The Exposure Profile Method

If counterparty credit risk metrics are to be useful in ways beyond an approach to decision making on credit limits for counterparties or calculating regulatory capital, they should also describe how one should price that exposure — this is the idea behind the exposure profile method. The assumption normally made in-order to obtain practical expressions for CVA is the independence of market prices from defaults. Giovan et al in [21] suggest that the CVA should be valued using the expression: CVA = Z T 0 EPE(s)D(0, s)Cs(s) ds≈ X i EPEi(Ti−Ti−1)B(0, Ti)Csi, (3.14)

where EPE(t) is the expected positive exposure assumed to be piecewise constant betweenTi−1 andTi. TheCsiis the credit spread of a forward starting CDS starting

atTi−1 and maturing atTi. Equation (3.14) implies that the counterparty exposure could be hedged by buying forward starting CDS’s with notional being determined by the expected positive exposure profile. This is very useful, especially with con- tracts where closed form valuations are not possible or the existence of a liquid market on the contract is not available. This removes the logic of viewing CVA as the cost of replacing the contract with an identical one at the time of default τ of the counterparty. The formula (3.14) is similar to (3.13), with one notable differ- ence, the measures the expectations are under are different, P and ˜P respectively for the formulas, see (3.6). The fundamental advantage with dealing OTC is that the deal can often be designed to suit specific requirements that are non-standard, however finding another counterparty to replace the defaulted counterparty may be impossible even at a fee.

A similar approach adopted by Pykhtin and Zhu [69] is to look at the loss L that would be incurred at a default time τ, given a constant recovery rateRi

twhich would be given by,

L=Iτ≤T(1−R)D(0, τ)CE, (3.15) where CE is the counterparty exposure. The definition of the CVA would then be the cost of hedging the loss L that could be incurred. They then deduce that the CVA is given by,

CVA =E˜P[L] = (1−R) Z T

0

E˜P[D(0, t)CE(t)|τ =t]d˜P(0, t), (3.16) where ˜P(s, t) are the risk neutral probabilities of counterparty default between times

3.5. QUANTIFYING CCR 36

sandt, which are normally backed out from CDS spreads. If the usual independence assumption is made, we have that

CVA = (1−R) Z T

0

D(0, t)EE(t)d˜P(0, t), (3.17) with EE(t) being the expected exposure.

An important thing to note is that the EE(t) is normally calculated at the counterparty portfolio level, thus individual CVA contributions from trades are not immediately obvious, especially when netting agreements are in place. This might not be a problem for risk management purposes but may lead to problems when the front office wants to quote a price on a trade with CVA taken into account. Pykhtin and Rosen [67] approach this issue and show how the problem of calculat- ing individual CVA contributions actually reduces to allocating expected exposure contributions to individual trades.

Calculating the Exposure Profiles

The calculation of exposure profiles is well understood having been the backbone of capital requirements for regulatory compliance. The algorithm for calculating these profiles can be decomposed into three steps, scenario generation, instrument valuation and aggregation.

• Scenario Generation : the underlying risk factors are simulated through

time normally under the physical measure. This implies that one needs to specify the stochastic models that the risk factors are assumed to obey. The risk factors may include stocks, foreign exchange, interest rates, volatility mod- els, hazard rates, etc. This phase involves the discretization of time between inception and the maturity of the latest trade in the portfolio. For practical applications banks use daily or weekly intervals up to a month, then monthly intervals up to a year and yearly up to five years [68]. There are generally two ways to perform the simulations, one known as path dependent simulations

(PDS) and the other known asdirect jump to simulation date (DJS). The for- mer involves the simulation of the risk factor as a path over all the discrete time points. The latter simulates the risk factor to a fixed date ti without any record of the path that was taken by the risk factor to get to ti. The DJS method relies on the existence of a closed form solution for the SDE describing the evolution of a particular risk factor.

• Instrument Valuation : the valuation of individual instruments given the

3.5. QUANTIFYING CCR 37

on the existence of a risk neutral measure may be inappropriate as credit expo- sures are normally calculated under the physical measure. There are technical difficulties that instrument valuations under simulations may introduce espe- cially when valuing path dependent derivatives that by definition require the knowledge of the full path. This leads to conditional valuations. For path independent derivatives, the valuations are simplified and the direct jump to simulation method is preferred. Another technical difficulty that occurs with Monte Carlo valuations is their bias when valuing American type derivatives. However, an algorithm first introduced by Broadie and Glasserman [19], which is now popularly known as American Monte Carlo gives highly reliable prices within modest error bounds, but it is an exponential algorithm as a function of exercise times.

• Aggregation : this phase requires the exposure profile formulas to be applied

at each discrete future time Ti to obtain the profile deemed necessary. The profiles would then be used for CVA calculations as suggested above.