Financial modelling with jump processes

 Rama Cont, Peter TANKOV. 
 CRC Press, 2003.  
 2nd edition forthcoming, 2008.

Frontiers in Quantitative Finance:
credit risk and volatility modeling

Rama CONT (ed.): Wiley Series in Financial Engineering, 2008.

With contributions from:

A d'Aspremont, S Benaim, L Bergomi, A Cousin, P Friz, K Giesecke, P Henry-Labordere, JP Laurent, Roger Lee, LCG Rogers, I Savescu, E Schlogl, P Tankov, Ph Very, E Voltchkova.

 Credit Derivatives and structured credit products

 Richard BRUYERE, Rama CONT, Christophe JAECK,
 Loic FERY, Thomas SPITZ.

  Wiley, 2005.

 Les produits dérivés de crédit

 Richard BRUYERE, Rama CONT, Christophe JAECK,
 Loic FERY, Thomas SPITZ.   Economica, 2004.

Mathematical finance: theory and practice

Edited by: Rama CONT and Jiongmin YONG.
Higher Education Press, Beijing, 2001.
 

 A Consistent Pricing Model for Index Options and Volatility Derivatives

Rama CONT and  Thomas KOKHOLM  (2009)
 

We propose a flexible modeling framework for the joint dynamics of an index and a set of forward variance swap rates written on this index, allowing options on forward variance swaps and options on the underlying index to be priced consistently. Our model reproduces various empirically observed properties of variance swap dynamics and allows for jumps in volatility and returns.
An affine specification using Levy processes as building blocks leads to analytically tractable pricing formulas for options on variance swaps as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European option on SP500 across strikes and maturities as well as options on the VIX volatility index. The calibration of the model is done in two steps, first by matching VIX option prices and then by matching prices of options on the underlying.

 Mimicking the marginal distributions of a semimartingale

Rama CONT and  Amel BENTATA  (2009)
Presented at: Workshop on Statistics and Levy Processes (EURANDOM, July 2009), Conference on Partial differential equations and Mathematical Finance (KTH, Stockholm, Aug 2009).  

 A Stochastic Model for Order Book Dynamics

Rama CONT, Sasha STOIKOV and  Rishi TALREJA  (2008)
To appear in: Operations Research.  

We propose a stochastic model for the continuous-time dynamics of a limit order book. The model strikes a balance between two desirable features: it captures key empirical properties of order book dynamics and its analytical tractability allows for fast computation of various quantities of interest without resorting to simulation. We describe a simple parameter estimation procedure based on high-frequency observations of the order book and illustrate the results on data from the Tokyo stock exchange. Using Laplace transform methods, we are able to efficiently compute probabilities of various events, conditional on the state of the order book: an increase in the mid-price, execution of an order at the bid before the ask quote moves, and execution of both a buy and a sell order at the best quotes before the price moves. Comparison with high-frequency data shows that our model can capture accurately the short term dynamics of the limit order book.

 Matching marginal distributions of a semimartingale with a Markov process

Rama CONT and  Amel BENTATA  (2009)
Comptes Rendus Mathematique, Vol 347, 2009.  

We give conditions under which the flow of marginal distributions of a discontinuous semimartingale X can be matched by a Markov process whose infinitesimal generator is expressed in terms of the local characteristics of X. Our results extend previous results of Gyongy (1986) to discontinuous semimartingales. As an application, we show that under some regularity conditions the transition densities of a semimartingale solves a forward partial integro-differential equation.

