Wim Schoutens

Chaotic and predictable representations for Levy processes

The only normal martingales which posses the chaotic representation property and the weaker predictable representation property and which are at the same time also Levy processes, are in essence Brownian motion and the compensated Poisson process. For a general Levy process (satisfying some moment conditions), we introduce the power jump processes and the related Teugels martingales. Furthermore, we orthogonalize the Teugels martingales and show how their orthogonalization is intrinsically related with classical orthogonal polynomials. We give a chaotic representation for every square integral random variable in terms of these orthogonalized Teugels martingales. The predictable representation with respect to the same set of orthogonalized martingales of square integrable random variables and of square integrable martingales is an easy consequence of the chaotic representation.

Lévy processes, polynomials and martingales

We study an unusual connection between orthogonal polynomials and martingales. We prove that all classical orthogonal polynomials from the Meixner class, when evaluated at a corresponding Levy process, are martingales. This result is well known for the case of Hermite polynomials evaluated in Brownian motion. Our results provide similar analogues for the Poisson process, for the Gamma process and for two less familiar processes related to Meixner polynomials

Backward stochastic differential equations and Feynman-Kac formula for Levy processes, with applications in finance

In this paper we show the existence and uniqueness of a solution for backward stochastic differential equations driven by a Levy process with moments of all orders. The results are important from a pure mathematical point of view as well as in the world of finance: an application to Clark-Ocone and Feynman-Kac formulas for Levy processes is presented. Moreover, the Feynman-Kac formula and the related partial differential integral equation provide an analogue of the famous Black-Scholes partial differential equation and thus can be used for the purpose of option pricing in a Levy market.

Pricing credit default swaps under Lévy models

Most structural models for credit pricing assume Geometric Brownian motion to describe the firm asset value. However, the underlying lognormal distribution does not match empirical distributions, typically skewed and leptokurtic. Moreover, defaults are usually driven by shocks, which are not captured by the continuous paths of Brownian motion. We assume the asset price process is driven by a pure-jump Levy process and default is triggered by the crossing of a preset barrier. Our model incorporates asymmetry, fat-tail behaviour, jumps and instantaneous defaults. Under this model we price Credit Default Swaps, detailing the calculations for the Variance Gamma process.

A Generic One-Factor Lévy Model for Pricing Synthetic CDOs

The one-factor Gaussian model is well known not to fit the prices of the different tranches of a collateralized debt obligation (CDO) simultaneously, leading to the implied correlation smile. Recently, other one-factor models based on different distributions have been proposed. Moosbrucker [12] used a one-factor Variance-Gamma (VG) model, Kalemanova et al. [7] and Guegan and Houdain [6] worked with a normal inverse Gaussian (NIG) factor model, and Baxter [3] introduced the Brownian Variance-Gamma (BVG) model. These models bring more flexibility into the dependence structure and allow tail dependence. We unify these approaches, describe a generic one-factor Levy model, and work out the large homogeneous portfolio (LHP) approximation. Then we discuss several examples and calibrate a battery of models to market data.

Static hedging of Asian options under Lévy models: the comonotonicity approach

The Asian option pricing problem is a lot like the American put problem in the 1970s. An Asian payoff is a rather simple, and common, option feature, but it messes up our clean, closed-form valuation equations. This situation is apparently a persistent source of annoyance to mathematicians and other quants, who respond with an outpouring of creativity, in the form of theory, algorithms, and approximate solutions. Although this may seem like overkill for the specific problem at hand, it produces useful new ideas and techniques for our general derivatives valuation toolkit. In this article, Albrecher et al, introduce a new approach to pricing Asian options, based on the principle of comonotonicity and the “stop-loss transform.” They derive tight bounds on the value, even when the underlying assets price follows a Lévy process, rather than a Gaussian diffusion. As with many of the solutions to the American put problem, this technique can potentially be applied to a much broader class of valuation problems.

General lower bounds for arithmetic Asian option prices

This paper provides model‐independent lower bounds for prices of arithmetic Asian options expressed through prices of European call options on the same underlying that are assumed to be observable in the market, and the corresponding subreplicating strategy is identified. The first bound relies on the no‐arbitrage assumption only and turns out to perform satisfactorily in various situations. It is shown how the bound can be tightened under mild additional assumptions on the underlying market model. This considerably generalizes lower bounds in the literature, which are only available in the Black–Scholes world. Furthermore, it is illustrated how to adapt the procedure to the case where only a finite number of strikes is available in the market. As a by‐product, the finite strike upper bound on the Asian call price of Hobson et al. (2005a), who considered basket options, is rederived. Numerical illustrations of the bounds are given together with comparisons to bounds resulting from model specifications.

Contingent Capital: An In-Depth Discussion

Regulators have embraced the idea of pre-arranging bank recapitalizations through (funded or unfunded) contingent capital issuance. Contingent capital is intended to be triggered when a bank is headed toward failure in order to provide an automatic equity injection that keeps the bank out of distress. This note discusses counterparty risk, effectiveness, moral hazard, contagion and systemic risk, as well as death-spiral issues arising from the hedging strategies of the investors. We pay attention to important design issues with respect to the trigger and conversion ratio and comment on their pricing from an equity and credit derivative perspective.

Completion of a Lévy market by power-jump assets

Abstract.Except for the geometric Brownian model and the geometric Poissonian model, the general geometric Lévy market models are incomplete models and there are many equivalent martingale measures. In this paper we suggest to enlarge the market by a series of very special assets (power-jump assets) related to the suitably compensated power-jump processes of the underlying Lévy process. By doing this we show that the market can be completed. The very particular choice of the compensators needed to make these processes tradable is delicate. The question in general is related to the moment problem.

THE PRICING OF EXOTIC OPTIONS BY MONTE–CARLO SIMULATIONS IN A LÉVY MARKET WITH STOCHASTIC VOLATILITY

Recently, stock price models based on Levy processes with stochastic volatility were introduced. The resulting vanilla option prices can be calibrated almost perfectly to empirical prices. Under this model, we will price exotic options, like barrier, lookback and cliquet options, by Monte–Carlo simulation. The sampling of paths is based on a compound Poisson approximation of the Levy process involved. The precise choice of the terms in the approximation is crucial and investigated in detail. In order to reduce the standard error of the Monte–Carlo simulation, we make use of the technique of control variates. It turns out that there are significant differences with the classical Black–Scholes prices.