Stéphane Crépey

Stéphane Crépey
University of Évry Val d'Essonne · Département de Mathématiques

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107
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Introduction
Skills and Expertise

Publications

Publications (107)
Preprint
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Emissions markets play a crucial role in reducing pollution by encouraging firms to minimize costs. However, their structure heavily relies on the decisions of policymakers, on the future economic activities, and on the availability of abatement technologies. This study examines how changes in regulatory standards, firms' abatement costs, and emiss...
Preprint
We present a unified framework for computing CVA sensitivities, hedging the CVA, and assessing CVA risk, using probabilistic machine learning meant as refined regression tools on simulated data, validatable by low-cost companion Monte Carlo procedures. Various notions of sensitivities are introduced and benchmarked numerically. We identify the sens...
Preprint
Invariance times are stopping times $\tau$ such that local martingales with respect to some reduced filtration and an equivalently changed probability measure, stopped before $\tau$ , are local martingales with respect to the original model filtration and probability measure. They arise naturally for modeling the default time of a dealer bank, in t...
Preprint
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Crépey, Frikha, and Louzi (2023) introduced a nested stochastic approximation algorithm and its multilevel acceleration to compute the value-at-risk and expected shortfall of a random financial loss. We hereby establish central limit theorems for the renormalized estimation errors associated with both algorithms as well as their averaged versions....
Preprint
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For vanilla derivatives that constitute the bulk of investment banks' hedging portfolios, central clearing through central counterparties (CCPs) has become hegemonic. A key mandate of a CCP is to provide an efficient and proper clearing member default resolution procedure. When a clearing member defaults, the CCP can hedge and auction or liquidate...
Preprint
Darwinian model risk is the risk of mis-price-and-hedge biased toward short-to-medium systematic profits of a trader, which are only the compensator of long term losses becoming apparent under extreme scenarios where the bad model of the trader no longer calibrates to the market. The alpha leakages that characterize Darwinian model risk are undetec...
Article
We propose a bottom-up quantitative reverse stress testing framework that identifies forward-looking fragilities tailored to a bank's portfolio, credit and funding strategies, models, and calibration constraints. Thus, instead of relying on historical events, we run a Monte Carlo simulation, and we mine those future states that contribute the most...
Preprint
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We propose a multilevel stochastic approximation (MLSA) scheme for the computation of the Value-at-Risk (VaR) and the Expected Shortfall (ES) of a financial loss, which can only be computed via simulations conditional on the realization of future risk factors. Thus, the problem of estimating its VaR and ES is nested in nature and can be viewed as a...
Preprint
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We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization...
Article
We consider the computation by simulation and neural net regression of conditional expectations, or more general elicitable statistics, of functionals of processes (X,Y)$(X,Y)$. Here an exogenous component Y (Markov by itself) is time‐consuming to simulate, while the endogenous component X (jointly Markov with Y) is quick to simulate given Y, but i...
Preprint
We consider the computation by simulation and neural net regression of conditional expectations, or more general elicitable statistics, of functionals of processes $(X, Y )$. Here an exogenous component $Y$ (Markov by itself) is time-consuming to simulate, while the endogenous component $X$ (jointly Markov with $Y$) is quick to simulate given $Y$,...
Preprint
We propose a non-asymptotic convergence analysis of a two-step approach to learn a conditional value-at-risk (VaR) and expected shortfall (ES) in a nonparametric setting using Rademacher and Vapnik-Chervonenkis bounds. Our approach for the VaR is extended to the problem of learning at once multiple VaRs corresponding to different quantile levels. T...
Preprint
We revisit Burnett (2021b,a)'s notion of hedging valuation adjustment (HVA) in the direction of model risk. The resulting HVA can be seen as the bridge between a global fair valuation model and the local models used by the different desks of the bank. However, model risk and dynamic hedging frictions, such as transaction costs {\`a} la Burnett (202...