 Default Intensities implied by CDO Spreads: Inversion Formula and Model Calibration

Rama CONT, Romain DEGUEST and  Yu Hang KAN  (2009)
 

We propose a simple computational method for constructing an arbitrage-free CDO pricing model which matches a pre-specified set of CDO tranche spreads. The key ingredient of the method is a formula for computing the local default intensity function of a portfolio from its expected tranche notionals. This formula can be seen as an analog, for portfolio credit derivatives, of the well-known Dupire formula. Together with a quadratic programming method for recovering expected tranche notionals from CDO spreads, our inversion formula leads to an efficient non-parametric method for calibrating CDO pricing models. Comparing this approach to other calibration methods, we find that model-dependent quantities such as the forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class. On the other hand, comparing the local default intensities implied by different credit portfolio models reveals that apparently very different models such as static Student-t copula models and reduced-form affine jump-diffusion models, lead to similar marginal loss distributions and tranche spreads.

Dynamic hedging of portfolio credit derivatives

Rama CONT and Yu Hang KAN (2008)

We study hedging of synthetic CDO with the underlying CDS index in various aggregate loss models which account for default contagion and spread risk. In particular, we compare sensitivity-based hedging with hedging strategies based on quadratic risk minimization. Numerical results obtained in models calibrated to iTraxx market
data reveal significant differences in the hedge ratios and show, unlike what had been previously suggested in the literature by comparing copula-based models, that hedging strategies are subject to substantial model risk. Finally, we perform an empirical
comparison of hedging performance using ITRAXX time series. Our study reveals in particular that delta-hedging of spread risk using a Gaussian Copula model does not appear to be an effective hedging strategy, especially after the subprime crisis.

Model uncertainty and its impact on the pricing of derivative instruments

Rama CONT

Published in: Mathematical Finance , Vol 16 Page 519-542,  July 2006.

Abstract: Model uncertainty, in the context of derivative pricing, can be defined as the uncertainty on the value of a contingent claim resulting from the lack of precise knowledge of the pricing model to be used for its valuation. We introduce here a quantitative framework for defining model uncertainty in option pricing models. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk measurement and management, we  propose a method for measuring model uncertainty which verifies these properties and yields numbers which are comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. We illustrate the difference between model uncertainty and the more common notion of "market risk" through examples. Finally, we illustrate the connection between our proposed measure of model uncertainty and  the recent literature on coherent and convex risk measures.
 

  Small world graphs: characterization and alternative constructions

Rama CONT and Emily TANIMURA.
Advances in Applied Probability, Volume 40, no 4 (December 2008).

Abstract:  Small world graphs are examples of random graphs which mimic empirically observed features of social networks. We propose an intrinsic definition of small world graphs, based on a probabilistic formulation of scaling properties of graph properties, which does not rely on an underlying lattice nor on any particular construction. Our definition is shown to encompass existing models of small world graphs, proposed by Watts and studied by Barbour & Reinert, which are based on random perturbations of a regular lattice. We also propose alternative constructions of small world graphs which are not based on lattices and study their scaling properties.

Robustness and sensitivity analysis of risk measurement procedures

Rama Cont, Romain Deguest and Giacomo Scandolo

Presented at: QMF 2006 (Sydney), Humboldt Univ. Berlin (May 2006) Cornell ORIE seminar (Feb 2007) Harvard Statistics Dept (2007) Torino (2007).

Abstract:   Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a ``risk measure" which summarizes the risk of the portfolio. We formulate and study the notion of ``risk measurement procedure", which includes both of these steps, focusing in particular on their robustness and sensitivity to a change in the data set. After introducing a rigorous definition of 'robustness' of a risk measurement procedure, we illustrate the presence of a conflict between subadditivity and robustness of risk measurement procedures. We propose a measure of sensitivity for risk measurement procedures and compute the sensitivity function of various examples of risk estimators used in financial risk management, showing that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate in particular that using historical Value at Risk leads to a more robust procedure for risk measurement than recently proposed alternatives like CVaR. We also propose other risk measurement procedures which possess the robustness property.

Recovering portfolio default intensities implied by CDO tranches

Rama Cont, Andreea MINCA
 
Presented at :Workshop on PDEs in finance (Stockholm Aug 2007), Stanford Financial Mathematics seminar (Sept 07), S&P Credit Risk Summit (New York Nov 2007), CCCP conference (Princeton 2007), AMS 2008 meeting (San Diego Jan 08)  S&P Credit Risk Summit (London 2008), Bachelier Congress 08.

