Peter
Tankov
Professor of Quantitative Finance, ENSAE ParisTech
5, avenue Henry Le Chatelier  91120 Palaiseau
Phone: +33170266873, Office 4122, Click
to email me


Associate editor of:
Finance and
Stochastics
Mathematical Finance
SIAM Journal on
Financial Mathematics
Statistics and Risk
Modeling
Electronic Journal of Probability /
Electronic Communications in Probability (for 20152017)
Coordinator of the ANR
project FOREWER (FOrecasting and Risk Evaluation of
Wind Energy Production), 20142018
Research
Publications
By Rama Cont and Peter Tankov
Publications in refereed journals
We develop generic and efficient importance sampling estimators for Monte Carlo evaluation of prices of single and multiasset European and pathdependent options in asset price models driven by Lévy processes, extending earlier works which focused on the BlackScholes and continuous stochastic volatility models. Using recent results from the theory of large deviations on the path space for processes with independent increments, we compute an explicit asymptotic approximation for the variance of the payoff under an Esscherstyle change of measure. Minimizing this asymptotic variance using convex duality, we then obtain an easy to compite asymptotically efficient importance sampling estimator of the option price. Numerical tests for European baskets and for Asian options in the variance gamma model show consistent variance reduction with a very small computational overhead.
We study arbitrage opportunities, market viability and utility maximization in market models with an insider. Assuming that an economic agent possesses from the beginning an additional information in the form of a random variable G, which only becomes known to the ordinary agents at date T, we give criteria for the No Unbounded Profits with Bounded Risk property to hold, characterize optimal arbitrage strategies, and prove duality results for the utility maximization problem faced by the insider. Examples of markets satisfying NUPBR yet admitting arbitrage opportunities are provided for both atomic and continuous random variables G.
We study the optimal trading policies for a wind energy producer who aims to sell the future production in the open forward, spot, intraday and adjustment markets, and who has access to imperfect dynamically updated forecasts of the future production. We construct a stochastic model for the forecast evolution and determine the optimal trading policies which are updated dynamically as new forecast information becomes available. Our results allow to quantify the expected future gain of the wind producer and to determine the economic value of the forecasts.
33.  Asymptotic Lower Bounds for Optimal Tracking: a
Linear Programming Approach (with Jiatu Cai and Mathieu Rosenbaum)Annals of Applied Probability Vol. 27(4), pp. 24552514 (2017) 
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We consider the problem of tracking a target whose dynamics is modeled by a continuous Ito semimartingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the timeaverage control of Brownian motion, which is characterized as a deterministic linear programming problem. A comprehensive list of examples with explicit expressions for the lower bounds is provided.
32.  Asymptotic Optimal Tracking: Feedback Strategies
(with J. Cai and M. Rosenbaum)
Stochastics vol. 89(67), pp. 943966 (2017) 
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This is a companion paper to (Cai, Rosenbaum and Tankov, Asymptotic lower bounds for optimal tracking: a linear programming approach, arXiv:1510.04295). We consider a class of strategies of feedback form for the problem of tracking and study their performance under the asymptotic framework of the above reference. The strategies depend only on the current state of the system and keep the deviation from the target inside a timevarying domain. Although the dynamics of the target is nonMarkovian, it turns out that such strategies are asympototically optimal for a large list of examples.
Motivated by the assetliability management of a nuclear power plant operator, we consider the problem of finding the least expensive portfolio, which outperforms a given set of stochastic benchmarks. For a specified loss function, the expected shortfall with respect to each of the benchmarks weighted by this loss function must remain bounded by a given threshold. We consider different alternative formulations of this problem in a complete market setting, establish the relationship between these formulations, present a general resolution methodology via dynamic programming in a nonMarkovian context and give explicit solutions in special cases.
Financial markets based on Lévy processes are typically incomplete and option prices depend on risk preferences of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the nonlinear partial integrodifferential equation associated to the indifference price. In this work, we develop closed form approximations to exponential utility indifference prices in exponential Lévy models. To this end, we first establish a new nonasymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity. Next, we use this formula to derive a closedform approximation of the indifference price by treating the Lévy model as a perturbation of the BlackScholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of Lévy processes (A. Cerny, S. Denkl and J. Kallsen, arXiv:1309.7833) to nonlinear and nonsmooth functionals. Our closed formula represents the indifference price as the linear combination of the BlackScholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying Lévy process, and the derivatives of the BlackScholes price. As a byproduct, we obtain a simple explicit formula for the spread between the buyer's and the seller's indifference price. This formula allows to quantify, in a modelindependent fashion, how sensitive a given product is to jump risk in the limit of small jump size.
