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Abstract:
We present a non-parametric method for calibrating jump-diffusion models to a finite set of observed option prices. We show that the usual formulations of the inverse problem via nonlinear least squares are ill-posed and 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. Our approach allows to conciliate the idea of calibration by relative entropy minimization with the notion of risk neutral valuation in a continuous time model. We discuss the numerical implementation of our method using a gradient based optimization algorithm and show via simulation tests on various examples that the entropy penalty resolves the numerical instability of the calibration problem. Finally, we apply our method to datasets of index options and discuss the empirical results obtained.

levy process, jump-diffusion models, implied volatility, option pricing, model calibration, non-parametric methods, inverse problems, relative entropy, regularization

Abstract:
Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. 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, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.

Abstract:
We propose a stochastic model for the continuous-time dynamics of a limit order book. The model strikes a balance between three desirable features: it can be estimated easily from data, 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. Using high-frequency data, we show that our model can effectively capture the short-term dynamics of a limit order book. We also evaluate the performance of a simple trading strategy that is based on our results.

High frequency data, limit order book, financial engineering, Laplace transform

Abstract:
This paper contains a statistical description of the whole U.S. forward rate curve (FRC), based on data from the period 1990-1996. We find that the average deviation of the FRC from the spot rate grows as the square-root of the maturity, with a proportionality constant 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 behavior 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, suggesting an analogy with the motion of a vibrating string subject to random perturbations. 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.

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 of the "cushion," the difference between the current portfolio value and the guaranteed amount. Whereas in diffusion models with continuous trading, this strategy has no downside risk, 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. Our framework leads to analytically tractable expressions for the probability of hitting the floor, the expected loss and the distribution of losses. This allows to measure the gap risk but also leads to a criterion for adjusting the multiplier based on the investor's risk aversion. Finally, we study the problem of hedging the downside risk of a CPPI strategy using options. The results are applied to a jump-diffusion model with parameters estimated from returns series of various assets and indices.

Portfolio insurance, CPPI, Levy process, hedging, CPPI option, Value at Risk, jump-diffusion models

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.

CDO, portfolio credit derivatives, model calibration, default risk, inverse problem

Abstract:
We present a simple model of a stock market where a random communication structure between agents 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. In particular, our study suggests a relation between the excess kurtosis observed in asset returns, the market order flow and the tendency of market participants to imitate each other.

Abstract:
The concepts of scale invariance and scaling behavior are now increasingly applied outside their traditional domains of application, the physical sciences. Their application to financial markets, initiated by Mandelbrot in the 1960s, has experienced a regain of interest in the recent years, partly due to the abundance of high-frequency data sets and availability of computers for analyzing their statistical properties. This lecture is intended as an introduction and a brief review of current research in a field which is increasingly applied in the study of time aggregation properties of financial data. We will try to show how the concepts of scale invariance and scaling behavior may be usefully applied in the framework of a statistical approach to the study of financial data, pointing out at the same time the limits of such an approach.

Abstract:
We propose a market-based approach to the modelling 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 the use of 'Vegas' for measuring volatility risk and provides a decomposition of volatility risk as a sum of independent contributions from empirically identifiable factors.

Implied volatility, volatility risk, risk management, portfolios of options

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 Levy 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 for the Merton jump-diffusion model and for the Variance Gamma model with smooth and non-smooth payoff functions. 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.

Jump-diffusion models, integro-differential equations, finite difference methods, Levy process, jump risk, option pricing, viscosity solutions

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 stochastic optimization 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 according to pricing performance. 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.

Option pricing, inverse problems, nonparametric estimation, stochastic modeling, derivatives

Abstract:
Uncertainty on the choice of an option pricing model can lead to "model risk" in the valuation of portfolios of options. 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 management of derivative instruments, we introduce a quantitative framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure compatible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. Finally, we discuss some implications for
the management of "model risk."

model uncertainty, Knightian uncertainty, option pricing, model risk, coherent risk measures, convex risk measures

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 define the notion of "risk measurement procedure", which includes both of these steps, and study the robustness of risk measurement procedures and their 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.

risk measure, Value at Risk, statistical estimation, robustness

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.

credit risk, portfolio credit derivatives, CDO, derivative pricing, tranche

Abstract:
As shown by the recent turmoil in credit markets, much remains to be done for the proper risk management of credit derivatives. In particular, the static copula-based models commonly used for pricing portfolio credit derivatives appear to be inappropriate for hedging and risk management. We study hedging of index CDO tranches with the underlying index default swap using various portfolio loss models which account for default contagion and spread risk. Numerical results obtained from models calibrated to iTraxx Europe data reveal significant differences in hedge ratios across models 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. An empirical analysis based on recent market data shows that strategies based on delta-hedging of spread movements have poorly performed during the 2007-2008 sub prime crisis, while variance-minimizing hedges led to significantly smaller losses. Our empirical study also reveals that, while significantly large moves - "jumps" - do occur in index spreads, these jumps do not necessarily occur on default dates of index constituents, an observation which contradicts the intuition conveyed by some recently proposed credit risk models.

hedging, portfolio credit derivatives, index default swaps, collateralized debt obligations, top-down credit risk models, default contagion, spread risk, sensitivity-based hedging, risk minimization

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.

