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Abstract:
In this paper we will explain how to perfectly hedge under Heston's stochastic volatility model with jump to default, which is in itself a generalization of the Merton jump-to-default model and a special case of the Heston model with jumps. The hedging instruments we use to build the hedge will be as usual the stock and the bond, but also the Variance Swap (VS) and a Credit Default Swap (CDS). These instruments are very natural choices in this setting as the VS hedges against changes in the instantaneous variance rate, while the CDS protects against the occurrence of the default event. First, we explain how to perfectly hedge a power payoff under the Heston model with jump to default. These theoretical payoffs play an important role later on in the hedging of payoffs which are more liquid in practice such as vanilla options. After showing how to hedge the power payoffs, we show how to hedge newly introduced Gamma payoffs and Dirac payoffs, before turning to the hedge for the vanillas. The approach is inspired by the Post-Widder formula for real inversion of Laplace Transforms. Finally, we will also show how power payoffs can readily be used to approximate any payoff only depending on the value of the underlier at maturity. Here, the theory of orthogonal polynomials comes into play and the technique is illustrated by replicating the payoff of a vanilla call option.

hHeston, hedging, variance swaps, CDS, jump to default

Abstract:
In this paper, we propose a multivariate financial model for nancial assets which incorporates jumps, skewness, kurtosis and stochastic volatility, and discuss its applications in the context of equity and credit risk. In the former case we describe the stochastic behavior of a series of stocks or indexes, in the latter we apply the model in a multi-firm, value-based default model. Starting from a independent Brownian world, we will introduce jumps and other deviations from normality, as well as non-Gaussian dependence, by the simple but very strong technique of stochastic time-changing. We work out the details in the case of a Gamma time-change, thus obtaining a multivariate Variance Gamma (VG) setting. We are able to characterize the model from an analytical point of view, by writing down the joint distribution function of the assets at any point in time and by studying their association via the copula technique.

The model is also computationally friendly, since numerical results require a modest amount of time and the number of parameters grows linearly with the number of assets. The main feature of the model however is the fact that - opposite to other, non jointly Gaussian settings - its risk neutral dependence can be calibrated from univariate derivative prices. Examples from the equity and credit market show the goodness of fit attained.

Abstract:
In the sixties Mandelbrot already showed that extreme price swings are more likely than some of us think or incorporate in our models. A modern toolbox for analyzing such rare events can be found in the field of extreme value theory. At the core of extreme value theory lies the modelling of maxima over large blocks of observations and of excesses over high thresholds. The general validity of these models makes them suitable for out-of-sample extrapolation. By way of illustration we assess the likeliness of the crash of the Dow Jones on October 19, 1987, a loss that was more than twice as large as on any other single day from 1954 until 2004.

exceedances, extreme value theory, heavy tails, maxima

Abstract:
In this paper, we study a new class of tractable diffusions suitable for model primitives of interest rates. We consider scalar diffusions with scale s'(x) and speed m(x) densities discontinuous at the level x*. We call that family of processes Self Exciting Threshold (SET) diffusions. Following Gorovoi and Linetsky (2004), we obtain semianalytical expressions for the transition density of SET (killed) diffusions. We propose several applications to interest rates modeling. We show that SET short rate processes do not generate arbitrage possibilities and we adapt the HJM procedure to forward rates with discontinuous scale density. We also extend the CEV and the shiftedlognormal Libor market models. Finally, the models are calibrated to the U.S. market. SET diffusions can also be used to model stock price, stochastic volatility, credit spread, etc.

SETAR, State-price density, Skew Brownian motion, Eigenfunction expansions, Interest rates, Market models

Abstract:
In a recent paper, Salminen and Yor (2004b) relate the distribution of the Dufresne's reflected perpetuity to the hitting time of a reflected Bessel process. In this contribution, we adapt the results of Salminen and Yor (2004b) in several ways. First, we use spectral theory to obtain a series expansion for the distribution that renders this quantity applicable to actuarial purposes. We also investigate perpetuities when the rate of return is modelled by a more general skew Brownian motion with drift.

