• Selected
• Entire List

Abstract:
This work focuses on the swaptions automatic cascade calibration algorithm (CCA) for the LIBOR Market Model (LMM) first appeared in Brigo and Mercurio (2001). This method induces a direct analytical correspondence between market swaption volatilities and LMM parameters, and allows for a perfect recovery of market quoted swaption volatilities if a common industry swaptions approximation is used.

We present explicitly an extension of the CCA to calibrate the entire swaption matrix rather than its upper triangular part. Then, while previous tests on earlier data showed the appearance of numerical problems, we present here different calibration cases leading to acceptable results. We analyze the characteristics of the configurations used and concentrate on the effects of different exogenous instantaneous historical or parametric correlation matrices.

We also investigate the influence of manipulations in input swaptions data for missing quotes, and devise a new algorithm maintaining all the positive characteristics of the CCA while relying only on directly quoted market data. Empirical results on a larger range of market situations and instantaneous covariance assumptions show this algorithm to be more robust and efficient than the previous version. Calibrated parameters are in general regular and financially satisfactory, as confirmed by the analysis of various diagnostics implied structures.

Finally we Monte Carlo investigate the reliability of the underlying LMM swaption analytical approximation in the new context, and present some possibilities to include information coming from the semi-annual tenor cap market.

Libor Market Model, swaptions, calibration, cascade calibration

Abstract:
In this document we show how to handle counterparty risk for Interest Rate Swaps (IRS). First we establish a general formula, showing that counterparty risk adds one level of optionality to the contract. Then we introduce the default probabilities using a deterministic intensity model where the default time is modeled as the first jump of a time-inhomogeneous Poisson process. We consider Credit Default Swaps as liquid sources of market default probabilities.

We then apply the general formula to a single IRS. The IRS price under counterparty risk turns out to be the sum of swaption prices with different maturities, each weighted with the probability of defaulting around that maturity. Then we consider a portfolio of IRS's in presence of a netting agreement. The related option cannot be valued as a standard swaption, and we resort both to Monte Carlo simulation and to two analytical approximations, investigating them by Monte Carlo simulation under several configurations of market inputs. We find a good agreement between the formula and the simulations in most cases. The approximated formula is well suited to risk management, where the computational time under each risk factors scenario is crucial and an analytical approximation contains it.

Interest Rate Swap, Counterparty Risk Pricing, Netting Agreements, Analytical Tractability, Simulation, Libor Model

Abstract:
In risk management it is desirable to grasp the essential statistical features of a time series representing a risk factor. This tutorial aims to introduce a number of different stochastic processes that can help in grasping the essential features of risk factors describing different asset classes or behaviors. This paper does not aim at being exhaustive, but gives examples and a feeling for practically implementable models allowing for stylised features in the data. The reader may also use these models as building blocks to build more complex models, although for a number of risk management applications the models developed here suffice for the first step in the quantitative analysis. The broad qualitative features addressed here are fat tails and mean reversion. We give some orientation on the initial choice of a suitable stochastic process and then explain how the process parameters can be estimated based on historical data. Once the process has been calibrated, typically through maximum likelihood estimation, one may simulate the risk factor and build future scenarios for the risky portfolio. On the terminal simulated distribution of the portfolio one may then single out several risk measures, although here we focus on the stochastic processes estimation preceding the simulation of the risk factors Finally, this first survey report focuses on single time series. Correlation or more generally dependence across risk factors, leading to multivariate processes modeling, will be addressed in future work.

Risk Management, Stochastic Processes, Maximum Likelihood Estimation, Fat Tails, Mean Reversion, Monte Carlo Simulation

Abstract:
In this paper we start by introducing the standard moment-matching procedure that one can apply to simulate the average price of a basket of basic assets. The basic idea is that of approximating the actual process of the basket value by a sufficiently simple stochastic process. The expression "sufficiently simple" should be interpreted as "simple enough to allow for analytic solutions to the pricing problem at hand".