Discrete choice models, social interactions, random utility models, aggregation, limit theorems, heterogeneity, externalities

Abstract:
The present paper examines various statistical properties of high-frequency market data by examining their unconditional distributions and correlation structure. The first part focuses on the distributional properties of high frequency data: we study the shape of the unconditional distribution of price changes and introduce a class of distributions, termed truncated Levy distributions, to model them. We compare the truncated Levy model with the Gaussian and the stable law alternatives. The second part examines the correlation properties - autocorrelation functions and temporal dependence in the amplitude - of price changes, with an emphasis on the scaling properties of price changes.

Abstract:
We propose and study 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 Lévy 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 conditional correlations in large and small spot/volatility moves.

We show that our model can simultaneously fit prices of European options on S&P 500 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.

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 variation based 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.

financial econometrics, jumps, exchange rates, high-frequency data, time series

Abstract:
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.

Portfolio credit derivatives, collateralized debt obligation, inverse problem, local intensity, default intensity, expected tranche notionals, calibration, CDO tranche

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. CPDO coupons and principal notes received high initial credit ratings from the major rating agencies, based on complex models for the joint transition of ratings and spreads for all names in the underlying portfolio. We propose a parsimonious model for analyzing the performance 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 small, the ratings obtained strongly depend on the credit environment -- high spread or low spread. More importantly, CPDO loss distributions are found to be bimodal and 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. By calculating rating transition probabilities we find that CPDO ratings can be quite unstable during the lifetime of the strategy.

CPDO, credit risk, top down models, credit rating, structured product, credit derivatives

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.

arbitrage, pricing rule, martingale, fundamental theorem of asset pricing

Abstract:
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.

volatility clustering, long range dependence, fractal, fractional Brownian motion, artbitrage, agent-based models

Abstract:
Uncertainty on the choice of an option pricing model can lead to "model risk" in the valuation of portfolios of options. 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 management of derivative instruments, we introduce a quantitative framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure compatible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of "model risk".

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 of the cushion, the difference between the current portfolio value and the guaranteed amount. Whereas in diffusion models with continuous trading, this strategy has no downside risk, in real markets this risk is nonnegligible and grows with the multiplier value. We study the behavior of CPPI strategies in models where the price of the underlying portfolio may experience downward jumps. Our framework leads to analytically tractable expressions for the probability of hitting the floor, the expected loss, and the distribution of losses. This allows to measure the gap risk but also leads to a criterion for adjusting the multiplier based on the investor's risk aversion. Finally, we study the problem of hedging the downside risk of a CPPI strategy using options. The results are applied to a jump-diffusion model with parameters estimated from returns series of various assets and indices.

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 stochastic optimization 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 independently and identically distributed 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" according to pricing performance. 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.

probabilistic, option pricing model, observed option prices, stochastic, algorithm, global minima, calibrated models, vanilla

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 "Vegas" for measuring volatility risk and provides a decomposition of volatility risk as a sum of contributions from empirically identifiable factors.

Implied volatility, options, volatility risk, stochastic volatility, principal component analysis, random field, factor model

Abstract:
This paper reviews various methods for extracting statistical information implicit in market prices of options. We present several methods for state price densities from options panel
data: lognormal Edgeworth expansions, cumulant expansions, Hermite polynomial expansions, non-parametric estimators, and maximum entropy methods. Parametric methods such as implied binomial trees and mixtures of lognormals are also briefly discussed.

We discuss the advantages and drawbacks of each method, the interpretation of their results in economic terms, their theoretical consequences and their relevance for applications.

The present text is an augmented version of a lecture presented at Eotvos University, Budapest in July 1997. The style is introductory and self contained.

Abstract:
Options markets offer an interesting example of the adaptation of a population to a complex environment, through trial and error and by 'natural' selection. Guided by the Black-Scholes theory but constrained by the fact that mispricing leads to arbitrage opportunities, options markets agree on prices which are close but significantly and systematically different from those given by the Black- Scholes formula. We re-examine the informational content of option prices in the light of the notion of implied kurtosis, analogous to that of implied volatility but taking into account the non-Gaussian character of the fluctuations of the underlying asset. We conclude by a detailed empirical study of market prices for options on German Bund futures, showing very good agreement between implied kurtosis calculated from option prices and empirical kurtosis calculated using prices of the underlying asset. Our results show that the market has adapted itself to incorporate more information on the statistical properties of returns than that conveyed by the Black-Scholes model.