Preprint
We present a one-period XVA model encompassing bilateral and centrally cleared trading in a unified framework with explicit formulas for most quantities at hand. We illustrate possible uses of this framework for running stress test exercises on a financial network from a clearing member's perspective or for optimizing the porting of the portfolio o...
Article
Full-text available
Since the 2008 crisis, derivative dealers charge to their clients various add-ons, dubbed XVAs, meant to account for counterparty risk and its capital and funding implications. As banks cannot replicate jump-to-default related cash flows, deals trigger wealth transfers and shareholders need to set capital at risk. We devise an XVA policy, whereby s...
Preprint
We devise a neural network based compression/completion methodology for financial nowcasting. The latter is meant in a broad sense encompassing completion of gridded values, interpolation, or outlier detection, in the context of financial time series of curves or surfaces (also applicable in higher dimensions, at least in theory). In particular, we...
Article
XVAs denote various counterparty risk related valuation adjustments that are applied to financial derivatives since the 2007–2009 crisis. We root a cost-of-capital XVA strategy in a balance sheet perspective which is key to identifying the economic meaning of the XVA terms. Our approach is first detailed in a static setup that is solved explicitly....
Preprint
Full-text available
XVAs denote various counterparty risk related valuation adjustments that are applied to financial derivatives since the 2007--09 crisis. We root a cost-of-capital XVA strategy in a balance sheet perspective which is key in identifying the economic meaning of the XVA terms. Our approach is first detailed in a static setup that is solved explicitly....
Article
Full-text available
Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for...
Preprint
Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for...
Chapter
Since the 2008–2009 financial crisis, banks have introduced a family of XVA metrics to quantify the cost of counterparty risk and of its capital and funding implications: the credit/debt valuation adjustment (CVA and DVA), the costs of funding variation margin (FVA) and initial margin (MVA), and the capital valuation adjustment (KVA). We revisit fr...
Article
Based on an XVA analysis of centrally cleared derivative portfolios, we consider two capital and funding issues pertaining to the efficiency of the design of central counterparties (CCPs). First, we consider an organization of a clearing framework, whereby a CCP would also play the role of a centralized XVA calculator and management center. The def...
Article
Full-text available
Since the 2008–2009 financial crisis, banks have introduced a family of X-valuation adjustments (XVAs) to quantify the cost of counterparty risk and of its capital and funding implications. XVAs represent a switch of paradigm in derivative management, from hedging to balance sheet optimization. They reflect market inefficiencies that should be comp...
Preprint
Full-text available
Modeling counterparty risk is computationally challenging because it requires the simultaneous evaluation of all the trades with each counterparty under both market and credit risk. We present a multi-Gaussian process regression approach, which is well suited for OTC derivative portfolio valuation involved in CVA computation. Our approach avoids ne...
Preprint
Modeling counterparty risk is computationally challenging because it requires the simultaneous evaluation of all the trades with each counterparty under both market and credit risk. We present a multi-Gaussian process regression approach, which is well suited for OTC derivative portfolio valuation involved in CVA computation. Our approach avoids ne...
Article
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We prove that the default times (or any of their minima) in the dynamic Gaussian copula model of Cr{\'e}pey, Jeanblanc, and Wu (2013) are invariance times in the sense of Cr{\'e}pey and Song (2017), with related invariance probability measures different from the pricing measure. This reflects a departure from the immersion property, whereby the def...
Preprint
We prove that the default times (or any of their minima) in the dynamic Gaussian copula model of Cr{\'e}pey, Jeanblanc, and Wu (2013) are invariance times in the sense of Cr{\'e}pey and Song (2017), with related invariance probability measures different from the pricing measure. This reflects a departure from the immersion property, whereby the def...
Article
On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$ we consider two filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time $\theta$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$ predictable processes on $(0,\theta]$. In this setup it is well-known that, for any $\mathbb{F}$ semimartingale $X$,...