The approximation happens on the basis of a moment matching principle, which can be stated as follows: set the parameters of the approximating process so that as many moments of the actual basket-price process as possible are exactly reproduced. With the usual lack of fantasy, the market choice of an approximating process seems to have fallen onto the lognormal one. The distinctive parameters of such a process being only two (the average return and the return's standard deviation over the time horizon set by the option to price) the moment matching procedure can only match the first two moments of the original distribution. The lengthy calculations can be performed so as to take into account the effect of dividends, either continuous or discrete (but in any case deterministic, both in payment dates and in amounts). A more compact formulation of this method is obtained by resorting to forward prices, which incorporate interest rates and dividends. We describe this basic framework in detail and then move to the three moments matching procedure, obtained by shifting the approximating Lognormal basket by a deterministic constant parameter. This new parameter allows to fit the first three moments without losing analytical tractability, in that we can immediately characterize the distributional properties of the resulting process trivially.

We then move to an empirical analysis of the two and three moments matching approximations, where we study the case of a basket of two equities in the Italian stock exchange and compare results by resorting to a Monte Carlo simulation to obtain the "true" distribution and statistics of the basket.

We subsequently move to analyze specifically the implications of the three moments method as far as a call option pricing is concerned.

The second part of the paper address the problem of computing a synthetic but at the same time rigorous measure of the deviation of the approximated baskets distributions from the real basket distribution. To characterize rigorously this distributional discrepancy, we introduce both the Kullback-Leibler information and the Hellinger distances in suitable spaces of probability densities, and explain how this can help us in our investigation. We compute the distances of the real basket from the parametric families of densities being used in the two and three moments approximations through Monte Carlo simulation. The two families are respectively the lognormal and shifted lognormal families. Finally, we try and isolate the variables and the situations causing this distance to increase drastically via a case study, thus characterizing the case where the two and three moments approximations can fail.

Abstract:
In this paper we develop a tractable structural model with analytical default probabilities depending on some dynamics parameters, and we show how to calibrate the model using a chosen number of Credit Default Swap (CDS) market quotes. We essentially show how to use structural models with a calibration capability that is typical of the much more tractable credit-spread based intensity models.

We apply the structural model to a concrete calibration case and observe what happens to the calibrated dynamics when the CDS-implied credit quality deteriorates as the firm approaches default. Finally we provide a typical example of a case where the calibrated structural model can be used for credit pricing in a much more convenient way than a calibrated reduced form model: The pricing of counterparty risk in an equity swap.

Credit Derivatives, Structural Models, Black Cox Model, Credit Default Swaps, Calibration, Analytical Tractability, Monte Carlo Simulation, Equity Swaps, Counterparty Risk, Barrier Options

Abstract:
We consider the risk neutral loss distribution as implied by index CDO tranche quotes through a "scenario default rate" model as opposed to the objective measure loss distribution based on historical analysis. The risk neutral loss distribution turns out to privilege large realizations of the loss with respect to the objective distribution, thus implying the well known presence of a risk premium. We quantify this difference numerically by pricing CDO tranches and indices under the two distributions. En passant we analyze the implied risk neutral default rate distributions calibrated from April-2004 throughout April-2006, pointing out its distinctive "bump feature" in the tail.

Default Rate distribution, CDO, CDO tranches, Perfect Copula, Transition Matrices, Rating Classes, Risk Premium, Recovery Rate

Abstract:
We illustrate the two main types of implied correlation one may obtain from market CDO tranche spreads. Compound correlation is more consistent at single tranche level but for some market CDO tranche spreads cannot be implied. Base correlation is less consistent but more flexible and can be implied for a much wider set of CDO tranche market spreads. Furthermore, base correlation is more easily interpolated and leads to the possibility to price non-standard detachments. Even so, base correlation may lead to negative expected tranche losses, thus violating basic no-arbitrage conditions. We illustrate these features with numerical examples.

