Peter Tankov

## Publications

### Publications in refereed journals

 36 Optimal importance sampling for Lévy Processes (with A. Genin) Stochastic Processes and their Applications (in press) [PDF] [Hide]
We develop generic and efficient importance sampling estimators for Monte Carlo evaluation of prices of single- and multi-asset European and path-dependent options in asset price models driven by Lévy processes, extending earlier works which focused on the Black-Scholes 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 pay-off under an Esscher-style 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.
 35 Arbitrage and utility maximization in market models with an insider (with H. N. Chau and W. Runggaldier), Mathematics and Financial Economics (published online) [PDF] [Hide]
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.
 34 Optimal trading policies for wind energy producer (with Z. Tan) SIAM Journal on Financial Mathematics (to appear). [PDF] [Hide]
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. 2455-2514 (2017) [PDF] [Hide]
We consider the problem of tracking a target whose dynamics is modeled by a continuous Ito semi-martingale. 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 time-average 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(6-7), pp. 943-966 (2017) [PDF] [Hide]
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 time-varying domain. Although the dynamics of the target is non-Markovian, it turns out that such strategies are asympototically optimal for a large list of examples.
 31 Hedging under multiple risk constraints (with Y. Jiao and O. Klopfenstein) Finance and Stochastics (to appear) [PDF] [Hide]
Motivated by the asset-liability 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 non-Markovian context and give explicit solutions in special cases.
 30 Asymptotic indifference pricing in exponential Lévy models (with Clément Ménassé) Applied mathematical finance vol. 23(3), pp. 197-235 (2016) [PDF] [Hide]
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 integro-differential 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 non-asymptotic 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 closed-form approximation of the indifference price by treating the Lévy model as a perturbation of the Black-Scholes 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 non-smooth functionals. Our closed formula represents the indifference price as the linear combination of the Black-Scholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying Lévy process, and the derivatives of the Black-Scholes price. As a by-product, 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 model-independent fashion, how sensitive a given product is to jump risk in the limit of small jump size.
 29 Tails of weakly dependent random vectors Journal of Multivariate Analysis, Vol. 145, 73-86 (2016). [PDF] [Hide]
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.
 28 Optimal discretization of hedging strategies with directional views (with Jiatu Cai, Masaaki Fukasawa and Mathieu Rosenbaum) SIAM Journal on Financial Mathematics vol. 7(1), pp. 34-69 (2016) [PDF] [Hide]
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 expectation-error 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 linear-quadratic optimal control problem. In specific contexts such as in the Black-Scholes model, explicit expressions for the optimal rebalancing times can be derived.
 27 Market models with optimal arbitrage (with Huy N. Chau), SIAM Journal on Financial Mathematics Vol. 6(1), 66-85 (2015). . [PDF] [Hide]
We construct and study market models admitting optimal arbitrage. We say that a model admits optimal arbitrage if it is possible, in a zero-interest 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 log-normal random variables (with A. Gulisashvili), Bernoulli Vol. 22 (1), 444-493 (2016). [PDF] [Hide]
We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated log-normal 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 Black-Scholes model.
 25 Finite-dimensional representations for controlled diffusions with delay (with S. Federico), Applied Mathematics and Optimization, Vol. 71 (1), 165-194 (2015). [PDF] [Hide]
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 finite-dimensional Markovian representation. As a second contribution, we show how approximate finite-dimensional 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.
 24 Numerical methods for the quadratic hedging problem in Markov models with jumps (with C. De Franco and X. Warin), The Journal of Computational Finance, Vol. 19 (2), 1-39 (2015). [PDF] [Hide]
We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integro-differential equations (PIDE), which can be shown to possess unique smooth solutions in our setting. The first equation is non-linear, but does not depend on the pay-off 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 time-inhomogeneous pure jump Markov processes appear in a natural manner.
 23 A new look at short-term implied volatility in asset price models with jumps (with A. Mijatovic) Mathematical Finance (to appear) [PDF] [Hide]
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 non-constant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal-Getoor) 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 model-independent slope. This makes infinite variation Lévy models better suited for calibration based on short-maturity option prices.
 22 Asymptotically optimal discretization of hedging strategies with jumps (with M. Rosenbaum) Annals of Applied Probability Vol. 24 (3), 1002-1048 (2014). [PDF] [Hide]
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 Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias (with J. E. Figueroa-Lopez) Bernoulli Vol. 20(3), 1126-1164 (2014). [PDF] [Hide]
We characterize the small-time asymptotic behavior of the exit probability of a Lévy process out of a two-sided 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. Kohatsu-Higa and S. Ortiz-Latorre), Mathematics of Computation Vol. 83, 2293-2324 (2014). [PDF] [Hide]
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.
 19 A finite dimensional approximation for pricing moving average options (with M. Bernhart and X. Warin), SIAM Journal on Financial Mathematics, Vol. 2, 989-1013 (2011). [PDF] [Hide]
