IMES DISCUSSION PAPER SERIESA Review of New Developments in Finance with Deep Learning:
Deep Hedging and Deep Calibration
Yuji Shinozaki
Discussion Paper No. 2024-E-2

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
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IMES Discussion Paper Series 2024-E-2
April 2024

A Review of New Developments in Finance with Deep Learning:
Deep Hedging and Deep Calibration
Yuji Shinozaki*
Abstract
The application of machine learning to the field of finance has recently become thesubject of active discussions. In particular, the deep learning is expected tosignificantly advance the techniques of hedging and calibration. As these twotechniques play a central role in financial engineering and mathematical finance, theapplication to them attracts attentions of both practitioners and researchers. Deep
hedging, which applies deep learning to hedging, is expected to make it possible toanalyze how factors such as transaction costs affect hedging strategies. Since theimpact of these factors was difficult to be assessed quantitatively due to thecomputational costs, deep hedging opens possibilities not only for refining andautomating hedging operations of derivatives but also for broader applications inrisk management. Deep calibration, which applies deep learning to calibration, is
expected to make the parameter optimization calculation, which is an essential
procedure in derivative pricing and risk management, faster and more stable. This
paper provides an overview of the existing literature and suggests future research
directions from both practical and academic perspectives. Specifically, the paper
shows the implications of deep learning to existing theoretical frameworks andpractical motivations in finance and identifies potential future developments thatdeep learning can bring about and the practical challenges.
Keywords: Financial engineering; Mathematical finance; Derivatives; Hedging;
Calibration; Numerical optimization
JEL classification: C63, G12, G13* Deputy Director, Institute for Monetary and Economic Studies, Bank of Japan (currently,
Associate Professor, Musashino University, E-mail:y-shino@musashino-u.ac.jp)
The author appreciates comments from and discussions with Yasufumi Osajima (SMBC Nikko
Securities), Yoshihiro Tawada (MUFG Securities EMEA), Toshiyuki Nakayama (MUFG Bank),
and Syoiti Ninomiya (Tokyo Institute of Technology). The author also thanks FORTE Science
Communications (https://www.forte-science.co.jp/) for English language editing. The views
expressed in this paper are those of the author and do not necessarily reflect the official views of
the Bank of Japan.

Contents
1
2

Introduction

1

Derivative pricing and risk management: Hedging and calibration
2.1 Development history of derivative pricing and risk management . . . . . .
2.2 Formulation of hedging: Convex risk minimization problem . . . . . . . .
2.3 Formulation of calibration: Estimation of model parameters in risk-neutral
valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3

Deep learning
3.1 Background of success . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Neural network notation . . . . . . . . . . . . . . . . . . . . . . . . . . .

14
15
20

4

Deep hedging
4.1 Algorithm proposed in Buehler et al. [2019a] . . . . . . . . . . . . . . . .
4.2 Research trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21
21
26
29

5

Deep calibration
5.1 Algorithm proposed in Hernandez [2017] . . . . . . . . . . . . . . . . . .
5.2 Research trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Practical advantages and challenges . . . . . . . . . . . . . . . . . . . . .

31
32
33
34

6

Concluding remarks

3
4
7

35

1 Introduction
In recent years, deep learning has been applied in a variety of ﬁelds, including image andnatural language processing. The ﬁeld of ﬁnance is no exception. Indeed, many innovative
methods using deep learning have been proposed to solve problems in ﬁnancial engineeringand mathematical ﬁnance. In particular, research on hedging and calibration, which are crucial in practical applications, has garnered attention both theoretically and practically. Given
this trend, this paper summarizes the application of deep learning in the ﬁeld of ﬁnance,with a focus on research related to hedging and calibration. It also discusses possible future
developments deep learning might bring to the ﬁeld of ﬁnance.
One of the main objectives of ﬁnance theory is to guide decision-making of market participants under uncertainty such as price ﬂuctuation risks of ﬁnancial assets. Among the prominent concerns in the ﬁeld of ﬁnance is how to price derivatives (ﬁnancial products whose
payoﬀs depend on the future prices of underlying assets such as stocks or foreign exchange)
and manage their risk. Financial engineering and mathematical ﬁnance, which mainly focus
on these issues, have developed under strong inﬂuence of practical applicability. In particular, the problem of hedging, which addresses how to mitigate the future price ﬂuctuation
risks of ﬁnancial assets held, and calibration, which is the process of adjusting model parameters to ensure consistency between observed market prices and theoretical model prices, arecentral research themes in ﬁnancial engineering and mathematical ﬁnance.
Deep learning is expected to signiﬁcantly advance techniques related to hedging and calibration. First, with regard to hedging, deep learning has the potential to derive hedging
strategies that take into account the impact of transaction costs and other actual market frictions. While numerous theoretical studies have been conducted under the traditional framework of risk-neutral valuation, in practice there are various constraints that prevent the full
implementation of hedging strategies that are derived theoretically. Recent approaches using
deep learning for hedging aim to overcome these constraints, potentially leading to more reﬁned hedging strategies and to the automation of hedging transactions. Second, with regard
to calibration, it has been pointed out that deep learning has the potential to signiﬁcantly
accelerate the process of parameter adjustments, which would lead to faster and more stablevaluation of derivatives. In this paper, we summarize such prospects in the ﬁeld of ﬁnanceand clarify the implications of deep learning to existing theoretical frameworks and practicalmotivations.
The reminder of the paper is organized as follows: First, in section 2, we summarize
the concepts of ﬁnancial engineering and mathematical ﬁnance related to the pricing andrisk management of derivatives, and formulate the problems of hedging and calibration. The
application of deep learning to hedging, on which we focus in this paper, is formulated undera framework referred to as convex risk minimization. Since this framework diﬀers from theconventional risk-neutral valuation framework, it is beneﬁcial to understand the relationshipbetween these two frameworks in order to understand the current trend of research on theapplication of deep learning to ﬁnancial engineering and mathematical ﬁnance. With this in1mind, we summarize the history of the development of each framework and subsequently
formulate the hedging and calibration problems.
In section 3, we provide an overview of deep learning. A building block of deep learning is
the neural network, a system of functions in which linear and non-linear functions are alternately composed. Deep learning is one of machine learning methods that involve deep layersof neural networks and has achieved remarkable results in various ﬁelds in recent years. To
provide an overview of these recent developments, we discuss the positioning of deep learning within the broader machine learning ﬁeld and explain its historical development. We
also provide brief explanations of the techniques used for eﬃciently training deep learningmodels and of the theoretical characteristics of neural networks. Subsequently, the notation
adopted in the following sections is introduced.
In section 4, we review the literature on deep hedging, a technique that applies deep learning to the problem of hedging. As explained in section 2, the conventional approach incurrent practice is built on the risk-neutral valuation framework, where hedging strategies
are derived based on the concept of the replication of derivatives. However, this approachhas diﬃculty in quantitatively incorporating real-world market frictions, such as transaction
costs, into pricing and hedging strategies because it assumes hypothetical frictionless markets. An alternative to the replication-based approach is the framework of convex risk minimization. In this approach, the agent seeks to minimize the risk associated with proﬁt andloss taking realistic elements such as transaction costs and hedging errors into consideration.
Deep hedging employs deep learning to perform the optimization calculations in the convex risk minimization framework, making it practically feasible to derive eﬀective hedging
strategies. Speciﬁcally, it derives optimal hedging strategies by describing a hedging strategy
using a neural network and training its parameters to minimize loss risk (measured based ona convex risk measure) under given future market scenarios. Note that deep hedging is a
general method to derive portfolio management strategies that minimize proﬁt and loss risk
and thus has potential applications beyond issues related to derivatives, such as asset liability management. In section 4, after explaining the deep hedging framework, we summarize
current research trends and the outlook for potential future developments.
In section 5, we review the literature on deep calibration, a technique that applies deeplearning to calibration in derivative pricing. Calibration is a calculation to adjust model
parameters so that the model prices (the prices of the instruments based on risk-neutral valuation) match the market prices of highly liquid ﬁnancial instruments. Thus, it is an important
process to ensure that a model does not violate the no-arbitrage condition among instruments, that is, to ensure that the pricing of illiquid complex ﬁnancial instruments is consistent with the prices of liquid instruments. Conventionally, iterative optimization methods
such as the Newton–Raphson method have been used in calibration; however, these methods come with computational challenges, especially when calibrating numerous ﬁnancial
products. In contrast, deep calibration bypasses iterative calculations by pre-training the relationship between model parameters and model prices using neural networks. Some studies
have reported that deep calibration accelerates and stabilizes derivative price calculations,2and, accordingly, practitioners have been paying close attention to it, as it may open thepossibility of using computationally expensive pricing models. Deep calibration essentiallyreplaces the optimization calculations within the risk-neutral valuation framework with deeplearning techniques, aiming to improve the computational techniques within current derivative practices. In section 5, we describe the deep calibration framework, summarize current
research trends, and outline the practical advantages and challenges of the method.
To summarize, deep hedging focuses on the convex risk minimization framework, while
deep calibration applies deep learning within the risk-neutral valuation framework. The factthat these two techniques are formulated under the two distinct frameworks indicates that
there is a following signiﬁcant diﬀerence in terms of their impact on practice. Deep hedging
has the potential to signiﬁcantly transform derivative practices in the long run, although
it requires further theoretical and practical studies as there are hurdles to use it in ﬁnancepractices, such as how to measure loss risk and how to generate future market scenarios.
On the other hand, deep calibration primarily targets improving computational techniques
within the risk-neutral valuation framework. Therefore, it is expected to provide promisingsolutions to speciﬁc challenges in current derivative practices.
This paper can be read in various manners depending on the readers’ knowledge and interest. Section 2 summarizes the background of the ﬁnance issues addressed in sections 4 and 5.
Thus, the readers familiar with ﬁnancial engineering and derivative practices may choose to
skip it. Similarly, section 3 provides a basic discussion of the deep learning techniques usedin sections 4 and 5, and thus the readers familiar with deep learning may ﬁnd it unnecessary
to read. Sections 4 and 5 can be read independently, so that the readers can select only thecontent that matches their interest.

2 Derivative pricing and risk management: Hedging andcalibration
In this section, we summarize the conceptual framework of ﬁnancial engineering andmathematical ﬁnance related to derivative pricing and risk management, and then formulate the problems of hedging and calibration.
Historically, the conventional theories of ﬁnancial engineering and mathematical ﬁnancehave evolved around the framework of risk-neutral valuation. On the other hand, deep hedging, one of main focuses of this paper, is formulated based on a framework known as convex
risk minimization. To understand the trend of applying deep learning to ﬁnancial engineering and mathematical ﬁnance, it is important to understand the relationship between these
two frameworks.
To this end, in this section, we ﬁrst summarize the historical development of the frameworks of risk-neutral valuation and convex risk minimization, keeping in mind the issues ofderivative pricing and risk management. We subsequently address the speciﬁc issues related
to convex risk minimization and formulate the problem of deep hedging which is discussed3

in section 4. Using the Black–Scholes model as an example, we then outline the methods
based on risk-neutral valuation and formulate the calibration problems discussed later in section 5. In the present section, our principal aim is to clarify the practical motivations behind
hedging and calibration. Therefore, we introduce each concept in an inductive way, starting from practical motivations, taking an approach that diﬀers from the usual explanations
provided in ﬁnancial engineering and mathematical ﬁnance.
For standard explanations of topics in mathematical ﬁnance, such as the framework of
risk-neutral valuation, readers may refer to introductory books such as Björk [2009], Shreve
et al. [2004], and Baxter et al. [1996]. For literature featuring practical challenges, books
such as Andersen and Piterbarg [2010] and Hull [2014] are also useful references. Föllmer
and Schied [2016] provides a detailed explanation of the theory of incomplete markets, inwhich the framework of convex risk minimization is covered.

