x) = x is concave, x ≥ u(x, t) ≥ u(x, t) forall (t, x), and there exists a t such that u(x, t) < u(x, t) < x. Hence, S is a strict localmartingale.

5.1.2. Illustrative examples. To illustrate this procedure, we use market price data fromthe alleged Internet dot-com bubble (and beyond), from 1999 to 2005. As explained above,we can use the previous theorem as follows: first we choose a parametric form for the diffusioncoefficient and estimate the parameters as explained in [9] by Genon-Catalot and Jacod afterchoosing a contrast function to minimize. That is, we choose a parametric form σ(ν, x), whereν is the multidimensional parameter that needs to be estimated, and a contrast f(G,x). Theirestimator is defined as νn = argmin 1

n

∑ni=1 f(σ

2(ν, Sti−1), Sni ), where Si

n =√n(Sti − Sti−1).

Usual choices for the contrast function f in our one dimensional setting are f1(G,x) = ln(G)+x2

G or f2(G,x) = (x2 − G)2. We do not provide further details in this paper and refer theinterested reader to [9] for a detailed description of the estimation procedure.

Then we estimate the volatility function using our nonparametric estimators. If the twovolatility function estimates are similar, then by applying the criteria of Theorem 7 to theparametric estimator, we can test for the existence of a price bubble using the comparisontheorem as in Theorem 8.

A conclusive test: When applied to the stock Lastminute.com, the methodology isDow

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Figure 1. Lastminute.com stock prices during the alleged dot-com bubble.

Figure 2. Lastminute.com. Estimates of σ(x).

conclusive.

Intuitively, given the stock price time series in Figure 1, one suspects the existence of a pricebubble. Our test confirms this belief. This can be seen from Figure 2 above, which displaysthe estimators of Florens-Zmirou (F-Z), the smoothed kernel (J-K-P), and the parametricestimator of Genon-Catalot and Jacod (GC-J), using the power parametric form σ(σ0, α, x) =σ0x

α (here ν = (σ0, α) is a two dimensional parameter) and the loglikelihood-like contrastf1(G,x). Using this estimation technique, we find an estimate σ(x) = σ(σ0, α, x), whose tailbehavior leads to the convergence of the integral

∫∞ε

xσ(x)2

dx. Also our estimator (J-K-P) lies

above the estimated function (GC-J); hence Theorem 8 guarantees that the price process is astrict local martingale, and we have bubble pricing.

An inconclusive test: A weakness of this procedure is that the comparison test usingparametric estimators might be inconclusive, even when intuitively one suspects a bubble. Anexample of this phenomenon is that of eToys; see Figure 3.

This stock price graph suggests the existence of a price bubble. Our methodology is il-lustrated in Figure 4. The three curves included in this figure and labelled GC-J1, GC-J2,D

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Figure 3. eToys stock prices during the alleged dot-com bubble.

Figure 4. eToys. Estimates of σ(x).

and GC-J3 represent the estimators of Genon-Catalot and Jacod, with different paramet-ric forms and contrasts to minimize. GC-J1 and GC-J3 are obtained from the parametricform σ(σ0, α, x) = σ0x

α and the contrasts f1 and f2, respectively. GC-J2 is obtained fromσ(σ0, α, β, x) = σ0x

α lnβ |x| and the contrast f1. In theory, we have a bubble if α > 1 or ifα = 1 and β > 1; however, the estimated parameters lie in these boundary values and we seethat the curves are so close to linear that we cannot conclude either convergence or divergenceof the integral in Theorem 8. Using this methodology, our test is inconclusive as to whetheror not there was a bubble in the stock price of eToys during the 1999–2001 period.

5.2. Method 2: RKHS theory. This section presents our second method for extrapola-tion. This method is different, in that it is based on RKHS theory. Previously in the paperwe have considered parameterized families of functions, so that once the parameter is chosen,the tail behavior is determined. We can observe the volatility coefficient σ only on a boundedinterval, of course, so it is a leap of faith to assume that (a) it is of the form of the parame-terized family of functions considered, and (b) its behavior continues unchanged into the tail.Nevertheless, this is more or less the standard technique in situations such as this.D

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Our second method is a bit more subtle. Our procedure here consists of two steps:

• We first interpolate an estimate of σ within the bounded interval where we haveobservations, and in this way we lose the irregularities of nonparametric estimators.

