Detecting asset price bubbles using deep learning Francesca BiaginiLukas GononAndrea MazzonThilo Meyer-Brandis June 21 2024

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Detecting asset price bubbles using deep learning

Francesca Biagini∗Lukas Gonon†Andrea Mazzon‡Thilo Meyer-Brandis ∗

June 21, 2024

Abstract

In this paper we employ deep learning techniques to detect ﬁnancial asset bubbles

by using observed call option prices. The proposed algorithm is widely applicable and

model-independent. We test the accuracy of our methodology in numerical experiments

within a wide range of models and apply it to market data of tech stocks in order to assess

if asset price bubbles are present. Under a given condition on the pricing of call options

under asset price bubbles, we are able to provide a theoretical foundation of our approach

for positive and continuous stochastic asset price processes. When such a condition is not

satisﬁed, we focus on local volatility models. To this purpose, we give a new necessary

and suﬃcient condition for a process with time-dependent local volatility function to be

a strict local martingale.

1 Introduction

The study of methodologies for detecting asset price bubbles has attracted an increasing

interest over the last years in order to prevent, or mitigate, the ﬁnancial distress often following

their burst. A ﬁnancial bubble is deﬁned as the temporary deviation of the market price of

an asset from its fundamental value. In the context of the martingale theory of bubbles (see

among others Cox and Hobson (2005), Loewenstein and Willard (2000), Jarrow et al. (2007),

Jarrow et al. (2010), Jarrow et al. (2011b), Biagini et al. (2014), Protter (2013)), the market

price of a (discounted) asset represents a bubble if it is given by a strict local martingale. In

this framework, detecting an asset price bubble amounts to determine if the stochastic process

modelling the discounted asset price is a true martingale or a strict local martingale.

Our contribution is part of a wide range of works on asset price bubbles detection. The

problem is tackled within the framework of local volatility models in the papers Jarrow (2016),

∗Workgroup Financial and Insurance Mathematics, Department of Mathematics, Ludwig-Maximilians

Universit¨at, Theresienstrasse 39, 80333 Munich, Germany. Emails: biagini@math.lmu.de, meyer-

brandis@math.lmu.de.

†Department of Mathematics, Imperial College London, London, SW7 1NE UK. Email:

l.gonon@imperial.ac.uk

‡Department of Economics, University of Verona, 37129 Verona, Italy. Email: andrea.mazzon@univr.it

arXiv:2210.01726v3 [q-fin.MF] 19 Jun 2024

Jarrow et al. (2011b) and Jarrow et al. (2011c), via volatility estimation techniques. In Fusari

et al. (2020), under the joint assumption of no free lunch with vanishing risk (NFLVR) and

no-dominance, the authors pursue this goal by looking at the diﬀerential pricing between put

and call options, with the motivation that traded call and put options reﬂect price bubbles

diﬀerently. On the other hand, a nonparametric estimator of asset price bubbles with the

only assumption of NFLVR is proposed in Jarrow and Kwok (2021), using a cross section

of European option prices. The approach is based on a nonparametric identiﬁcation of the

state-price distribution and the fundamental value of the asset from option data. In Piiroinen

et al. (2018), the authors introduce a statistical indicator of asset price bubbles based on the

bid and ask market quotes for the prices of put and call options. This methodology is in

particular applied to the detection of asset price bubbles for SABR dynamics. Finally, an

asymptotic expansion of the right wing of the implied volatility smile is used in Jacquier and

Keller-Ressel (2018) to determine the strict local martingale property of the underlying.

In this paper, we propose a new approach that formulates bubble detection as a supervised

learning problem. The learning problem is solved using neural networks that take as inputs

observed call option prices. In this way we provide a theoretical foundation and a model-free

deep learning approach for bubble detection, which is new in the literature. More precisely,

we feed the network with a set of training data whose i-th data point consists of analytical

call option prices for diﬀerent strikes K1, . . . , Knand maturities T1, . . . , Tmwritten on an

underlying process Xiwith known dynamics, together with an indicator specifying if Xiis a

strict local martingale or a true martingale. To assess if a new stock (possibly corresponding to

market data) has a bubble or not, we evaluate the trained neural network at (market) prices of

call options on this underlying for strikes K1, . . . , Knand maturities T1, . . . , Tm. The output

of the network is in (0,1) and indicates the probability that the underlying exhibits a bubble.

Slightly changing the algorithm, we are also able to estimate the size of the bubble.

