Evaluating market risk from leveraged derivative exposures.
Date: 2022-09-21
Institutions: European Central Bank
Subjects: financial derivative, financial institution, financial market, financial risk, market

Working Paper Series 

Audrius Jukonis 

Evaluating market risk from leveraged 
derivative exposures 

No 2722 / September 2022 

Disclaimer: This paper should not be reported as representing the views of the European Central Bank 
(ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB. 

Abstract

Market participants use leveraged derivatives to gain access to equity market exposure

through broker banks. Leverage and interconnectedness via overlapping portfolios of dealer

banks can amplify adverse market movements, potentially causing sizeable losses. I propose

a model, based on granular data, to simulate losses from a banks’ trading book in case of an

adverse market scenario. Following a move in asset prices, banks mark their positions and

issue margin calls; some (leveraged) counterparties fail to pay their margins, forcing banks

to liquidate their positions causing a pressure on asset prices due to market impact. The

impact is amplified because of the leverage and when counterparties are exposed to multiple

banks over the same underlying. I employ the model to assess current capital and margin

rules in covering risks from broker’s exposure to highly leveraged clients.

Keywords: EMIR, market risk, leverage, Initial margin, Variation margin

JEL codes: C60, G23, G13, G17.

ECB Working Paper Series No 2722 / September 20221Non-technical summary

After the global financial crisis, bank stress tests have become a key tool of bank prudential

supervision in EU and in other jurisdictions. At the same time, authorities have put a

considerable effort to collect daily data on derivatives markets to increase transparency,

mitigate systemic risk, and limit market abuse. Yet these two crucial reforms have had

limited feedback between each other in prudential sense, with the result that the wide amount

of data available on derivatives exposures of banks is not currently utilised in the context of

stress test. The main reasons for this gap are the challenges that comes with using such big

data, and the model complexity underlying the derivatives market.

Against this background, this paper proposes a model to effectively exploit activity-based

data to complement and enrich the picture from supervisory data and measure the market

risk from banks’ exposures in the derivatives market. The main application of the proposed

framework is to challenge banks’ self risk assessment in bottom-up stress tests, for which now

only limited data is available. In addition, authorities could employ the framework to cali-

brate macroprudential measures aimed at mitigating the impact of a disorderly liquidations

of banks positions.

The proposed model estimates the losses banks will incur would some of their counterpar-

ties fail to pay their margins, forcing banks to liquidate their positions. The model considers

the potential amplification effect of overlapping exposures across dealer banks by including

the market impact of fire sales of hedge by banks.

Simulations focus on equity derivatives market and they show that effects are hetero-

geneous, with some banks being particularly hit, due their high interconnectedness in the

market and high concentration in their exposures.

ECB Working Paper Series No 2722 / September 20222Introduction

Banks play a central role in the derivatives market, acting as prime brokers,

liquidity

providers or clearing members. However, other financial institutions are increasingly ac-

tive and tend to take directional and often leveraged positions. Due to the complexity of the

exposure structure, potential spillovers and data limitations which can also restrict modelling

perimeter, estimating market risk from derivative exposures for financial stability assessment

and in stress tests remains a challenging task.

Recently, the failure of a family office Archegos (see ESMA (2022), SEC (2022)) has trig-

gered renewed regulatory attention to highly leveraged non-banks active in the derivative

markets, which pose counterparty risk to prime brokers. In particular, at the end of March

2021, a number of banks started to liquidate billions of dollars’ worth of various stocks on

positions tied to the total return swaps held by Archegos after it had failed to meet margin

calls. This sale caused an additional plunge of around 27% to some of the underlying stocks

and the share price of some broker banks themselves declined by around 15 % (CreditSuisse

(2021), ESMA (2022), SEC (2022)). Hence, any (long) position that had exposure to the

stocks whose price was falling owing to fire sales or stocks of the broker banks were immedi-

ately affected and potentially could have also triggered a default on margin requirements. In

this particular event, no other cases of unmet margin requirements were recorded; however,

the broker banks registered substantial quarterly losses (see Figure 1).

Figure 1: Reported losses of broker banks due to Archegos exposure (Source: ESMA (2022))

I focus on equity derivatives since prime brokerage is almost exclusively limited to this

derivative asset class. As of end year 2020, the notional outstanding in this market segment

in the Euro Area (EA) totaled EUR 25 trillion, which represents around 10% of the total

EA derivatives market in terms of notional. At the same time, elevated periods of volatility

and common leverage practices embeds increased exposure to market, liquidity and credit

risk for investors and banks providing broker services. Within the equity derivatives, this

ECB Working Paper Series No 2722 / September 20223paper focuses on the non-centrally cleared OTC1 equity market segment - equity swaps. The

reason for this choice is twofold. Firstly, as these contracts are mostly traded OTC, they

are less standardised and allow clients to potentially increase exposure beyond their liquidity

capacity across multiple dealer banks, such as in the case of Archegos. This type of syn-

thetic prime financing, where broker banks only pay-out the return (also negative) on stock

performance and receive an agreed rate in exchange, offer benefits to both parties: in some

jurisdictions clients do not need to disclose their exposures to the banks, while brokers enjoy

a more favorable regulatory treatment in comparison to a traditional prime financing (ESMA

(2022)). Secondly, other types of contracts present either additional modelling complexity,

e.g.

futures require modelling clearing network, default fund contributions and collateral

transformation channels, or, as in the case of options, do not entail a liability for the buyer,

typically a non-bank financial institution or a corporate. In the latter case the bank would

not be exposed to the type of risk I am trying to capture with the model.