We introduce a new functional measure of tail dependence for weakly dependent (asymptotically independent) random vectors, termed weak tail dependence function. The new measure is defined at the level of copulas and we compute it for several copula families such as the Gaussian copula, copulas of a class of Gaussian mixture models, certain Archimedean copulas and extreme value copulas. The new measure allows to quantify the tail behavior of certain functionals of weakly dependent random vectors at the log scale.
We consider the hedging error of a derivative due to discrete trading in the presence of a drift in the dynamics of the underlying asset. We suppose that the trader wishes to find rebalancing times for the hedging portfolio which enable him to keep the discretization error small while taking advantage of market trends. Assuming that the portfolio is readjusted at high frequency, we introduce an asymptotic framework in order to derive optimal discretization strategies. More precisely, we formulate the optimization problem in terms of an asymptotic expectationerror criterion. In this setting, the optimal rebalancing times are given by the hitting times of two barriers whose values can be obtained by solving a linearquadratic optimal control problem. In specific contexts such as in the BlackScholes model, explicit expressions for the optimal rebalancing times can be derived.
We construct and study market models admitting optimal arbitrage. We say that a model admits optimal arbitrage if it is possible, in a zerointerest rate setting, starting with an initial wealth of 1 and using only positive portfolios, to superreplicate a constant c>1. The optimal arbitrage strategy is the strategy for which this constant has the highest possible value. Our definition of optimal arbitrage is similar to the one in Fernholz and Karatzas (2010), where optimal relative arbitrage with respect to the market portfolio is studied. In this work we present a systematic method to construct market models where the optimal arbitrage strategy exists and is known explicitly. We then develop several new examples of market models with arbitrage, which are based on economic agents' views concerning the impossibility of certain events rather than ad hoc constructions. We also explore the concept of fragility of arbitrage introduced in Guasoni and Rasonyi (2012), and provide new examples of arbitrage models which are not fragile in this sense.
26. 
Tail behavior of sums and differences of lognormal
random variables (with A. Gulisashvili),
Bernoulli Vol. 22 (1), 444493 (2016). 
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We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated lognormal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to be determined by a subset of components of the Gaussian vector, and we identify the relevant components by relating the asymptotics to a tractable quadratic optimization problem. As a corollary, we characterize the limiting behavior of the conditional law of the Gaussian vector, given a linear combination of the exponentials of its components. Our results can be used either to estimate the probability of tail events directly, or to construct efficient variance reduction procedures for precise estimation of these probabilities by Monte Carlo methods. They lead to important insights concerning the behavior of individual stocks and portfolios during market downturns in the multidimensional BlackScholes model.
We study stochastic delay differential equations (SDDE) where the coefficients depend on the moving averages of the state process. As a first contribution, we provide sufficient conditions under which a linear path functional of the solution of a SDDE admits a finitedimensional Markovian representation. As a second contribution, we show how approximate finitedimensional Markovian representations may be constructed when these conditions are not satisfied, and provide an estimate of the error corresponding to these approximations. These results are applied to optimal control and optimal stopping problems for stochastic systems with delay.
We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the HamiltonJacobiBellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integrodifferential equations (PIDE), which can be shown to possess unique smooth solutions in our setting. The first equation is nonlinear, but does not depend on the payoff of the option to hedge (the pure investment problem), while the other two equations are linear. We propose convergent finite difference schemes for the numerical solution of these PIDEs and illustrate our results with an application to electricity markets, where timeinhomogeneous pure jump Markov processes appear in a natural manner.
23. 
A new look at shortterm implied volatility in asset
price models with jumps (with A. Mijatovic) Mathematical
Finance (to appear) 
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We analyse the behaviour of the implied volatility smile for options
close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (BlumenthalGetoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the point of view of option prices. For infinite variation processes, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the finite variation case, the wings have a constant modelindependent slope. This makes infinite variation Lévy models better suited for calibration based on shortmaturity option prices.
In this work, we consider the hedging error due to discrete trading in models with jumps. Extending an approach developped by Fukasawa (2011) for continuous processes, we propose a framework enabling to (asymptotically) optimize the discretization times. More precisely, a discretization rule is said to be optimal if for a given cost function, no strategy has (asymptotically, for large cost) a lower mean square discretization error for a smaller cost. We focus on discretization rules based on hitting times and give explicit expressions for the optimal rules within this class.
21. 
Smalltime asymptotics of stopped Lévy bridges and simulation schemes
with controlled bias (with J. E. FigueroaLopez)
Bernoulli Vol. 20(3), 11261164 (2014). 
