Yong Wang, PhD, CFA, FRM

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Counterparty Risk Pricing

• Yong Wang, PhD, CFA, FRM
• Managing Director, Quantitative Analysis
• Group Risk Management, Royal Bank of Canada

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• All the views expressed in this presentation are
  those of my own and do not necessarily represent
  the views of RBC.

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Outline

• Credit Exposure (Derivatives Credit Risk)
• Special character of derivatives credit risk
• Exposure calculation
• Migration techniques
• Pricing Counterparty Risk
• How to price CCIRS
• Adding more risk factors
• How the netting should be treated

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Credit Exposure Derivatives Credit Risk

• Risk of loss due to counter-party default on a
  derivative contract.
• The loss due to a default is the cost of
  replacing the contract with a new one
• The replacement cost at the time of default is
  equal to the present value of the expected future
  cash flows

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OTC Derivatives Markets

• Where is most of the counterparty risk?

Source Bank of International Settlements
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Derivative Credit Risk

• The credit risk of a derivative transaction
  fluctuates over time with the underlying
  variables that determine the value of the
  contract.
• To determine the credit risk, we need
• current exposure, mark-to-market
• potential exposure determined by finding the
  future probability distribution of interest and
  exchange rates. The policy is to measure the
  maximum loss with certain confidence.

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Credit Exposure

• The one sided payoff means that the exposure at
  default is

M2M
Time
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Maximum Potential Exposure
Credit Limit

Mark to Market Value
0
-
t0
t1
t2
t3
Time
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Credit Migration Techniques

• Mark-to-market caps
• Bilateral netting
• Early termination clauses

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Payoff of the Credit Contingent IRS (CCIRS)

• Payoff when the reference obligor defaults at
  time
• where
• the default time of the reference obligor
• the maturity of an underlying swap
• the into-forward swap rate
  observed at time the swap strike
• the into-forward annuity
  observed at time 1(-1)for
  pay/receive swaps
• notional amount of the underlying swap
• recovery rate of the reference obligor
• The value seen at time t

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Risk factors in a CCIRS

• Interest rate risks
• interest rate and volatility risk
• Credit risks
• credit spread risk, default risk, recovery rate
  risk, credit spread volatility
• Correlation risk
• correlation between interest rate risk and credit

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Modeling of Default Arrival of Underlying Obligor

• Structural Approach
• Mertons model in 1974
• Default happens asset value goes below certain
  threshold
• Asset process can be modeled as a lognormal
  process
• Reduced Form
• Default probability is described by a Poisson
  intensity (or hazard rate )

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Reduced Form

• Risk Free Zero Coupon Bond

• Default-risky Zero Coupon Bond/Price of
  Default Risk

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Theoretical (toy) Modeling Framework

• Short Process for both hazard rate and Interest
  Rate

• HJM framework for both credit spread
• See Philipp Schonbucher
• can be set up but not aware of any practical
  models

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Issues for Short Rate Model -Analytically
tractable but not practically useful

• Market Implied Volatilities for forward swaps a
  two dimensional issue
• Parameters for hazard rate process
• Correlation between IR and credit
• Negative IR and default probability

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Solutions

• Negative default probability can be solved by
• CIR process
• Lognormal process (BK)
• Implied vol issues can be (partially) dealt with
  using
• Shifted CIR (CIR) (Damiano Brigo)
• Shifted lognormal (Peter Jackel)

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Remaining Issues

• Lost analytical tractability need an efficient
  algorithm
• Volatilities will still be an issue what vol to
  use? Interpolation with IR swaption expiry at the
  default time, which can any time before maturity
• Hazard rate calibration with correlation
  assumption

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Numerical Algorithms Better Algorithm Always in
Demand

• Analytical solution with convexity adjustments
• Semi-analytical approach - Spectral quadrature
  scheme
• Two-dimensional tree approaches
• Monte-Carlo simulation

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Practical Solution

• Two dimensional BK tree approach
• Calibration to IR term structure and CDS term
  structure
• Calibration directly to implied volatilities

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Implementation

• IR tree algorithm can be found in Interest Rate
  Models Theory and Practice by Brigo and Mercurio
• Default tree algorithm can be found in Credit
  Derivatives Pricing Models by Schnobucher
• Correlation in two-dimensional tree proposed by
  Hull and White

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Default Branching
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IR Tree and Hazard Rate Tree
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Correlation
   
 
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An Example Trade

• Underlying Swap pay fixed quarterly at 5.6 and
  receive 3m Libor
• Three sets of market information shown below IR
  term structure, implied vol with strike 5.6, and
  credit spread curve and recovery

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Input Data
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Results with different correlation, mean
reversion rate, and volatility
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Results with a different credit spread curve
(50bps flat)
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Some Issues need to be solved

• Underlying swap may not be a vanilla swap
• Upon default, the settlement is MTM of underlying
  swap, which may have to including outstanding
  accrual
• easy for fixed but non-trivial for floating
  because the libor is determined before default
  time

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Extension to a general CCDS case

• More risk factors such as FX
• For the example shown, it is difficult to add
  more risk factors
• More complicated valuation implication of the
  underlying correlation assumption
• Portfolio and netting
• A portfolio may have many types of trades
• Maybe large number of risk factors

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Bibliographic

• Counterparty Risk for Credit Default Swap,
  Damiano Brigo Kyriakos Chourdakis
• A tree implementation of a credit spread model
  for credit derivatives, Philipp Schonbucher
  1999
• Semi-analytic valuation of credit linked swaps in
  a Black-Karasinski framework, Peter Jackel