The modelling approach in this paper complements and could enrich the modelling of

market risk in standard banking stress tests. Derivative exposures and, in general, the

bank’s trading book is subject to market risk capital requirements. During the EBA 2021

stress test, a third and largest (36.58bn EUR or 31%) of total market risk impact (118bn

EUR in the first year of adverse scenario) was estimated to originate from Net trading

income (NTI) and 17bn EUR of which were from financial assets and liabilities in the trading

book. However, the estimate is based on the static balance sheet assumption and mostly

relying on bank submitted data to the standardized templates (see EBA (2020, 2021)).

Under the current setup of EBA stress-testing (EBA (2020)) banks, on a highest group

consolidated level, are requested to provide supervisors with their projected losses that would

occur under an adverse scenario – a bottom-up estimate. The supervisor, on the other hand,

may employ a top-down2 and horizontal peer comparison approaches to challenge the bank’s

beliefs. For market risk exposures, the stress test covers full revaluation of various accounting

items in the balance sheet (such as items held for trading, hedging or other comprehensive

income), CCR3 and CVA4 provisions, revaluation of liquidity reserves and projection of

client revenues. While some aggregate information on the composition of the trading book

would be available in the provided templates, the type of concentrated derivative exposures,

such as stemming from prime brokerage business, and potential further spillovers cannot be

addressed as it requires additional and highly granular data set that would contain details

1Over-the-Counter
2A high-level assessment of overall exposures using internally developed models
3Counterparty Credit Risk
4Credit Valuation Adjustment

ECB Working Paper Series No 2722 / September 20224on contract characteristics.

Against this background, I propose a model to simulate losses from the derivatives trading

book in an adverse scenario, that can be used to challenge banks bottom-up estimates and to

enhance market risk treatment in stress tests. In short, I focus on losses in the trading book

from leveraged equity derivatives and related margin calls, by utilising granular transaction

data on euro area investors’ derivatives positions. The model considers the potential ampli-

fication effect of overlapping exposures across dealer banks by including the market impact

of fire sales of hedge by banks. In addition, the model also speaks to (synthetic) leverage

and liquidity risk in the financial system and in non-banks. Thus, it can be utilized more

generally for financial stability risk assessment and enhancing the macroprudential frame-

work for non-banks. This approach has several advantages. Firstly, it considers explicitly the

interaction between the banking system with its counterparties via an OTC market, which

is currently only partially considered in stress testing. Secondly, it highlights the importance

of overlapping exposures and how these can create spillover in the market, thus amplifying

individual losses. Thirdly, the approach is consistent with existing methodologies to assess

current capital and margin rules.

This paper contributes to a number of streams of literature. One category is related to

the use of micro-data to conduct stress test. Sydow et al. (2021) shows how the combined

endogenous reaction of banks and investment funds to an exogenous shock can amplify or

dampen losses to the financial system compared to results from single-sector stress testing

models. Fukker et al. (2022) study how overlapping portfolios provide a channel for financial

contagion induced by the market price impact of asset deleveraging. Bardoscia et al. (2019b)

analyse the network of exposures constructed by using the UK trade repository data and

study how liquidity shocks related to variation margins propagate across the network and

translate into payment deficiencies across different derivative markets. They find that in

extreme theoretical scenarios where liquidity buffers are small, a handful of institutions may

experience significant spillover effects due to the directionality of their portfolios. Similar

findings are shown in Jukonis et al. (2022a), who assess whether the current levels of funds’

holdings of cash and other highly liquid assets would be adequate to meet variation margin

calls on derivatives under a range of stress scenarios. The work of Jukonis and Thorin

(2022b) relate to the methodologies employed in the EBA stress test (EBA (2020)) and how

can micro-data be used in calibrating challenger views for assessing the non-linear equity

derivative exposures. Next, it also contributes to a growing body of research using TR5 data

5Trade repository

ECB Working Paper Series No 2722 / September 20225such as Abad et al. (2016), Bank of England (2018), Bardoscia et al. (2019a), Duffie et al.

(2015), FSB (2018), Hau et al. (2019), Jensen and Achord (2019).

Other closely related are the studies by Fricke (2021) and Molestina Vivar et al. (2020)

on synthetic leverage among funds. The former proposes a new measure of synthetic leverage

and finds that synthetically leveraged funds tend to underperform and display higher levels

of fragility while the latter demonstrates that outflows are larger during stressed periods

and after bad performance compared with unleveraged funds. The work of Aramonte et al.

(2021) describes a framework for the key channels of systemic-risk propagation in the presence

of NBFIs6 through margin requirements - leverage enables greater leverage, and spikes in

margins can lead to system-wide deleveraging (see also Schrimpf et al. (2020)).

The rest of the paper is organised as follows. Section 1 describes the data employed and

provides some stylised fact on the equity derivatives market. Section 2 introduces the model

set up and Section 3 describes the results of the simulations. Sections 4 concludes.

6Non-bank financial intermediaries

ECB Working Paper Series No 2722 / September 202261 Equity derivatives market: stylised facts

In this section I introduce the data that is employed throughout the paper, and describe how

the sample is constructed. Further, I present some stylised facts on the equity derivatives

market, clarifying how investors can build synthetic leverage by entering equity swaps.