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We characterize the smalltime asymptotic behavior of the exit probability of
a Lévy process out of a twosided interval and of the law of its overshoot,
conditionally on the terminal value of the process. The asymptotic expansions
are given in the form of a first order term and a precise computable error
bound. As an important application of these formulas, we develop a novel
adaptive discretization scheme for the Monte Carlo computation of functionals
of killed Lévy processes with controlled bias. The considered functionals
appear in several domains of mathematical finance (e.g. structural credit risk
models, pricing of barrier options, and contingent convertible bonds) as well
as in natural sciences. The proposed algorithm works by adding discretization
points sampled from the Lévy bridge density to the skeleton of the process
until the overall error for a given trajectory becomes smaller than the maximum
tolerance given by the user. As another contribution of particular interest on
its own, we also propose two simple methods to simulate from the Lévy bridge
distribution based on the classical rejection method.
20.  Optimal simulation schemes for Lévy driven
stochastic differential equations (with A. KohatsuHiga and
S. OrtizLatorre), Mathematics of Computation
Vol. 83, 22932324 (2014). 
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We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied between the jump times. The overall approximation is analyzed using a stochastic splitting argument. The resulting error bound involves separate contributions of the compound Poisson approximation and of the discretization scheme for the Brownian part, and allows, on one hand, to balance the two contributions in order to minimize the computational time, and on the other hand, to study the optimal design of the approximating compound Poisson process. For driving processes whose Lévy measure explodes near zero in a regularly varying way, this procedure allows to construct discretization schemes with arbitrary order of convergence.
We propose a method for pricing American options whose payoff depends on the moving average of the underlying asset price. The method uses a finite dimensional approximation of the infinitedimensional dynamics of the moving average process based on a truncated Laguerre series expansion. The resulting problem is a finitedimensional optimal stopping problem, which we propose to solve with a least squares Monte Carlo approach. We analyze the theoretical convergence rate of our method and present numerical results in the BlackScholes framework.
We provide asymptotic results for timechanged Lévy processes sampled at
random instants. The sampling times are given by first hitting
times of symmetric barriers whose distance with respect to the
starting point is equal to ε. For a wide class of Lévy processes, we
introduce a renormalization depending on ε, under
which the Lévy process converges in law to an αstable
process as ε goes to 0. The convergence is extended
to moments of hitting times and overshoots. These results can be used
to build high frequency statistical procedures. As examples we construct consistent estimators of the time change
and, in the case of the CGMY process, of the BlumenthalGetoor
index. Convergence rates and a central limit theorem for suitable
functionals of the increments of the observed process are
established under additional assumptions.
We compute the improved bounds on the copula of a bivariate random vector when partial information is available, such as the values of the copula on the subset of $[0,1]^2$, or the value of a functional of the copula, monotone with respect to the concordance order. These results are then used to compute modelfree bounds on the prices of twoasset options which make use of extra information about the dependence structure, such as the price of another twoasset option.
There is vast empirical evidence that given a set of assumptions on the realworld dynamics of an asset, the European options on this asset are not efficiently priced in options markets, giving rise to arbitrage opportunities. We study these opportunities in a generic stochastic volatility model and exhibit the strategies which maximize the arbitrage profit. In the case when the misspecified dynamics is a classical BlackScholes one, we give a new interpretation of the classical butterfly and risk reversal contracts in terms of their (near) optimality for arbitrage strategies. Our results are illustrated by a numerical example including transaction costs.
We analyze the errors arising from discrete readjustment of the hedging portfolio when hedging options in exponential Lévy models, and establish the rate at which the expected squared error goes to zero when the readjustment frequency increases. We compare the quadratic hedging strategy with the common market practice of delta hedging, and show that for discontinuous option payoffs the latter strategy may suffer from very large discretization errors. For options with discontinuous payoffs, the convergence rate depends on the underlying Lévy process, and we give an explicit relation between the rate and the BlumenthalGetoor index of the process.
We study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold. If, at maturity, the portfolio value is below the guaranteed level, a third party will refund the investor up to the guarantee. In exchange for this protection, the third party imposes a limit on the risk exposure of the fund manager, in the form of a convex monetary risk measure. The fund manager therefore tries to maximize the investor's utility function subject to the risk measure constraint.We give a full solution to this nonconvex optimization problem in the complete market setting and show in particular that the choice of the risk measure is crucial for the optimal portfolio to exist. Explicit results are provided for the entropic risk measure (for which the optimal portfolio always exists) and for the class of spectral risk measures (for which the optimal portfolio may fail to exist in some cases).