Skew Brownian motion, Bessel processes, local time, spectral theory, perpetuities

Abstract:
We employ a Levy process subject only to negative jumps to describe the motion of asset values. This specification permits fast computation of first passage probabilities. As a result we are able to calibrate all CDS curves for the 125 ITRAXX underliers weekly and develop a time series for the implied parameter values. A variety of models are investigated for the process, gamma, inverse gaussian, and the one sided CGMY here referred to as CMY.

Default Probabilities, Levy Models, CDS pricing

Abstract:
Cox & Leland (2000) used techniques from the field of stochastic control theory to show that in the particular case of a Brownian motion for the asset log-returns risk averse decision makers with a fixed investment horizon prefer path-independent pay-offs over path-dependent ones.

In this note we provide a novel and simple proof for the Cox & Leland result and we will extend it to general Levy markets in case pricing is based on the Esscher transform (exponential tilting). It is also shown that in these markets optimal path-independent pay-offs are increasing with the underlying final asset value. We provide examples that allow explicit verification of our theoretical findings and also show that the inefficiency cost of path-dependent pay-offs can be significant.

Our results indicate that path-dependent investment pay-offs, the use of which is widespread in financial markets, do not offer good value from the investor's point of view.

Financial Structured Product, CPPI, Asian Option, Optimal investment, Mean Variance, Markowitz, Lévy Process, Exponential tilting, CAPM, Esscher transform

Abstract:
In this paper, we introduce a new robust model for modelling and pricing LCDX tranches. We extend the generic one-factor model of [1], which was developed for modelling and pricing of a synthetic CDO of CDSs, to a model for tranched portfolio of loan-only CDSs (LCDSs). The essential difference is that now also the possibility of prepayments is built in. As a main advantage, the proposed model allows to trade LCDX tranches expressed in base correlations.

LCDX tranches, LCDS, loan-only credit default swap, credit risk, credit derivatives, one-factor model, base correlations, tranche pricing, recursive formula, Levy processes, default risk, prepayment risk, alpha-stable process, variance gamma process

Abstract:
This paper describes a dynamic multivariate jump driven model in a credit setting. We set up a dynamic Levy model, more precisely a Multivariate Variance Gamma (VG) model, for a series of correlated spreads. The parameters of the model come from a two step calibration procedure. First, a joint calibration on swaptions on the spreads is performed and second, a correlation matching procedure is applied. For the first calibration step, we make use of equity-like pricing formulas for payer and receiver swaptions, based on the characteristic function and the Fast Fourier Transform (FFT) method. In the second calibration step, we fix the correlation in the model to match the prescribed (in casu historically observed) correlation. This can be done fast since a closed form expression is readily available. The resulting jump driven dynamic model generates correlated spreads very fast. This model can be used to price a whole range of exotic structures. We illustrate this by pricing the currently popular credit Constant Proportion Portfolio Insurance (CPPI) structures. Because of the built in jump dynamics a better assessment of gap risk is possible.

CPPI, Levy, Variance Gamma, CDO, Credit Derivatives

Abstract:
Although market is busy today working on the bullet LCDS contract to remove the cancellation feature from syndicated secured loan derivatives, in their current form LCDSs and LCDX tranches are still exposed to the cancellation risk. Until recently, in lack of proper modelling framework, market practitioners neglected the cancellation risk and they priced and hedged these products as simple CDSs and CDO tranches. However, cancellation risk does matter! Especially in the current market situation. As we show here, it is more than important to take into account the cancellation risk while marking-to-market and hedging syndicated secured loan derivatives. For this purpose, we present here an easy and robust way to model the cancellation.

cancellation risk, bullet LCDS, LCDX, tranche pricing, marking-to-market, hedging, one-factor models, base correlation, L¿vy copulas, stochastic recovery