In this manner the methodology does not attempt to detect a bubble based on the trajectory

of the underlying, but instead uses the information contained in all call option prices available

at a given point in time for diﬀerent maturities and strikes to recover information on the asset’s

price probability distribution. In fact, the (theoretical) observation of call option prices for

all strikes and maturities determines full information about the distribution of the underlying

by the Breeden-Litzenberger formula. The method aims to exploit the information contained

in call option prices in the best possible way to provide an estimate for the probability that

an asset price bubble is present.

Formulating bubble detection as a supervised learning problem leads to a purely sample-based

procedure. The only information used when training the neural network is a collection of call

option prices associated to the process Xiand an indicator stating if these call option prices

correspond to a price process with a bubble or not. Thus, we can arbitrarily combine diﬀerent

models for bubbles and could also enhance synthetic model training data by historical data.

One of the main advantages of our approach is that it does not require the direct estimation of

any parameter or quantity related to the asset price process, and that it is model independent.

In particular, we show via numerical experiments that our method works not only within a

class of models, but also if the network is trained using a certain kind of stochastic processes

(like local volatility models) and tested within another class (for example, stochastic volatility

models). We also test the method with market data associated to assets that have been

suspected to be involved in the new tech bubble (see among others Fusari et al. (2020),

Piiroinen et al. (2018), Kolakowski (2019), Libich et al. (2021), Bercovici (2017), Ozimek

(2017), Serla (2017), Sharma (2017)). According to the neural network, output bubbles were

present in these assets with high probability at certain points in time, and these dates match

retrospectively expected results.

Our motivation for this method is manifold: on the one hand, the price of call options on a

bubbly underlying has an additional term which is added to the usual risk neutral valuation

due to a collateral requirement represented by a constant α∈[0,1], see Cox and Hobson

(2005) and Theorem 2.4 in our paper. Looking at call option prices, it is then theoretically

possible to assess if the underlying has a bubble by identifying the presence of this term.

On the other hand, in the case when the underlying price follows a local volatility model,

a modiﬁed version of Dupire’s formula stated in Theorem 2.3 of Ekstr¨om and Tysk (2012)

permits to theoretically recover the local volatility function, which is crucial to determine if

the process is a strict local martingale or a true martingale, from the observation of call option

prices.

In this way we are able to provide a theoretical foundation for our method. In particular, in

Theorems 2.10 and 2.24 we prove the existence of a sequence of neural networks approximating

a theoretical “bubble detection function” F, under the assumption α < 1 for underlyings given

by continuous and positive stochastic processes and in the general case α∈[0,1] for local

volatility models. The function Fmaps from the space Xof call option prices as functions

of strike and maturity to {0,1}, being 1 if and only if the underlying process is a strict local

martingale. Several steps are required in order to prove Theorems 2.10 and 2.24. First,

we show the existence of the function Fintroduced above and show that it is measurable

with respect to a natural topology on X, see Propositions 2.6 and 2.20. In the case when

α= 1, we consider the class of local volatility models under a fairly general assumption on

the local volatility function. Speciﬁcally, we use some new ﬁndings providing a suﬃcient and

necessary condition for a stochastic process with time dependent local volatility function to

be a strict local martingale. The main result of this analysis is stated in Theorem 2.18 and

is of independent interest, since it is a generalisation of Theorem 8 of Jarrow et al. (2011b).

In Propositions 2.7 and 2.23 we then construct a sequence of measurable functions (Fn)n≥1

approximating Fpointwise, where Fn:Rn→[0,1] takes the prices of a call option with

ﬁxed underlying for ndiﬀerent strikes and maturities and outputs the likelihood that the

underlying has a bubble according to the observation of these prices. These propositions are

then used together with the universal approximation theorem, see Hornik (1991), to prove

Theorems 2.10 and 2.24, respectively.

The paper is structured as follows. Section 2 is devoted to the theoretical foundation of our

approach. In particular, in Section 2.1 we brieﬂy recall the martingale theory of asset price

bubbles, in Section 2.2 we introduce call option pricing in presence of ﬁnancial asset bubbles,

and in 2.3 we prove existence of the test function Fand of its neural network approximation

when α < 1, when the asset price is given by a continuous and positive stochastic process.

On the other hand, in Section 2.4 we focus on local volatility models to cover the general

case α∈[0,1]: in Section 2.4.1 we provide suﬃcient and necessary conditions for the strict

local martingale property of local volatility models, whereas in Section 2.4.2 we give results

analogous to the ones in Section 2.3 under this new setting.

