Perturbation Methods in Credit Derivatives. Colin Turfus

so as to facilitate implementation, and fully general to allow real market data to be used without any modification or re‐working of results.

NOTE

      1 1 Approximate solutions are sometimes presented as being valid for short times to maturity, but this limitation usually serves to limit the magnitude of the term variance which by construction tends to be a monotonically increasing function of time to maturity.

CHAPTER 2 Some Representative Case Studies

      Consider the following plausible scenarios where the methods set out in the remainder of this book are found to address the types of challenge faced by risk management groups, looking to capture risk more effectively and accurately under regulatory and other pressures without increasing computational overheads unduly or engaging in costly new model development.

2.1 QUANTO CDS PRICING

      A Korean client of a US bank wishes to sell protection on KRW‐denominated sovereign Korean debt and/or that of some systemically important Korean corporation. Providing a KRW‐based CDS rate is available, this can be used to price the protection (in KRW) according to the well-known formula

(2.1)

      with

the KRW short rate,

the instantaneous KRW‐denominated credit spread and

the expected recovery level on the debt post‐default. This can, if wished, be converted to a USD‐based price at today's spot exchange rate.

However, it may be that the KRW‐based spread is less liquid than the USD‐based alternative and the desk (or risk management department) prefer to use the latter. There is likely to be a so‐called quanto spread betweeen the USD CDS rate and the local KRW equivalent (reflecting the expected reduced value of a KRW protection payment after the default of a systemically important credit). This can be addressed by assuming a downwards jump‐at‐default in the value of KRW/USD exchange rate. Such remains amenable to analytic calculation. However, if the trader wants to take into account the additional possibility that the KRW/USD rate is negatively correlated with the credit spread, she appears to have no choice but to resort to Monte Carlo simulation of both the FX rate and the instantaneous credit spread (or credit default intensity).

      However, referring to Chapter 10, she sees that a highly accurate modification to the above analytic formula is available for exactly this situation. This consists in simply replacing

in the above with the effective credit intensity

defined in (10.17).

      If it happens that the coupons paid by the US bank are USD‐denominated (as will often be the case), there is a quanto effect here also which prevents the coupon leg being priced straightforwardly by analytic means. However, recourse to Monte Carlo simulation can again be avoided if, in the discount factor

used to price the coupon payment at time

,

is replaced by an effective credit intensity given this time by (10.9). In this way the trade can be priced and risk‐managed entirely using analytic formulae.

2.2 WRONG‐WAY INTEREST RATE RISK

      Another issue arises shortly after at the same bank, this time raised by the market risk department. While the credit trading desk for developed markets uses analytic pricing for most vanilla credit products, the emerging markets desk, in recognition of the possibility of significant “wrong‐way” risk associated with correlation between credit default risk and the local interest rate on foreign‐denominated floating rate notes, uses a Monte Carlo approach with short‐rate models representing both the credit intensity and the local rates processes. Market risk currently use the same (analytic pricing‐based) risk engine for the trades of both desks. However, auditors have suggested, and market risk are now concerned, that there may be problems with back‐testing of the Internal Model for market risk as a result of the discrepancy between the risk model and the emerging markets model, with only the latter capturing the wrong‐way risk. They would prefer not to incur the significant cost of migrating part of the bank's credit portfolio to be priced by a Monte Carlo engine instead of an analytic approach.

However, it turns out that there is no need for the risk model to be changed to perform Monte Carlo pricing of floating rate notes. From results presented in Chapter 8, rather than (2.1), the protection leg can be priced using (8.54) which incorporates the wrong‐way risk to a high degree of accuracy. Likewise the value of floating rate payments denominated in the local (foreign) currency can be priced using (8.58) to the same high level of accuracy. Here again, recourse to Monte Carlo simulation is avoided and computational resources are spared.

In the wake of this discussion, a question arises as to whether the quanto CDS trades currently priced with an analytic model taking account of FX‐credit risk might likewise face the possibility of wrong‐way rates‐credit correlation risk, potentially in relation to both domestic and foreign currencies. It turns out from the extended calculations set out in Chapter 12 that the impact of wrong‐way risk, from correlation of credit with the FX rate and with both interest rates on the value of a protection leg, can be taken into consideration by use of an effective credit intensity given by (12.46). Similarly the value of foreign currency coupon payments priced with a domestic currency credit curve can be obtained by making use of an effective credit intensity given by (12.38). Furthermore, it is seen that the value of a foreign currency float leg can be obtained to good accuracy by use of (12.39). In this case too, no recourse is needed to Monte Carlo methods.

2.3 CONTINGENT CDS PRICING AND CVA

      Encouraged by the successful migration of a large number of trades away from Monte Carlo models to more efficient analytic models, attention falls on a portfolio of contingent CDS trades offering counterparty default protection on interest rate (including cross‐currency) underlyings. Calculations