• We next extrapolate our function σ by choosing an RKHS from a family of Hilbertspaces in such a way as to remain as close as possible (on the bounded interval ofobservations) to the interpolated function provided in the previous step.

This represents a new methodology which allows us to choose a good extrapolation method.We do this via the choice of a certain extrapolating RKHS, which—once chosen—again de-termines the tail behavior of our volatility σ. If we let (Hm)m∈N denote our family of RKHSs,then any given choice of m (call it m0) allows us to interpolate perfectly the original estimatedpoints and thus provides a valid RKHS Hm with which we extrapolate σ. But this representsa choice of m0 and not an estimation. So if we stop at this point, the method would be asarbitrary as parametric estimation. That is, choosing m0 is analogous to choosing the pa-rameterized family of functions which fits σ best. The difference is that we do not arbitrarilychoose m0. Instead we choose the index m given the data available. In this sense we are usingthe data twice. To do this we evaluate different RKHSs in order to find the most appropriateone given the arrangement of the finite number of grid points from our observations.

The RKHS method (see [14]) is intimately related to the reconstruction of functions fromscattered data in certain linear functional spaces. The reproducing kernel Q(x, x

′) that is

associated with an RKHS H(D) in the spatial domain D, over the coordinate x, is unique andpositive and thus constitutes a natural basis for generic interpolation problems.

5.2.1. Reproducing kernel Hilbert spaces. Let H(D) be a Hilbert space of continuousreal-valued functions f(x) defined on a spatial domain D. A reproducing kernel Q possessesmany useful properties for data interpolation and function approximation problems.

Proposition 1. There exists a kernel function Q(x, x′), the reproducing kernel, in H(D)

such that the following properties hold:

(i) Reproducing property. For all x and y,

f(x) = 〈f(x′), Q(x, x

′)〉′ ,

Q(x, y) = 〈Q(x, x′), Q(y, x

′)〉′ .

The prime indicates that the inner product 〈·, ·〉′ is performed over x′.

(ii) Uniqueness. The RKHS H(D) has one and only one reproducing kernel Q(x, x′).

(iii) Symmetry and positivity. The reproducing kernel Q(x, x′) is symmetric, i.e., Q(x

′, x) =

Q(x, x′), and positive definite, i.e.,

n∑i=1

n∑k=1

ciQ(xi, xk)ck ≥ 0,

for any set of real numbers ci and for any countable set of points (xi)i∈[1,n].In this framework, interpolation is seen as an inverse problem. The inverse problem is the

following. Given a set of real-valued data (fi)i∈[1,M ] at M distinct points SM = xi, i ∈ [1,M ],in a domain D, and an RKHS H(D), find a suitable function f(x) that interpolates these dataD

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points. Using the reproducing property, this interpolation problem is reduced to solving thefollowing linear inverse problem:

(8) ∀i ∈ [1,M ], f(xi) = 〈f(x′), Q(xi, x

′)〉′ ,

where we need to invert this relation and exhibit the function f(x) in H(D). We refer thereader to [14] for a detailed discussion.

We first present the normal solution that allows an exact interpolation, and second theregularized solution that yields quasi-interpolative results, accompanied by an error boundanalysis. Then, in the next section, we will construct a family of RKHSs that enable us tointerpolate not σ(x) but 1

σ(x)2. This transformation makes natural the choice of the family of

RKHSs. Note that for every choice of an RKHS one can construct an interpolating functionusing the input data. For this reason, we define a family of RKHSs that encapsulate differentassumptions on the asymptotic forms and smoothness constraints. From this set, we choosethat RKHS which best fits the input data in the sense explained below.

Normal solutions: The most straightforward interpolation approach is to find the nor-mal solution that has the minimal squared norm ||f ||2 = 〈f(x′

), f(x′)〉′ subject to the inter-

polation condition (8).That is, given a set of real-valued data {fi}, 1 ≤ i ≤ K, specified at K distinct points in

a domain D, we wish to find a function f that is the normal solution:

f(x) =M∑i=1

ciQ(xi, x),

where the coefficients ci satisfy the linear relation

(9) ∀k ∈ [1,M ],M∑i=1

ciQ(xi, xk) = fk.