We present new algorithms for weak approximation of stochastic differential
equations driven by pure jump Lévy processes. The method is built upon
adaptive nonuniform discretization based on the jump times of the driving
process coupled with suitable approximations of the solutions between these jump times. Our technique avoids the costly simulation of the increments of the
Lévy process and in many cases achieves better convergence rates than the
traditional schemes with equal time steps. To illustrate the method, we consider applications to simulation of portfolio strategies and option pricing in the Libor market model with jumps.
We analyze a new class of exotic equity derivatives called gap options or gap risk swaps. These products are designed by major banks to sell off the risk of rapid downside moves, called gaps, in the price of the underlying. We show that to price and manage gap options, jumps must necessarily be included into the model, and present explicit pricing and hedging formulas in the single asset and multiasset case. The effect of stochastic volatility is also analyzed.
We investigate optimal consumption policies in the liquidity risk model introduced in Pham and Tankov (2007). Our main result is to derive smoothness results for the value functions of the portfolio/consumption choice problem. As an important consequence, we can prove the existence of the optimal control (portfolio/consumption strategy) which we characterize both in feedback form in terms of the derivatives of the value functions and as the solution of a secondorder ODE. Finally, numerical illustrations of the behavior of optimal consumption strategies between two trading dates are given.
Most authors who studied the problem of hedging an option in
incomplete markets, and, in particular, in models with jumps,
focused on finding the strategies that minimize the residual hedging
error. However, the resulting strategies are usually unrealistic
because they require a continuously rebalanced portfolio, which is
impossible in practice due to transaction costs. In reality, the
portfolios are rebalanced discretely, which leads to a 'hedging
error of the second type', due to the difference between the optimal
strategy and its discretely rebalanced version. In this paper, we
analyze this second hedging error and establish a limit theorem for
the renormalized error, when the discretization step tends to zero,
in the framework of general Itô processes with jumps. Theses results
are applied to hedging options with discontinuous payoffs in
jumpdiffusion models.
We study a portfolio/consumption choice problem in a market model with liquidity
risk. The main feature is that the investor can trade and observe stock prices only
at exogenous Poisson arrival times. He may also consume continuously from his cash
holdings, and his goal is to maximize his expected utility from consumption. This is
a mixed discrete/continuous stochastic control problem, nonstandard in the literature.
We show how the dynamic programming principle leads to a coupled system of Integro
Differential Equations (IDE), and we prove an analytic characterization of this control
problem by adapting the concept of viscosity solutions. This coupled system of IDE may
be numerically solved by a decoupling algorithm, and this is the topic of a companion
paper: A model of optimal consumption under liquidity risk with random trading times.
8.  Jumpdiffusion models: a practitioner's guide (with
E. Voltchkova), Banque et Marchés, No. 99, MarchApril 2009 
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The goal of this paper is to show that the jumpdiffusion models are an essential and
easytolearn tool for option pricing and risk management, and that they provide an adequate description of stock price fluctuations and market risks.
We try to give an overview of the field without focusing on technical details. After introducing several widely used jumpdiffusion models, we discuss
Fourier transform based methods for European option pricing, partial differential equations for barrier and American options, and the existing approaches to calibration and hedging.
The recent deregulation of electricity markets has led to the
creation of energy exchanges, where the electricity is traded like any other commodity. In this paper, we study the most salient statistical features of electricity prices with a particular attention to the European energy exchanges. These features can be adequately reproduced by the sumOU model: a model representing the price as a sum of Lévydriven OrnsteinUhlenbeck (OU) processes. We present a new method for filtering out the different OU components and develop a statistical procedure for estimating the sumOU model from data.
6.  Constant Proportion Portfolio Insurance in presence
of Jumps in Asset Prices (with R. Cont), Mathematical Finance, Vol. 19, No. 3, 379401 (2009) 
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Constant proportion portfolio insurance (CPPI) allows an investor
to limit downside risk while retaining some upside potential by
maintaining an exposure to risky assets equal to a constant multiple
m>1 of the cushion, the difference between the current
portfolio value and the guaranteed amount. In diffusion models with
continuous trading, this strategy has no downside risk, whereas in
real markets this risk is nonnegligible and grows with the
multiplier value. We study the behavior of CPPI strategies in
models where the price of the underlying portfolio may experience
downward jumps. This allows to quantify the ``gap risk" of the
portfolio while maintaining the analytical tractability of the
continuoustime framework. We establish a direct relation between
the value of the multiplier m and the risk of the insured
portfolio, which allows to choose the multiplier based on the risk
tolerance of the investor, and provide a Fourier transform method
for computing the distribution of losses and various risk measures
(VaR, expected loss, probability of loss) over a given time horizon.