Abstract:  We propose a stable non-parametric algorithm for the calibration of pricing models for portfolio credit derivatives: given a set of observations of market spreads for CDO tranches, we construct a risk-neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk neutral loss process 'closest' to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality methods to solve the problem: the dual problem is shown to be an intensity control problem, characterized in terms of a Hamilton--Jacobi system of differential equations, for which we present an analytical solution. Given a set of observed CDO tranche spreads, our method allows to construct an implied intensity process consistent with the observed spreads. We illustrate our method on ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the past number of defaults, thus offering quantitative evidence for contagion effects in the risk--neutral loss process.
 

Model-free representation of pricing rules as conditional expectations

Sara Biagini, Rama Cont.

Published in : Stochastic processes and applications to mathematical finance,
edited by Jiro Akahori, Shigeyoshi Ogawa & Shinzo Watanabe, World Scientific, 2006, pages 53-66.

Abstract:  We introduce a distinction between model-based and model-free arbitrage and formulate an operational definition for absence of model-free arbitrage in a financial market, in terms of a set of minimal requirements for the pricing rule prevailing in the market. We show that any pricing rule verifying these properties can be represented as a conditional expectation operator with respect to a probability measure under which prices of traded assets follow martingales. Our result can be viewed as a model-free version of the fundamental theorem of asset pricing, which does not require any notion of ``reference" probability measure.
 

Recovering exponential Lévy models from option prices: regularization of an ill-posed inverse problem

Rama Cont, Peter Tankov.

Published in : SIAM Journal on Control and Optimization, Vol 45, No 1, 1-25 (2007)

Abstract:  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. 
 

Constant Proportion Portfolio Insurance in presence of jumps in asset prices

Rama Cont, Peter TANKOV.

Published in: Mathematical Finance, Vol. 19, Issue 3, pp. 379-401, July 2009.

Abstract: 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 market data.
 

Forward equations for portfolio credit derivatives

Rama Cont &  Ioana SAVESCU

Presented at: S&P Credit Risk Summit New York 2006, London 2007.
Published in: Frontiers in Quantitative Finance: Credit Risk and Volatility Modeling, Wiley 2008.

Abstract: We introduce an alternative approach for computing the values of CDO tranche spreads in reduced-form models for portfolio credit derivatives ("top-down" models), which allows for efficient computations and can be used as an ingredient of an efficient calibration algorithm. Our approach is based on the solution of a system of ordinary differential equations, which is the analogue for portfolio credit derivatives of Dupire's famous equation for call option prices. It allows to efficiently price CDOs and other portfolio credit derivatives without Monte Carlo simulation.

Risk analysis of Constant proportion debt obligations(CPDOs)

Rama Cont, Cathrine Jessen (2008)

Abstract:  Constant Proportion Debt Obligations (CPDOs) are structured credit derivatives indexed on a portfolio of investment grade debt which generate high coupon payments by dynamically leveraging a position in an underlying portfolio of index default swaps. Existing CPDO coupons and principal notes have received AAA from the major rating
agencies, based on high-dimensional models for the joint transition of ratings and spreads for all names in the underlying portfolio. We propose a parsimonious model for analyzing  the risk of CPDO strategies using a top-down approach which captures the essential risk factors of the CPDO. Our analysis allows to compute default probabilities, loss distributions and other tail risk measures for the CPDO strategy and to analyze the dependence of these risk measures on various parameters describing the risk factors. Though the probability of the CPDO defaulting on its coupon payments is found to be very small, we find that the ratings obtained strongly depend on the credit environment -- high spread or low spread -- and that CPDOs are highly exposed to spread widening scenarios.  Our results also point to a heterogeneous range of tail risk measures inside a given rating category, suggesting that credit ratings for such complex
leveraged strategies should be suitably complemented by other risk measures for the purpose of performance analysis. A worst-case scenario analysis indicates that CPDO strategies have a high exposure to persistent spread-widening scenarios.