If the matrixQM whose entries are theQ(xi, xk) is “well conditioned,” then the linear algebraicsystem above can be efficiently solved numerically. Otherwise, we use regularized solutions.

Regularized solutions: When the matrix QM is “ill conditioned,” regularization pro-cedures may be invoked for approximately solving the linear inverse problem. In particular,the Tikhonov regularization procedure produces an approximate solution fα, which belongsto H(D) and which can be obtained via the minimization of the regularization functional

||Qf − F ||2 + α||f ||2

with respect to f(x). Note that here F is the data vector (fi), and the residual norm ||Qf−F ||2is defined as

||Qf − F ||2 =M∑i=1

(〈f(x′

), Q(xi, x′)〉′ − fi

)2.

The regularization parameter α is chosen to impose a proper balance between the residualconstraint ||Qf − F || and the magnitude constraint ||f ||. The regularized solution has theD

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form

(10) fα(x) =M∑i=1

cαi Q(xi, x),

where the coefficients cαi satisfy the linear relation

(11) ∀k ∈ [1,M ],M∑i=1

cαi (Q(xi, xk) + αδi,k) = fk,

where δi,k is the Kronecker delta function. Note that, for α > 0, QαM whose entries are

[Q(xi, xk) + αδi,k] is symmetric and positive definite, and the problem can now be solvedefficiently. Also, the RKHS interpolation method leads to an automatic error estimate of theregularized solution (see [14] for more details).

5.2.2. Construction of the reproducing kernels. We consider reciprocal power repro-ducing kernels that asymptotically behave as some reciprocal power of x, over the interval[0,∞[. We are interested in this type of RKHS because this is a reasonable assumption forf(x) = 1

σ2(x). The CEV model dSt = Sα

t dWt, where α > 0, is an important local volatility

model proposed in the literature and satisfies this assumption, with fcev(x) =1

x2α . We alsoassume that the function f(x) possesses the asymptotic property

limx→∞xkf (k)(x) = 0 ∀k ∈ [1, n − 1]

for some n ≥ 1 that controls the minimal required regularity. This property is often satisfied

by the volatility functions used in practice. For instance, xkf(k)cev(x) =

∏k−1i=0 (−2α−i)

x2α convergesto 0 as x tends to infinity for all k. The asymptotic property is also satisfied by many volatilityfunctions that explode faster than any power of x, for example σ(x) = xαeβx with α > 0 andβ > 0. The condition appears restrictive only when σ and its derivatives explode too slowlyor when σ is bounded; however, in these cases, it is likely that there is no bubble and that noextrapolation using this RKHS theory will be required. We would like to emphasize that theasymptotic property satisfied by f is the key point for the whole method to work, as may beseen from Proposition 3 below.

Concerning the degree of smoothness, in practice we usually take n to be 1, 2, or 3. Wecan now define our Hilbert space

Hn = Hn([0,∞[) ={f ∈ Cn([0,∞[) | lim

x→∞xkf (k)(x) = 0 ∀k ∈ [1, n − 1]}.

We now need to define an inner product. A smooth reproducing kernel qRP (x, x′) can be

constructed via the choice

〈f, g〉n,m =

∫ ∞

0

ynf (n)(y)

n!

yng(n)(y)

n!

dy

w(y),

where w(y) = 1ym is the asymptotic weighting function. From now on we consider the RKHS

Hn,m = (Hn, 〈, 〉n,m). The next proposition can be shown following the steps in [14].Dow

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Proposition 2. The reproducing kernel is given by

qRPn,m(x, y) = n2x

−(m+1)> B(m+ 1, n)F2,1

(−n+ 1,m+ 1, n +m+ 1,

x<x>

),

where x> and x< are respectively the larger and smaller of x and y, B(a, b) is the beta function,and F2,1(a, b, c, z) is Gauss’s hypergeometric function.

Remark 9. The integers n− 1 and m+ 1 are respectively the order of smoothness and theasymptotic reciprocal power behavior of the reproducing kernel qRP (x, y). This kernel is arational polynomial in the variables x and y and has only a finite number of terms, so it iscomputationally efficient.