The results are applied to a jumpdiffusion model with parameters
estimated from returns series of various assets.
5.  A model of optimal consumption under liquidity risk with random trading times (with H. Pham), Mathematical Finance, Vol. 18, No. 4, 613627 (2008) 
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We consider a portfolio/consumption choice problem in a market model with liquidity risk. The main feature is that the investor can trade and observe stock prices only at exogenous Poisson arrival times. He may also consume continuously from his cash holdings, and his goal is to maximize his expected utility from consumption. This is a mixed discrete/continuous stochastic control problem, nonstandard in the literature. We show how the dynamic programming principle leads to a coupled system of IntegroDifferential Equations (IDE), and we prove an analytic characterization of this control problem by adapting the concept of viscosity solutions. We also provide a convergent numerical algorithm for the resolution to this coupled system of IDE, and illustrate our results with some numerical experiments.
We propose a stable
nonparametric method for constructing an option pricing model of
exponential Lévy type, consistent with a given data set of
option prices. After demonstrating the illposedness of the usual and
least squares version of this inverse problem, we suggest to regularize
the calibration problem by reformulating it as the problem of finding
an exponential Lévy model that minimizes the sum of the pricing
error and the relative entropy with respect to a prior exponential
Lévy model. We prove the existence of solutions for the
regularized problem and show that it yields solutions which are
continuous with respect to the data, stable with respect to the choice
of prior, and converge to the minimumentropy least square solution of the
calibration problem.
In this paper we propose to use Lévy copulas to characterize the dependence among components of
multidimensional Lévy processes. This concept generalizes
a corresponding notion introduced in Tankov (2003) for Lévy processes with only positive jumps
in every component. We construct parametric families of Lévy copulas and prove a limit theorem,
which indicates how to obtain the Lévy copula of a multidimensional Lévy process X
from the ordinary copulas of the random vectors X_{t} for fixed t.
Lévy processes are now popular models for stock price behavior since they allow to incorporate jump risk and reproduce the implied volatility smile. In this paper, we focus on the tempered stable processes, also known as CGMY processes, which form a flexible 6parameter family of Lévy processes with infinite jump intensity. It is shown that under an appropriate equivalent probability measure a tempered stable process becomes a stable process whose increments can be simulated exactly. This provides a fast Monte Carlo algorithm for computing expectation of any functional of tempered stable process. We apply our method to the pricing of European options and compare the results to a recent approximate simulation method for tempered stable process by Madan and Yor (2005).
We present a nonparametric method for calibrating jumpdiffusion
models to a set of observed option prices. We show that the usual
formulations of the inverse problem via nonlinear least squares are
illposed. In the realistic case where the set of observed prices is
discrete and finite, we propose a regularization method based on
relative entropy: we reformulate our calibration problem into a problem
of finding a risk neutral jumpdiffusion model that reproduces the
observed option prices and has the smallest possible relative entropy
with respect to a chosen prior model. We discuss the numerical
implementation of our method using a gradient based optimization and
show via simulation tests on various examples that using the entropy
penalty resolves the numerical instability of the calibration problem.
Finally, we apply our method to empirical data sets of index options
and discuss the empirical results obtained.
Refereed book chapters and conference proceedings
We consider the problem of a wind producer who has access to the spot
and intraday electricity markets and has the possibility of partially
storing the produced energy using a battery storage facility. The aim
of the producer is to maximize the expected gain of selling in the market the
energy produced during a 24hour period. We propose and calibrate
statistical models for the power production and the intraday
electricity price, and compute the optimal strategy of the producer
via dynamic programming.
7.  Approximate Option Pricing in the Lévy Libor Model
(with Z. Grbac and D. Krief) In: Advanced Modelling in Mathematical Finance,
In Honour of Ernst Eberlein, J. Kallsen and A. Papapantoleon (eds.), Springer, 2016 
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In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the Lévy Libor model developed by Eberlein and Ozkan (2005). This model is an extension to Lévy driving processes of the classical lognormal Libor market model (LMM) driven by a Brownian motion. Option pricing is significantly less tractable in this model than in the LMM due to the appearance of stochastic terms in the jump part of the driving process when performing the measure changes which are standard in pricing of interest rate derivatives. To obtain explicit approximation for option prices, we propose to treat a given Lévy Libor model as a suitable perturbation of the lognormal LMM. The method is inspired by recent works by Cerny, Denkl and Kallsen (2013) and Ménassé and Tankov (2015). The approximate option prices in the L\'evy Libor model are given as the corresponding LMM prices plus correction terms which depend on the characteristics of the underlying Lévy process and some additional terms obtained from the LMM model.
