Fast Monte Carlo CVA using Exposure Sampling Method Alexander Sokol Numerix RiskMinds Conference 2010 (Geneva)
Definitions Potential Future Exposure (PFE) PFE(T) is maximum loss due to counterparty default at time T with a given confidence level Computed by simulating without credit events (market evolution simulation) Credit Value Adjustment (CVA) Change in portfolio value due to the possibility of counterparty default (unilateral or bilateral) Computed by simulating both market evolution and credit events (credit event simulation)
Credit Simulation Performance Performance challenge of credit simulation Credit events are rare, requiring > 1m paths Trade valuation is slow Performing them together is inefficient
More Performance Problems The resulting CVA estimator is subject to poor convergence The cashflows being averaged to compute CVA are discontinuous due to credit events This is similar to poor convergence of a barrier option under Monte Carlo simulation due to discontinuous option payoff
Two Optimization Approaches Importance sampling Perform market evolution and credit event simulation together, but make credit event simulation more effective by making credit events more likely through measure change, or eliminating paths without credit events altogether Separate market and credit simulation Perform market evolution simulation only Evaluate CVA using semi-analytical method without performing credit event simulation
Problem with Importance Sampling American Monte Carlo reuses market evolution simulation paths for pricing Eliminates performance bottleneck due to Monte Carlo within Monte Carlo Essential element of any practical CVA approach Importance sampling interferes with the use of American Monte Carlo Gives up the best available technique for calculating exposures of complex derivatives
Analytical Calculation of CVA Analytical calculation of CVA formula assuming counterparty default is uncorrelated with exposure where EPE(t) is expected positive exposure at time t PD(t,t+dt) is forward probability of default per dt R is recovery rate
CVA for Wrong Way Risk CVA is strongly dependent on the correlation between default probability PD(t,t+dt) and exposure Capture default-exposure correlation by replacing EPE(t) in CVA formula by conditional expected exposure (Redon 2006) Positive correlation between the probability of default and exposure (wrong way risk) increases CVA
Limitations of Analytical Calculation Common situations where using expected exposures prevents accurate modeling of CVA Modeling credit events using a structural default model (SDM) instead of a reduced form model based on the Poisson process Modeling credit insurance on counterparty Modeling collateral agreement dependent on credit rating
Introducing Exposure Sampling Exposure sampling method steps Perform market evolution simulation first Tabulate the distribution of exposures Draw samples from this distribution during credit event simulation using one or several additional random shocks
Full MC vs. Exposure Sampling
Exposure Sampling Advantages Advantages of Exposure Sampling Capable of modeling complex default models, collateral rules, credit transitions, and other aspects of the first principles approach which the analytical method is not able to capture Fast because drawing samples from the precomputed distribution of exposures is much faster than full valuation
Markovian 1-factor Exposure Sampling Implementation of Markovian one factor Exposure Sampling For each time T, tabulate an inverse cumulative distribution function for exposure(t) from the results of market evolution simulation Perform credit event simulation by using endpoint(t) of an additional random factor correlated with credit event model factors for drawing samples from the tabulated distribution of exposure(t)
Markovian 1-factor Exposure Sampling
Relationship to Redon 2006 formula Exposure sampling reproduces Redon 2006 wrong way risk formula if the following additional assumptions are made Path independent, reduced-form credit event model (must assume forward default probability with Poisson process for default) No ratings dependence in collateral agreement No path dependence in portfolio assumption required for Redon 2006 but only made in Markovian exposure sampling models
Exposure Sampling Example Example of modeling rating dependent collateral agreement using exposure sampling Portfolio of CDS on large financials done with a major dealer which is itself a member of large financials group a classic wrong way risk example (Lipton, Shelton 2010) Collateral agreement requires to post more collateral in case of downgrade The effect of ratings-dependent collateral cannot be modeled by the analytical method
Exposure Sampling Example Our intuition that wrong way risk will be reduced by ratings-dependent collateral threshold is confirmed Rigorous calculation with calibration to market observables Not just the intuition anymore - can be used for CVA reporting
Calibration of Correlation Calibration of exposure sampling correlation to the historical data Compute exposures for each day of a historical time interval using only the cashflows that did not yet occur as of today Calculate structural or reduced form default model factors for each day of same interval Compute historical correlation
Non-Markovian / Multifactor Sampling Exposure sampling method is not limited to the simple one-factor Markovian variant Can also be used to calibrate significantly more complex models of the exposure process, albeit with greater computational effort
Exposure Sampling for Bilateral CVA Uncorrelated credit events of the firm and the counterparty Two separate credit event simulations using exposure sampling, one each for the firm s and the counterparty s credit events Correlated credit events The firm s and counterparty s credit events simulated together with a single set of factors for exposure sampling
Non-Gaussian Exposure Sampling Non-Gaussian structural credit event models Can be used with sophisticated structural default models including the models with jumps and other non-gaussian distribution types (Lipton, Sepp 2009) Fat tails in exposure distribution Market evolution model with jump diffusion Or calibrated directly to the historical exposure distribution tails
Summary Introducing exposure sampling method Performs fast credit event simulation by drawing samples from the distribution of exposures computed during market evolution simulation Compatible with a wide variety of sophisticated credit event models, including both reducedform and structural default model types Compatible with non-gaussian / fat tail market and credit models Permits path dependence in credit simulation, e.g. rating transition modeling
Summary Even the simplest Markovian one-factor exposure sampling method offers advantages over the analytical CVA calculation in modeling: Correlation between exposures and credit model factors for both structural default and reduced form model types Counterparty rating transitions Ratings-dependent collateral agreement terms Non-Markovian / multifactor variants offer even more flexibility