1.1 Sample definition

The results presented in this paper are based on granular transaction-by-transaction EMIR

data on derivatives. EMIR data have been reported by counterparties resident in the EU

since February 2014 and include more than 120 data fields for each individual derivative

transaction.

I employ a sub-sample of the data set restricted to the trades reported by

counterparties located in the euro area (data which is available at the ECB). I use the set of

paired and deduplicated transactions in the trade state reports as defined in Perez-Duarte

and Skrzypczynski (2019). This is further enriched with data from the ECB’s Centralised

Securities Database (CSDB) on characteristics of individual securities and information from

Anna Derivatives Service Bureau (AnnaDSB) on characteristics of individual derivative con-

tracts.

From the EMIR data, I retrieve the derivatives representing equity asset class and with

underlying an equity instrument traded by euro area investors. The data set is highly

granular and complex, thus I performed extensive data manipulation and cleaning. Despite

this processing, the final data are still subject to some quality limitations. Most results

presented in this paper are based on a cleaned sub-sample using the reference date of 31

December 2020 to be able to compare it to EU-wide banking stress test templates, but also

a longer time series is considered to gauge the market structure. In general, the results of

this paper fall into a sample period that starts at Q4-2019 and ends in Q4-2021. When

selecting the sample from EMIR, I identify contracts by a number of criteria - the reporting

of contract type by the counterparties and its characteristics (maturity date, execution date,

payment frequencies, underlying reference rate, underlying security ISIN, etc), classification

of financial instrument (CFI) code, product ISIN as issued by AnnaDSB. In addition, I also

use fields that relate to collateral - Variation and Initial margins (VM and IM, respectively)

and overcollateralization. It is important to note that EMIR reporting does not allow to

identify the type collateral that is posted, but only it’s value. However, according to ISDA

(2022) survey, the regulatory IM collected by 20 largest market participants included 5.5% of

cash, 73.6% of government securities and 20.9% of other securities at year-end 2021. While

cash is more widely used for VM - the same firms reported receiving 43.8% of cash, 17.9%

government securities and 38.2% other securities to cover VM calls (same source).

ECB Working Paper Series No 2722 / September 20227Furthermore, I source the month end market prices of the underlying instruments using

CSDB database. To identify the trades conducted by euro area banks and their subsidiaries,

I filter the EMIR data based on the list of banks directly supervised by the SSM7. All

subsidiaries of the parent bank, together with the parent bank itself, are considered as one

entity when computing or aggregating the exposures. This approach is in-line with the

EBA stress-test requirements where banks submit their reports based on the highest level

of consolidation. In addition, I drop transactions that are marked as intragroup and create

additional flags if the direct or ultimate parent of both reporting sides are the same or have

any subordinated relationship. This identification is done with the help of GLEIF database.

Finally, I employ the classification from (Lenoci and Letizia, 2021) to identify the sector of

the counterparty.

1.2 Market overview

The EA equity derivatives market is smaller in terms of notional than other segments, but its

higher volatility embeds higher exposure to market risk for investors. This translates in high

margin requirements, especially in times of market turmoil, see e.g. Jukonis et al. (2022a),

and can lead to higher credit risk, especially for banks providing broker service to investors.

This justifies looking in more detail at this segment.

As of end year 2020, the notional outstanding in this market totalled EUR 25 trillion,

which represents around 10% of the EA derivatives market in terms of notional (see Figure

2). A significant part of the market - EUR 10 trillion notional outstanding - is represented

by Delta-1 derivatives, i.e. contracts that have no optionality, hence give investors the same

exposure as they would own the underlying asset. These are futures, which are standardised,

traded on exchanges (ETD) and cleared; and swaps, forwards and contracts-for-differences

(CFDS), which are traded OTC and typically not cleared. While options represent the bulk

of the total outstanding notional, they do not entail a liability for the buyer, in this context

typically a non-bank financial institution or a corporate. Hence the bank would not be

exposed to the type of risk I am trying to capture with the model, i.e. the client defaulting

on the margin account and thus triggering unwinding of the hedge by the bank.

Euro area banks play a crucial role in the market, owing to their activities of market

makers, and brokers in the OTC segment. Overall, 8 out of 10 EUR trillion of notional

outstanding sees a euro area bank as one of the counterparty in Delta-1 equity derivatives (as

reported by all euro area banks in the ECB EMIR sample). This figure includes both cleared

and non-cleared market segments and in case of the former, banks enjoy the almost exclusive

7https://www.bankingsupervision.europa.eu/banking/list/html/index.en.html

ECB Working Paper Series No 2722 / September 20228Figure 2: Derivatives market overview. Amounts are expressed in EUR trillion. (Source: EMIR
data available to the ECB. Reporting includes all transactions where at least one counterparty
is from euro area or where an underlying security is from euro area. For aggregation, paired
transactions in the trade state reports are used (as in Perez-Duarte and Skrzypczynski (2019))
in order to avoid double counting.)

status of being clearing members with the central counterparties (CCPs) and intermediating

the trades. As for swaps, banks trade with investors from the financial and non-financial

sectors, with EA investment funds being increasingly more active (around 25% yearly increase

in outstanding notional in Q2021) and with the exposures being mainly concentrated in funds

from three countries - see left panel in Figure 3. Banks acting as dealers typically hedge their

position in these contracts in two ways:

• Buy/Sell underlying stock

• Buy/sell Opposite swap (futures) for non-index (index) underlying.