2.1 Development history of derivative pricing and risk management
2.1.1 Framework of risk-neutral valuation
Financial engineering and mathematical ﬁnance, whose main topics include derivative
pricing and risk management, have evolved around two concepts: replication and no-arbitrage.
Replication refers to generating cash ﬂows identical to a derivative by appropriately trading
the underlying assets. No-arbitrage means that one cannot proﬁt without initial capital andwithout bearing any risk of making losses. If the market satisﬁes the no-arbitrage condition and a derivative is replicable, then the fair value of that derivative is equivalent to theinitial cost required for replication. This is a fundamental principle in ﬁnancial engineeringand mathematical ﬁnance.1 From a practical standpoint, the equivalence between replicability and hedging of derivatives is crucial. Namely, if a derivative is replicable, taking an
opposite position through replication can fully eliminate the uncertainty in proﬁt and losses(P&Ls). Consequently, the trading strategy for replication (replication strategy) is identicalto a hedging strategy for the derivative.
The principles of replication and no-arbitrage were theoretically developed within theframework of risk-neutral valuation. In this framework, the fair value of a derivative equals
the expected value of future cash ﬂows under a probability measure called the risk-neutral
measure, discounted by the risk-free interest rate. The following is an intuitive explanationof why derivatives with uncertain future cash ﬂows can be priced by discounting them at therisk-free interest rate. First, for a derivative that can be replicated or hedged, we can eliminate the risks of underlying price ﬂuctuations and construct a risk-free portfolio that consistsof the derivative and the hedging instrument. Moreover, if the no-arbitrage condition holds,
1

For instance, if the price of a derivative exceeds the initial cost required to replicate it, selling the derivative
would enable one to make proﬁt without bearing any risk through replication, thereby violating the no-arbitrage
condition. Similarly, it can be demonstrated that the opposite scenario also leads to the violation of the noarbitrage condition. Hence, it becomes evident that the price of the derivative must be equal to the initial costrequired for replication.

4

the return rate on the risk-free portfolio must be equal to the risk-free interest rate. Thus,
based on these discussions of replication and no-arbitrage, derivatives can be priced underrisk-neutral valuation.
In the seminal work on the Black–Scholes model (Black and Scholes [1973]), the option
pricing formula was derived in a heuristic way based on the ideas of replication and noarbitrage. In section 2.3, we formulate the framework of risk-neutral valuation from thisperspective. Following the emergence of the Black–Scholes model, Harrison and Kreps[1979] and Harrison and Pliska [1981] established a rigorous mathematical proof for the factthat an arbitrage-free market admits risk-neutral valuation, speciﬁcally, that a risk-neutral
probability measure exists when the no-arbitrage condition is satisﬁed.
There are two reasons why the theoretical framework of risk-neutral valuation based on
the principles of no-arbitrage and replication has been accepted by practitioners in the ﬁnancial industry and developed further. First, in the risk-neutral valuation framework, one
can obtain replication strategies explicitly based on rigorous mathematical tools such as Itô
calculus. For instance, models in mathematical ﬁnance, including the Black–Scholes model,
describe asset prices using stochastic diﬀerential equations,2 which is the theoretical basis
for the techniques used in practice. Second, there is no need to estimate expected returns
on the underlying assets. The expected returns on ﬁnancial assets are notoriously volatileand challenging to estimate.3 Therefore, derivative models requiring estimates of expected
growth rates of underlying asset prices were impractical for pricing and risk management. In
contrast, within the risk-neutral valuation framework, the estimation of expected returns is
unnecessary, because observable risk-free interest rates are used for pricing. As elaborated
in section 2.3, complex derivative pricing relies on observable market prices of highly liquid
ﬁnancial instruments. This implies that with risk-neutral valuation, one can derive pricing
and hedging strategies based solely on currently observable information (see Table 1 for acomparison with deep hedging, which will be discussed in section 4). This is a signiﬁcantadvantage in option pricing and risk management.

2.1.2 Framework of convex risk minimization
As mentioned above, while the framework of risk-neutral valuation has signiﬁcant advantages, there are also practical diﬃculties. Most theoretical models, including the Black–
Scholes model, are based on the assumption of an ideal ﬁnancial market, where no market
frictions exist (e.g., risk-free assets can be borrowed and lent freely, and the underlying
asset can be bought or sold without any transaction cost). However, in a real market, transaction costs exist, and various constraints such as short-selling constraints and borrowing
constraints are present. Particularly, after the ﬁnancial crisis of 2007-08, the impact of counterparty credit risks and funding costs on transactions has become signiﬁcant, making valu2

For instance, in the Black–Scholes model, it is assumed that the price of the underlying asset follows astochastic process called geometric Brownian motion. Under this stochastic process, the price of the underlying
asset at a certain point in time follows a log-normal distribution.
3
For example, see Cochrane [2011] and Martin [2016].

5

ation adjustments like xVAs (x-Valuation Adjustments) crucial.4 These market frictions andassociated transaction terms aﬀect optimal hedging strategies and also inﬂuence the pricing
of derivatives through their impact on hedging costs.
In situations where market frictions exist, replication and hedging errors are inevitable
as perfect replication is not possible, which induces uncertainty into the total P&L after
hedging. Therefore, it is necessary to consider frameworks other than risk-neutral valuation,
leading to numerous attempts to develop theories for incomplete markets, where factors suchas transaction costs prevent perfect replication. Leland [1985] extended the Black–Scholesmodel to propose methods for modifying replication strategies according to the amount oftransaction costs and constraints such as trading frequency, demonstrating that the fair valueof derivatives remains within a certain range even when market frictions exist. Building on
this, Hodges and Neuberger [1989] incorporated the utility functions to formulate the optimality of hedging strategies, quantifying the uncertainty in total P&Ls after hedging.5 In
addition, Davis et al. [1993] reﬁned the work of Hodges and Neuberger [1989] and formulated a setting consistent with the Black–Scholes model, and many studies have examined
price behavior under this setting (e.g., Shreve [1995], Rogers [2004], Whalley and Wilmott[1997], and Barles and Soner [1998]). More recently, Xu [2006] and Ilhan et al. [2009] introduced the idea of minimizing a convex risk measure instead of maximizing utility. Thisapproach oﬀers the advantage of providing a more realistic representation of trader preferences and a clearer relationship between price and hedging strategies, as described in section
2.3.1.
This paper employs the formulation by Hodges and Neuberger [1989], which deals withthe framework of convex risk minimization. Speciﬁcally, it aims to quantify the uncertainty
in total P&Ls after hedging using convex risk measures and seeks hedging strategies thatminimize this risk. For instance, when adopting expected shortfall (ES) as a convex risk
measure, the problem becomes that of minimizing the area under the distribution of total
losses (i.e., minimizing the potential for signiﬁcant losses). In this framework, the derivative
price can be set as the price at which an agent has the same expected utility (or risk measure)when trading the derivative and when not trading it, known as the indiﬀerence price.
Despite certain theoretical advancements, the derivation of hedging strategies considering
4xVAs are adjustments to the market price of derivatives that take into account various costs associated with
derivative trading, such as counterparty credit risks and funding costs.
5
In the context of incomplete markets, the primary formulations for the fair pricing of derivatives include
approaches based on utility functions, dominant replication, and the minimal equivalent martingale measure(MEMM). Dominant replication involves trading the underlying asset in such a way that the total P&Ls atmaturity are non-negative regardless of the outcome. The fundamental idea is to set the fair value of thederivative as the minimum cost required to achieve dominant replication. However, strategies for dominant
replication often become trivial (for instance, holding one unit of the underlying asset from the start to replicate
a single call option), making this approach impractical due to resulting high fair prices (see Kramkov [1996]
and Föllmer and Kabanov [1997]). The MEMM approach involves searching for a measure within the set of
equivalent martingale measures (or risk-neutral probability measures) that minimizes hedging errors. However,challenges arise, such as the lack of practically feasible hedging strategies under the identiﬁed measure (seeFöllmer and Schweizer [1991] and Miyahara [2001]). For a survey on the impact of transaction costs onderivatives and other ﬁnancial assets, Muhle-Karbe et al. [2017] is a valuable reference.

6

transaction costs requires complex optimization calculations, preventing the practical application of incomplete market theories for a long time. Thus, in derivative practices relying
on risk-neutral valuation as a benchmark, it is customary to make adjustments to cover errors incurred from market frictions and numerical inaccuracies in model calculations. Since
most of these adjustment methods rely on simpliﬁed formulas and trader experience, moreobjective methods based on incomplete market models have long been called for.
In this historical context, deep hedging, a hedging technique which utilizes deep learning, has been rapidly progressing from both theoretical and practical perspectives in recent
years. Deep hedging, proposed by Buehler et al. [2019a], performs optimization calculations
under the framework of an incomplete market such as convex risk minimization using deeplearning, allowing for pricing and risk management that considers transaction costs and otherfactors.
There is a signiﬁcant diﬀerence between deep hedging based on convex risk minimizationand hedging derived from risk-neutral valuation. In the latter approach, one does not need to
specify the traders’ risk preferences explicitly, because under the assumption that a complete
replication is possible, uncertainty in payoﬀs is eliminated and thus traders’ risk preferences
toward uncertainty do not matter. As a result, the hedge strategy is uniquely determinedirrespective of trader preference and utility function. On the other hand, deep hedging solves
the convex risk minimization problem by explicitly considering trader utility function (orrisk measure that quantiﬁes loss risk). As mentioned above, it is necessary to specify how
traders evaluate uncertainty regarding possible losses in the case where complete replications
and hedges are not possible. One may regard deep hedging as a revival of conventional asset
pricing theory in the sense that it derives optimal trading strategies by taking the traders’utility function (risk preference) into consideration.6

2.2 Formulation of hedging: Convex risk minimization problem
In this section, we formulate the hedging problem for derivatives as a convex risk minimization problem. In derivative hedging, the objective is to mitigate the uncertainty of thetotal P&L at maturity by trading hedging instruments (assets used for hedging) up to thederivative’s expiration. Here, we formulate the P&L at maturity as a random variable. The
convex risk minimization problem aims to minimize the risk of the P&L at maturity, whichis quantiﬁed by a convex risk measure. Subsequently, following Buehler et al. [2019a], we
formulate this concept in a discrete-time model.

6

Among the foundational results in modern ﬁnance theory, Markowitz [1952] portfolio selection theory
and the Capital Asset Pricing Model (CAPM) by Sharpe [1964]; Lintner [1965]; Mossin [1966] stand out.
Markowitz’s theory analyzes investors who optimally choose their portfolio based on mean-variance utility,
considering the expected return (mean) and risk (variance) of portfolios. CAPM identiﬁes the determinants ofrisk premiums for individual risky assets under the assumption of a representative investor maximizing expectedutility. Thus, research that assumes investors maximize expected utility (or minimize expected loss) has becomea primary approach in ﬁnance, providing insights into optimal investment behavior and asset pricing.