We propose a method for pricing American options whose pay-off depends on the moving average of the underlying asset price. The method uses a finite dimensional approximation of the infinite-dimensional dynamics of the moving average process based on a truncated Laguerre series expansion. The resulting problem is a finite-dimensional 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 Black-Scholes framework.
 18 Asymptotic results for time-changed Lévy processes sampled at hitting times (with M. Rosenbaum), Stochastic Processes and their Applications Vol. 121, 1607-1632 (2011) [PDF] [Hide]
We provide asymptotic results for time-changed 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 α-stable 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 Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.
 17 Improved Frechet bounds and model-free pricing of multi-asset options, Journal of Applied Probability , Vol. 43, 389-403 (2011). [PDF] [Hide]
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 model-free bounds on the prices of two-asset options which make use of extra information about the dependence structure, such as the price of another two-asset option.
 16 Arbitrage opportunities in misspecified stochastic volatility models (with Rudra P. Jena), SIAM Journal on Financial Mathematics, Vol. 2, 317-341 (2011) [PDF] [Hide]
There is vast empirical evidence that given a set of assumptions on the real-world 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 Black-Scholes 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.
 15 Tracking errors from discrete hedging in exponential Lévy models (with Mats Brodén), International Journal of Theoretical and Applied Finance, Vol. 14, 1-35 (2011). [PDF] [Hide]
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 pay-offs the latter strategy may suffer from very large discretization errors. For options with discontinuous pay-offs, the convergence rate depends on the underlying Lévy process, and we give an explicit relation between the rate and the Blumenthal-Getoor index of the process.
 14 Portfolio Insurance under a risk-measure constraint (with C. De Franco), Insurance Mathematics and Economics, Vol. 49, 361-370 (2011). [PDF] [Hide]
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).
 13 Jump-adapted discretization schemes for Lévy-driven SDEs (with A. Kohatsu-Higa), Stochastic Processes and their Applications, Vol. 120, 2258-2285 (2010). [PDF] [Hide]
We present new algorithms for weak approximation of stochastic differential equations driven by pure jump Lévy processes. The method is built upon adaptive non-uniform 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.
 12 Pricing and hedging gap risk, The Journal of Computational Finance, Vol. 13 (3), (2010) [PDF] [Hide]
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 multi-asset case. The effect of stochastic volatility is also analyzed.
 11 Optimal consumption policies in illiquid markets (with A. Cretarola, F. Gozzi and H. Pham ), Finance and Stochastics , Vol. 15, 85-115 (2011). [PDF] [Hide]
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 second-order ODE. Finally, numerical illustrations of the behavior of optimal consumption strategies between two trading dates are given.
 10 Asymptotic analysis of hedging errors in models with jumps (with E. Voltchkova), Stochastic Processes and their Applications, Vol. 119, 2004-2027 (2009) [PDF] [Hide]
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 jump-diffusion models.
 9 A coupled system of integrodifferential equations arising in liquidity risk model (with H. Pham), Applied Mathematics and Optimization, Vol. 59, No. 2, 147-173 (2009) [PDF] [Hide]
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 Jump-diffusion models: a practitioner's guide (with E. Voltchkova), Banque et Marchés, No. 99, March-April 2009 [PDF] [Hide]
The goal of this paper is to show that the jump-diffusion models are an essential and easy-to-learn 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 jump-diffusion 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.
 7 Multi-factor jump-diffusion models of electricity prices (with T. Meyer-Brandis), International Journal of Theoretical and Applied Finance, Vol. 11, No. 5, 503 - 528 (2008) [PDF] [Hide]
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 sum-OU model: a model representing the price as a sum of Lévy-driven Ornstein-Uhlenbeck (OU) processes. We present a new method for filtering out the different OU components and develop a statistical procedure for estimating the sum-OU model from data.
 6 Constant Proportion Portfolio Insurance in presence of Jumps in Asset Prices (with R. Cont), Mathematical Finance, Vol. 19, No. 3, 379-401 (2009) [PDF] [Hide]
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 non-negligible 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 continuous--time 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 jump-diffusion 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, 613-627 (2008) [PDF] [Hide]
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 Integro-Differential 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.
 4 Retrieving Lévy processes from option prices: regularization of an ill-posed inverse problem (with R. Cont), SIAM Journal on Control and Optimization, Vol. 45, No. 1, 1-25 (2006) [PDF] [Hide]
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 ill-posedness 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 minimum-entropy least square solution of the calibration problem.
 3 Characterization of dependence of multidimensional Lévy processes using Lévy copulas (with Jan Kallsen), Journal of Multivariate Analysis, Vol. 97, 1551-1572 (2006) [PDF] [Hide]
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 Xt for fixed t.
 2 Monte Carlo option pricing for tempered stable (CGMY) processes (with J. Poirot), Asia Pacific Financial Markets, Vol. 13-4 (2006) [PDF] [Hide]
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 6-parameter 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).
 1 Calibration of jump-diffusion option pricing models: a robust non-parametric approach (with Rama Cont), Journal of Computational Finance, Vol. 7, No. 3, 1-49 (2004) [PDF] [Hide]
We present a non-parametric method for calibrating jump-diffusion models to a set of observed option prices. We show that the usual formulations of the inverse problem via nonlinear least squares are ill-posed. 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 jump-diffusion 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

