A multivariate Lévy process model with linear correlation

In this paper, we develop a multivariate risk-neutral Lévy process model and discuss its applicability in the context of the volatility smile of multiple assets. Our formulation is based upon a linear combination of independent univariate Lévy processes and can easily be calibrated to a set of one-dimensional marginal distributions and a given linear correlation matrix. We derive conditions for our formulation and the associated calibration procedure to be well-defined and provide some examples associated with particular Lévy processes permitting a closed-form characteristic function. Numerical results of the option premiums on three currencies are presented to illustrate the effectiveness of our formulation with different linear correlation structures.     

An intensity model for credit risk with switching Lévy processes

We develop a switching regime version of the intensity model for credit risk pricing. The default event is specified by a Poisson process whose intensity is modeled by a switching Lévy process. This model presents several interesting features. First, as Lévy processes encompass numerous jump processes, our model can duplicate the sudden jumps observed in credit spreads. Also, due to the presence of jumps, probabilities do not vanish at very short maturities, contrary to models based on Brownian dynamics. Furthermore, as the parameters of the Lévy process are modulated by a hidden Markov chain, our approach is well suited to model changes of volatility trends in credit spreads, related to modifications of unobservable economic factors.     

Skewness premium with Lévy processes

We study the skewness premium (SK) introduced by Bates [J. Finance, 1991, 46(3), 1009–1044] in a general context using Lévy processes. Under a symmetry condition, Fajardo and Mordecki [Quant. Finance, 2006, 6(3), 219–227] obtained that SK is given by Bates' x% rule. In this paper, we study SK in the absence of that symmetry condition. More exactly, we derive sufficient conditions for the excess of SK to be positive or negative, in terms of the characteristic triplet of the Lévy process under a risk-neutral measure.     

On a Sparre Andersen risk model perturbed by a spectrally negative Lévy process

In this paper, we consider a Sparre Andersen risk model perturbed by a spectrally negative Lévy process (SNLP). Assuming that the interclaim times follow a Coxian distribution, we show that the Laplace transforms and defective renewal equations for the Gerber–Shiu functions can be obtained by employing the roots of a generalized Lundberg equation. When the SNLP is a combination of a Brownian motion and a compound Poisson process with exponential jumps, explicit expressions and asymptotic formulas for the Gerber–Shiu functions are obtained for exponential claim size distribution and heavy-tailed claim size distribution, respectively.     

Multivariate Lévy processes with dependent jump intensity

In this work we propose a new and general approach to build dependence in multivariate Lévy processes. We fully characterize a multivariate Lévy process whose margins are able to approximate any Lévy type. Dependence is generated by one or more common sources of jump intensity separately in jumps of any sign and size and a parsimonious method to determine the intensities of these common factors is proposed. Such a new approach allows the calibration of any smooth transition between independence and a large amount of linear dependence and provides greater flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. The model is analytically tractable and a straightforward multivariate simulation procedure is available. An empirical analysis shows an accurate multivariate fit of stock returns in terms of linear and nonlinear dependence. A numerical illustration of multi-asset option pricing emphasizes the importance of the proposed new approach for modeling dependence.     

Lévy flight-based real-time bird strike risk assessment for airports

Bird strike is a terrible but common incident in aviation. There is, however, a lack of systematic approaches for real-time bird strike risk assessment at present. This paper provides a novel method for bird strike risk assessment at airports with the detected data (e.g. data from radar systems), including the estimation of bird strike probability and collision severity. The Lévy flight model, an influential random walk model in bird foraging behaviour research, is adopted for the bird strike probability estimation. After dividing the area around the airport into a square matrix, the Lévy flight model is modified by the Chapman Kolmogorov equation. Meanwhile, the estimation of collision severity is based on the bird mass. The proposed method is applied to Dalian Zhoushuizi Airport with simulated bird data. The simulated results demonstrate the efficiency and real-time performance of our method.     