72.2.1 P&L at maturity
Let t0 = 0 be the initial time and T be the maturity of a derivative. We assume thatmarket transactions are possible at times t0 < · · · < tk < · · · < tn = T . Let (Ω, F , P)
be a probability space and {Fk }k=0,...,n be a discrete ﬁltration. For simplicity, we consider
only one underlying asset, and let its asset price {S k }k=0,...,n be an Fk -adapted real-valued
stochastic process. Furthermore, let the payoﬀ ZT of the derivative be an Fn -measurable
real-valued random variable. For example, for a European call option with a strike price K,
ZT = max{S n − K, 0}.7
For simplicity, we consider a setting where only the underlying asset is used as a hedging
instrument.8 In this case, a hedging strategy is determined by the holding amounts of theunderlying asset at each time, which is an Fk -adapted real-valued stochastic process δ ={δk }k=0,1,...,n . Moreover, the transaction costs determined by the trading volume of the hedging
instrument at each time tk are denoted as ck (δk − δk−1 ).9 For instance, if transaction costs areproportional to trading volume, with a constant percentage ε of the transaction amount, thenck (δk − δk−1 ) = ε|δk − δk−1 |S k . Let p0 denote the current price of this derivative.
Given this setup, the P&L at maturity for an investor with a short position in the derivativeis as follows:PL(ZT ,p0 ,δ)= p 0 − ZT +n−1
Xδk (S k+1 − S k ) −

k=0

n
X

ck (δk − δk−1 ) .

(1)

k=0

The ﬁrst and second terms on the right-hand side represent the proﬁt from selling thederivative at the initial time t0 and the payoﬀ that must be paid at maturity T , respectively.
The third term on the right-hand side represents the cumulative hedging proﬁt or loss resulting from the hedging transactions. The ﬁnal term on the right-hand side represents the
cumulative transaction costs incurred for adjusting the hedge position at each possible trading point.10

2.2.2 Risk measurement of P&L
The P&L at maturity, formulated in the previous section, is a random variable. A realvalued function that maps this random variable to a measurement of risk is simply calleda risk measure. Particularly in mathematical ﬁnance, convex risk measures that satisfy the
The subsequent discussions also hold even when ZT is an Fn -measurable random variable, making themsuitable for cases of payoﬀs depending on other than S n , including path-dependent payoﬀ cases. The framework is also easily extendable to scenarios with multiple underlying assets.
8
The approach can also be extended to settings where other assets serve as hedge instruments.
9Here, we set δ−1 = δn = 0, so that transaction costs for purchasing δ0 units of the hedge instrument at the
beginning of the hedge operation and selling δn−1 units at the maturity are reﬂected in the P&L.
10
In Buehler et al. [2019a], the notation PLT (ZT , p0 , δ) is used for the P&L at maturity. In this paper, since
the P&L at maturity is formulated as a random variable, we denote the P&L at maturity as PL(ZT ,p0 ,δ) (ω) forω ∈ Ω. Note that the arguments to be modiﬁed in subsequent discussions are (ZT , p0 , δ), and given that only
the P&L at maturity are considered, we adopt the notation PL(ZT ,p0 ,δ) .
78

following deﬁnition are frequently considered. For a more detailed explanation of convex
risk measures, see Föllmer and Schied [2016].
Deﬁnition 1 (Convex risk measure). Let χ be the space of all random variables on Ω. A
function ρ : χ → R is called a convex risk measure if it satisﬁes the following conditions:
1. Monotonic Decrease: For X1 , X2 ∈ χ, if X1 ≥ X2 , then ρ(X1 ) ≤ ρ(X2 ).
2. Convexity: For α ∈ [0, 1] and X1 , X2 ∈ χ, ρ(αX1 + (1 − α)X2 ) ≤ αρ(X1 ) + (1 − α)ρ(X2 ).
3. Cash Invariance: For X ∈ χ and c ∈ R, ρ(X + c) = ρ(X) − c.
The intuitive meaning of each condition for a convex risk measure is as follows: Firstly, amonotonic decrease implies that if the P&L of portfolio 1, X1 , is greater than or equal to the
P&L of portfolio 2, X2 , then the risk measure of portfolio 1 will be less than or equal to that of
portfolio 2 (indicating that portfolio 1 is less risky). Secondly, convexity formulates what is
known as the diversiﬁcation eﬀect, stating that the risk of a portfolio formed by diversifying
between two portfolios is less than or equal to a weighted average of the individual portfolio
risks. Lastly, cash invariance means that a certain payoﬀ c will reduce the risk measure by
the same amount. These properties are deemed essential for a risk measure, and convex risk
measures, including ES, have long been a topic of study in ﬁnance; see Föllmer and Schied[2016].11

2.2.3 Convex risk minimization problem
Given the above setting, the convex risk minimization problem is formulated as follows.
Deﬁnition 2 (Convex risk minimization problem). Given a convex risk measure ρ, the convex
risk minimization problem for hedging a derivative with payoﬀ ZT using a hedging instrument S is deﬁned as the problem of ﬁnding a hedging strategy δ∗ = {δ∗k }nk=0 that solves thefollowing minimization problem:hiδ∗ = arg min ρ PL(ZT ,p0 ,δ) .
(2)
δ∈H̄

Here, the minimum is sought over the appropriate set of hedging strategies H̄ (a subset ofthe space of real-valued stochastic processes).
Remark 1 (Exploration space of hedging strategies). In Deﬁnition 2, it was assumed
that 

(ZT ,p0 ,δ)∗within the appropriate set of hedging strategies H̄, there exists a δ that minimizes ρ PL.
12
However, the conditions for H̄ to satisfy this assumption are not trivial. Therefore, in
The expected shortfall ESα (X) is deﬁned as ESα (X) = −E [X | X ≤ VaRα (X)]. It is essential to note thatwhile Value at Risk (VaR) is a commonly used risk measure in ﬁnancial practice, it does not satisfy convexity(i.e., it is not a convex risk measure). For research examining the properties of risk measures, refer to works
such as Föllmer and Leukert [2000] and Föllmer and Schied [2002].
12
Lemma 2.1 in Godin [2016] provides the conditions in the case of the ES risk measure.
119Buehler et al. [2019a], the authors deﬁned the set of selectable hedging strategiesH with(ZT ,p0 ,δ)out assuming the existence of a minimum and discuss the inﬁmum inf H ρ PL
ratherthan the minimum (refer tosection 4.1). In formulating deep hedging, the existence of a δ∗that minimizes ρ PL(ZT ,p0 ,δ) is not an issue. Hence, in this paper, we place strong assumptions on H̄ to explicitly formulate the convex risk minimization problem.
2.2.4 Utility-indiﬀerence price
Under the framework of convex risk minimization, it is common to price derivatives asutility-indiﬀerence prices
deﬁned below. Here, the minimized risk quantity is denoted by(ZT ,p0 ,δ)π(ZT , p0 ) = minδ∈H̄ ρ PL.
Deﬁnition 3 (Utility-indiﬀerence price). The utility-indiﬀerence price p∗0 for a derivativewith payoﬀ function ZT satisﬁesπ(ZT , p∗0 ) = π(0, 0).
It is thus the case that the utility-indiﬀerence price is a price such that the risk quantity
when one does not trade the derivative at all equals the risk quantity when one trades thederivative with an optimal hedging trade. Further, using equation (1) and the cash invarianceof the convex risk measure (Condition 3 of Deﬁnition 1), we can derive


n−1
n−1
X
X



π(ZT , p∗0 ) = min ρ  p∗0 − ZT +δk (S k+1 − S k ) −ck (δk − δk−1 )
δ∈H̄

k=0

k=0



n−1
n−1
X
X



= −p∗0 + min ρ −ZT +δk (S k+1 − S k ) −ck (δk − δk−1 )
δ∈H̄

k=0

k=0

= −p∗0 + π(ZT , 0).
From Deﬁnition 3, it follows thatp∗0 = π(ZT , 0) − π(0, 0).

2.3 Formulation of calibration: Estimation of model parameters in riskneutral valuation
Calibration refers to the process of adjusting model parameters to ensure consistency between observed market prices and theoretical prices based on the model. In general derivatives practice, after calibrating the model parameters to align with highly liquid ﬁnancial
instruments whose prices are considered to be informative, models are used for pricing andderiving hedging strategies for complex ﬁnancial instruments that are not actively tradedin the market or have low liquidity. Therefore, the development of techniques that achieve
rapid and stable calibration is crucial for derivatives practice and risk management. In this10

paper, we focus on calibration within the framework of risk-neutral valuation. Speciﬁcally,
we address the problem of adjusting model parameters to ensure that the model prices (therisk-neutral price) are consistent with market prices. In the following sections, we ﬁrst discuss the intuitive concept of risk-neutral valuation, considering its relationship with convex
risk minimization. Subsequently, we formulate the calibration problem.

2.3.1 Derivation of risk-neutral valuation
In mathematical ﬁnance, risk-neutral valuation employs continuous-time models, often
under the assumption of an hypothetical frictionless market. This ideal setup postulates thattransactions, including short selling, can be executed at any time without incurring trading
costs. To distinguish the P&L in the discrete-time model, PL(ZT ,p0 ,δ) , discussed in the previous
T ,pt ,δ)f (Zsection, we denote the P&L in the continuous model as PL. Here, for t ∈ [0, T ],t
(ZT ,pt ,δ)13ft
PL
represents the P&L from time t to T .
In this notation, replication is formulated as a pair consisting of a risk-neutral price p(RN)t(RN)
e
and a corresponding replication strategy {δu }u∈[0,T ] to hedge the uncertainty associated withthe derivative’s payoﬀ that satisﬁes the following condition:!
,eδ(RN) )(ZT ,p(RN)tf= 0 = 1 for t ∈ [0, T ].
(3)
P PLt
In other words, if replication is feasible, the uncertainty associated with the derivative’s
payoﬀ can be eliminated through hedging, ensuring that the P&L at maturity T can be almost
surely guaranteed to be zero. Thus, replication serves as a method to eliminate loss risk
related to the derivative’s payoﬀ without explicitly assuming a risk measure.
The pioneering work by Black and Scholes [1973] and Merton [1973] utilized Itô calculus
to formulate a speciﬁc replication technique for derivatives. The derivation approach proposed by them diﬀers somewhat from recent standard explanations in ﬁnancial engineeringand mathematical ﬁnance. However, it is useful for understanding the relationship between
hedging and calibration, so we present an outline as follows. Consider a European option
based on an underlying asset price Set . Namely, the payoﬀ at maturity T is ZT = f (SeT ), where
f is a deterministic payoﬀ function and SeT is the underlying asset price at maturity T . Let
us denote the option price at time t as pt . The underlying asset price follows the stochastic
diﬀerential equation:
Set = Se0 +Z tµSeu du +0

Z t

σSeu dWu ,0

where Wt represents the standard Brownian motion under the probability measure P, andµ and σ are constant parameters denoting drift and volatility, respectively. This stochastic
process is known as geometric Brownian motion, where the price of the underlying asset at aﬁxed time follows a log-normal distribution. Then, the dynamics of the P&L are formulated
13

In discrete-time models, PL(ZT ,p0 ,δ) represents the P&L occurring from time 0 to T .