 8 Optimal management of a wind power plant with storage capacity (with J. Collet and O. Féron) In: Renewable Energy: Forecasting and Risk Management, P. Drobinski et al., Eds., Springer, 2018. [PDF] [Hide]
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 24-hour 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 [PDF] [Hide]
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 log-normal 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 log-normal 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 Barndorff-Nielsen, Springer (to appear). [PDF] [Hide]
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 non-parametric 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 easy-to-use 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. [PDF] [Hide]
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 Black-Scholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional Black-Scholes 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 model-free tail-wing 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 Lévy-driven SDEs , in: Monte Carlo and Quasi Monte Carlo methods 2010, Plaskota, Leszek; Wozniakowski, Henryk (Eds.), Springer, 2012. [PDF] [Hide]
We propose new jump-adapted weak approximation schemes for stochastic differential equations driven by pure-jump 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 stable-like 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). [PDF] [Hide]
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 discrete-time 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, Paris-Princeton Lecture Notes in Mathematical Finance, Springer (2010). [PDF] [Hide]
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, 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) [PDF] [Hide]
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

 Lévy processes in finance and risk management, Wilmott magazine, Sept-Oct 2007 [PDF] [Hide]
We start with an accessible "practitioner's introduction" to Lévy processes and jump-diffusion 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 Lévy-based models provide a better understanding of risk while preserving a high level of mathematical tractability.

### Recent preprints

 Optimal wind-solar 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) [PDF] [Hide]
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 1989-2012 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 mean-risk 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 cross-region 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 low-frequency climate variability.
 Mean-field games of optimal stopping: a relaxed solution approach (with Géraldine Bouveret and Roxana Dumitrescu,) [PDF] [Hide]
We consider the mean-field 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.
 Long-time trajectorial large deviations for affine stochastic volatility models and application to variance reduction for option pricing (with Zorana Grbac and David Krief) [PDF] [Hide]
This work extends the variance reduction method for the pricing of possibly path-dependent derivatives, which was developed in (Genin and Tankov, 2016) for exponential Lévy models, to affine stochastic volatility models (Keller-Ressel, 2011). We begin by proving a pathwise large deviations principle for affine stochastic volatility models. We then apply a time-dependent 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 Monte-Carlo estimator. We test the method on the Heston model with and without jumps to demonstrate the numerical efficiency of the method.
 Long-time large deviations for the multi-asset Wishart stochastic volatility model and option pricing (with Aurélien Alfonsi and David Krief) [PDF] [Hide]
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 Monte-Carlo methods to price derivatives.
 Volatility options in rough volatility models (with Blanka Horvath and Antoine Jacquier) [PDF] [Hide]
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 log-volatility 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) [PDF] [Hide]
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 McKean-Vlasov SDEs (with G. Dos Reis and G. Smith) [PDF] [Hide]
This paper deals with the Monte-Carlo methods for evaluating expectations of functionals of solutions to McKean-Vlasov Stochastic Differential Equations (MV-SDE) with drifts of super-linear growth. We assume that the MV-SDE 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 2--3 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 [PDF] [Hide]
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 long-term timescales, the variability of the surface wind speed is strongly influenced by the atmosphere large-scale situation. Our aim is to predict the daily mean wind speed distribution at a specific location using the information on the atmosphere large-scale 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 large-scale 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 [PDF] [Hide]
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 proc©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 [PDF] [Hide]
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.

## Teaching and lecture notes

#### Gestion des risques d'énergie

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#### Mathématiques financières

Polycopié du cours "Mathématiques Financières" du Master 2 ISIFAR à l'Université Paris-Diderot (Paris 7)
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et ici pour les exercices du cours

#### Surface de volatilité

Polycopié du cours "Surface de volatilité" du Master Modélisation Aléatoire à l'Université Paris-Diderot (Paris 7)
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#### Asset pricing in derivatives market

Lecture notes for the course "Asset pricing in derivatives market" (MAP568) at Ecole Polytechnique, written jointly with Nizar Touzi

#### Calibration de modèles et couverture de produits dérivés

Polycopié du cours des masters "Modélisation Aléatoire" de l'Université Paris-Diderot (Paris 7) et "Probabilités et Finances" de l'Université Pierre et Marie Curie (Paris 6), 2006-2010
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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