Implied Lévy volatility

We study the cause of large fluctuations in prices on the London Stock Exchange. This is done at the microscopic level of individual events, where an event is the placement or cancellation of an order to buy or sell. We show that price fluctuations caused by individual market orders are essentially independent of the volume of orders. Instead, large price fluctuations are driven by liquidity fluctuations, variations in the market's ability to absorb new orders. Even for the most liquid stocks there can be substantial gaps in the order book, corresponding to a block of adjacent price levels containing no quotes. When such a gap exists next to the best price, a new order can remove the best quote, triggering a large midpoint price change. Thus, the distribution of large price changes merely reflects the distribution of gaps in the limit order book. This is a finite size effect, caused by the granularity of order flow: in a market where participants place many small orders uniformly across prices, such large price fluctuations would not happen. We show that this also explains price fluctuations on longer timescales. In addition, we present results suggesting that the risk profile varies from stock to stock, and is not universal: lightly traded stocks tend to have more extreme risks.     

Optimal portfolios when stock prices follow an exponential Lévy process

We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Lévy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Lévy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Lévy process and its stochastic exponential are investigated.Received: January 2003Mathematics Subject Classification: Primary: 60F05, 60G51, 60H30, 91B28; secondary: 60E07, 91B70JEL Classification: C22, G11, D81We would like to thank Jan Kallsen and Ralf Korn for discussions and valuable remarks on a previous version of our paper. The second author would like to thank the participants of the Conference on Lévy Processes at Aarhus University in January 2002 for stimulating remarks. In particular, a discussion with Jan Rosinski on gamma processes has provided more insight into the approximation of the variance gamma model.     

Option pricing and hedging under a stochastic volatility Lévy process model

In this paper, we discuss a stochastic volatility model with a Lévy driving process and then apply the model to option pricing and hedging. The stochastic volatility in our model is defined by the continuous Markov chain. The risk-neutral measure is obtained by applying the Esscher transform. The option price using this model is computed by the Fourier transform method. We obtain the closed-form solution for the hedge ratio by applying locally risk-minimizing hedging.     

Tangent Lévy market models

In this paper, we introduce a new class of models for the time evolution of the prices of call options of all strikes and maturities. We capture the information contained in the option prices in the density of some time-inhomogeneous Lévy measure (an alternative to the implied volatility surface), and we set this static code-book in motion by means of stochastic dynamics of It?’s type in a function space, creating what we call a tangent Lévy model. We then provide the consistency conditions, namely, we show that the call prices produced by a given dynamic code-book (dynamic Lévy density) coincide with the conditional expectations of the respective payoffs if and only if certain restrictions on the dynamics of the code-book are satisfied (including a drift condition à la HJM). We then provide an existence result, which allows us to construct a large class of tangent Lévy models, and describe a specific example for the sake of illustration.     

A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options

Vanilla (standard European) options are actively traded on many underlying asset classes, such as equities, commodities and foreign exchange (FX). The market quotes for these options are typically used by exotic options traders to calibrate the parameters of the (risk-neutral) stochastic process for the underlying asset. Barrier options, of many different types, are also widely traded in all these markets but one important feature of the FX options markets is that barrier options, especially double-no-touch (DNT) options, are now so actively traded that they are no longer considered, in any way, exotic options. Instead, traders would, in principle, like to use them as instruments to which they can calibrate their model. The desirability of doing this has been highlighted by talks at practitioner conferences but, to our best knowledge (at least within the realm of the published literature), there have been no models which are specifically designed to cater for this. In this paper, we introduce such a model. It allows for calibration in a two-stage process. The first stage fits to DNT options (or other types of double barrier options). The second stage fits to vanilla options. The key to this is to assume that the dynamics of the spot FX rate are of one type before the first exit time from a ‘corridor’ region but are allowed to be of a different type after the first exit time. The model allows for jumps (either finite activity or infinite activity) and also for stochastic volatility. Hence, not only can it give a good fit to the market prices of options, it can also allow for realistic dynamics of the underlying FX rate and realistic future volatility smiles and skews. En route, we significantly extend existing results in the literature by providing closed-form (up to Laplace inversion) expressions for the prices of several types of barrier options as well as results related to the distribution of first passage times and of the ‘overshoot’.     