This is evident when splitting individual banks’ portfolios by the side they take in the

contract (long or short), with major banks having fairly balanced positions. The same

picture emerges when looking into the templates reported by banks for the EU-wide banking

stress test8. For this purpose I utilised the templates for Full revaluation accounting items,

where banks report instruments held for trading (HfT) and their related economic hedges.

In making this comparison, I assume that all equity derivatives reported in EMIR belong the

the HfT portfolio - while this assumption could be strong for certain asset classes, notably

interest rate derivative, as they can be used to hedge the banking book, this seems more

reasonable when dealing with equity instruments, which seldom are reported as part of other

8See https://www.eba.europa.eu/risk-analysis-and-data/eu-wide-stress-testing

ECB Working Paper Series No 2722 / September 20229accounting portfolios. Focusing on the equity instruments, it is important to note that for the

stress test, banks have to report not only their derivatives positions, but also their exposures

via linear equity instruments (i.e. holding of stocks) - these are reported as L1 instrument

- which give a measure of the hedge. Further, templates include a breakdown for cleared

instruments, which allow to map L2 cleared as equity futures and L2 other and L3 as equity

swaps and CFDS.

Figure 3: EA bank role in the equity derivatives market. Left panel: EA fund exposure in
the equity swap (aggregate of all types, i.e. total return, variance, dividend, etc.) derivatives
market. Amounts are expressed in EUR billion. (Source: EMIR data available to the ECB)
Right panel: A snippet of bank positions in equity derivatives, by side. Banks selected based
on 2021 EBA Stress Test data submissions. The full data and amounts are not shown for
confidentiality reasons. (Source: 2021 EBA Stress test data)

The collected and reconciled data (a snippet is shown on the right panel of Figure 3) shows

that equity swaps have the largest notional exposure among Delta-1 derivatives traded by

sample banks by far. On the other hand, while futures also have some leverage arrangements,

they require modelling clearing network, default fund contributions and collateral transfor-

mation channels which are out of scope of this study. Hence the paper will focus only on

equity swaps.

1.3 Equity swaps in more detail

An equity swap is conceptually very similar to the more commonly used interest rate swap,

i.e. two counterparties accepts to exchange payments at predefined dates, each indexing a

certain rate.

In an equity swap the buyer gets a cash flow related to equity stock/index

performance and pays a reference rate (e.g. EURIBOR) on the notional. Depending on

the type of swap, the performance of the stock can measured in several ways, including

difference in stock price, dividend rate, volatility, variance, or total return (return on stock

ECB Working Paper Series No 2722 / September 202210plus dividend payouts). Notably, buyer’s cash flows can be negative depending on underlying

performance.

Despite being part of the non-cleared market segment and traded OTC, equity swaps are

typically fully collateralized and require both, initial and variation margin, which is a usual

practice among broker banks rather than a regulatory constraint.9 Variation margin (VM)

covers changes in the contract value, while initial margin (IM) is required at execution and

constitute the margin account. Similarly to most contracts, which require initial margin to

reflect the potential loss over 2 to 10 days in a tail event, depending on broker agreement,

the initial margin for equity swaps can be static or dynamic and reflects the tail risk of

the overall portfolio. For a non-model based IM, EMIR allows 15 % flat rate for equity

derivatives10 which is applicable to a total portfolio notional. This standard schedule was

slightly adjusted to take into account net-to-gross ratio, which is applicable to netting sets

(see Annex IV of EU2016/2251 (2016)).

1.4 Leveraged trading and overlapping portfolios

The fact that equity swaps are traded on margin accounts makes these instruments very ap-

pealing for clients that aim to build synthetic leverage. By entering an equity swap, clients

obtain synthetic exposure to the underlying stocks without owning them. This exposure is

leveraged as clients pay only a fraction of the notional value into the margin account (see

Figure 4 for a schematic representation). However, not all equity swap positions found in

the EMIR data can be attributed to synthetic leverage. To assess the leverage of market

participants (non-banks), the derivative (synthetic) positions would need to be linked with

on-balance sheet positions to see how derivatives positions are used - to gain synthetic expo-

sure or to hedge. For hedging purposes, market participants (bank clients) would need to sell

equity swap contracts in order to have a delta neutral exposure against their stock holdings

and the market value of their derivative exposures would increase (rather than decrease) in

case of falling stock prices, which would not result in an additional liquidity pressure. This

paper emphasizes the opposite situation - when market participants use brokerage services

to buy the equity swap contracts, this way obtaining the synthetic exposure.

In general, clients need to replenish their margin account when market moves against

them, by using either cash or HQLA securities. Depending on broker agreement, positions

9For market participants (financial counterparties and non-financial counterparties that exceed the clearing
threshold pursuant to Article 4a or 10 of EMIR) with a nominal volume of non-centrally cleared OTC derivatives
of more than € 3 billion at group level, the obligation to exchange initial and variation margins started on
4 February 2017. The requirement to exchange initial margins will apply to counterparties with an aggregate
average notional amount exceeding EUR 8 bln starting 1st Sep 2022, current threshold is EUR 50 bln.

10The rate varies per asset class, for Interest rate derivatives it is between 1% to 4%, Credit 2% to 10% (in

both cases depending on the maturity of the contracts), 10% for Commodities and 6% for FX.