11as a stochastic process, withef t(ZT ,pt ,δ) = pt − ZT +
PL

Z T

e
δu dSeu .

t

Let us assume that the option price pt at time t is expressed as a function of t and Set as
pt = v(t, Set ).
(ZT ,pt ,eδ)ft
The dynamics of PLwith respect to t can be described using Itô formula as

∂
∂
1 ∂2
v(t, Set ) dt + v(t, Set ) dSet +v(t, Set ) dSet · dSet − e
δt dSet
∂t
∂x
2 ∂x2
!
!
2
∂
∂
1
∂
∂
2
2
=
v(t, Set ) + v(t, Set )µSet +v(t, Set )σ Set − e
δt µSet dt +v(t, Set ) − e
δt σSet dWt .
∂t
∂x
2 ∂x2
∂x
(ZT ,pt ,e
δ)

ft
dPL

=

∂
Here, ∂x
represents partial diﬀerentiation with respect to the second argument S t of v. Then,
as a condition for eliminating the risk of this P&L process, that is, setting the diﬀusion term
to zero, we obtain the following:


∂ 
e
δDel
= v t, Set .
t∂xThis is the hedging strategy known as delta hedging. The right-hand side of the above equation is the sensitivity of the derivative price to changes in the price of the underlying asset.
Therefore, the intuitive idea behind delta hedging is to hold an amount of the underlying
asset equal to the sensitivity (delta) per one unit of the derivative, so that the price changes
in the underlying and hedge position completely oﬀset each other.14 .
Further, given that (1) a risk-free asset is traded, (2) the market is arbitrage-free, and (3)δ)
T ,pt ,ef (Zthe P&L at maturity are zero (PL= 0), it can be demonstrated that v satisﬁes thetBlack–Scholes partial diﬀerential equation:∂∂
σ2 x2 ∂2v (t, x) = rv (t, x) , v (t, x) + rx v (t, x) dSet +
∂t
∂x
2 ∂x2




 v(T, x) = f (x) .

(4)Here, r represents the continuously-compounded risk-free interest rate of the risk-free assets.
For the solution v of this equation, setting pt = v t, Set , we have
 (Z ,p ,eδDel )

T t
f
P PLt
= 0 = 1 for t ∈ [0, T ].

If we assume more complex models for the underlying asset price Set , this argument does not hold, andthus condition (3) can fail. In such a case, one seeks a hedging strategy {eδ(RN)}u∈[0,T ] such that condition (3) isuapproximately satisﬁed, taking into account delta, vega, gamma, and other higher-order greeks. Such a strategy
is referred to as greeks hedging or parameter hedging.
14

12

Furthermore, v can be expressed in terms of expectation using the Feynman–Kac formula
as follows:hiv(t, Set ) = e−r(T −t) E Q f (SeT ) Set .
(5)
Here, the probability measure Q is the risk-neutral probability measure. Under this probability measure, Set follows the stochastic diﬀerential equation
Z t
Z t
e
e
e
St = S0 +rS u du +σSeu dWuQ .
00

Equation (5) shows that the option value v is equal to the present value of the expected payoﬀ
under the risk-neutral probability measure discounted by the risk-free interest rate. This iscalled the risk-neutral valuation framework because derivative prices can be obtained basedon this equation.
When computing option prices based on risk-neutral valuation, it is essential to predetermine parameters like r and σ, which describe the ﬂuctuations of S t under the probability
measure Q. Moreover, in models more complex than the Black–Scholes model, determining
multiple model parameters becomes necessary. This parameter determination is discussed in
the subsequent section on calibration.

2.3.2 Calibration problem
When pricing derivatives and deriving hedging strategies based on risk-neutral valuation,
it is necessary to determine the values of the parameters (for instance, r and σ in the Black–
Scholes model).
As mentioned at the beginning of section 2, the general approach in current derivative
practice for determining these parameters is to assume that the market prices of ﬁnancial
instruments with high market liquidity (the instruments to be calibrated) are correct and to
set the parameters so that the theoretical model prices ﬁt those market prices. Adjusting theparameters of the model in this way is referred to as calibration in the theory of derivative
pricing. Using calibrated models, practitioners price more complex derivative instruments
whose market prices are unobservable. Such procedures allow for pricing complex derivatives so as to be consistent with the market prices of highly liquid instruments; that is, there
is no-arbitrage between the complex derivatives and the highly liquid instruments. Thus,
accurate and eﬃcient calibration is extremely important in derivative pricing and risk management.
Below, we formulate the calibration problem. First, consider the following three parametergroups:• Model parameters (parameters of the stochastic diﬀerential equation governing theunderlying asset price): σ ∈ RQ
• Product-speciﬁc parameters (such as maturity and strike price): τ ∈ R p

13

• External factors (e.g., risk-free interest rates): ϕ ∈ Rm
Under this framework, let QP(τ) denote the market price of a ﬁnancial product with productspeciﬁc parameters τ, and let MP(σ; τ, ϕ) represent the model price under risk-neutral valuation.
Given R calibration target products with market prices {QP(τi )}i=1,...,R , the calibration problem can be formulated as a minimization problem to determine the model parameters:σ∗ = argmin Error σ; {QP (τi )}i=1,...,R , ϕ ,
(6)
σ∈RQ

where Error is an error function, that is, a function that penalizes the error between the market
price QP(τ) and the model price MP (σ; τ, ϕ). For example, the following function is used:

Error σ; {QP (τi )}i=1,...,R , ϕ =R
X(MP (σ; τi , ϕ) − QP(τi ))2 .

i=1

A typical practical problem is to adjust about 10 model parameters for tens to hundredsof calibration targets.15 For such high-dimensional optimization problems, methods like the
Newton–Raphson method are commonly employed. However, these optimizations are often
computationally expensive and may not necessarily converge. Especially when convergenceissues arise, it is diﬃcult to determine whether the problem lies with the optimization algorithm or the set of model parameters that can represent the market prices with in an acceptable error range is empty. Hence, calibration stands as one of the most intricate stages inthe development of derivative pricing models. To address this, deep calibration, introduced
in section 5, aims to replace these optimization computations with deep learning techniques,
thereby eliminating the need for optimization calculations with each valuation, leading tofaster and more stable calculations.

3 Deep learning
In this section, we provide an overview of deep learning. Neural networks utilize a systemof functions with a vast number of parameters, composed alternately of linear and nonlinear functions. Machine learning methods that utilize functions with deep layers of neuralnetworks are referred to as deep learning. While deep learning has empirically proven to behighly eﬀective in numerous practical applications, recent studies have gradually elucidated
its theoretical foundations.
In section 3.1, we provide a brief overview of deep learning and recent research trends
in the ﬁeld. In section 3.2, we introduce the notation used in the explanations in sections 4
and 5. For more detailed information on deep learning, see monographs such as Goodfellow
et al. [2016].
15

A typical example of calibration in practice is a calibration of the volatility of a Hull-White model, which
has roughly 10 parameters, calibrated to tens to hundreds of swaptions data.

143.1 Background of success3.1.1 Classiﬁcation of machine learning and the positioning of deep learning
Deep learning is a kind of machine learning. Therefore, to understand the characteristics
and potential applications of deep learning, it is beneﬁcial to comprehend its positioning
within machine learning as a whole. In the following, we provide a general classiﬁcationof machine learning methods. We also provide detailed explanations of the positioning andcharacteristics of deep learning within the broader context of machine learning.16
In machine learning, one ﬁrst trains a statistical model, called a machine learning model,using training data. These trained models are then employed for tasks such as decisionmaking, classiﬁcation, and prediction. Here, training refers to estimating the parameters ofa function system (statistical model) that links input and output data. Generally, machinelearning is said to oﬀer a more inductive and precise understanding of the relationships between data points as compared to classical statistical models, especially when vast amountsof data are used for training.
Given the above, machine learning can be classiﬁed from several perspectives. The ﬁrst
axis of classiﬁcation is the type of objective function set in the learning (i.e., parameter
estimation). In supervised learning, given a training dataset consisting of known output data,
the model parameters are trained to replicate the given outputs as closely as possible. Here,for instance, a machine learning model might be trained with images of dogs and images ofanimals that are not dogs, with each image correctly labeled (e.g., as a dog or not a dog).
The trained model is then used to judge whether a given arbitrary image is that of a dog.
In unsupervised learning, tasks like data clustering are performed without labeled output.
A third objective function-type category of machine learning is reinforcement learning. In
reinforcement learning, there is an agent who seeks to maximize the rewards that will bereceive in the future; the rewards depend on the agent’s actions and the state of the model,
which changes sequentially. The maximization result implies an optimal policy, which isa mapping from the state to the agent’s optimal action. The second axis of classiﬁcationis the purpose of the machine learning. Machine learning models can be developed forany number of purposes; for example, one might use a model to judge whether an emailis spam, to classify handwritten numbers as 0 to 9, or to predict future stock prices andunemployment rates from various kinds of feature variables. The third axis of classiﬁcationis the function system employed in the model’s formulation. In machine learning, dependingon the objective function of the optimization and on the purpose of the model, a speciﬁcfunction system with a suitable structure for the given model setup is employed. For example,
logistic regression is used to predict relatively simple phenomena, random forest and knearest neighbor methods are used for classiﬁcation and regression, and neural networks(multilayer perceptrons) are used to describe more complex phenomena.
16

Machine learning is a vast and rapidly evolving ﬁeld that encompasses various models, learning approaches, and applications. Consequently, there might be instances where the deﬁnitions and classes are notentirely clear.

15

Deep learning is a category under the third axis, in which a function system with a verylarge number of parameters that repeatedly synthesizes linear and nonlinear functions alternately (e.g., neutral network) is used. In particular, deep reinforcement learning, which uses
neural networks in reinforcement learning, has solved many problems in recent years and is
attracting widespread attention.

3.1.2 History and current status of deep learning
The history of deep learning began in the 1950s, and is said to currently be in its third
wave of enthusiasm. The ﬁrst boom (in the 1950s) was propelled by the development of
the perceptron (Rosenblatt [1958]), which is often considered to be the prototype of neuralnetworks. The perceptron modeled a nervous system, sparking high expectations for artiﬁcial
intelligence. However, by the late 1960s, Minsky and Papert [1969] had highlighted thelimitations of the perceptron in recognizing even simple shapes such as linearly separable
ﬁgures, leading to a decline in interest.
The second boom began in the 1980s with the introduction of backpropagation (Rumelhart et al. [1988]). As explained in detail later, backpropagation is a method for eﬃciently
performing the calculations required to update the parameters in the training of a multilayer
neural network. This technique made learning possible for neural networks with several
layers and greatly advanced the study of deep learning. Despite these advancements, however, issues like overﬁtting and the challenges in parameter updates led to another ebb inenthusiasm.17
The ongoing third boom gained momentum when Hinton et al. [2006] successfully applied pre-training to multilayer neural networks. Multilayer neural networks have since attracted widespread attention, leading to signiﬁcant developments. For example, AlexNet
by Krizhevsky et al. [2017] achieved high performance in the ImageNet Large Scale Visual Recognition Challenge (ILSVRC), and further improvement ensued. Deep learning hasbrought about breakthroughs such as AlphaGo’s victory over the Go world champion (Silver
et al. [2016]) and the development of data generation techniques (Goodfellow et al. [2020]).
These achievements indicate that deep learning has expanded its applications to now include
control problems and data generation tasks, in addition to traditional prediction and regression tasks.
Multiple reasons have been given for the success of deep learning in the current thirdboom. First are the advancements in the techniques and algorithms used for learning. As
noted above, the second boom lost momentum because of the diﬃculties encountered in
multilayer neural network learning due to such problems as overﬁtting. In the third boom,
the development of breakthrough technologies that overcome this issue have played an important role. Second is rapid development of infrastructure to support numerical computations for deep learning. This includes the availability of powerful Graphics Processing
17

Overﬁtting occurs when a learning model becomes overly tailored to the training data, leading to a decreasein performance when applied to unknown data.