6. 
Lévy copulas: review of recent results, in: The
Fascination of Probability, Statistics and their Applications, in
honour of Ole BarndorffNielsen, Springer (to appear). 
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We review and extend the now considerable literature on Lévy
copulas. First, we focus
on Monte Carlo methods and present a new robust algorithm for the simulation of multidimensional Lévy
processes with dependence given by a Lévy copula. Next, we review
statistical estimation techniques in a parametric and a nonparametric
setting. Finally, we discuss the
interplay between Lévy copulas and multivariate regular
variation and briefly review the applications of Lévy copulas in risk management. In particular, we provide a new easytouse sufficient condition for
multivariate regular variation of Lévy measures in terms of their
Lévy copulas.
5.  Implied volatility of basket options at extreme
strikes (with Archil Gulisashvili), in: Large Deviations and
Asymptotic Methods in Finance, Friz, P., J. Gatheral,
A. Gulisashvili, A. Jacqier and J. Teichmann (Eds.), Springer Proceedings in Mathematics and Statistics, Vol. 110, 2015. 
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In the paper, we characterize the asymptotic behavior of the implied volatility of a basket call option at large and small strikes in a variety of settings with increasing generality. First, we obtain an asymptotic formula with an error bound for the left wing of the implied volatility, under the assumption that the dynamics of asset prices are described by the multidimensional BlackScholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional BlackScholes model with time change by an independent increasing stochastic process. Finally, we deal with a general situation in which the dependence between the assets is described by a given copula function. In this setting, we obtain a modelfree tailwing formula that links the implied volatility to a special characteristic of the copula called the weak lower tail dependence function.
4.  High order weak approximation schemes for
Lévydriven SDEs , in: Monte Carlo and Quasi Monte Carlo
methods 2010, Plaskota, Leszek; Wozniakowski, Henryk (Eds.),
Springer, 2012. 
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We propose new jumpadapted weak approximation schemes for stochastic
differential equations driven by purejump Lévy
processes. The idea is to replace the driving Lévy process Z
with a finite intensity process which has the same Lévy
measure outside a neighborhood of zero and matches a given number of
moments of Z. By matching 3 moments we construct a scheme which
works for all Lévy measures and is superior to the existing
approaches both in terms of convergence rates and easiness of
implementation. In the case of Lévy processes with
stablelike behavior of small jumps, we construct schemes with
arbitrarily high rates of convergence by matching a sufficiently
large number of moments.
3.  Swing Options Valuation: a BSDE with Constrained
Jumps Approach (with M. Bernhart, H. Pham and X. Warin), Numerical Methods in Finance, R. Carmona et al. (eds), Springer (2012). 
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We introduce a new probabilistic method for solving a class of impulse control problems based on their representations as Backward Stochastic Differential Equations (BSDEs for short) with constrained jumps. As an example, our method is used for pricing Swing options. We deal with the jump constraint by a penalization procedure and apply a discretetime backward scheme to the resulting penalized BSDE with jumps. We study the convergence of this numerical method, with respect to the main approximation parameters: the jump intensity, the penalization parameter and the time step. Combining this approach with Monte Carlo techniques, we then work out the valuation problem of (normalized) Swing options in the Black and Scholes framework. We present numerical tests and compare our results with a classical iteration method.
2.  Pricing and hedging in exponential Lévy
models: review of recent results, ParisPrinceton
Lecture Notes in Mathematical Finance, Springer (2010). 
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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 2122, 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, properties of implied volatility, and modern approaches to hedging in these models.
1.  Hedging with options in models with jumps (with R. Cont and E. Voltchkova), appeared in Stochastic Analysis and Applications  the Abel Symposium 2005, Springer (2007) 
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We consider the problem of hedging a contingent claim, in a market
where prices of traded assets can undergo jumps, by trading in the
underlying asset and a set of traded options. We give a general
expression for the hedging strategy which minimizes the variance
âof the hedging error, in terms of integral representations of the
options involved. This formula is then applied to compute hedge
ratios for common options in various models with jumps, leading to
easily computable expressions. The performance of these hedging
strategies is assessed through numerical experiments.
Other publications
We start with an accessible "practitioner's introduction" to Lévy processes and jumpdiffusion models. Next, we discuss the calibration of exponential Lévy models from traded option prices. Without going into details of every specific algorithm we focus on different approaches for determining the qualitative properties of the model.
Finally, we review two recent applications which emphasize the importance of jumps in stock price modeling, namely construction of optimal hedging portfolios and computation of risk measures for dynamically insured portfolios in presence of jumps in asset prices. Both examples show that Lévybased models provide a better understanding of risk while preserving a high level of mathematical tractability.