ECB Working Paper Series No 2722 / September 202211Figure 4: Schematic representation of leveraged margin account

can be one-way collateralised, i.e., only client pays, but not the dealer bank. Importantly,

the amount to be paid in the margin account depends on the whole notional amount of the

contract so also a relatively modest market movement can generate a material margin call,

depending on the leverage. For investors with low liquid holdings, timing is crucial to allow

for collateral transformation, as failing to meet the call in a timely manner (typically one

day) can cause the position to be partially or entirely liquidated. Empirical observations

from EMIR data suggest that the leverage of investors in the euro area varies by sector but

on average is around 5 : 1. This is shown in Figure 5 as the ratio between the initial margin

received by the broker against the underlying notional (i.e. 1/Leverage) on each derivative

portfolio traded by the counterparties of various sectors.

Figure 5: Leverage in non-cleared equity derivative market embedded in portfolios held by
different sectors (Source: EMIR data available to the ECB). How to read the chart: for example,
20% means 5:1 leverage, so for a counterparty to have exposure of EUR 1000 to a certain security,
it needs to post EUR 200 in the margin account.

An investor may also seek to build its leveraged portfolios across multiple dealers banks,

ECB Working Paper Series No 2722 / September 202212e.g. to overcome limits imposed by each bank on the size of the portfolio. While banks should

in theory be aware of exposures towards other dealers, the level of disclosure is unclear and

recent evidence, see e.g. Branzoli et al. (2021), shows that the market is still very opaque.

In addition, EMIR data allows only partial view of the exposure (limited to EU or EA)

and granular data sharing across jurisdictions would need to be set up to identify leveraged

portfolios across multiple dealer banks. Nevertheless, despite limitations, granular data still

allows identifying the concentration in certain stocks by the same counterparty via different

broker banks (Figure 6). In such configuration, in the case of adverse market movement a

counterparty will receive several concurrent margin calls. Failure to meet such margin calls

will prompt the banks to unwind their hedge, potentially causing fire sales of the underlying

stock.

Figure 6: Portfolio overlap among EA banks - selected example. ISIN refers to an underlying
security of equity swap. Data shows that some counterparties have long exposures of the same
underlying across multiple dealers.

2 Model

This section describes the model I developed to measure the market risk from banks’ portfolio

in the equity derivatives market. In a nutshell, the model simulates several clients defaulting

on their margin calls following a market shock, to which banks react by liquidating, in part

or in total, the hedge, to keep their exposure delta neutral. When several banks trade

simultaneously on the same underlying, due to overlapping portfolio, the market impact

generates extra losses on their HfT portfolios, hence impacting the bank profit via the net

trading income (NTI).

ECB Working Paper Series No 2722 / September 202213Each run of the simulation uses as starting point the portfolios reported at end year 2020,

by d ∈ D dealer banks with their clients k ∈ K and consists of the following steps:

i. The equity market suffers an exogenous shock u on prices S of each underlying µ in

the sample:

t+1 = Sµ
Sµ

t × (1 + uµ) ∀µ

(1)

The shocks are calibrated to be within range of those applied in the last EU-wide

banking stress test where they depend on the geography and are within the interval

[−0.64, −0.32] (relative change in stock price, %), see (EBA, 2020).

In the model,

assuming an overall adverse scenario, for each individual stock µ a shock u is drawn from

a uniform distribution U(−0.55, −45), with shocks being uncorrelated across stocks.11

ii. Each bank d marks its positions on equity swaps and calculate margin requirements

(VM ) for its clients k.

VM µ

d,k = VM (Sµ

t+1, Sµ

0 , Qµ

d,k) = [∆MtMt+1(Sµ

t+1, Sµ

0 , Qµ

d,k)]+

where

MtMt(Sµ

t , Sµ

0 , Qµ

d,k) = w((L(0, δ) + r)N δ − Qµ

d,k(Sµ

t − Sµ

0 ))

(2)

(3)

where S0 is the underlying stock price at contract trade date, Qd,k is the quantity of

underlying stock/contracts between bank d and client k and N = Qd,k × S0 denotes

underlying notional amount. L(0, δ) is the reference rate that remunerates floating rate

leg (such as EURIBOR, LIBOR, etc), r is a spread and δ denotes the length of the

contract in years. The specification (3) refers to the so called bullet swap where cash

flows are exchanged on a single date. It is straightforward to generalise this formula

for the case with more resetting dates and cash flows.

In addition, an equivalent

formulation is when return is expressed in relative and not absolute terms. In this case
equity leg would be (Sµ

0 × N . Finally w ∈ {−1, 1} denotes the side that the
bank takes in the contract, with w = 1 being short (in which case it would sell the

t − Sµ

0 )/Sµ

contract by paying out the stock performance and receiving fixed rate on notional).

When repricing the contracts, all contracts that have more than one cash flow reset

11The range of impacts is within the bounds of market risk scenario for equities as defined EU-wide banking
stress test. This scenario of static shocks is calibrated based on a multivariate copula approach which already
embeds a historical correlation structure (for details see Rancoita and Ojea Ferreiro (2019))

ECB Working Paper Series No 2722 / September 202214date and maturity longer than 1 year were valued as 1 year bullet swaps12. Since

the current architecture of EMIR reporting does not allow to consistently identify the

spread r, it was set at 2%.