Estimating large covariance matrices: insights from random matrix theory

Rama Cont, Sandrine Péché (forthcoming).

Abstract:  Small sample sizes and large number of assets in financial portfolios can lead to the failure of classical limit theorems for classical estimators of the covariance matrix of asset returns, leading to large biases or even non-consistency of the sample covariance matrix, even in presence of IID data. Using insight from random matrix theory, we investigate the size of the resulting estimation errors for typical portfolio and sample sizes and assess their impact on portfolio optimization and the pricing of multi-asset options. We then propose improved estimators for the covariance matrix and demonstrate the efficiency of the proposed estimators through numerical experiments.

Hedging with options in presence of jumps

Rama Cont, Peter Tankov, Ekaterina VOLTCHKOVA.

Published in : Benth, F.E.; Di Nunno, G.; Lindstrom, T.; Øksendal, B.; Zhang, T. (Eds.) Stochastic Analysis and Applications: The Abel Symposium  2005 in honor of Kiyosi Ito,  Springer 2007, pages 197-218.

Abstract:  We study the problem of hedging options when the underlying asset is described by a process with jumps. We compare various hedging strategies using the underlying asset and a set of liquid options and examine the properties of the hedging error, both theoretically and through numerical experiments.
 We illustrate in particular that using  sensitivities to compute Delta-neutral and Gamma-neutral hedge ratios can lead to a large hedging error and illustrate how such strategies can be improved by using a risk-minimizing approach to hedging and by taking positions in options.

Nonparametric tests for analyzing the fine structure of price fluctuations

Rama Cont &  Cecilia Mancini (November 2007).

Abstract:  We consider a semimartingale model where (the logarithm of) an asset price is modeled as the sum of a Levy process and a general Brownian semimartingale. Using a nonparametric threshold estimator for the continuous component of the quadratic variation ("integrated variance"), we design a test for the presence of a continuous component in the price process and a test for establishing whether the jump component has finite or infinite variationbased on observations on a discrete time grid. Using simulations of stochastic models commonly used in finance, we confirm the performance of our tests and compare them with analogous tests constructed using multipower variation estimators of integrated variance. Finally, we apply our tests to investigate the fine  structure of the DM/USD exchange rate process and of SPX futures prices. In both cases, our tests reveal the presence of a non-zero
Brownian component, combined with a finite variation jump component.

Option pricing models with jumps: integro-differential equations and inverse problems. 

Rama Cont, Peter Tankov, Ekaterina Voltchkova.

Published in : P. Naittaanmäki, T. Rossi, S. Korotov, E. Onate, J. Périaux and D. Knorzer (Eds.) European Congress on Computational Methods in Applied Sciences (ECCOMAS 2004)}, Jyväskylä.

Abstract:  Observation of sudden, large movements in the prices of financial assets has led to the use of stochastic processes with discontinuous trajectories - jump processes -- as models for financial assets. Exponential Lévy models provide an analytically tractable subclass of models with jumps and the flexibility in choice of the Lévy  process allows to calibrate the model to market prices of options and reproduce a wide variety of implied volatility skews/smiles. We discuss the characterization of prices of European and barrier options in exponential Lévy  models in terms of solutions of partial integro-differential equations (PIDEs). These equations involve, in addition to a second-order differential operator, a non-local integral term which requires specific treatment both at the theoretical and numerical level. The study of regularity of option prices in such models shows that, unlike the diffusion case, option price can exhibit lack of smoothness. The proper relation between option prices and PIDEs is then expressed using the notion of viscosity solution. Numerical solution of the PIDE allows efficient computation of option prices. The identification of exponential Lévy  models from option prices leads to an inverse problem for such PIDEs. We describe a regularization method based on relative entropy and its numerical implementation. This inversion algorithm, which allows to extract an implied Lévy  measure from a set of option prices, is illustrated by numerical examples.

