A Study On The Reduced Pricing Model Of Credit Derivatives Swaps Based On The Lévy Processes

Being the earliest designed credit derivative,Credit Default Swap(CDS)is an efficient tool of credit risk management.It is a significant revolution in credit risk management domain,which allows financial institutions separating the credit risk from other risks and transferring it to investors through market pricing on the premise of retaining ownership of assets.Credit Default Swaps holds more than 50 percent in current credit derivatives market.Even Credit Derivatives Market was heavily defeated in financial crisis in 2008,the Credit Default Swap market was recovered quickly especially the single-asset Credit Default Swaps whose market share increased rapidly from 56% at the end of 2007 to 67% at the end of 2009.Above truth reflects strong vitality and huge market demand for single-asset CDS.After financial crisis,researchers discovered that,during pricing procedure of Credit Derivatives,some factors like market condition,product structure & model were simplified,is one of most important cause of the financial crisis.Lots of evidence and research proves existence of financial market data jumps.Before the financial crisis,for the sake of simplifying market condition and product structure,Gaussian-process based Geometric-Brownian-motion was utilized in modelling asset value & default intensity process.Unfortunately,such continuous distribution can NOT accurately fit the “sharp peak” and “heavy tail” in financial market data which resulted in errors on pricing of credit derivatives.Therefore,financial market data jumps related modeling attracted the attention of researchers after the crisis.Among them,the Lévy process created by the French mathematician Paul Lévy was focused,which is able to accurately and flexibly model the jumps.Based on stochastic analysis and Lévy processes,this paper studies the reduced pricing model of CDS based on individual Lévy process—subordinator Lévy process,a general Lévy process and a Markov switching Lévy process.Main conclusions are reached as below:Firstly,according to the reduced credit risk model,this paper assumes the default process of the reference asset of a single-asset CDS being determined by an exogenously jump process,considers the default as the first jump of jump process,hence the default intensity process can be defined by the cumulative intensity process.It is assumed that the intensity process is a Cox process.Considering only positive jump will trigger default event,the default intensity process is assumed to be a subordinator Lévy process.The conditional survival probabilities and unconditional survival probabilities are calculated by the martingale method along with Laplace transform,subsequently established a reduced credit risk model based on the subordinator Lévy process.Meanwhile,suppose the value process of the reference asset obeys the geometric Brownian motion,start with intrinsic functional relationship between the asset's value process & the debt,together with probability theory,we can deduce endogenous function expression between recovery rate and the default probability,and establish an endogenous recovery rate model.Finally by applying no-arbitrage principle,we construct CDS reduced pricing model with endogenous recovery rate based on subordinator Lévy processSecondly,considering real financial market has both negative & positive jump,subordinator Lévy process does not fit because ONLY positive jump was allowed.We naturally extend the subordinator-Lévy-process based pricing model to general-Lévy-process driven pricing model.By analyzing the property of the default intensity,according to the law of value,it is found that the default intensity shall display the mean reversion property.In additional,analysis on the characteristics of the CIR process indicates it can describe the mean reversion property of default intensity appropriately.In order to accurately capture the jumps in default intensity,it's better to use Lévy process instead of Wiener process to drive CIR process.Since default process is a Cox process whose default intensity is subjected to Lévy-process-driven CIR process,relevant conditional survival probabilities & unconditional survival probabilities can be calculated by applying following 3 methods(operation,martingale and Laplace transform).Also the reduced credit risk model based on the CIR process driven by the Lévy process can be constructed subsequently.Furthermore,the CDS reduced pricing model based on the CIR process driven by Lévy process can be modelled according to no-arbitrage pricing principle.Finally,extending single-asset CDS pricing model with constant parameters to parameters changing with the macroeconomic cycle case.Due to macroeconomic cycle and being a part of the market,for the reference asset of single-asset CDS,its drift rate,volatility,risk-free interest rate and other parameters will vary with the economic cycle.The Markov regime switching is introduced into the pricing model of the single-asset CDS.The Markov regime switching model is also used to describe the macroeconomic cycle,and the Lévy process is used to describe financial market data jumps.Assuming the default intensity process is a Cox process,the default intensity is a Markov regime switching mean-reverting process driven by Lévy process,thus reduced credit risk model based on a Markov regime switching Lévy process can be constructed.Moreover CDS reduced pricing model based on a Markov regime switching Lévy process is modelled by the no-arbitrage pricing principle.The reduced single-asset CDS pricing model based on Lévy process in this study can solve the deficiently description in the geometric-Brown-motion based pricing model.It can match the “sharp peak” and “heavy tail” in financial market data better,provide tools for the accurate pricing of single-asset CDS contracts.What's more,it also provides a theoretical basis for the financial innovation in China.