The total margin call from bank d to client k is the sum across the entire portfolio:

VMd,k =

VM µ
d,k

(cid:88)

µ

while client k needs to meet margin calls for a total amount given by

VMk =

(cid:88)

d

VMd,k

(4)

(5)

iii. With probability ps a client k fails to meet a fraction θs of the new margin call VMk. I

assume both p and θ depend only the sector s = s(k) of the client, hence all dealers of

a defaulting investor are impacted proportionally to their exposure, i.e. all banks are
treated equally by the debtor. I define the set of defaulting clients as ¯K.

iv. When considering the loss, I distinguish 2 cases - full and partial liquidation. In the

first case the amount of stock sold is equal to Qµ

d,k. In the second case, banks liquidate

part of their hedges and sell an amount of stock X such that:

VM (Sµ

t+1, Sµ

0 , Qµ

d,k − X µ

d,k) = θs(k)VM (Sµ

t+1, Sµ

0 , Qµ

d,k) .

The amount X µ

d,k can be approximated by

X µ

d,k ≈ min


Qµ

d,k,

(cid:32)

Qµ

d,k −

d,k + MtMt(Sµ
θs(k)VM µ
Sµ
0 (1 + r) − Sµ

t+1

t , Sµ

0 , Qµ

d,k)

(6)

(7)

(cid:33)+


which is obtained by simply substituting (3) into (6), setting L(0, δ) = 0 (which is

not unrealistic assumption given the current low rate environment) and solving for X.
Clearly, if X µ

d,k, the position is fully sold.

d,k = Qµ

v. Price impact for stock µ, denoted as P µ, applies to securities being liquidated, which

suppresses realisation values for dealers and increases marked losses in the balance

sheet. The impact is based on Fukker et al. (2022), where it is derived by applying a

quantile regression for securities at ISIN level. One of the main advantages of this ap-

proach is the inclusion of the system-level return as a main component in security-level

price changes defined for different security buckets, this way solving a high degree of

12Note that δ ≤ 1 in all cases.

ECB Working Paper Series No 2722 / September 202215sensitivity to potential outliers. In addition, it allows to obtain a concave representa-

tion with a full distribution of impacts for each security. Nevertheless, the technique

employed here is heavily data dependent and relies on a number of sources which often

can be scarce or not available. The missing cases are handled by averaging and pro-

jecting the impacts based on geographical, sector and capitalization level. The impact

on indices is aggregated based on geographical criteria by computing the average im-

pact. Alternative modelling approaches could include linear (Kyle (1985)), square-root

(Bouchaud et al. (2009)) or other concave (Bouchaud et al. (2009), T´oth et al. (2011))

representations that would offer a relatively more flexible and less data dependant in-

ference. However, this does not offer the benefits of a quantile regression approach

that allows to repeat the impact assessment for the full distribution which is neces-

sary in modelling fire sales dynamics with deleveraging pressures and assessing policy

implications on controlled liquidation.

Importantly, when computing the price impact, the volumes sold by each bank con-

nected to the defaulting clients are considered jointly, also those not located in the

euro area. In this way the model captures the loss amplification due to overlapping

portfolios across dealers banks, and partially the cross-border contagion. The exposure

to non-EA dealers is available only when the client is domiciled in the EA, due to its

reporting obligation. Similarly, by considering all clients of EA dealer bank is possible

to capture also those located outside EA, and gain additional insight on the potential

contagion. This is is one the advantage of using an activity based dataset as opposed

to supervisory data alone, as it allow to overcome, at least in part, the limit imposed

by jurisdictions and supervisory perimeters.

vi. Banks recognise the loss. Losses can stem from both types of hedging, i.e. when the

dealer bank holds the underlying stock referenced on defaulted contracts, or it entered

another derivatives contract in the opposite direction13. If banks initially fully hedged

by holding a stock at price S0, the loss due to liquidation (denoted as LL - Liquidation

Loss) is recognised in the profit and loss (PnL) statement via the NTI, and it is given

by:

LLµ

d =

(cid:88)

k∈ ¯K

X µ

d,k(Sµ

t+1 − Sµ

0 ) < 0 .

(8)

On the other hand, if banks hedge fully by taking opposite trade (in which case they

13While excluded from the sample, it is important to note that the return on securities as offered by prime

brokers often includes performance of basket of securities.

ECB Working Paper Series No 2722 / September 202216do not own the physical stock, but only exchange opposite financial cash flows), the

reduction of short position, would create a mismatch between the hedge legs, and the

loss would materialise in the PnL of their balance sheet (where the sum of long position

is restricted to related hedges):

HLµ

d =

(cid:88)

k∈ ¯K

MtM short

t+1 (P µSµ

t+1, Sµ

0 , Qµ

d,k − X µ

d,k) +

MtM long

t+1 (P µSµ

t+1, Sµ

0 , Qµ

d,k) < 0 .

(cid:88)

k /∈ ¯K

(9)

The total loss depends on how the banks built the hedge for each underlying and the

decision to unwind one or both hedges in case θ < 1, which is an information difficult

to reconstruct from the datasets employed here. Further, a bank can decide to use

a different but correlated stock to hedge one exposure, e.g.

if the exact hedge is not

available or illiquid. However, EMIR data do not indicate the purpose of a trade or

what accounting category it is assigned to. To partially overcome this shortcoming, if

one indicates with αµ ∈ [0, 1] the fraction of exposure in equity µ hedged via holding

the stock itself, the total loss for the bank can be approximated by the following:

Ld =

(cid:88)

µ

αµLLµ

d + (1 − αµ)HLµ
d .