Recovering volatility from option prices by evolutionary optimization 

Rama Cont, Sana BenHamida

Appeared in : Journal of Computational Finance, Vol 8, Number 4, Summer 2005.

Abstract:  We propose a probabilistic approach for estimating parameters of an option pricing model from a set of observed option prices. Our approach is based on a random search algorithm which generates a random sample from the set of global minima of the in-sample pricing error and allows for the existence of multiple global minima. Starting from an IID population of candidate solutions drawn from a prior distribution of the set of model parameters, the population of parameters is updated through cycles of independent random moves followed by ``selection" using the calibration criterion. We examine conditions under which such an evolving population converges to a sample of calibrated models. The heterogeneity of the obtained sample can then be used to quantify the degree of ill--posedness of the inverse problem: it provides a natural example of a coherent measure of risk, which is compatible with observed prices of benchmark (``vanilla") options and takes into account the model uncertainty resulting from incomplete identification of the model. We describe in detail the algorithm in the case of a diffusion model, where one aims at retrieving the unknown local volatility surface from a finite set of option prices, and illustrate its performance on simulated and empirical data sets of index options. 

Social distance, heterogeneity and social interactions.

 Rama Cont, Mathias Loewe (2001).

Presented at: Humboldt Universitat Berlin, Santa Fe, Ecole Polytechnique, WEHIA 2000, Ecole des Hautes Etudes en Sciences Sociales, Universite de Paris X, Workshop on Social Interactions (Paris, 2002) .

Abstract: A crucial ingredient in social interaction models is the structure of peer groups with which individuals interact. We argue that this structure can vary from one individual to another and thus should be modeled as randomly distributed across individuals. We propose and study a dynamic binary choice model with social interactions in which heterogeneity is introduced at two different levels: at the level of agents preferences by introducing an agent-specific random component in the utility function, and at the level of the interaction structure by taking into account affinities between agents with similar characteristics. Our framework allows for positive as well as negative interactions as well as a heterogeneous structure of peer groups across individuals. Dynamic equilibria are studied in this framework using large deviation techniques adopted from the statistical mechanics of disordered systems, in the limit when the number of agents is large. We show that the model exhibits multiple equilibria, with behavior which can be identified as resulting from conflicts between various group pressures the individuals are subjected to. We study in particular the correlation in the population at equilibrium between the characteristics of the agents and their decisions: we show that this quantity has an interesting empirical interpretation and solves a simple analytical equation when the number of agents is large. Finally we discuss the empirical content of this model and present an estimator of the model parameter which is consistent for any typical population regardless of the structure of individual characteristics.  
 

Integro-differential equations for option prices in exponential Lévy models. 

Rama Cont & Ekaterina Voltchkova.
Published in: Finance and Stochastics, Volume 9, Number 3, pages 299-325 (2005).

Abstract:  In stochastic models where the random evolution of the underlying asset is driven by a Lévy process or a time-inhomogeneous jump-diffusion process, European and barrier options in models with jumps are formally expressible as solution of partial integro-differential equations. We study the precise relation between such partial integro-differential equations (PIDEs) and the values of European and barrier options in exponential Lévy models. After giving sufficient conditions under which options prices are classical solutions of the PIDEs, we illustrate that these conditions may fail in pure-jump models where the option prices can have non-smooth dependence on the underlying. In such cases the option prices are solutions of the PIDE in the viscosity sense, using an appropriate extension of the notion of viscosity solution to such nonlocal boundary value problems. We give sufficient conditions on the Lévy triplet for the option price to be continuous in the underlying asset and show that in this case it is the unique viscosity solution of the PIDE.

Nonparametric calibration of jump-diffusion option pricing models.

Rama Cont & Peter Tankov.
Journal of Computational Finance, Vol. 7, No. 3, 1 - 49.

Abstract:
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.

Modeling term structure dynamics: an infinite dimensional approach.