Recent preprints
Optimal windsolar energy mix in Italy: Impact of climate variability
(with Tantet Alexis, Stéfanon Marc, Drobinski Philippe, Badosa Jordi, Concettini Silvia, Creti Anna, D'Ambrosio Claudia and Thomopulos Dimitri) 
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In the context of the 2009 EU directive promoting the use of energy from renewable sources, Italy has reached its 2020 target of a 17% share of renewables in the final energy consumption 6 years in advance. In this study, we evaluate the existing renewable energy mix in Italy at regional scale by comparing it to an optimized mix taking into account climate variability and allowing full decommissioning of the currently installed plants. The variability of the production and of the demand over the 19892012 period is resolved by plugging regional climate simulations of this period into a model simulating the renewable energy production as well as the Italian electrical consumption at regional scale. The optimal mix is then inferred from a meanrisk analysis with as objectives both to maximize the mean of the total renewable production and to minimize the variance, or risk, of the latter. We consider two cases: in the first one the analysis takes crossregion correlations in the production and the demand into account and in the second one the analysis is local to each region. The optimal mix maximizing the ratio of the total mean penetration over the total risk for the same renewable capacity as installed in 2015 consists of about two thirds wind and one third solar, i.e. twice as much wind as the actual 2015 Italian renewable mix. The spatial distribution also differs significantly from the actual mix and from what would be obtained ignoring the risk and lowfrequency climate variability.
Meanfield games of optimal stopping: a relaxed solution approach
(with Géraldine Bouveret and Roxana Dumitrescu,) 
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We consider the meanfield game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and show its convergence.
Longtime trajectorial large deviations for affine stochastic volatility models and application to variance reduction for option pricing
(with Zorana Grbac and David Krief) 
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This work extends the variance reduction method for the pricing of possibly pathdependent derivatives, which was developed in (Genin and Tankov, 2016) for exponential Lévy models, to affine stochastic volatility models (KellerRessel, 2011). We begin by proving a pathwise large deviations principle for affine stochastic volatility models. We then apply a timedependent Esscher transform to the affine process and use Varadhan's lemma, in the fashion of (Guasoni and Robertson, 2008) and (Robertson, 2010), to approximate the problem of finding the Esscher measure that minimises the variance of the MonteCarlo estimator. We test the method on the Heston model with and without jumps to demonstrate the numerical efficiency of the method.
Longtime large deviations for the multiasset Wishart stochastic volatility model and option pricing
(with Aurélien Alfonsi and David Krief) 
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We prove a large deviations principle for the class of multidimensional affine stochastic volatility models considered in (Gourieroux, C. and Sufana, R., J. Bus. Econ. Stat., 28(3), 2010), where the volatility matrix is modelled by a Wishart process. This class extends the very popular Heston model to the multivariate setting, thus allowing to model the joint behaviour of a basket of stocks or several interest rates. We then use the large deviation principle to obtain an asymptotic approximation for the implied volatility of basket options and to develop an asymptotically optimal importance sampling algorithm, to reduce the number of simulations when using MonteCarlo methods to price derivatives.
Volatility options in rough volatility models
(with Blanka Horvath and Antoine Jacquier) 
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We discuss the pricing and hedging of volatility options in some rough volatility models. First, we develop efficient Monte Carlo methods and asymptotic approximations for computing option prices and hedge ratios in models where logvolatility follows a Gaussian Volterra process. While providing a good fit for European options, these models are unable to reproduce the VIX option smile observed in the market, and are thus not suitable for VIX products. To accommodate these, we introduce the class of modulated Volterra processes, and show that they successfully capture the VIX smile.
Regression Monte Carlo for Microgrid Management
(with Clemence Alasseur, Alessandro Balata, Sahar Ben Aziza, Aditya Maheshwari and Xavier Warin) 
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We study an islanded microgrid system designed to supply a small village with the power produced by photovoltaic panels, wind turbines and a diesel generator. A battery storage system device is used to shift power from times of high renewable production to times of high demand. We introduce a methodology to solve microgrid management problem using different variants of Regression Monte Carlo algorithms and use numerical simulations to infer results about the optimal design of the grid.