Implied Correlation, Base Correlation, Compound Correlation, Expected Tranche Loss, DJ iTraxx, CDX, CDO Tranche, Back-Test, No-Arbitrage Conditions

Abstract:
In this work we derive an approximated no-arbitrage market valuation formula for Constant Maturity Credit Default Swaps (CMCDS). We move from the CDS options market model in Brigo (2004), and derive a formula for CMCDS that is the analogous of the formula for constant maturity swaps in the default free swap market under the LIBOR market model.

A convexity adjustment-like correction is present in the related formula. Without such correction, or with zero correlations, the formula returns an obvious deterministic-credit-spread expression for the CMCDS price. To obtain the result we derive a joint dynamics of forward CDS rates under a single pricing measure, as in Brigo (2004). Numerical examples of the convexity adjustment impact complete the paper.

CDS Options, CDS Options Market Model, Constant Maturity CDS, Convexity Adjustment, Participation Rate, CDS rates volatility, CDS rates correlation

Abstract:
We consider counterparty risk for interest rate payoffs in presence of correlation between the default event and interest rates. The previous analysis of Brigo and Masetti (2006), assuming independence, is further extended to interest rate payoffs different from simple swap portfolios. A stochastic intensity model with possible jumps is adopted for the default event. We find that correlation between interest-rates and default has a relevant impact on the positive adjustment to be subtracted from the default free price to take into account counterparty risk. We analyze the pattern of such impacts as product characteristics and tenor structures change through some fundamental numerical examples.

We find the counterparty risk adjustment to decrease with the correlation for receiver payoffs, while the analogous adjustment for payer payoffs increases. The impact of correlation decreases when the default probability increases. Finally, our analysis applies naturally also to Contingent Credit Default Swaps.

counterparty risk, contingent credit default swap, hybrid products, interest-rate default correlation, risk-neutral valuation, default risk, interest-rate models, default intensity models

Abstract:
We introduce a general class of analytically tractable models for the dynamics of an asset price based on the assumption that the asset-price density is given by the mixture of known basic densities. We consider the lognormal-mixture model as a fundamental example, and for the first time we derive the related explicit dynamics and show that it leads to a stochastic differential equation admitting a unique strong solution. We also provide closed form formulas for option prices and analytical approximations for the implied volatility function. We then introduce the asset-price model that is obtained by shifting the previous lognormal-mixture dynamics and investigate its analytical tractability. We finally consider a specific example of calibration to real market option data.

Abstract:
We propose a general setting for pricing single-name knock-out options. Examples include Credit Default Swaps (CDS), European and Bermudan CDS options. The default of the underlying reference entity is modeled within a doubly stochastic framework where the default intensity follows a CIR++ process. We estimate the model parameters through a combination of a calibration-based method and a historical approach. We propose a numerical procedure based on dynamic programming and a piecewise linear approximation to price American-style knock-out credit options. Our numerical investigation shows consistency, convergence and efficiency. We find that American-style CDS options can complete the credit derivatives market by allowing the investor to focus on spread movements and not credit default.

Credit Derivatives, Credit Default Swaps, Bermudan Options, Dynamic Programming, Doubly Stochastic Process, Cox Process

Abstract:
In this work we consider three problems of the standard market approach to the pricing of credit index options: the definition of the index spread is not valid in general, the payoff considered leads to a pricing which is not always defined, and the candidate numeraire to define a pricing measure is not strictly positive, which would lead to an inequivalent pricing measure. We give to the three problems a general mathematical solution, based on a novel way of modelling the flow of information through the definition of a new subfiltration. Using this subfiltration, we take into account consistently the possibility of default of all names in the portfolio, that is neglected in the standard market approach. We show that, while this mispricing can be negligible for standard options in normal market conditions, it can become highly relevant for different options or in stressed market conditions.