16

Units (GPUs) and libraries like PyTorch and TensorFlow; these developments in hardware
and software make it possible to execute large-scale, computationally expensive numericalcalculations. Furthermore, accumulating experience in solving various problems with deeplearning has fostered a positive feedback loop, encouraging further exploration and innovation. In other words, the accumulation of rules of thumb for solving problems using deeplearning has made it easier to generate ideas for solving new problems with deep learning.
Recent theoretical research has shed further light on why deep learning has been so successful in practical terms. Speciﬁcally, as we will shortly discuss in section 3.1.4, it is becoming
clear that neural networks have theoretically desirable natural performance with respect tolearning and function representation. Such theoretical ground might serve to eﬀectively distinguish deep learning from its machine learning predecessor.
This trend has led to a growing body of research on the application of deep learning to ﬁnancial engineering and mathematical ﬁnance. The main reasons for this are rooted in threeproperties that are speciﬁc to problems in these ﬁelds. First, as discussed in section 2, theproblem of hedging derivatives is a control problem in that it requires a hedging strategy that
controls the uncertainty of a P&L. For this reason, empirical rules of reinforcement learning, which have developed rapidly in recent years, are useful.18 Second, derivative prices
and hedging strategies are often described by functions with nonlinearity due to nonlinear
payoﬀ functions and other factors. Therefore, they are compatible with neural networks
that have high representational capacity for the functions of interest for estimation purposes.
Third, given that techniques in the derivatives domain are grounded in ﬁnancial engineeringand mathematical ﬁnance theory, precise numerical calculations and theoretical assurances
are paramount. With recent advancements in deep learning technologies and infrastructure,achieving the required precision for practical applications in the derivatives domain is becoming feasible, although it still requires careful validation, as discussed in sections 4 and5.

3.1.3 Technique for learning
As previously discussed, in machine learning, a statistical model called a machine learningmodel is ﬁrst trained with appropriate data and then used for various tasks. Here, the term
training refers to solving an optimization problem to estimate the parameter θ in the function
y = f (x; θ) that connects the input data x to the output data y.
Speciﬁcally, parameter θ is estimated by minimizing the objective function, which is referred to as the loss function J. For instance, in supervised learning, where training datacomprising pairs of input data xi and corresponding output data yi denoted as {(xi , yi )}i=1,...,q
are provided, the error (diﬀerence) between the training data and the model’s output is captured as loss and the problem is formulated so as to minimize its value. If the loss function J

18

Indeed, the deep hedging introduced in section 4 draws inspiration from the policy gradient method of
reinforcement learning.

17

is deﬁned in terms of the squared error, it can be expressed as follows:
1X
J(θ) =e (ωi ; θ) ,q i=1qe (ωi ; θ) = (yi − f (xi ; θ))2 .

(7)

For model learning for speciﬁc problems such as calibration or hedging, it is essential to set
an appropriate loss function.
The basic algorithm to solve the aforementioned minimization problem is to search for the
optimal solution θ∗ by iteratively updating the parameters and terminating the iteration when
the update in θi becomes smaller than a certain threshold. Speciﬁcally, the update is given by
θi+1 = θi + ∇J(θi ).


∂J(θ)
Here, ∇J(θ) represents the gradient, deﬁned by ∇J(θ) = ∂J(θ)
,
.
.
.
,
, where θ(k) repre(1)
(k)
∂θ
∂θ
sents the k-th element of θ. This approach follows the idea of the Newton–Raphson method,where the gradient of the loss function is computed, and the parameters are updated in the
direction in which the loss decreases.
In machine learning, θ often becomes high-dimensional, ranging from thousands to trillions of dimensions, making this iterative approach extremely challenging. For instance,
issues such as the computational burden of computing the gradient ∇J(θ) and convergenceto saddle points or local minima (gradient vanishing problem) frequently arise (Figure 1).

Figure 1: saddle point and optimal point
To address these challenges, various techniques have been proposed. The following three
are among the most prominent of these techniques:
1. Setting the learning rate: Multiplying ∇J(θ) by an appropriate constant νi preventsgetting stuck in local minima or making excessive updates.
2. Stochastic gradient descent (SGD): Rather than using all the training data to compute18

∇J(θ), SGD approximates it using a randomly sampled subset of the training data.19
Here, the amount of training data extracted is qB , referred to as the number of batches.
For example, if theis given by equation (7), one stochastically samples
n loss functionoqB data points as (xi( j) , yi( j) )⊂ {(xi , yi )}i=1,...,q , and then approximates as follows.
j=1,...,qB

q
qB


1X
1 X
∇J(θ) =∇e (ωi ; θ) ≈∇e ωi( j) ; θ .
q i=1
qB j=1(8)Note that in doing so, we take advantage of the fact that the loss function J(θ) can be
represented in terms of the sum of the errors e (ωi ; θ) with respect to individual data
samples.20
3. Backpropagation: This is a method to eﬃciently compute ∇J(θ) using the fact thatneural networks are layered or composite functions. By using the chain rule of composite functions, partial diﬀerential coeﬃcients for each parameter can be eﬃciently
calculated in the order from the layer on the output side to the layer on the input side(in the opposite direction of input-output). Note that, if the error e (ωi ; θ) with respect
to individual training data samples is diﬀerentiable with respect to the parameter θ (i.e.,
if the analytical solution of the derivative exists), the method becomes more eﬃcient.
Additionally, various techniques such as skip connections, momentum methods, automaticlearning rate adjustments, regularization layers, data augmentation, and dropout have been
introduced to address the issue of not converging to global optima. Notably, the Adam
method proposed by Kingma and Ba [2015] combines the principles of SGD, momentum,
and RMSprop, representing one of the most widely used learning algorithms. For detailed
insights, the readers are referred to the aforementioned literature.
3.1.4 Natural performance of neural networks
As discussed in section 3.1.2, theoretical research on neural networks has progressed inrecent years, gradually revealing the reasons why deep learning works so eﬀectively. Specifically, it has become evident that neural networks possess desirable properties in terms ofrepresentation, optimization, and generalization abilities. This section provides an overview
of these properties.
Firstly, representation ability refers to how well a neural network can approximate thefunction it aims to represent. Studies on representation ability have a long history, with theuniversal approximation property of neural networks (i.e., functions that are important for applications can generally be approximated by neural networks) being particularly renowned.
19

If one can select a sample of suﬃcient size from the training data, using only a subset of the training data
can signiﬁcantly reduce the computational cost of ∇J(θ). Additionally, by stochastically sampling the training
data, the problem of vanishing gradients is less likely to occur.
20
As mentioned in section 4.1.2, in deep hedging where the risk measure is learned as a loss function,eﬃcient learning can be achieved by representing the loss function as a sum over samples of the training data,
as given in equation (7), and applying techniques such as SGD and backpropagation.

19

Recent research has highlighted the adaptive approximation capability of neural networks,
demonstrating that they can eﬀectively adjust the resolution based on the input, thereby eﬃciently approximating functions (Suzuki [2018]). Intuitively, adaptive approximation meansthat the network can adjust its resolution depending on the input, especially increasing theresolution where the function has large variations.
Secondly, optimization ability relates to the predictive power on training data and the capability to explore global solutions θ∗ . As previously mentioned, J(θ) is a high-dimensional
function, making it susceptible to becoming stuck in local minima or saddle points. However, neural networks exhibit favorable properties in the shape of the objective function J(θ),
making such issues less likely. For instance, Garipov et al. [2018] experimentally demonstrated the connectedness of regions containing optimal solutions, and Kawaguchi [2016]
theoretically showed that if the activation functions are restricted to linear functions, all local minima are also global minima.
Lastly, generalization ability pertains to the predictive power on unseen data, ensuring thatthe network does not overﬁt to training data and can adaptively represent functions basedon problem complexity. Neural networks are known to possess an implicit regularization
capability that adjusts model complexity based on problem complexity. Concrete examplesof this include norm minimization (Hastie et al. [2022]), ﬂat minima (Soudry et al. [2018]),
and the lottery ticket hypothesis (Frankle and Carbin [2018]). This automatic adjustment isrealized, for instance, by certain parameters θ∗ approaching zero, eﬀectively nullifying some
input values.

3.2 Neural network notation
Deﬁnition 4. Let L ∈ N with L ≥ 2 be the number of layers in a neural network, and let
N0 , N1 , . . . , NL ∈ N denote the dimensions of the intermediate layers. Let Wl : RNl−1 → RNl
represent an aﬃne transformation. In other words, there exist matrices Al ∈ RNl ×Nl−1 andvectors bl ∈ RNl such that Wl (x) = Al x + bl . Further, let η : R → R be an activation function.
Then, the composite function F θ : RN0 → RNL deﬁned by the following is referred to as a(feed-forward) neural network:F θ (x) = WL ◦ F L−1 ◦ . . . ◦ F1 ,where Fl = η ◦ Wl (with η applied element-wise). Here, θ represents the parameters of theneural network, that is, the set of components of matrices Al and vectors bl for l = 1, . . . , L.
Figure 2 shows a schematic diagram of a neural network. Each layer is composed of
an aﬃne function followed by a non-linear activation function. By stacking these layers L
times, the function operates on input data x by alternating between linear and non-linear
transformations, producing an output y.
Hereafter, we denote the set of feed-forward neural networks with input dimension N0 ,output dimension NL , parameter dimension M as NN M;N0 ,NL , and the set of parameters as
20

Figure 2: Neural NetworkΘ M .21

4 Deep hedging
In this section, we introduce the framework of deep hedging presented by Buehler et al.
[2019a]. Deep hedging is a framework that utilizes deep learning to numerically solve convex risk minimization problems. As mentioned in section 2, it bypasses risk-neutral valuation and directly solves the convex risk minimization problem. This approach has gained
attention due to its ability to quantify the impact of market frictions such as trading costs.22
In section 4.1, we introduce the framework of deep hedging proposed by Buehler et al.
[2019a]. In section 4.2, we discuss the subsequent research trends in this area. In section
4.3, we conclude with future prospects.

4.1 Algorithm proposed in Buehler et al. [2019a]
Deep hedging consists of a series of algorithms to determine an appropriate hedging strategy using a neural network. A hedging strategy is ﬁrst modeled by a neural network. The
model is then trained to minimize the loss risk using deep learning. The learning results areused to determine the hedging strategy. Using the notation introduced in section 2, we canformulate this procedure as shown below. For simplicity, we consider an underlying asset(hedging instrument) price process {S k }k=0,...,n−1 that is a Markov process; that is, the future
price depends only on the current price and not on past prices.