Importance sampling for McKeanVlasov SDEs
(with G. Dos Reis and G. Smith) 
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This paper deals with the MonteCarlo methods for evaluating expectations of functionals of solutions to McKeanVlasov Stochastic Differential Equations (MVSDE) with drifts of superlinear growth. We assume that the MVSDE is approximated in the standard manner by means of an interacting particle system and propose two importance sampling (IS) techniques to reduce the variance of the resulting Monte Carlo estimator. In the \emph{complete measure change} approach, the IS measure change is applied simultaneously in the coefficients and in the expectation to be evaluated. In the \emph{decoupling} approach we first estimate the law of the solution in a first set of simulations without measure change and then perform a second set of simulations under the importance sampling measure using the approximate solution law computed in the first step. For both approaches, we use large deviations techniques to identify an optimisation problem for the candidate measure change. The decoupling approach yields a far simpler optimisation problem than the complete measure change, however, we can reduce the complexity of the complete measure change through some symmetry arguments. We implement both algorithms for two examples coming from the Kuramoto model from statistical physics and show that the variance of the importance sampling schemes is up to 3 orders of magnitude smaller than that of the standard Monte Carlo. The computational cost is approximately the same as for standard Monte Carlo for the complete measure change and only increases by a factor of 23 for the decoupled approach. We also estimate the propagation of chaos error and find that this is dominated by the statistical error by one order of magnitude.
Probabilistic forecasting of the wind energy resource at the monthly to seasonal scale (with B. Alonzo, P. Drobinski and R. Plougonven 
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We build and evaluate a probabilistic model designed for forecasting the distribution of the daily mean wind speed at the seasonal timescale in France. On such longterm timescales, the variability of the surface wind speed is strongly influenced by the atmosphere largescale situation. Our aim is to predict the daily mean wind speed distribution at a specific location using the information on the atmosphere largescale situation, summarized by an index. To this end, we estimate, over 20 years of daily data, the conditional probability density function of the wind speed given the index. We next use the ECMWF seasonal forecast ensemble to predict the atmosphere largescale situation and the index at the seasonal timescale. We show that the model is sharper than the climatology at the monthly horizon, even if it displays a strong loss of precision after 15 days. Using a statistical postprocessing method to recalibrate the ensemble forecast leads to further improvement of our probabilistic forecast, which then remains sharper than the climatology at the seasonal horizon.
Old preprints, which were never published or became parts of other papers
Simulation and option pricing in Lévy copula model 
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Lévy copulas are functions that completely
characterize the law of a multidimensional Lévy process given the laws
of its components. In this paper, after recalling the basic properties
of Lévy copulas, we discuss the simulation of multidimensional Lévy
processes with dependence structure given by a Lévy copula. Being able
to describe the dependence structure of a Lévy procĀ©ess in terms of its
Lévy copula allows us to quantify the effect of dependence on the
prices of basket options in a multidimensional exponential Lévy model.
We conclude that these prices are highly sensitive not only to the
linear correlation between assets but also to the exact type of
dependence beyond linear correlation.
Dependence structure of spectrally positive multidimensional Lévy processes 
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We
propose a general characterization of the dependence among components
of multidimensional Lévy processes admitting only positive jumps in
every component, by introducing Lévy copulas. These objects have the
same properties as ordinary copulas but are defined on a different
domain. They can be used to separate dependence from the behavior of
the components of a multidimensional Lévy process. We construct
parametric families of Lévy copulas and develop an algorithm for
simulating multidimensional Lévy processes via series representation,
using their Lévy copulas. Finally, we illustrate our method by showing
how it can be used to build multivariate models with jumps for finance
and insurance.
PhD thesis: Lévy Processes in Finance: Inverse Problems and Dependence Modelling
Teaching and lecture notes
Gestion des risques d'énergie
Cliquez ici pour accéder à la page web du cours
Mathématiques financières
Polycopié du cours "Mathématiques Financières" du
Master 2 ISIFAR à l'Université ParisDiderot (Paris 7)
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télécharger le fichier
et ici pour les exercices du cours
Surface de volatilité
Polycopié du cours "Surface de volatilité" du
Master Modélisation Aléatoire à l'Université ParisDiderot (Paris 7)
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télécharger le fichier
Asset pricing in derivatives market
Lecture notes for the course "Asset pricing in derivatives market"
(MAP568) at Ecole Polytechnique, written jointly with Nizar Touzi
Click here to download the PDF file
Calibration de modèles et couverture de produits dérivés
Polycopié du cours des masters "Modélisation
Aléatoire" de l'Université ParisDiderot (Paris 7) et
"Probabilités et Finances" de l'Université Pierre et
Marie Curie (Paris 6), 20062010
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Financial Modeling with Lévy Processes
Notes of lectures I gave at the Institute of Mathematics of the Polish Academy of Sciences in October 2010
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go to the download page
Curriculum Vitae
Click here to download my CV (pdf file last updated in September 2017).
You can also download my Habilitation
thesis.