(10)

In case of full liquidation and due to leveraged margin account, the broker would keep

a fraction (or full amount, depending on a loss) of posted initial margin. In particular,
the loss in NTI will be adjusted net of received collateral, indicated as IM µ

d,k (Initial

Margin).

ˆLL

µ
d =

(cid:88)

k∈ ¯K

(Qµ

d,k(Sµ

0 − Sµ

t+1) − IM µ

t,d,k)+

(11)

Based on the empirical observation from Figure 5, it is assumed that Initial Margin

account is always at 20% of the maximum between underlying notional and current

market value of the position, i.e.

IM µ

t,n,k = 20% × max(N µ

n,k, MtMt(Sµ

t , Sµ

0 , Qµ

n,k))

(12)

In terms of stress test templates, such loss would need be compared to that reported

in the Full revaluation table, as Total impact HfT for linear equity instruments and

related derivatives under the category HfT and their related economic hedges.

ECB Working Paper Series No 2722 / September 202217vii. Finally, I define a metric for loss amplification by

LA =

(cid:80)
d

where Ld is defined in Eq. (10).

(cid:80)

d Ld

(cid:80)

k∈ ¯K θs(k)VMk

(13)

The advantages of this set up are several. Fist, it allows to enhance the quality assurance

process during the stress test with granular data and challenge bank’s potential capital

depletion in the integrated system of exposures and beyond the self risk assessment. Secondly,

it provides a background for further integration to the more general system-wide stress

testing framework such as Sydow et al. (2021) where banks would be interconnected beyond

investment funds.

3 Results

This section presents the results of the simulations, where the model is run with different set

of values for the parameters θs and ps. Each simulation comprises 10000 runs with 75 banks

and results in two distributions of losses for dealers, LLd and HLd. The results will show

1% quantile and mean of these loss distributions across banks. As mentioned in Section 2,

it is difficult, with the data at hand, to have the precise composition of the dealers’ hedge,

so both LLd and HLd are reported separately. While model outcomes are available at bank

level, here I present distributions across all banks to preserve confidentiality.

Subsection 3.1 presents the results for the case in which bank fully liquidate their hedge

in response to the missed margin call. Here only ps will vary.

Subsection 3.2 considers partial liquidation in which banks can decide to keep the position

open in case of missed margin call by the client. This means both θs and ps can vary.

Subsection 3.3 maps the results to actual observed losses of banks during the Archegos

incident and market risk impact during the EBA stress test.

3.1 Full liquidation

Figure 7 shows results for full liquidation for a different set of default probabilities across

sectors. In particular, x -axis shows a pair of default probabilities (PDs) assigned for counter-

parties from certain sectors during each simulation run. The first entry in the pair indicates

the PD of a counterparty that is classified either Non-financial, Pension fund, Insurance

company or Other (not including Investment fund or OFI ), while the second entry applies

ECB Working Paper Series No 2722 / September 202218to a counterparty labelled Investment fund or OFI 1415.

Results suggest, and as can be observed by consistent outliers in the boxplot distributions,

that the losses are concentrated on large banks which have also significantly larger positions.

The most adverse case is for the 1% quantile and PD set (0.07, 0.09) where the losses for

biggest dealers in NTI range from around EUR -1 bln to EUR -2.4 bln and from EUR -0.8

bln to EUR -4 bln due to the mismatch between hedge positions. Hedge loss is somewhat

larger than the liquidation loss (NTI) in all considered cases and this difference is precisely

the price impact after the initial sell-off amplified by the interconnection between clients and

brokers. The dealer is left with partially hedged position which it needs to mark against

the prevailing adverse market prices. In turn, it is affected by the common fire sale from all

involved banks that close positions with a defaulted counterparty.

Figure 7: Bank losses due to full liquidation. Each box plot shows distributions of 1 % quantile
(left panel) and mean (right panel) losses for each bank due to fire sale and hedge accounting.

3.2 Partial liquidation

Figure 8 shows results for the case of partial liquidation. The parameter θ is shown on the

x -axis and varies between 0 and 0.8. The default probabilities were fixed at (0.07, 0.09) for

all counterparty sector cases as described in the previous section.

When θ = 0 results are comparable to the case of full liquidation. This observation

suggests that it could be beneficial to design policies that would force controlled liquidation

in case of large mutual losses across banks on certain securities. In particular, already in the

case when θ = 0.4 the losses are approximately cut by half. However, this requires a more

careful study of potential permanent or transient price impact behaviour.

14Other Financial Intermediary
15For example, for a pair (0.07, 0.09), at each simulation run, the probability to default for a counterparty
representing Non-financial, Pension fund, Insurance company or Other (not including Investment fund or OFI )
sectors is 7%. Similarly, the probability to default for a counterparty which is an Investment fund or OFI is 9%.

ECB Working Paper Series No 2722 / September 202219Figure 8: Bank losses due to partial liquidation. Each box plot shows distributions of 1 %
quantile (left panel) and mean (right panel) losses for each bank due to fire sale and hedge
accounting.