PL(Ni−1 × Ni + Ni ) because it is the sum of the numbers of the
The dimension M is given as M = l=1
elements of matrix Al and vector bl for l = 1, . . . , L.
22
According to Risk Magazine, an industry publication, some ﬁnancial institutions appear to be actively
utilizing deep hedging. Furthermore, Hans Buehler (XTX Markets), one of the authors of Buehler et al. [2019a],
was awarded Quant of the year 2022 for this achievement (Risk.net [2022]).
21

21

Hedging strategies are modeled using the following neural network functions:
 θ
θ0


 δ̂0 = F (0, S 0 ) , δ̂θ = F θk δ̂θ , S k (k = 1, . . . , n − 1) .
k
k−1

(9)

Here, F θk ∈ NN M;2,1 (with two inputs δ̂θk−1 , S k and one output δ̂θk ) typically consists of ashallow neural network with two to three layers. We assume the parameters θk ∈ Θ M for eachneural network F θk describing the hedging strategy are distinct for each period. The neuralnetwork is trained using data across all periods. Furthermore, the parameters describingQthe entire hedging strategy up to maturity are denoted as θ = (θ0 , . . . θn−1 ) ∈ n−1
k=0 Θ M . In
essence, the hedging strategy of each period is modeled by a shallow-layer neural network
whose input data are the hedge strategy in the previous period and the current underlying
asset price (as shown in one block in Figure 3), and the strategy across periods is modeled bya recursive deep-layer neural network (as shown in the entire Figure 3). In section 4.1.1, we
explain the validity (convergence) of modeling the hedging strategy by such a neural network
to minimize the loss risk.

Figure 3: Modeling of hedging strategies in deep hedging
Next, in the learning of parameters θ, given the future scenarios of(probabilityn {S k }k=0,...,no(ZT ,p0 ,δθ )distribution and its realization values), {S k (ωi )}k=0,...,n , the P&Ls PL
(ωi )
for
i=1,...,q

i=1,...,q

scenarios i = 1, ..., q are calculated.
Then, we calculate the optimal θ∗ that minimizes thenoθloss risk ρ PL(ZT ,p0 ,δ ) (ωi ). Here, the deep learning technique introduced in section
i=1,...,q3.1.3 is used. Importantly, the loss function needs to be represented in an appropriate form,
which is discussed in section 4.1.2.
ω)}k=0,...,n are sequentiallyfed
In the execution of the hedging, the realized asset prices {S k (e
n ∗o
θk∗
θ
into the trained neural network F . Thus, we produce the hedging strategy δ̂kby
k=1,...,n−1
deep hedging as follows:
 θ∗
θ∗


 δ̂0 = F 0 (0, S 0 ) , δ̂θ∗ = F θk∗ δ̂θ∗ , S k (eω) (k = 1, . . . , n − 1) .
k−1
k
22

4.1.1 Approximation theorem
In the framework described above, the process is described, for simplicity, for the casewhere the asset price {S k }k=0,...,n follows a one-dimensional Markov process. In this subsection, following Buehler et al. [2019a], we discuss the validity (approximation theorem) ofmodeling hedging strategies using neural networks for more general Fk -adapted Rd -valued
stochastic processes {S k }k=0,...,n .
Before presenting the approximation theorem, additional notation and assumptions are
required. Firstly, the transaction cost function ck in the PL(ZT ,p0 ,δ) (equation (1)) is assumed tobe right-continuous; that is, for x ∈ R, lim x↓a ck (x) = ck (a) holds.23 Additionally, {Ik }k=0,...,n−1
represents market information at each time and Ik is assumed to generate Fk . Moreover, let
H be a subset of Rd -valued stochastic processes representing possible hedging strategies.24
H M is deﬁned as the set of hedging strategies expressible by neural networks NN M;N0 ,N1 as
follows:n o
H M = δ̂θk| δ̂θk = F θk (δk−1 , I0 , . . . , Ik ) , θk ∈ Θ M , F θk ∈ NN M;r(k+1)+d,d .
k=0,...,n−1

Note that when {S k }k=0,...,n is a Markov process, this deﬁnition aligns with equation (9). Let
X be an Fn -measurable random variable, and deﬁneπ (X) = inf ρ PL(X,0,δ) , π M (X) = inf ρ PL(X,0,δ) .
δ∈H

δ∈H M

Here, π(X) represents the risk when choosing the optimal hedging strategy from all possible
strategies, while π M (X) represents the risk when choosing the optimal strategy from the setof strategies expressible by neural networks.
Theorem 1 (Approximation theorem for deep hedging).
lim π M (X) = π (X) .

M→∞

The proof of Theorem 1 can be established using the universal approximation theorem ofneural networks under assumptions such as the right-continuity of ck . For more details, refer
to Proposition 4.2 in Buehler et al. [2019a]. It is worth mentioning that this theorem only
guarantees convergence; showing the precision or rate of convergence analytically remains
challenging.
Theorem 1 indicates that the risk π M (X) minimized within the hedging strategy H M approximated by a neural network converges to the risk π(X) minimized within the set of possible hedging strategies H as the number of parameters M (parameters describing the neuralnetwork for a one-period hedging strategy) increases. This observation suggests that model-

In this case, S k for k = 0, . . . , n is considered as part of the market information ({S k }k=0,...,n generating aσ-algebra contained within the σ-algebra generated by {Fk }k=0,...,n ).
24
In section 2.2 of Buehler et al. [2019a], the set of possible hedging strategies is deﬁned, taking into accountthe transaction constraints. For more details, refer to the original paper.
23

23

ing hedging strategies using neural networks to minimize loss risk is a viable approach.25

4.1.2 A key in learning neural network weight parameters
To eﬃciently learn the weight parameters θ∗ that minimize the loss function J(θ), techniques such as SGD and backpropagation, as introduced in section 3, are employed. It is
desirable that the loss function J(θ) be the sum of errors concerning a single training data
sample and that it be diﬀerentiable with respect to θ. Therefore, in the deep hedging, one considers a class of risk measures for whichcann the loss function
 be represented as the expectedoθvalue of a random variable J(θ) = ρ PL(ZT ,p0 ,δ ) (ωi ).26i=1,...,q
Attempts to represent risk measures such as ES as expected values began with Rockafellar
et al. [2000]; and were further developed by Ben-Tal and Teboulle [2007] through the introduction of the concept of Optimized Certainty Equivalence (OCE).27 OCE is formulated
as follows: Let l : R → R be a continuous, non-decreasing, convex function, and for areal-valued random variable X,
κ(X) = inf {w + E[l(−X − w)]} ,w ∈ R.

Then, κ satisﬁes the deﬁnition of a convex risk measure and is called a risk measure with
OCE. For instance, ES can be expressed by setting l(x) = (1/(1 − α)) max {x, 0} as below,
and thus ES has the property of OCE:#)"(
1
max {−X − w, 0} .
ESα (X) = inf w + E
w∈R
1−αA risk measure with OCE can be eﬃciently learned. Namely, using the representations via
OCE, we transform the argument in optimization from θ to a two-dimensional variable θ × w.
Using the formulation of OCE, eﬃcient learning is possible by transforming the variable ofthe minimization problem from θ to θ × w:
n


h 
io
θ
θ
inf ρ PL(ZT ,p0 ,δ̂ ) = inf inf w + E l PL(ZT ,p0 ,δ̂ ) − w
θ
θ w∈R
n
h 
io
θ
= inf w + E l PL(ZT ,p0 ,δ̂ ) − w .
θ̄=θ×w

By this transformation, the loss function for learning is set as


q

X
  
h 
i


θ


Z
,p
,
δ̂
(
)
T 0
(ω
)
J θ̄ = 
w
+
l
PL
−
w
P(ω
)
.

i
i




i=1

As mentioned in section 3.1.3, representing the loss function in this form allows the use ofn o
In Buehler et al. [2019a], it is not directly stated that the modeled hedge strategy δ̂θkconverges tok=0,...,n−1∗
the solution δ of the convex risk minimization problem. As mentioned in Remark 1, the reason for this is to
avoid discussing the existence of hedge strategies that minimize risk.
26
It is known that Value-at-Risk cannot be represented as the expectation of a random value.
27
In decision theory, OCE was introduced as an extension of certainty equivalence.
25

24

SGD. Moreover, if the derivative with respect to θ on the right-hand side is analytically available, it can be used to accelerate backpropagation (refer to Buehler et al. [2019a] Proposition
4.6).

4.1.3 Summary of numerical results
In section 5 of Buehler et al. [2019a], the authors provide empirical evidence of the validityof the learning outcomes through numerical experiments. Here, we present a summary ofthis evidence.
Setup
EuropeanCall Option (1-month maturity, at-the-money strike) ZT =• Targetn option:o(1)30
max S T − K, 0 , T = 365 .
• Distribution for hedging instruments: Heston model (Heston [1993])
q


(1)

e

Set(2) Set(1) dWt(1) ,
d
S
=

t



q 





p
(2)
(2)
(1)
(2)

2 dW (2) ,
e
e
e

d
S
=
α
b
−
S
dt
+
σ
+
S
ρdW
1
−
ρ

t
t
t
t
t



 (1) (2)   (1) (2)  Se , Se = s , s ,
0
0
0
0

(10)

(2)
with parameters set as α = 1, b = 0.04, ρ = −0.7, σ = 2, s(1)0 = 100, and s0 = 0.04.
k• Daily hedging: tk = 365, n = 30

• Risk measure: ES• Generation of future scenarios: By discretizing(10)n (1) equation
o and using Monte Carlo(2)
simulation, generate q future scenarios S i (ω j ), S i (ω j ) i=1,...,n .
j=1,...,q• Learning neural network weights: Using a learning rate νi = 0.0005 and batch sizenoθqB = 256, compute the optimal weights θ∗ that minimize ρ PL(ZT ,p0 ,δ ) (ωi )via
i=1,...,q
Adam optimization.
Results
1. Comparison of deep hedging and delta hedging (no transaction costs)
Figure 2 in Buehler et al. [2019a] shows the frequency distribution of the realized
P&Ls for both deep hedging and delta hedging in no-transaction-cost setting.28 Here,the risk measure is set as 50%-ES. The graph shows that the behavior of the realized
P&Ls in deep hedging closely aligns with the P&Ls in delta hedging in the case of noThis is the frequency distribution obtained from an out-of-sample test with a sample size q′ = 106 , namely,
the frequency distribution of the P&Ls when applying the hedging strategy derived from the learning outcomesunder scenarios {S k (eωi )} k=0,...,n as distinct from those used for learning.
28

i=1,...,q′25

transaction costs.29 In addition, the greeks are also closely align.30 This indicates thevalidity of deep hedging in the case of no transaction costs.31
2. Impact of risk measure parameter setting on hedging strategy learning
Figure 7 in Buehler et al. [2019a] displays the frequency distribution of the realized
P&Ls for deep hedging when the risk measures are set as 50% and 99%-ES, respectively. From the graph in the upper part of this ﬁgure, in the case of 50%-ES, thedistribution of the realized P&Ls is higher around the median; that is, the hedging
strategies are trained to avoid losses below the mean. The table at the bottom of theﬁgure also shows that in the case of 99%-ES, the distribution of the realized P&Ls issmaller than the case of 50%-ES; that is, hedging strategies that avoid large losses arelearned. These facts indicate that the learning outcome of deep hedging changes as
expected depending on the parameter setting of the risk measure.
3. Impact on utility-indiﬀerence pricing due to transaction costs
Figure 11 in Buehler et al. [2019a] shows the utility-indiﬀerence prices pε for diﬀerent
P2
(i)
(i)
ε, setting transaction cost as ck (δk − δk−1 ) = ε i=1
δ(i)k − δk−1 S k . The graph indicatesthatpε − p0 = O ε0.71
holds. This is consistent with the theoretical results (Whalley and Wilmott [1997],
Rogers [2004]) for the case in which the price process of the underlying asset follows
a one-dimensional diﬀusion process such as the Black–Scholes model. This fact indicates that the utility-indiﬀerence prices calculated through deep hedging vary withrespect to the transaction costs, as theoretically predicted, and that the hedging error
converges with a reasonable degree of accuracy.