In Section 3 we test the performance of our methodology via numerical experiments. Speciﬁ-

cally, we introduce our methodology in Section 3.1 and in Section 3.2 we both feed and test the

network within local volatility models, with diﬀerent, randomly chosen parameters, whereas

in Section 3.3 we feed the network with data coming from stochastic volatility models and test

it with local volatility models, and vice-versa. In Section 3.4 we comment on the choice of the

network architecture. In Section 4 we apply our approach to the analysis of four stocks that

have been claimed to be aﬀected by a bubble in recent years, namely Nvidia, Meta, GameStop

and Tesla. In Section 5 we present some conclusions.

2 Theoretical foundation

2.1 Asset price bubbles

In the literature there have been many approaches proposed for the mathematical modelization

of asset price bubbles.

From an economic point of view, the main challenge consists in explaining how bubbles are

generated at the microeconomic level by the interaction of market participants; see Tirole

(1982), Harrison and Kreps (1978), DeLong et al. (1990), Scheinkman and Xiong (2003),

Abreu and Brunnermeier (2003), F¨ollmer et al. (2005) and the references therein. From a

mathematical point of view, asset price bubbles are mainly studied by using the local martin-

gale framework: Loewenstein and Willard (2000), Cox and Hobson (2005), Jarrow and Madan

(2000), Jarrow et al. (2007), Jarrow et al. (2010), Jarrow et al. (2011a), Jarrow and Protter

(2009), Jarrow and Protter (2011), Jarrow and Protter (2013), Pal and Protter (2010), Kar-

daras et al. (2015), Biagini et al. (2014). Another deﬁnition has been proposed in Herdegen

and Schweizer (2016), where the fundamental value is assumed to be the superreplication price

of the asset. In Jarrow et al. (2012), Biagini and Nedelcu (2015) and Biagini et al. (2018)

constructive models are proposed to take in account triggering factors for bubble formation

and bubble bursting. For a survey, we refer to Protter (2013).

Here we follow the approach of Jarrow and Protter (2009), Jarrow et al. (2011a). The notion

of an asset price bubble has two ingredients: the observed market price Sof a given ﬁnancial

asset and the asset’s fundamental value. Assume discounting equal 1. Given an equivalent

martingale measure Qfor S, the fundamental value SQis usually deﬁned as the expected

sum of future dividends under Q. The bubble perceived under Qis deﬁned as the diﬀerence

between the two:

βQ=S−SQ.

In this setting, we have a bubble if and only if Sis a strict local martingale under Q. In

the sequel, we present a new method to detect ﬁnancial asset prices’ bubbles by testing for

strict local martingale properties based on observations of call option prices and by applying

machine learning techniques.

2.2 European call options under asset price bubbles

Let (Ω, P, F) be a probability space endowed with a ﬁltration F= (Ft)t≥0satisfying the usual

hypothesis. It is well known that the risk-neutral valuation of European options when the

underlying asset price has a bubble (i.e., when it is a strict local martingale) does not satisfy

the put-call parity, see for example Jarrow et al. (2010). An alternative way to price call

options for a bubbly underlying, also supported by market data, has been proposed in Cox

and Hobson (2005).

We start with the following deﬁnition.

Deﬁnition 2.1. Introduce an underlying given by a continuous, positive stochastic process

X= (X)t≥0, and ﬁx a pricing measure Q∼Punder which Xis a local martingale. The

price Cα,X (T, K)at time t= 0 of a call option of maturity Tand strike Kwritten on Xwith

collateral constant α∈[0,1] is given by

Cα,X (T, K) = EQ[(XT−K)+] + αm(T),(1)

where

m(T) := X0−EQ[XT].(2)

The term m(T) in (2) is called the martingale defect of X. Note that m(T)>0 if Xis a

strict local martingale bounded from below and m(T) = 0 if Xis a martingale.

Remark 2.2. Note that if for t∈[0, T ]we set

Cα,X

t(T, K) := EQ[(XT−K)+|Ft] + αmt(T)

for mt(T) = Xt−EQ[XT|Ft], then Cα,X

t(T, K),t∈[0, T ], is a non-negative local martingale,

which equals (XT−K)+at ﬁnal time. Hence Cα,X

t(T, K)represents an arbitrage-free price

at time tfor the call option (XT−K)+.

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DetectingassetpricebubblesusingdeeplearningFrancescaBiagini∗LukasGonon†AndreaMazzon‡ThiloMeyer-Brandis∗June21,2024AbstractInthispaperweemploydeeplearningtechniquestodetectfinancialassetbubblesbyusingobservedcalloptionprices.Theproposedalgorithmiswidelyapplicableandmodel-independent.Wetesttheaccuracy...

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Detecting asset price bubbles using deep learning Francesca BiaginiLukas GononAndrea MazzonThilo Meyer-Brandis June 21 2024

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