3.3 Impact materiality

As it can be observed in Figure 1, the total impact during the Archegos incident for the

involved banks amounted to around EUR 10 bln and ranged from EUR 0.3 bln to EUR

4.7 bln. In this case, banks fully unloaded their securities which were used for hedging the

equity swap exposures. In the model setup that is described in this paper, this is illustrated

in Figure 7 and is labeled as NTI - full liquidation. The fire-sale from involved banks in

Archegos case caused an additional plunge of around 27 % to some of the underlying stocks

and the share price of some broker banks themselves declined by around 15 % (CreditSuisse

(2021), ESMA (2022), SEC (2022)). However, when a partial liquidation is employed, losses

are substantially reduced as is shown in Figure 8. Here θ illustrates the fraction of position

that is not sold immediately and this approach can be used to find optimal liquidation

strategy with least market and price impact.

As observed by ESMA (2022), due to some difference in disclosure and reporting regula-

tions of that time between EU and non-EU jurisdictions, Archegos had less opportunities to

build these positions with dealers from EA. Consequently, similar practice would be also ap-

plicable to other clients that want to enter similar type of contracts. Hence one would expect

that within the EA there is less space for this type of scenario to materialize. Nevertheless,

results show that losses across dealer banks are still material while the data available to the

ECB gives only a partial picture. The 2021 EBA stress test estimates that the total impact

to market risk segment from NTI was around EUR 36.58 bln (EBA (2021)). However, some

of the losses in the trading book might not be marked as this type of assessment does not

utilize granular data and assumes a static balance sheet. Hence, the model described here

could be used to enrich the market risk assessment during the bank stress tests also from a

ECB Working Paper Series No 2722 / September 202220system perspective.

4 Conclusions

This paper introduces a model to estimate market risk for bank’ derivatives exposures in an

adverse scenario.

The proposed model utilises granular transaction data on euro area investors’ derivatives

positions, and it allows to estimate the losses banks will incur would some of their coun-

terparties fail to pay their margins, forcing banks to liquidate their positions. The model

considers the potential amplification effect of overlapping exposures across dealer banks by

including the market impact of fire sales of hedge by banks. Simulations show that effects

are heterogeneous, with some banks being particularly hit, due their high interconnectedness

in the market and high concentration in their exposures.

Despite the high level of insights that this framework provides, it still bears some limi-

tations. Firstly, the framework is data and computational intensive, primarily for the need

to filter and reprice thousands of exposures and for the simulation setup. Secondly, the

computation of the actual loss depends on the exact composition of the hedge, which is

not considered here. Finally, a more accurate probability of default for clients would be

based on their individual liquidity holdings, rather than their sector. While the framework

is flexible enough to accommodate certain improvements, and could be further extended to

capture other market segments (e.g. FX, interest rate), granular data sharing across juris-

dictions (EU and non-EU) would need to be set up to identify leveraged portfolios across

multiple dealer banks and to better capture the price impact effect. In fact, one goal of the

OTC derivatives reform was to make the market more transparent but individual regulators

continue to have limited view on this global market.

In the model setup the losses for banks with largest concentration of equity swap expo-

sures range from around EUR -1 bln to EUR -2.4 bln in a full liquidation scenario (losses

during the Archegos incident for the involved banks ranged from EUR 0.3 bln to EUR 4.7

bln and totalled around EUR 10 bln). In addition, the 2021 EBA stress test estimates that

the total impact to market risk segment from NTI was around EUR 36.58 bln (EBA (2021)).

However, some of the losses in the trading book might not be marked as this type of assess-

ment does not utilize granular data. Hence, the model described here could be used to enrich

the market risk assessment during the bank stress tests also from a system perspective.

In addition, this approach could potentially be used to calibrate macroprudential mea-

sures in case of severe market turmoil, to mitigate the effect of a disorderly liquidation of their

hedge by affected dealers. One such measure could consider a controlled liquidation, where,

ECB Working Paper Series No 2722 / September 202221following the default of an interconnected clients, the hedges against its positions are pooled

together and liquidated in smaller tranches, hence reducing the overall market impact and

removing first-move advantage among dealers in need to re-balance their positions. Another

measure could envisage limits on the leverage of investors, which decline with the size of the

exposures, taking into account the overlaps among dealers over the same counterparties and

underlying.

Finally, although the model presented here is bank-centered, it also speaks to (synthetic)

leverage and liquidity risk in the financial system and in non-banks. This sector has grown in

size over the last decades and still lacks a comprehensive macroprudential policy framework.

Developing and enhancing such a framework would in turn limit the counterparty risk for

banks.

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ECB Working Paper Series No 2722 / September 202225Acknowledgements 

I am kindly grateful for comments and suggestions received from Elisa Letizia. I also thank anonymous referee, Matthias Sydow, Gabor 
Fukker and the participants of seminar organised by the Working Group on Stress Testing. 
This paper should not be reported as representing the views of the European Central Bank (ECB). The views expressed are those of the 
author and do not necessarily reflect those of the European Central Bank or the Eurosystem. This paper was completed while Audrius 
Jukonis was at the European Central Bank. 

Audrius Jukonis 
European Central Bank, Frankfurt am Main, Germany; email: audrius.jukonis@ecb.europa.eu 

© European Central Bank, 2022 

Postal address  60640 Frankfurt am Main, Germany 
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This paper can be downloaded without charge from www.ecb.europa.eu, from the Social Science Research Network electronic library or 
from RePEc: Research Papers in Economics. Information on all of the papers published in the ECB Working Paper Series can be found 
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