4.2 Research trends
4.2.1 Future scenario generation
In section 4.1, we presented the framework of deep hedging, without specifying how the
future scenarios of the underlying assets (hedging instruments) are generated.32 Developing
methods for generating future scenarios is a central issue in the subsequent research on deep
hedging, since generating realistic future scenarios is essential for the practical implementation of deep hedging.
29

The horizontal axis of this graph represents the hedging error as a percentage of one unit of the underlying
asset. As mentioned above, this is because the current price of the underlying asset is set to s(1)0 = 100.
30
Refer to Figure 3 in Buehler et al. [2019a].
31
As explained in section 2, in the case of no transaction costs, a delta hedge can eliminate hedging errors
through replication. However, due to minimal numerical computation errors, the frequency distribution of therealized P&Ls from the delta hedge also has a slight spread.
32
In Buehler et al. [2019a], future scenarios are generated using a stochastic diﬀerential equation with ﬁxed
parameters.

26

Among the various methodologies, techniques employing generative models based on
deep learning have been extensively investigated. For instance, Wiese et al. [2019] proposeda method to generate future scenarios for vanilla options33 using Generative Adversarial Networks (GANs).34 When evaluated against the quasi-maximum likelihood method and vector
autoregressive models, the scenarios generated by GANs better captured the characteristicsof the given past data across multiple metrics. Additionally, Buehler et al. [2020, 2021a]
proposed an approach using a deep generative model called the variational autoencoder andfeatures termed signatures to generate future scenarios from limited market data.35
It is apparent that if we use a scenario that gives a hedging strategy where the expected
return genuinely becomes positive upon increasing the number of scenario samples, that is,a statistical arbitrage strategy, the hedging strategy learned via deep hedging can be significantly inﬂuenced by this arbitrage strategy. This inﬂuence may result in a biased hedging
strategy, such as monotonically increasing the holdings of speciﬁc assets. Buehler et al.
[2022a, 2021b] extended the notion of risk-neutral valuation in the presence of transaction
costs and introduced a method to remove statistical arbitrage from future scenarios. Essentially, this method transforms the given future scenarios to ensure the absence of hedging
strategies with genuinely positive expected returns. Moreover, numerical results demonstrate that by eliminating statistical arbitrage from the future scenarios, the performance ofhedging strategies trained via deep hedging improves; that is, hedging errors are reduced.
Furthermore, in empirical studies, Zhang and Huang [2021] and Mikkilä and Kanniainen[2021] reported numerical experiments using historical data for option prices to generate
future scenarios. Horvath et al. [2021b] discussed scenarios that follow non-Markovian
stochastic processes, that is, the rough volatility model (Gatheral et al. [2018]). Additionally,Lütkebohmert et al. [2021] examined scenarios governed by stochastic diﬀerential equations
with uncertain coeﬃcients.

4.2.2 Validation of deep hedging under diﬀerent settings
In this paper, following the framework of Buehler et al. [2019a], we formulate the hedging
problem as a minimization problem of the risk quantiﬁed by convex risk measures of the
P&Ls at maturity, and use utility indiﬀerence pricing. In practice, it is essential to modify
and extend the formulation of the hedging problem depending on the speciﬁc situation or
objective.
Kolm and Ritter [2019] and Cao et al. [2021] formulated the risk measure using the mean
and variance of the P&Ls at maturity as an optimization objective. Additionally, Carbonneau [2021] incorporated hedge errors as the optimization objective without resorting to
33

These are basic derivatives such as European options.
This is the generative model of deep learning proposed by Goodfellow et al. [2020]. The method uses
neural networks to generate pseudo datasets with features similar to the input data. By training two networks
for generation and discrimination in competition, high-quality datasets can be generated.
35
Signatures are features that generalize high-order correlation structures between the prices of multiple
assets.
34

27

convex risk measures. The conditions of the derivatives change daily due to factors such as
shortening expiration times. Buehler et al. [2022b] considered this aspect and proposed aframework to formulate the hedging problem using the Bellman equations and solve it using
the actor-critic method of reinforcement learning.
Moreover, Carbonneau and Godin [2021] discussed the concept of Equal Risk Pricing as
an alternative to utility indiﬀerence pricing (Deﬁnition 3).36
Note that even before Buehler et al. [2019a], researchers had explored the derivation ofhedging strategies using deep learning and reinforcement learning, under speciﬁc conditions
such as the absence of transaction costs. For instance, Halperin [2019] proposed a method
to derive hedging strategies under the Black–Scholes model in a cost-free setting using the
Q-learning framework.37 Speciﬁcally, the author formulated the replication via adjustmentsin the portfolio, the basic idea of the Black–Scholes model, in a discrete time setting. Using
the Markov decision process in reinforcement learning, the author then proposed a method
for formulating a hedging strategy as a solution to the dynamic optimization problem andcalculating an optimal hedging strategy analytically. In Halperin [2019], the author discussed
the numerical computations for this method. Notably, research works such as Mikkilä andKanniainen [2021] and Buehler et al. [2019b] have adopted the formulation using Q-learning.

4.2.3 Numerical experiments for more complex derivatives
Whereas the numerical experiments in Buehler et al. [2019a] were conducted on relatively
simple short-dated derivatives, there are a number of studies on more complex derivatives.
For example, Carbonneau and Godin [2021] and Imaki et al. [2021] dealt with exotic derivatives such as look-back options and Asian options. Carbonneau [2021] dealt with long-dated
derivatives. In another study, Shi et al. [2021] compared the method proposed in Buehler
et al. [2019a] with the method based on numerical solutions of forward-backward stochastic
diﬀerential equations (Weinan et al. [2017]), and concluded that the method in Buehler et al.
[2019a] is more eﬀective for long-term optimization problems.

4.2.4 Modiﬁcations in modeling hedging strategies
In Buehler et al. [2019a], a method was introduced to model hedging strategies using
feedforward neural networks as described in equation (9). As discussed in section 4.2.3, it
is considered that this model can adequately learn hedging strategies for speciﬁc problems.
However, for more intricate problems, where an eﬃcient derivation of hedging strategies isrequired, it is beneﬁcial to modify the hedging strategies model.

36This is the price formulated by Guo and Zhu [2017], where the risk amount for the seller of a derivativeequals that of the buyer. Generally, while utility indiﬀerence prices may diﬀer between the seller and the buyer,
risk-equivalent prices are the same for both sides.
37
Q-learning is a method in reinforcement learning that represents future cumulative rewards as a functionof the current environment’s state and action and estimates this function.

28

Zhang and Huang [2021] presented a model using Long Short Term Memory Recurrent Neural Networks (LSTM-RNNs) that utilizes historical hedging strategy data to determine future-period hedging strategies. Through numerical experiments, they veriﬁed thatthe hedging error in P&Ls is improved, compared to classical methodologies like Leland[1985]. Similarly, Carbonneau [2021] adopted the use of LSTM-RNNs and discussed longterm hedging for derivatives in settings where the underlying asset price process includes
jumps.
Tawada and Sugimura [2020] proposed a technique using Gaussian Process Regression(GPR) rather than neural networks for modeling hedging strategies. GPR oﬀers the advantage of high interpretability for the trained parameters, rendering the derived hedging strategies more comprehensible. Making use of this property, the authors proposed a method totransition to alternative hedge techniques (such as greeks hedging) when the scenarios used
for learning signiﬁcantly deviate from the realized scenarios. Imaki et al. [2021] proposed amodel that includes the additional hypothesis that trading does not occur if the trading volume remains below a certain threshold and oﬀered a method to estimate this threshold byusing deep learning. The authors not only theoretically demonstrated that the hedging strategy under this model becomes optimal even with its approximation, but they also showed asigniﬁcant reduction in the computational costs required for learning hedging strategies innumerical experiments. Additionally, Son and Kim [2021] conducted numerical experiments
utilizing various deep learning models, including regression neural networks and convolutional neural networks, among others.

4.2.5 Application to general risk management
Since the convex risk minimization problem is a general optimization problem given P&L
measures, we can apply the same solution approaches not only to derivative hedging but also
to various risk management situations. Consequently, there is a growing number of discussions on the application of deep hedging frameworks to broader risk management problems.
For instance, the application of deep hedging to risk management in the banking sector has
been discussed in Krabichler and Teichmann [2020], while its application to risk management in insurance companies is explored in Fernandez-Arjona and Filipović [2022]. Additionally, Shimizu [1998] utilized neural networks within a supervised learning framework to
consider feedback eﬀects in stress tests. It seems possible to conduct such stress tests basedon deep hedging.

4.3 Future prospects
4.3.1 Development potential
Deep hedging oﬀers a framework for determining optimal trading strategies given theamount of P&L risk. Due to its ﬂexibility, it can be applied to a wide range of problems,

29including derivative hedging and risk management. Below, we summarize its development
potential in the ﬁeld of derivatives and elsewhere.
In the ﬁeld of derivatives, both the reﬁnement and automation of hedging strategies are
expected. Regarding reﬁnement, deep hedging makes it possible to incorporate market frictions such as transaction costs. Additionally, by considering information other than ﬁnancial
market observations—for example, news—in generating future scenarios for hedging instruments, we can reﬂect such information in the hedging strategy. Importantly, by setting risk
measures according to the risk preference of the entity hedging the derivative, a strategy tailored to that risk preference can be derived. These could represent new analytical methods
in derivative risk management. The automation of hedging operations may also be advancedby the reﬁnement of hedging strategies. As discussed in section 2.1.2, in hedging strategies
derived via risk-neutral valuation, adjustments such as additional cost charges are usually applied to cover inevitable errors in replicating the derivative. Such price adjustments are oneof the main obstacles to automating hedging operations, as the size of the price adjustments
are often determined in an ad hoc manner based on the subjective judgements of the traders.
On the other hand, by virtue of deep hedging, such qualitative adjustments become unnecessary because deep hedging can directly take into account market frictions, thus making theautomation of hedging operations more feasible. Additionally, by using deep learning models for future scenario generation, it becomes possible to repeatedly perform the learning ofhedging strategies and scenario generation, which opens the possibility of deriving hedging
strategies purely from data without any model dependency.
Outside the ﬁeld of derivatives, as discussed in section 4.2.5, applications to general risk
management and broader decision-making based on data such as corporate management
strategy become feasible. However, compared to the hedge problems in the ﬁeld of derivatives, the formulation and validation of such broader problems represent a bigger challenges.
Thus, it would seem appropriate that we ﬁrst develop the techniques of deep hedging forderivatives.