Preprint
On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$ we consider two filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time $\theta$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$ predictable processes on $(0,\theta]$. In this setup it is well-known that, for any $\mathbb{F}$ semimartingale $X$,...
Chapter
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Two nonlinear Monte Carlo schemes, namely, the linear Monte Carlo expansion with randomization of Fujii and Takahashi (Int J Theor Appl Financ 15(5):1250034(24), 2012 [9], Q J Financ 2(3), 1250015(24), 2012, [10]) and the marked branching diffusion scheme of Henry-Labordère (Risk Mag 25(7), 67–73, 2012, [13]), are compared in terms of applicability...
Article
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We develop an XVA analysis of centrally cleared trading, parallel to the one that has been developed in the last years for bilateral transactions. A dynamic framework incorporates the sequence of the cash-flows involved in the waterfall of resources of the CCP. The total cost of the clearance framework for a member of the clearinghouse, called CCVA...
Article
We study a BSDE with random terminal time that appears in the modeling of counterparty risk in finance. We model the terminal time as an invariant time, i.e. a time such that local martingales with respect to a reduced filtration and a possibly changed probability measure, once stopped right before that time, stay local martingales with respect to...
Article
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We develop a multi-curve term structure set-up in which the modelling ingredients are expressed by rational functionals of Markov processes. We calibrate to London Interbank Offer Rate swaptions data and show that a rational two-factor log-normal multi-curve model is sufficient to match market data with accuracy. We elucidate the relationship betwe...
Article
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The recent publication of the IFRS 9 norms related to collective provisions for non defaulted instruments has settled a new vision to banking book portfolios. In this paper we show that the IFRS 9 provision measured through the Expected Credit Loss (ECL), inspired from a market vision on loan books, is very similar to the Credit Value Adjustment (C...
Article
We consider the problem of valuation of interest rate derivatives in the post-crisis setup. We develop a multiple-curve model, set in the HJM framework and driven by a L�evy process. We proceed with joint calibration to caps and swaptions of different tenors, the calibration to caps guaranteeing that the model correctly captures volatility smile e�...
Article
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URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/documents-de-travail/
Article
From a broad perspective, this work deals with the question of reduction of filtration, i.e., given a stopping time θ relative to a full model filtration G, when and how to separate the information that comes from θ from a reference filtration in order to simplify the computations. Toward this aim, some kind of local martingale invariance property...
Book
Solve the DVA/FVA Overlap Issue and Effectively Manage Portfolio Credit Risk Counterparty Risk and Funding: A Tale of Two Puzzles explains how to study risk embedded in financial transactions between the bank and its counterparty. The authors provide an analytical basis for the quantitative methodology of dynamic valuation, mitigation, and hedging...
Article
We devise a bottom-up dynamic model of portfolio credit risk where instantaneous contagion is represented by the possibility of simultaneous defaults. Due to a Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a two-step procedure, much like in a standard static copula setup....
Article
Full-text available
In order to dynamize the static Gaussian copula model of portfolio credit risk, we introduce a model filtration made of a reference Brownian filtration progressively enlarged by the default times. This yields a multidimensional density model of default times, where, as opposed to the classical situation of the Cox model, the reference filtration is...
Article
We consider a bottom-up Markovian copula model of {portfolio} credit risk where instantaneous contagion is possible in the form of simultaneous defaults. Due to the Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a two-steps procedure, much like in a standard static copula s...
Article
In "Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model", the authors introduced a Markov copula model of portfolio credit risk where pricing and hedging can be done in a sound theoretical and practical way. Further theoretical backgrounds and practical details are developed in "A Bottom-Up Dynamic Model of Portfolio Credit Risk. Part...
Article
In this paper, we prove that the conditional dependence structure of default times in the Markov model of "A Bottom-Up Dynamic Model of Portfolio Credit Risk. Part I: Markov Copula Perspective" belongs to the class of Marshall-Olkin copulas. This allows us to derive a factor representation in terms of "common-shocks", the latter being able to trigg...