In particular, we show on 2007 market data that after the subprime credit crisis the mispricing of the market formula compared to the no arbitrage formula we propose has become financially relevant even for the liquid Crossover Index Options.

credit option, subprime, correlation, market models, arbitrage

Abstract:
We consider counterparty risk for Credit Default Swaps (CDS) in presence of correlation between default of the counterparty and default of the CDS reference credit. Our approach is innovative in that, besides default correlation, which was taken into account in earlier approaches, we also model credit spread volatility. Stochastic intensity models are adopted for the default events, and defaults are connected through a copula function. We find that both default correlation and credit spread volatility have a relevant impact on the positive counterparty-risk credit valuation adjustment to be subtracted from the counterparty-risk free price. We analyze the pattern of such impacts as correlation and volatility change through some fundamental numerical examples, analyzing wrong-way risk in particular. Given the theoretical equivalence of the credit valuation adjustment with a contingent CDS, we are also proposing a methodology for valuation of contingent CDS on CDS.

Counterparty Risk, Credit Valuation adjustment, Credit Default Swaps, Contingent Credit Default Swaps, Credit Spread Volatility, Default Correlation, Stochastic Intensity, Copula Functions, Wrong Way Risk

Abstract:
We consider the standard Credit Default Swap (CDS) payoff and some alternative approximated versions, stemming from different conventions on the premium and protection legs. We consider standard running CDS (RCDS), upfront CDS and postponed-payments running CDS (PRCDS). Each different definition implies a different definition of forward CDS rate, which we consider with some detail. We introduce defaultable floating rate notes (FRN)'s. We point out which kind of CDS payoff produces a forward CDS rate that is equal to the fair spread in the considered defaultable FRN. An approximated equivalence between CDS's and defaultable FRN's is established, which allows to view CDS options as structurally similar to the optional component in defaultable callable notes. We briefly investigate the possibility to express forward CDS rates in terms of some basic rates and discuss a possible analogy with the LIBOR and swap default free models. Finally, we discuss the change of numeraire approach for deriving a Black-like formula for CDS options or, equivalently, defaultable callable FRN's. We also introduce an analytical formula for CDS option prices under the CDS-calibrated SSRD stochastic-intensity model, and discuss the impact of the different CIR++ dynamics parameters on the related CDS options implied volatilities. Hints on possible methods for smile modeling of CDS options are given for possible future developments of the CDS option market.

Credit Default Swaps, CDS Options, Callable Defaultable Floaters, CIR++ model, CDS options market models, CDS Calibration, Stochastic Intensity Models

Abstract:
In the first part we consider a dynamical model for the number of defaults of a pool of names. The model is based on the notion of generalized Poisson process, allowing for more than one default in small time intervals, contrary to many alternative approaches to loss modeling. We illustrate how to define the pool default intensity and discuss recovery assumptions. The models are tractable, pricing and simulation are straightforward, and consistent calibration to quoted index CDO tranches and tranchelets for several maturities is feasible, as we illustrate with numerical examples. In the second part we model directly the pool loss and we introduce extensions based on piecewise-gamma, scenario-based or CIR random intensities, leading to richer spread dynamics, investigating calibration improvements and stability.

Loss Distribution, Loss Dynamics, Calibration, CDO Tranches and Tranchelets, Generalized Poisson Processes, Gamma intensity, Spread Dynamics

Abstract:
In the present paper we show how to extend any time-homogeneous short-rate model and analytically tractable short-rate model (such as Vasicek (1977), Cox-Ingersoll-Ross (1985), Dothan (1978)) to a model which can reproduce any observed yield curve, through a procedure that preserves the possible analytical tractability of the original model. In the case of the Vasicek (1977) model, our extension is equivalent to that of Hull and White (1990), whereas in the case of the Cox-Ingersoll-Ross (1985) (CIR) model, our extension is more analytically tractable and avoids problems concerning the use of numerical solutions. Our approach can also be applied to the Dothan (1978) or Rendleman and Bartter (1980) model, thus yielding a "quasi" lognormal short-rate model which fits any given yield curve and for which there exist analytical formulae for prices of zero coupon bonds. We also consider the extension of time-homogeneous models without analytical formulae but whose tree-construction procedures are particularly appealing, such as the exponential Vasicek's. We explain why the CIR++ extended CIR model is the more interesting model obtained through our procedure. We also give explicit analytical formulae for bond options, hence swaptions, caps and floors, and we explain how the model can be used for Monte Carlo evaluation of European path-dependent interest-rate derivatives. We finally hint at the same extension for multifactor models and explain its strong points for concrete applications.