As pointed out above, any choice of n and m creates an RKHS Hn,m and allows one toconstruct an interpolating function fn,m(x) with a specific asymptotic behavior. The followingresult gives the exact asymptotic behavior.

Proposition 3. For every x, qRP (x, y) is equivalent to n2

ym+1B(m + 1, n) at infinity as afunction of y and

limx→∞xm+1fα(x) = n2B(m+ 1, n)

M∑i=1

cαi ,

where fα is defined as in (10) and the constants cαi are obtained as in (11). Hence, if∑Mi=1 c

αi = 0, then fα(x) is equivalent to n2B(m+1,n)

xm+1

∑Mi=1 c

αi .

5.2.3. Choosing the best m. The choice of m allows one to decide whether the integralin Theorem 7 converges or diverges. If m > 1, there is a bubble. This section explains howto choose m. Let us first summarize the idea. We choose the RKHS by optimizing over theasymptotic weight m that allows us to construct a function that interpolates the input datapoints and remains as close as possible to the interpolated function on the finite interval D.This optimization provides an m which allows us to construct σm(x). We employ a four stepprocedure as follows.

(A) Nonparametric estimation over D. Estimate σ(x) using our nonparametric estimatoron a fixed grid x1, . . . , xM of the bounded interval D = [minS,maxS], where minS andmaxS are the minimum and the maximum reached by the stock price over the estimation

time interval [0, T ]. In our illustrative examples, we use the kernel φ(x) = 1ce

14x2−1 for |x| < 1

2 ,where c is the appropriate normalization constant. The number of data available n and therestriction on the sequence (hn)n≥1 make the number of grid points M relatively small inpractice. In our numerical experiments, 7 ≤ M ≤ 25.

(B) Interpolate σ(x) over D using RKHS theory. Use any interpolation method on thefinite interval D to interpolate the data points (σ(xi))i∈[1,M ]. Call the interpolated function

σb(x). For completeness, we provide a methodology to achieve this using the RKHS theory.However, any alternative interpolation procedure for a finite interval could be used.

Define the Sobolev space Hn(D) ={u ∈ L2(D) | ∀k ∈ [1, n], u(k) ∈ L2(D)

}, where u(k)

is the weak derivative of u. The norm that is usually chosen is ||u||2 =∑nk=0

∫D(u

(k))2(x)dx.Due to Sobolev inequalities, an equivalent and more appropriate norm is ||u|| = ∫D u2(x)dx+1

τ2n

∫D(u

(n))2(x)dx. We denote by Ka,bn,τ the kernel function of Hn(]a, b[), where in this case

D = ]a, b[. This reproducing kernel is provided for n = 1 and n = 2 in the following lemma.Dow

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Lemma 10.

Ka,b1,τ (x, y) =

τ

sinh(τ(b− a))cosh(τ(b− x>)) cosh(τ(x< − a)),

Ka,b2,τ (x, y) = Lx>(x<),

and Lx(t) is of the form∑4

i=1

∑4k=1 likbi(τt)bk(τx).

We refer to [29, equation (22) and Corollary 3, p. 28] for explicit analytic expressions forlik and bk, which while simple, are nevertheless tedious to write. In both equalities, x> andx< respectively stand for the larger and smaller of x and y. In practice, one should check thequality of this interpolation and carefully study the outputs by choosing different τ ’s beforeusing the interpolated function σb = 1/

√f b in the algorithm detailed above, where f b(x) =∑M

i=1 cbiK

Dn,τ (xi, x) for all x ∈ D, and for all k ∈ [1,M ],

∑Mi=1 c

biK

Dn,τ (xi, xk) = fk = 1√

σest(xk).

(C) Deciding whether an extrapolation is required. If the extended form of the estimatedσ(x) implies that the volatility does not diverge to ∞ as x → ∞ and remains bounded onR+, no extrapolation is required. In such a case

∫∞ε

xσ2(x)

is infinite, and the process is a true

martingale. If one decides, however, that σ(x) diverges to ∞ as x → ∞, then the next step isrequired to obtain a “natural” candidate for its asymptotic behavior as a reciprocal power.