A Lévy HJM multiple-curve model with application to CVA computation

We consider the problem of valuation of interest rate derivatives in the post-crisis set-up. We develop a multiple-curve model, set in the HJM framework and driven by a Lévy process. We proceed with joint calibration to OTM swaptions and co-terminal ATM swaptions of different tenors, the calibration to OTM swaptions guaranteeing that the model correctly captures volatility smile effects and the calibration to co-terminal ATM swaptions ensuring an appropriate term structure of the volatility in the model. To account for counterparty risk and funding issues, we use the calibrated multiple-curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counterparty risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration, we study the case of a basis swap and a related swaption, for which we compute the counterparty risk and funding adjustments.     

A cross-currency Lévy market model

The Lévy Libor or market model which was introduced in Eberlein and Özkan (The Lévy Libor model. Financ. Stochast., 2005, 9, 327–348) is extended to a multi-currency setting. As an application we derive closed form pricing formulas for cross-currency derivatives. Foreign caps and floors and cross-currency swaps are studied in detail. Numerically efficient pricing algorithms based on bilateral Laplace transforms are derived. A calibration example is given for a two-currency setting (EUR, USD).     

Numerical methods for Lévy processes

We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy models.      

Time consistency of Lévy models

Time consistency of the models used is an important ingredient to improve risk management. The empirical investigation in this article gives evidence for some models driven by Lévy processes to be highly consistent. This means that they provide a good statistical fit of empirical distributions of returns not only on the timescale used for calibration but on various other timescales as well. As a result these models produce more reliable risk numbers and derivative prices.     

Asian options and meromorphic Lévy processes

One method to compute the price of an arithmetic Asian option in a Lévy driven model is based on an exponential functional of the underlying Lévy process: If we know the distribution of the exponential functional, we can calculate the price of the Asian option via the inverse Laplace transform. In this paper, we consider pricing Asian options in a model driven by a general meromorphic Lévy process. We prove that the exponential functional is equal in distribution to an infinite product of independent beta random variables, and its Mellin transform can be expressed as an infinite product of gamma functions. We show that these results lead to an efficient algorithm for computing the price of the Asian option via the inverse Mellin–Laplace transform, and we compare this method with some other techniques.     

Symmetry and duality in Lévy markets

The aim of this paper is to introduce the notion of symmetry in a Lévy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, which is also reviewed in the paper, and that we call put–call duality. Symmetric Lévy markets have the distinctive feature of producing symmetric smile curves, in the log of strike/futures prices.

Put–call duality is obtained as a consequence of a change of the risk neutral probability measure through Girsanov's theorem, when considering the discounted and reinvested stock price as the numeraire. Symmetry is defined when a certain law before and after the change of measure through Girsanov's theorem coincides. A parameter characterizing the departure from symmetry is introduced, and a necessary and sufficient condition for symmetry to hold is obtained, in terms of the jump measure of the Lévy process, answering a question raised by Carr and Chesney (American put call symmetry, preprint, 1996 Carr, P and Chesney, M. 1996. American put call symmetry. preprint [Google Scholar]). Some empirical evidence is shown, supporting that, in general, markets are not symmetric.     

Variance swaps on time-changed Lévy processes

We prove that a multiple of a log contract prices a variance swap, under arbitrary exponential Lévy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving Lévy process, subject to integrability conditions. We solve for the multiplier, which depends only on the Lévy process, not on the clock. In the case of an arbitrary continuous underlying returns process, the multiplier is 2, which recovers the standard no-jump variance swap pricing formula. In the presence of negatively skewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of time-changed Lévy processes to equity options data. Moreover, we show that discrete sampling increases variance swap values, under an independence condition; so if the commonly quoted multiple 2 undervalues the continuously sampled variance, then it undervalues even more the discretely sampled variance. Our valuations admit enforcement, in some cases, by hedging strategies which perfectly replicate variance swaps by holding log contracts and trading the underlying.     

Existence of Lévy term structure models

Lévy driven term structure models have become an important subject in the mathematical finance literature. This paper provides a comprehensive analysis of the Lévy driven Heath–Jarrow–Morton type term structure equation. This includes a full proof of existence and uniqueness in particular, which seems to have been lacking in the finance literature so far.