Article
We devise simulation/regression numerical schemes for pricing the CVA on CDO tranches, where CVA stands for Credit Valuation Adjustment, or price correction accounting for the defaultability of a counterparty in an OTC derivatives transaction. This is done in the setup of a continuous-time Markov chain model of default times, in which dependence be...
Article
We study the numerical approximation of doubly reflected backward stochastic differential equations with intermittent upper barrier (RIBSDEs). These denote reflected BSDEs in which the upper barrier is only active on certain random time intervals. From the point of view of financial interpretation, RIBSDEs arise as pricing equations of game options...
Chapter
Have you ever seen a financial contract which only pays one single cash-flow at a terminal maturity T? There are some (I heard “vanillas”!), but not so many. Ever seen a continuous-time coupon stream? I’d be surprised. More broadly, to account for many real-life practical features such as discrete dividends or discrete path-dependence, we need to e...
Chapter
Our purpose in this chapter is to give an overview of the basics of stochastic calculus, an important mathematical tool that is used in control engineering, in modern finance and insurance, and in modern management science, among other fields. The chain rule of stochastic calculus, the so-called Itô formula, is one of the most used mathematical (or...
Chapter
Prices of liquid financial instruments are given by the market and are determined by supply-and-demand. Calibrating a model means finding numerical values of its parameters such that the prices of market instruments computed within the model, at a given time, coincide with their market prices. Liquid market prices are thus actually used by models i...
Article
This and the follow-up paper deal with the valuation and hedging of bilateral counterparty risk on over-the-counter derivatives. Our study is done in a multiple-curve setup reflecting the various funding constraints (or costs) involved, allowing one to investigate the question of interaction between bilateral counterparty risk and funding. The firs...
Article
The correction in value of an over-the-counter derivative contract due to counterparty risk under funding constraints is represented as the value of a dividend-paying option on the value of the contract clean of counterparty risk and excess funding costs. This representation allows one to analyze the structure of this correction, the so-called Cred...
Article
The credit crisis and the ongoing European sovereign debt crisis have highlighted the native form of credit risk, namely the counterparty risk. The related Credit Valuation Adjustment, (CVA), Debt Valuation Adjustment (DVA), Liquidity Valuation Adjustment (LVA) and Replacement Cost (RC) issues, jointly referred to in this paper as Total Valuation A...
Article
The 2007 subprime crisis has induced a persistent disconnection between the LIBOR derivative markets of different tenors and the OIS swap market. Commonly proposed explanations for the corresponding spreads are a combination of credit risk and liquidity risk. However in these explanations the meaning of the terms credit and liquidity is typically n...
Article
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In the aftermath of the 2007–2009 financial crisis, a variety of spreads have developed between quantities that had been essentially the same until then, notably LIBOR–OIS spreads, LIBOR–OIS swap spreads, and basis swap spreads. By the end of 2011, with the sovereign credit crisis, these spreads were again significant. In this paper we study the va...
Article
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A Markov model is constructed for studying the counterparty risk in a CDS contract. The "wrong-way risk" in this model is accounted for by the possibility of the common default of the reference name and of the counterparty. A dynamic copula property as well as affine model specifications make pricing and calibration very efficient. We also consider...
Chapter
A Markov model is constructed for studying the counterparty risk in a CDS contract. The “wrong-way risk” in this model is accounted for by the possibility of the common default of the reference name and of the counterparty. A dynamic copula property as well as affine model specifications make pricing and calibration very efficient. We also consider...
Article
We deal with the issue of pricing convertible bonds numerically, using simulation. A convertible bond can be seen as a coupon-paying and callable American option. Moreover, call times are typically subject to constraints, called call protections, preventing the issuer from calling the bond in certain subperiods of time. The nature of the call prote...