4.3.2 Practical challenges
As noted above, discussions of the challenges associated with deep hedging is ongoing,as the method is still quite novel. Below, we summarize the practical challenges of deep
hedging by categorizing the problem settings of deep hedging and the technical nature ofdeep learning.
As discussed in section 4.2.2, for the practical use of deep hedging, expanding or modifying the problem settings according to the speciﬁc situation or purpose is necessary. Especially since there is no established method for setting the parameters of the risk measures,comparing strategies learned under diﬀerent risk measure parameters would seem practically useful for risk management. Moreover, when calculating derivative prices using deep
hedging (especially for counterparty transactions or fair value calculations), one needs to becareful since external parameters (like risk measure parameters) can signiﬁcantly aﬀect the30price.
In addition to the extension and modiﬁcation of the problem settings mentioned in section
4.2.2, there are other settings closely related to the practical application of deep hedging. For
instance, optimizing not only the risk amount at maturity T of the derivative but also the risk
amount at intermediate points t ∈ [0, T ] is, in practice, considered essential. Additionally,
incorporating value adjustments called xVAs, which have become important in derivative
practices post the 2007-08 ﬁnancial crisis, into the framework of deep hedging is anothersigniﬁcant challenge. Operationally, it is crucial to re-evaluate the hedging strategy after
signiﬁcant changes in one’s portfolio or market environment.
Regarding deep learning techniques, as mentioned in section 3, compared to establishedﬁnancial engineering techniques such as the Monte Carlo method, there is no theoreticalguarantee regarding numerical behavior, such as the accuracy of learning. Hence, we should
be careful in practical applications. Moreover, similar to applying deep learning in otherﬁelds, enhancing the speed and stability of learning and increasing the interpretability of
results present serious challenges.
In summary, while deep hedging has signiﬁcant development potential, it also poses numerous challenges. Thus, it is crucial that we explore how best to utilize deep hedging bycomparing it with the already established hedging approach based on risk-neutral valuation.
Especially when one takes a model-based approach in practical applications, it is criticallyimportant to understand the underlying data and calculation principles and to use the modelderived results cautiously. Table 1 summarizes the relationship between deep hedging andhedging based on risk-neutral valuation.
Table 1: Comparison of deep hedging and risk-neutral valuation
Hedging via risk-neutral valuation(Greeks hedging)

Deep hedging

Principle

Convex risk minimization(Incomplete market)— Physical measureMain
inputs

1. Future market scenario
2. Risk preference (Risk measure)

Replication / No-arbitrage(Risk neutral valuation)— Risk neutral measure
Information on
the current market environment(Market prices of liquid assets)

5 Deep calibration
In this section, we introduce deep calibration (Hernandez [2017], Horvath et al. [2021a],etc.). As noted in section 2.3, the calibration process is computationally intensive and prone
to problems such as with convergence issues in the optimization calculations. Deep calibration is an algorithm that trains a neural network to learn the relationship between the model31

parameters and model prices in advance, and then replaces the optimization calculations of
the model parameters which are a step in the pricing calculations (risk-neutral valuation).
This approach is expected to accelerate and stabilize the pricing calculations.
In section 5.1, we introduce the deep calibration algorithm proposed in Hernandez [2017].
In section 5.2, we discuss the trends in subsequent research. We then summarize, in section
5.3, the practical advantages and challenges.

5.1 Algorithm proposed in Hernandez [2017]
Deep calibration is executed in the following two steps: First, a neural network is trainedby taking the model prices of the calibration target products as the input and the model
parameters as the output. Then, in the daily pricing tasks, the market prices of the calibration
target products are input into the trained neural network to approximate the model parameters(Figure 4). Using the notation from section 2.3, we describe the algorithm below. Note thatthe algorithm belongs to the supervised learning classiﬁcation as described in section 3.1.1.

Figure 4: Process of deep calibration
Here, we consider the problem of determining model parameters σ ∈ RQ with given R
calibration target product market prices {QP(τi )} for i = 1, ..., R.
1. Training data generation
Randomly generate external
ϕ̂ j ∈oRm and model parameters σ̂ j ∈ RQ .38 Next,
n factorscompute the model price MP σ̂ j ; τi , ϕ̂ j
. Repeat this q times to generate traini=1,...,Rnoing data σ̂ j , ϕ̂ j , MP σ̂ j ; τi , ϕ̂ j i=1,...,R .
j=1,...,q

2. Learning

 n


o
Train K θ ∈ NN θM;m+R,Q with inputs ϕ̂ j , MP σ̂ j ; τi , ϕ̂ j
∈ Rm+R for i = 1, ..., R
i=1,...,Rand output σ̂ j . For instance, minimize the loss function deﬁned as
J(θ) =

q
X



2

σ̂ j − K θ ϕ̂ j , MP σ̂ j ; τi , ϕ̂ j

j=1

to determine θ∗ .

38

When generating external factors and model parameters randomly, upper and lower bounds are determinedby referencing past model parameters and market data.

32

3. ApplicationIn the daily valuation of derivatives, given the market price QPi i=1,...,R of
 the calimθ∗
bration target product and external factors ϕ ∈ R , use K ϕ, QPi i=1,...,R as model
parameters.
Hernandez [2017] applied the above algorithm to the Hull-White model (with swaptions
as calibration targets) and compared it with conventional calibration methods using historical
data. For the setup of this numerical experiment, refer to Table 2. Additionally, Hernandez[2017] reported that in the generation of pairs ϕ̂ j , σ̂ j , the calculations become stable if thecorrelations between model parameters and external factors using past data are taken into
account.

Reference

Table 2: Numerical examples of deep calibration
Calibration
Setting ofModelneural networks
targets

Acceleration

Num layers L = 5(4 interm. layers)
Swaption
Hull–White model
AroundDim of interm. layers
Hernandez
6M-Libor- Num params.
225
Nl = 64[2017]- Num targets
Q=2
times fasterNum training data
R = 200
q = 150, 000
Num layers L = 5(Rough) Bergomi(4 interm. layers)
Volatility surface
AroundHorvath etmodel
Dim of interm. layers- Num targets
9,000 to 16,000
al. [2021a] - Num params.
Nl = 30
R = 88times faster
Q=8
Num training dataq = 68, 000
Note: For more details, refer to the indicated references.

5.2 Research trends
5.2.1 Improvements in the algorithm
Horvath et al. [2021a], Bayer et al. [2019], and Liu et al. [2019a] proposed a method,
called here the two-step method, to enhance the speed and stability of the algorithm in Hernandez [2017]. This method consists of an algorithm that trains a neural network to learn
the forward function MP(·; τi , ϕ) of the model price and then calibrates the model parameters
using conventional optimization methods. According to Horvath et al. [2021a] and Bayeret al. [2019], the primary advantage is that by training the neural network on the forward33

function of the model price, the results are easier to validate and interpret relative to the casewhere the inverse function is trained.39
Subsequently, Itkin [2020] modiﬁed a part of the algorithm in Hernandez [2017] by introducing a method that uses the forward function of the model price for training data generation, akin to the two-step method. Itkin [2020] pointed out that by using a method which
ensures that the function maintains the no-arbitrage condition and smooth greeks during thelearning of the price function’s forward function, stable and rapid calibration becomes feasible.
Furthermore, Horvath et al. [2021a] and Bayer et al. [2019] discussed methods to minimize
errors across various maturities and strikes by carefully designing the error function. Thesemethods are useful for swaptions and the volatilities of foreign exchange rates among otherproducts, as the calibration targets of these products can be seen as a multi-dimensional
matrix of maturities and strikes.

5.2.2 Applications to various models
Various numerical experiments have been conducted with the aim of putting these algorithms into practice. In particular, in Horvath et al. [2021a], the calibration of the roughvolatility model using the two-step method was discussed．For the setting of the numericalexperiments in Horvath et al. [2021a], refer to Table 2.
Calibration problems for other models have also been discussed, including the Heston
model in Dimitroﬀ et al. [2018], the SABR model in Lokvancic [2020], the interest rate
models in Sabbioni et al. [2021], and the general HJM model in Benth et al. [2020]. In
addition, in Liu et al. [2019b], the generation of training data for deep calibration under theBlack–Scholes and Heston models is discussed. In Brigo et al. [2021], the interpretability of
deep calibration under the Heston model is addressed.

5.3 Practical advantages and challenges
Deep calibration oﬀers several desirable features for practical operations. Firstly, it operates within a supervised learning framework. This is advantageous for enhancing andverifying the stability of learning because one can prepare as much training data and test
data as needed by randomly generating the model parameters and associated model prices.
Moreover, several studies reported that deep calibration can achieve satisfactory accuracy ata relatively modest computational expense. In addition, by validating the learned function,
one can verify the attainable range of the calibration targets’ prices and the behavior of the
39

Note that research on learning model prices with neural networks began with Hutchinson et al. [1994]
and has been extensively studied since then. In particular, Garcia and Gençay [2000] and Dugas et al. [2009]
have presented methods for learning model prices using neural networks by considering important conditions
in ﬁnance such as the absence of arbitrage as constraint conditions. Ruf and Wang [2020], a review paper,provides a summary of these studies.

34

greeks prior to the daily valuations, which can render the calibration more robust compared
to existing optimization methods.40
It should be noted that it is crucial to regularly inspect the results obtained by deep calibration. In particular, if the market environment undergoes signiﬁcant changes, the relationshipstructure between the model parameters and external factors may also change signiﬁcantly,
therefore changing the shape of the learned function. In such cases, prompt validation of themodel or switching to existing optimization methods may be required.

6 Concluding remarks
This paper provides an overview of the trends in research related to deep hedging anddeep calibration, which are applications of deep learning in the ﬁeld of ﬁnance, from bothpractical and academic perspectives. Deep hedging applies deep learning to solve convex risk
minimization problems, thereby making it possible to solve problems that were not whollysolvable analytically and to quantify the impact of market frictions such as transaction costs
on derivative pricing and hedging strategies. On the other hand, deep calibration makes thecalculations in calibration faster and more stable, which is a part of the derivative pricing
process under the risk-neutral valuation framework. As such, deep learning has the potential
to advance a variety of techniques in the ﬁeld of ﬁnance over both the short and long term.
Beyond the applications to hedging and calibration which are discussed in this paper,
there have been extensive discussions of other potential applications of deep learning in ﬁnance. Many problems in ﬁnance, including hedging, are formulated as control problems,
which seek optimal actions under given constraints. The compatibility of control problems
with reinforcement learning has inspired numerous reinforcement learning-based methods tosolve problems such as portfolio optimization (Jiang et al. [2017]), optimal execution (Ninget al. [2021]), market-making (Spooner et al. [2018]), and algorithmic trading (Deng et al.
[2016]). In addition, studies on predicting asset prices (Patel et al. [2015]) and generating ﬁnancial time series data using Generative Adversarial Networks (Wiese et al. [2020])
have been progressing. There have also been studies on the application of deep learning
to general computational problems, including applications to forward-backward stochastic
diﬀerential equations (Weinan et al. [2017]) and optimal stopping problems (Becker et al.
[2019]). The former study (application to forward-backward stochastic diﬀerential equations) has direct applications in ﬁnance, including the calculation of adjustments to credit
valuation adjustment (CVA) and initial margin (Henry-Labordère [2020]), as well as xVA
calculations (Gnoatto et al. [2020]). Several survey papers have discussed the applicationof machine learning and deep learning in ﬁnance, including Hambly et al. [2021], Ruf andWang [2020], Emerson et al. [2019], Warin [2021], and Mashrur et al. [2020].
As both deep learning and ﬁnance are fast-developing research ﬁelds, it is crucially important to keep up with the most current cutting-edge discussions in these ﬁelds.
40

For instance, if anomalies are detected in the process of calibration, it becomes easier to identify the cause.

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