Short-rate models, Analytical tractability, Yield-Curve fitting, Vasicek's model, Dothan's model, Cox-Ingersoll-Ross' model, Longstaff and Schwartz's model, Monte Carlo evaluation

Abstract:
We explain how the payoffs of credit indices and tranches are valued in terms of expected tranched losses (ETL). ETL are natural quantities to imply from market data. No-arbitrage constraints on ETL's as attachment points and maturities change are introduced briefy. As an alternative to the temporally inconsistent notion of implied correlation we consider the ETL surface, built directly from market quotes given minimal interpolation assumptions. We check that the kind of interpolation does not interfere excessively. Instruments bid/asks enter our analysis, contrary to Walker's (2006) earlier work on the ETL implied surface. By doing so we find less and very few violations of the no-arbitrage conditions. The ETL implied surface can be used to value tranches with nonstandard attachments and maturities as an alternative to implied correlation.

expected tranche loss, loss surface, implied correlation, CDO, tranches, interpolation

Abstract:
In the present paper we construct stock price processes with the same marginal log-normal law as that of a geometric Brownian motion and also with the same transition density (and returns' distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes', in that option prices can be either arbitrarily close to the option intrinsic value or arbitrarily close to the underlying stock price. We also explain that this is due to the particular way one models the stock-price process in-between the grid time instants which are relevant for trading. The theoretical result concerning scalar stochastic differential equations with prescribed diffusion coefficient whose densities evolve in a prescribed exponential family, on which part of the paper is based, is presented in detail.

Stochastic Differential Equations, Fokker--Planck Equation, Exponential Families, Stock Price Models, Black and Scholes model, Option Pricing, Trading Time Grid, Delta-Markovianity, Market Incompleteness, Option replication error

Abstract:
In the present paper, given an evolving mixture of probability
densities, we define a candidate diffusion process whose marginal
law follows the same evolution. We derive as a particular case a
stochastic differential equation (SDE) admitting a unique strong
solution and whose density evolves as a mixture of Gaussian densities. We present an interesting result on the comparison
between the instantaneous and the terminal correlation between the obtained process and its squared diffusion coefficient.
As an application to mathematical finance, we construct diffusion
processes whose marginal densities are mixtures of lognormal densities. We explain how such processes can be used to model the
market smile phenomenon. We show that the lognormal mixture
dynamics is the one-dimensional diffusion version of a suitable
uncertain volatility model, and suitably reinterpret the earlier
correlation result. We explore numerically the relationship
between the future smile structures of both the diffusion and the
uncertain volatility versions.

Stochastic Differential Equations, Mixtures of Densities, Mixtures of Gaussians, Mixtures of Lognormals, Risk-Neutral Valuation, Option Pricing, Volatility-Underlying Correlation, Smile Modeling

Abstract:
It is commonly accepted that Commodities futures and forward prices, in principle, agree under some simplifying assumptions. One of the most relevant assumptions is the absence of counterparty risk. Indeed, due to margining, futures have practically no counterparty risk. Forwards, instead, may bear the full risk of default for the counterparty when traded with brokers or outside clearing houses, or when embedded in other contracts such as swaps. In this paper we focus on energy commodities and on Oil in particular. We use a hybrid commodities-credit model to assess impact of counterparty risk in pricing formulas, both in the gross effect of default probabilities and on the subtler effects of credit spread volatility, commodities volatility and credit-commodities correlation. We illustrate our general approach with a case study based on an oil swap, showing that an accurate valuation of counterparty risk depends on volatilities and correlation and cannot be accounted for precisely through a pre-defined multiplier.