(D) Extrapolate σb(x) to R+ using RKHS . Fix n = 2 and define

(12) m = argminm≥0

√∫[a,∞[∩D

|σm − σb|2ds,

where fm = 1σ2m

is in the RKHS H2,m = (H2,m([0,∞[), 〈, 〉RP ). By definition, all σm will

interpolate the input data points, and σm has the asymptotic behavior that best matchesour function on the estimation interval. a is the threshold determining closeness to the in-terpolated function. Choosing a too small is misleading since then it would account more(and unnecessarily) for the interpolation errors over the finite interval D than desirable. Weshould choose a large a since we are interested only in the asymptotic behavior of the volatil-ity function. In the illustrative examples below, the threshold a in (12) is chosen to bea = maxS − 1

3 (maxS −minS).

5.2.4. Illustrative examples. We illustrate our testing methodology for price bubblesusing the stocks that are often alleged [30], [25] as experiencing Internet dot-com bubbles. Weconsider those stocks for which we have tick data. The data was obtained from WRDS [31].We apply this methodology to four stocks: Lastminute.com, eToys, Infospace, and Geocities.The methodology performs well. The weakness of the method is the possibility of inconclusivetests, as illustrated by eToys. For Lastminute.com and Infospace our methodology supportsthe existence of a price bubble. For Infospace, we reproduce the methodology step-by-step.Finally, the study of Geocities provides a stock commonly believed to have exhibited a bubble(see, for instance, [30], [25]), but which our method says did not. We now provide our analyses.

Lastminute.com: Our methodology confirms the existence of a bubble. The stock pricesare given in Figure 5. The optimization performs as expected with the asymptotic behaviorgiven by m = 8.26, which means that σ(x) is equivalent at infinity to a function proportionalD

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Figure 5. Lastminute.com stock prices during the alleged dot-com bubble.

Figure 6. Lastminute.com. RKHS estimates of σ(x).

to xα with α = 4.63. We plot in Figure 6 the different extrapolations obtained using differentreproducing kernel Hilbert spaces H2,m and their respective reproducing kernels qRP

2,m.

Figure 6 shows that m is between 7 and 9, as obtained by the optimization procedure.The orange curve labeled sigma is the interpolation on the finite interval D obtained fromthe nonparametric estimation procedure where the interpolation is achieved using the RKHStheory as described in step (B) with the choice of the reproducing kernel Hilbert space H1(D)and the reproducing kernel KminS,maxS

1,6 . Then m is optimized as in step (D) so that theinterpolating function σm(x) is as close as possible to the orange curve in the last third of thedomain D; i.e., the threshold a in (12) is chosen to be a = maxS − 1

3(maxS −minS).

eToys: While the graph of the stock price of eToys as given in Figure 7 makes theexistence of a bubble plausible, the test nevertheless is inconclusive. Different choices of mgiving different asymptotic behaviors are all close to linear (see Figure 8).D

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Figure 7. eToys.com stock prices during the alleged dot-com bubble.

Figure 8. eToys. RKHS estimates of σ(x).

Because the estimates are so close to being linear, we cannot tell with any level of assurancethat the integral in question diverges or converges; we simply cannot decide which is the case.If it were to diverge, we would have a martingale (and hence no bubble), and were it toconverge, we would have a strict local martingale (and hence bubble pricing).

The estimated m is close to 1. In Figure 8, the powers α are given by 12(m+1), where m

is the weight of the reciprocal power used to define the Hilbert space and its inner product.We plot the extrapolated functions obtained using different Hilbert spaces H2,m together withtheir reproducing kernels qRP

2,m. Figure 9 shows that the extrapolated functions obtained usingthese different RKHS H2,m produce the same quality of fit on the domain D.

Infospace: Our methodology shows that Infospace exhibited a price bubble. We detailDow

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Figure 9. eToys. RKHS estimates of σ(x), quality of fit.

Figure 10. Infospace stock prices during the alleged dot-com bubble.

the methodology step by step in this example. The graph of the stock prices in Figure 10suggests the existence of a bubble.

(i) We compute the Florens-Zmirou’s estimator and our smooth kernel local time-basedestimator, using a sequence hn = 1

n1/3 . The result is not smooth enough, as seen inFigure 11.

