CDO term structure modelling with Levy processes and the relation to market models

Representation of infinite dimensional forward price models in commodity markets

We study the forward price dynamics in commodity markets realized as a process with values in a Hilbert space of absolutely continuous functions defined by Filipović. The forward dynamics are defined as the mild solution of a certain stochastic partial differential equation driven by an infinite dimensional Lévy process. It is shown that the associated spot price dynamics can be expressed as a sum of Ornstein-Uhlenbeck processes, or more generally, as a sum of certain stationary processes. These results link the possibly infinite dimensional forward dynamics to classical commodity spot models.…

基于CBI时间变换Lévy过程的多货币建模

CBI-time-changed Lévy processes for multi-currency modeling

本文提出基于CBI时间变换Lévy过程的多货币随机波动率框架，利用仿射过程理论保证解析可处理性和风险中性测度不变性，并通过Fourier方法与深度学习校准在货币三角数据上验证了模型的有效性。

Efficient and accurate log-Lévy approximations to Lévy driven LIBOR models

The LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a Lévy-driven LIBOR model and aim at developing accurate and efficient log-Lévy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps, swaptions and sticky rat…

Discrete tenor models for credit risky portfolios driven by time-inhomogeneous Lévy processes

The goal of this paper is to specify dynamic term structure models with discrete tenor structure for credit portfolios in a top-down setting driven by time-inhomogeneous Lévy processes. We provide a new framework, conditions for absence of arbitrage, explicit examples, an affine setup which includes contagion and pricing formulas for STCDOs and options on STCDOs. A calibration to iTraxx data with an extended Kalman filter shows an excellent fit over the full observation period. The calibration is done on a set of CDO tranche spreads ranging across six tranches and three maturities.

Rational term structure models with geometric Levy martingales

In the "positive interest" models of Flesaker-Hughston, the nominal discount bond system is determined by a one-parameter family of positive martingales. In the present paper we extend this analysis to include a variety of distributions for the martingale family, parameterised by a function that determines the behaviour of the market risk premium. These distributions include jump and diffusion characteristics that generate various properties for discount bond returns. For example, one can generate skewness and excess kurtosis in the bond returns by choosing the martingale family to be given by…

Approximate Option Pricing in the Lévy Libor Model

In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the Lévy Libor model developed by Eberlein and Özkan (2005). This model is an extension to Lévy driving processes of the classical log-normal Libor market model (LMM) driven by a Brownian motion. Option pricing is significantly less tractable in this model than in the LMM due to the appearance of stochastic terms in the jump part of the driving process when performing the measure changes which are standard in pricing of interest rate derivatives. To obtain explicit approximation for option price…

Levy Random Bridges and the Modelling of Financial Information

The information-based asset-pricing framework of Brody, Hughston and Macrina (BHM) is extended to include a wider class of models for market information. In the BHM framework, each asset is associated with a collection of random cash flows. The price of the asset is the sum of the discounted conditional expectations of the cash flows. The conditional expectations are taken with respect to a filtration generated by a set of "information processes". The information processes carry imperfect information about the cash flows. To model the flow of information, we introduce in this paper a class of …

The American put and European options near expiry, under Levy processes

We derive explicit formulas for time decay, for the European call and put options at expiry, and use them to calculate analytical approximations to the price of the American put and early exercise boundary near expiry. We show that for many families of non-Gaussian processes used in empirical studies of financial markets, the early exercise boundary for the American put without dividends is separated from the strike price by a non-vanishing margin on the interval [0,T). As the riskless rate vanishes and the drift decreases accordingly so that the stock remains a martingale, the optimal exercis…

Parameter Estimation using Empirical Likelihood combined with Market Information

During the last decade Levy processes with jumps have received increasing popularity for modelling market behaviour for both derviative pricing and risk management purposes. Chan et al. (2009) introduced the use of empirical likelihood methods to estimate the parameters of various diffusion processes via their characteristic functions which are readily avaiable in most cases. Return series from the market are used for estimation. In addition to the return series, there are many derivatives actively traded in the market whose prices also contain information about parameters of the underlying pr…

Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes

This paper introduces the class of volatility modulated Lévy-driven Volterra (VMLV) processes and their important subclass of Lévy semistationary (LSS) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structu…