Counterparty Risk, Credit Valuation adjustment, Commodities, Swaps, Oil models, Convenience Yield models, Stochastic Intensity models

Abstract:
In this work we develop a tractable structural model with analytical default probabilities depending on a random default barrier and possibly random volatility ideally associated with a scenario based underlying firm debt. We show how to calibrate this model using a chosen number of reference Credit Default Swap (CDS) market quotes. In general this model can be seen as a possible extension of the time-varying AT1P model in Brigo and Tarenghi (2004). The calibration capability of the Scenario Volatility/Barrier model (SVBAT1P), when keeping time-constant volatility, appears inferior to the one of AT1P with time-varying deterministic volatility. The SVBAT1P model, however, maintains the benefits of time-homogeneity and can lead to satisfactory calibration results, as we show in a case study where we compare different choices on scenarios and parameters.

Similarly to AT1P, SVBAT1P is suited to pricing hybrid equity/credit derivatives and to evaluate counterparty risk in equity payoffs, and more generally to evaluate hybrid credit/equity payoffs. We consider the equity return swap in Brigo and Tarenghi (2004) and show its valuation under SVBAT1P with the same CDS and equity calibration input used earlier for AT1P.

Credit Derivatives, Structural Models, Black Cox Model, Credit Default Swaps, Calibration, Analytical Tractability, Monte Carlo Simulation, Equity Return Swaps, Counterparty Risk, Barrier Options, Scenario Default Barrier, Scenario Volatility

Abstract:
We present a two-factor stochastic default intensity and interest rate model for pricing single-name default swaptions. The specific positive square root processes considered fall in the relatively tractable class of affine jump diffusions while allowing for inclusion of stochastic volatility and jumps in default swap spreads. The parameters of the short rate dynamics are first calibrated to the interest rates markets, before calibrating separately the default intensity model to credit derivatives market data. A few variants of the model are calibrated in turn to market data, and different calibration procedures are compared. Numerical experiments show that the calibrated model can generate plausible volatility smiles. Hence, the model can be calibrated to a default swap term structure and few default swaptions, and the calibrated parameters can be used to value consistently other default swaptions (different strikes and maturities, or more complex structures) on the same credit reference name.

Credit Derivatives, Credit Default Swap, Credit Default Swaption, Jump-Diffusion, Stochastic Intensity, Doubly Stochastic Poisson Process, Cox Process

Abstract:
In this paper we investigate implied volatility patterns in the Shifted Square Root Diffusion (SSRD) model as functions of the model parameters. We begin by recalling the Credit Default Swap (CDS) options market model that is consistent with a market Black-like formula, thus introducing a notion of implied volatility for CDS options. We examine implied volatilies coming from SSRD prices and characterize the qualitative behavior of implied volatilities as functions of the SSRD model parameters. We introduce an analytical approximation for the SSRD implied volatility that follows the same patterns in the model parameters and that can be used to have a first rough estimate of the implied volatility following a calibration. We compute numerically the CDS-rate volatility smile for the adopted SSRD model. We find a decreasing pattern of SSRD implied volatilities in the interest-rate/intensity correlation. We check whether it is possible to assume zero correlation after the option maturity in computing the option price and provide an upper bound for the Monte Carlo standard error in cases where this is not possible.

Abstract:
We introduce the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, including default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net value of the contract at the relevant default times. We allow for correlation between the default times of the investor, counterparty and underlying portfolio risk factors. We use arbitrage-free stochastic dynamical models. We then specialize our analysis to Credit Default Swaps (CDS) as underlying portfolio, generalizing the work of Brigo and Chourdakis (2008) [5] who deal with unilateral and asymmetric counterparty risk. We introduce stochastic intensity models and a trivariate copula function on the default times exponential variables to model default dependence. Similarly to [5], we find that both default correlation and credit spread volatilities have a relevant and structured impact on the adjustment. Differently from [5], the two parties will now agree on the credit valuation adjustment. We study a case involving British Airways, Lehman Brothers and Royal Dutch Shell, illustrating the bilateral adjustments in concrete crisis situations.