(ii) We use the sequence hn = 1n1/4 to compute our estimators (the number of points

where the estimation is performed is smaller, M = 11). Theoretically, we no longerhave the convergence of the Florens-Zmirou’s estimator. However, as seen in Fig-ure 12, this estimator is robust with respect to the constraint on the sequence hn.F-Z, LowerBound, and UpperBound are Florens-Zmirou’s estimator together with theD

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Figure 11. Infospace. Nonparametric estimation using hn = 1

n1/3 .

Figure 12. Infospace. Nonparametric estimation using hn = 1

n1/4 .

95% confidence bounds her estimation procedure provides. J-K-P is our estimator.(iii) We obtained in (ii) estimations on a fixed grid containing M = 11 points, and we

now construct a function σb(x) on the finite domain (see Figure 13), which perfectlyinterpolates those points. Here the RKHS used is H1(D), where D = [minS,maxS],together with the reproducing kernels KD

1,τ , where τ takes the values 1, 3, 6, and 9. Thefunctions obtained using these different reproducing kernels provide the same qualityof fit within D, and we can use any of the four outputs as the interpolated function,σb, over the finite interval D.

(iv) Finally, we optimize over m and find the RKHSH2,m that allows the best interpolationof the M = 11 estimated points and such that the extrapolated function σ(x) remainsas close as possible to σb(x) near the right edge of the region D. Of course, thereproducing kernels used in order to construct the functions σm and minimize thetarget error as in (12) are qRP

2,m. We obtain m = 6.17 (i.e., α = m+12 = 3.58), and we

can conclude that there is a bubble.Dow

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Figure 13. Infospace. Interpolation σb(x) on the compact domain.

Figure 14. Infospace. Final estimator and RKHS extrapolation.

Remark 10. One might expect α ≈ 1.8, as suggested by the green curve in Figure 14. Butthis is different from what the RKHS extrapolation has selected. Why? In Figure 14, we plotthe RKHS extrapolation obtained when α = 1.8. We have proved that

limx→∞

xm+1

σ2(x)= 4B(m+ 1, 2)

M∑i=1

ci.

The numerical computations give σ(x) ≈ x3.58

127009 when using optimization over m, and σ(x) ≈x1.8

5.66 when fixing α = 1.8. Independent of the power chosen, the ci’s and hence the constant ofproportionality are automatically adjusted to interpolate the input points. But, as can be seenin Figure 15, the power 3.58 is more consistent in terms of naturally extending the behaviorof σb(x) to R

+.Geocities: Our methodology shows that this stock did not have a price bubble. TheD

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Figure 15. Infospace.

Figure 16. Geocities stock prices during the alleged dotcom bubble.

stock prices are graphed in Figure 16.

This is an example where we can stop at step (C): we do not need to use RKHS theoryto extrapolate our estimator in order to determine its asymptotic behavior. As seen fromFigure 17, the volatility is a nice bounded function, and any natural extension of this behaviorimplies the divergence of the integral

∫∞ε

xσ2(x)

dx. Hence the price process is a true martingale.

6. Conclusion. Given the price process of a risky asset that follows an SDE under therisk neutral measure of the form

dXt = σ(Xt)dWt,

where W is a standard one dimensional Brownian motion, we provide methods for estimatingthe volatility coefficient σ(x) at the values where it is observed. If the behavior of σ(x) isD

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Figure 17. Geocities. Estimates of σ.

reasonable, we extend this estimator to all of R+ via the technology of reproducing kernelHilbert spaces. Having done this, we are then able to decide on the convergence or divergenceof the integral ∫ ∞

ε

x

σ(x)2dx,

for any ε > 0, which in turn determines whether or not the risky price process is experiencing,or has experienced, a bubble. Unfortunately, the test does not always work, since it dependson the behavior of σ(x).

We illustrated our methodology using data from the alleged Internet dot-com bubble of1998–2001. Not surprisingly, we find that all three eventualities occur: in one case we are ableto confirm the presence of a bubble, in a second case we confirm the lack of a bubble, and ina third case we find that the test is inconclusive. It is our hope that our methodology openssome new avenues for the detection of stock price bubbles in real time.

Acknowledgment. We wish to thank Jean Jacod for his help and advice with the revisionof this paper, and in particular for alerting us to reference [15], and further for indicating howwe could shorten and improve some of the proofs in the paper.

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http://en.wikipedia.org/wiki/Dot-com_bubble

https://wrds-web.wharton.upenn.edu/wrds

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