Rama Cont.
International Journal of Theoretical and Applied Finance, Vol 8, No 3 (2005), p 357- 380.

Abstract: We present a family of models for term structure dynamics in an attempt to describe several statistical features observed in empirical studies of forward rate curves by decomposing the deformations of the term structure into the variations of the short rate, the long rate and the fluctuations of the curve around its average shape. This fluctuation is then described as a solution of a stochastic evolution equation in an infinite dimensional space. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates and the structure of principal components of term structure deformations. Finally, we discuss calibration issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.

Dynamics of implied volatility surfaces

Rama Cont & Jose da Fonseca.
Quantitative Finance, Volume 2, No. 1, 45-60 (2002).

Abstract: The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. However the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modeling approaches, giving rise to 'Vega' risk in option portfolios. Using time series of option prices on the SP500 and FTSE indices, we study the deformation of this surface and show that it may be represented as a randomly fluctuating surface driven by a small number of orthogonal random factors. We identify and interpret the shape of each of these factors, study their dynamics and their correlation with the underlying index. Our approach is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data. A simple factor model compatible with the empirical observations is proposed. We illustrate how this approach model and improves the the well-known ``sticky moneyness'' rule used by option traders for updating implied volatilities. Our approach gives a justification for use of ``Vega''s for measuring volatility risk and provides a decomposition of volatility risk as a sum of contributions from empirically identifiable factors.

Stochastic models of implied volatility surfaces

Rama Cont, Jose da Fonseca, Valdo Durrleman.
Economic Notes, Vol. 31, No 2, 361-377 (2002).

Abstract: We propose a market-based approach to the modeling of implied volatility, in which the implied volatility surface is directly used as the state variable to describe the joint evolution of market prices of options and their underlying asset. We model the evolution of an implied volatility surface by representing it as a randomly fluctuating surface driven by a finite number of orthogonal random factors. Our approach is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data on SP500 and DAX options.
We illustrate how this approach extends and improves the accuracy of the well-known ``sticky moneyness'' rule used by option traders for updating implied volatilities. Our approach gives a justification for use of ``Vega''s for measuring volatility risk and provides a decomposition of volatility risk as a sum of independent contributions from empirically identifiable factors. 

Volatility clustering in financial markets: empirical facts and agent-based models.

Rama CONT
Appeared in: A Kirman and G Teyssiere (Eds.): Long Memory in Economics,  Springer, 2005.

Time series of financial asset returns often exhibit the volatility clustering property: large changes in prices tend to cluster together, resulting in persistence of the amplitudes of
price changes. After recalling various methods for quantifying and modeling this phenomenon, we discuss several economic mechanisms which have been proposed to explain the origin of this volatility clustering in terms of behavior of market participants and the news arrival process. A common feature of these models seems to be a switching between low and high activity regimes with heavy-tailed durations of regimes. Finally, we discuss a simple agent-based model which links such variations in market activity to threshold behavior of market participants and suggests a link between volatility clustering and investor inertia.

Long range dependence in financial time series.

Rama CONT

Appeared in: E Lutton and J Levy Véhel (Eds.): Fractals in Engineering,  Springer, 2005.

The notions of self-similarity, scaling, fractional processes and long range dependence have been repeatedly used to describe properties of financial time series: stock prices, foreign exchange rates, market indices and commodity prices. We discuss the relevance of these concepts in the
context of financial modelling, their relation with the basic
principles of financial theory and possible economic explanations
for their presence in financial time series.

Finite difference methods for option pricing in jump- diffusion and exponential Lévy models.

Rama Cont, Ekaterina Voltchkova.

Published in: SIAM Journal of Numerical Analysis, Vol 43, No. 4,  pp. 1596-1626.

Abstract: We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Lévy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit time-stepping scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Numerical tests are performed with smooth and non-smooth initial conditions. Our scheme can be used for European and barrier options, applies in the case of pure-jump models or degenerate diffusion coefficients, and extends to time-dependent coefficients.
 