Counterparty Risk, Arbitrage-Free Credit Valuation Adjustment, Credit Default Swaps, Contingent Credit Default Swaps, Credit Spread Volatility, Default Correlation, Stochastic Intensity, Copula Functions, Wrong Way Risk

Abstract:
We extend the common Poisson shock framework reviewed for example in Lindskog and McNeil (2003) to a formulation avoiding repeated defaults, thus obtaining a model that can account consistently for single name default dynamics, cluster default dynamics and default counting process. This approach allows one to introduce significant dynamics, improving on the standard bottom-up approaches, and to achieve true consistency with single names, improving on most top-down loss models. Furthermore, the resulting GPCL model has important links with the previous GPL dynamical loss model in Brigo, Pallavicini and Torresetti (2006a,b), which we point out. Model extensions allowing for more articulated spread and recovery dynamics are hinted at. Calibration to both DJi-TRAXX and CDX index and tranche data across attachments and maturities shows that the GPCL model has the same calibration power as the GPL model while allowing for consistency with single names.

Loss Distribution, Loss Dynamics, Single Name Default Dynamics, Cluster Default Dynamics, Calibration, Generalized Poisson Process, Stochastic Intensity, Spread Dynamics, Common Poisson Shock Models

Abstract:
We develop and test a fast and accurate semi-analytical formula for single-name default swaptions in the context of the shifted square root jump diffusion (SSRJD) default intensity model. The formula consists of a decomposition of an option on a summation of survival probabilities in a summation of options on the underlying survival probabilities, where the strike for each option is adjusted.

Credit derivatives, Credit Default Swap, Credit Default Swaption, Jump-diffusion, Stochastic intensity, Doubly stochastic poisson process, Cox process, Semi-Analytic formula, Numerical integration

Abstract:
Following the recent introduction of new forms of Credit Default Swap (CDS) contracts expressed as upfront payments plus a fixed coupon, this note examines the methodology suggested by Barclays Capital, Goldman Sachs, JPMorgan, Markit (BGJM)/ISDA (2009), for conversion of CDS quotes between upfront and running. The proposed flat hazard rate (FHR) conversion method is to be understood as a rule-of-thumb single-contract quoting mechanism rather than as a modelling device. For example, an hypothetical investor who would put the FHR converted running spreads into her old running CDS library would strip wrong hazard rates, inconsistent with those coming directly from the quoted term structure of upfronts.

This new methodology appears mostly as a device to transit the market towards adoption of the new upfront CDS as direct trading products while maintaining a semblance of running quotes for investors who may be suffering the transition. We caution though that
- the conversion done with proper hazard rates consistent across term would produce different results;
- the quantities involved in the conversion should not be used as modelling tools anywhere; and
- for highly distressed names with a high upfront paid by the protection buyer, the conversion to running spreads fails unless, as we propose, a third recovery scenario of 0% is added to the suggested 20% and 40%.

This paper is not meant as a criticism of the proposed standardization of the conversion method but as a warning on the confusion this may generate when the method is not used carefully.

Credit Default Swap, Upfront Credit Default Swap, Running Credit Default Swap, Hazard Rates, Conversion Running Upfront

Abstract:
The purpose of this paper is introducing rigorous methods and formulas for bilateral counterparty risk credit valuation adjustments (CVA's) on interest-rate portfolios. In doing so, we summarize the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, as developed more in detail in Brigo and Capponi (2008), including the default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net present value of the contract at the relevant default times. We allow for correlation between the default times of the investor and counterparty, and for correlation of each with the underlying risk factor, namely interest rates. We also analyze the often neglected impact of credit spread volatility. We include Netting in our examples, although other agreements such as Margining and Collateral are left for future work.

Counterparty Risk, Arbitrage-Free Credit Valuation Adjustment, Interest Rate Swaps, Interest Rate Derivatives, Credit Valuation Adjustment, Bilateral Risk, Credit Spread Volatility, Default Correlation, Stochastic Intensity, Short Rate Models, Copula Functions, Wrong Way Risk