Chapter
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In this paper, we study convertible securities (CS) in a primary market model consisting of: a savings account, a stock underlying a CS, and an associated CDS contract (or, alternatively to the latter, a rolling CDS more realistically used as an hedging instrument). We model the dynamics of these three securities in terms of Markovian diffusion set...
Article
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While the Gaussian copula model is commonly used as a static quotation device for CDO tranches, its use for hedging is questionable. In particular, the spread delta computed from the Gaussian copula model assumes constant base correlations, whereas we show that the correlations are dynamic and correlated to the index spread. It might therefore be e...
Article
We study mathematical aspects of dynamic hedging of Credit Valuation Adjustment (CVA) in a portfolio of OTC financial derivatives. Since the sub-prime crisis, counterparty risk and wrong way risk are a crucial issue in connection with valuation and risk management of credit derivatives. In this work we first derive a general, model free equation fo...
Article
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This paper discusses the main modeling approaches that have been developed for handling portfolio credit derivatives, with a focus on the question of hedging. In particular, the so-called top, top down and bottom up approaches are considered. We give some mathematical insights regarding the fact that information, namely the choice of a relevant mod...
Chapter
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In this article we study a decoupled forward backward stochastic differential equation (FBSDE) and the associated system of partial integro-differential obstacle problems, in a flexible Markovian set-up made of a jump-diffusion with regimes. These equations are motivated by numerous applications in financial modeling, whence the title of the paper....
Chapter
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This set of lecture notes is concerned with the following pair of ideas and concepts: 1. The Skorokhod Embedding problem (SEP) is, given a stochastic process X=(X t ) t≥0 and a measure μ on the state space of X, to find a stopping time τ such that the stopped process X τ has law μ. Most often we take the process X to be Brownian motion, and μ to...
Chapter
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These lecture notes cover a major part of the crash course on financial modeling with jump processes given by the author in Bologna on May 21–22, 2009. After a brief introduction, we discuss three aspects of exponential Lévy models: absence of arbitrage, including more recent results on the absence of arbitrage in multidimensional models, propertie...
Chapter
In this first chapter, we show that a CDO tranche payoff can be perfectly replicated with a self-financed strategy based on the underlying credit default swaps. This extends to any payoff which depends only upon default arrivals, such as basket default swaps. Clearly, the replication result is model dependent and relies on two critical assumptions....
Chapter
This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.
Article
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The goal of the paper is the numerical analysis of the performance of Monte Carlo simulation based methods for the computation of credit-portfolio loss-distributions in the context of Markovian intensity models of credit risk. We concentrate on two of the most frequently touted methods of variance reduction in the case of stochastic processes: impo...
Chapter
An important issue in quantitative finance is model calibration. The calibration problem is the inverse of the pricing problem. Instead of computing prices in a model with given values for its parameters, one wishes to compute the values of the model parameters that are consistent with observed prices. Now, it is well known to physicists that such...
Article
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In this paper we study the counterparty risk on a payer CDS in a Markov chain model of two reference credits, the firm underlying the CDS and the protection seller in the CDS. We first state few preliminary results about pricing and CVA of a CDS with counterparty risk in a general set-up. We then introduce a Markov chain copula model in which wrong...
Article
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The valuation and hedging of defaultable game options is studied in a hazard process model of credit risk. A convenient pricing formula with respect to a reference filteration is derived. A connection of arbitrage prices with a suitable notion of hedging is obtained. The main result shows that the arbitrage prices are the minimal superhedging price...
Article
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It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds...
Article
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This paper is the first in a series that we devote to studying the problems of valuation and hedging of defaultable game options in general, and convertible corporate bonds in particular. Here, we present mathematical foundations for our overall study. Specifically, we provide several results characterizing the arbitrage price of a defaultable game...
Article
This paper is a follow-up to “Valuation and Hedging of Defaultable Game Options in a Hazard Process Model” by the same authors. In the present paper we give user friendly assumptions ensuring that the general conditions in the previous paper are satisfied. We also give a systematic procedure to construct suitable intensity models of credit risk, an...