Heterogeneity and feedback in an agent-based market model

Rama CONT, Francois GHOULMIE and Jean-Pierre NADAL
Journal of Physics: Condensed Matter, Vol 17, No 14, 2005.

We propose an agent-based model of a single-asset financial market, described in terms of a small number of parameters, which generates price returns with statistical properties similar to the stylized facts observed in financial time series. Our agent-based model generically leads to absence of autocorrelation in returns, self-sustaining excess volatility, mean-reverting volatility, volatility clustering and endogenous bursts of market activity non-attributable to external noise. The parsimonious structure of the model allows to identify feedback and heterogeneity as the key mechanisms leading to these effects.

Empirical properties of asset returns: stylized facts and statistical issues.

Rama Cont.
Quantitative Finance, Vol. 1, No. 2 (March 2001) 223-236.

Abstract: We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studied of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

Phenomenology of the interest rate curve

Jean-Philippe Bouchaud, Rama CONT, Nicole El-Karoui, Marc Potters, Nicolas Sagna.
APPLIED MATHEMATICAL FINANCE, Vol 6, 209-232, September 1999.

Abstract: This paper contains a statistical description of the whole U.S. forward rate curve (FRC), based on an data in the period 1990-1996. We find that the average FRC (measured from the spot rate) grows as the square-root of the maturity, with a prefactor which is comparable to the spot rate volatility. This suggests that forward rate market prices include a risk premium, comparable to the probable changes of the spot rate between now and maturity, which can be understood as a `Value-at-Risk' type of pricing. The instantaneous FRC however departs from a simple square-root law. The distortion is maximum around one year, and reflects the market anticipation of a local trend on the spot rate. This anticipated trend is shown to be calibrated on the past behaviour of the spot itself. We show that this is consistent with the volatility `hump' around one year found by several authors (and which we confirm). Finally, the number of independent components needed to interpret most of the FRC fluctuations is found to be small. We rationalize this by showing that the dynamical evolution of the FRC contains a stabilizing second derivative (line tension) term, which tends to suppress short scale distortions of the FRC. This shape-dependent term could lead, in principle, to arbitrage. However, this arbitrage cannot be implemented in practice because of transaction costs. We suggest that the presence of transaction costs (or other market `imperfections') is crucial for model building, for a much wider class of models becomes eligible to represent reality.
 

Herd behavior and aggregate fluctuations in financial markets

Rama Cont and Jean-Philippe Bouchaud.
Macroeconomic dynamics, Vol 4, 170 - 196 (June 2000).
Abstract: We present a simple model of a stock market where a random communication structure between agents generically gives rise to a heavy tails in the distribution of stock price variations in the form of an exponentially truncated power-law, similar to distributions observed in recent empirical studies of high frequency market data. Our model provides a link between two well-known market phenomena: the heavy tails observed in the distribution of stock market returns on one hand and 'herding' behavior in financial markets on the other hand.

A Langevin approach to stock market fluctuations and crashes

Rama Cont and Jean-Philippe Bouchaud.
Appeared in: European Physical Journal B 6 (1998) 4, 543-550.

Abstract: We propose a non linear Langevin equation as a model for stock market fluctuation and crashes. This equation is based on an identification of the different processes influencing the demand and supply, and their mathematical transcription. We emphasize the importance of feedback effects of price variations onto themselves. Risk aversion, in particular, leads to an `up-down' symmetry breaking term which is responsible for crashes, where `panic' is self reinforcing. It is also responsible for the sudden collapse of speculative bubbles. Interestingly, these crashes appear as rare, `activated' events, and have an exponentially small probability of occurence. We predict that the `shape' of the falldown of the price during a crash should be logarithmic. The `normal' regime, where the stock price exhibits behavior similar to that of a random walk, however reveals non trivial correlations on different time scales, in particular on the time scale over which operators perceive a change of trend.