Condicionantes

A CVA tool needs to incorporate valuation models for the assets that it supports. Bj¨ork (2009) gives an account of the valuation of IRSs settled in arrears, which has been the basis of this exposition. In this section the price formula for a payer swap will be stated, i.e., a swap that pays a fixed rate and receives a floating rate. The price of a receiver swap is merely a matter of changing signs.

An IRS is tied to a floating rate, such as the 6 month LIBOR rate. At inception the swap rate, which is the fixed rate, is determined such that the PV of the swap is zero. Define two sets of dates: tf loat_{i i=0,...,N} is the set of dates that are relevant for the floating payments. Floating payments occur at the dates tf loat₁ , ...tf loat_N and the rate over the period (tf loat_i , tf loat_i+1 ] is fixed at date tf loat_i −offset, where the offset is typically two banking days. Thus the first floating rate is fixed at date tf loat₀ −offset.

tf ix_{j j=0,...,M}is the set of dates that are relevant for the fixed payments. Fixed payments occur at the dates tf ix₁ , ...tf ix_M . Furthermore, let K be the nominal amount, rf ix the swap rate, δ_jf ix the day count over the interval (tj−1, tj] according to the fixed rate convention, L(tf loat_i , tf loat_i+1 ) the floating rate over the time interval (tf loat_i , tf loat_i+1 ], δf loat_i

the day count over the interval (ti−1, ti] according to the floating rate convention and

D(s, t) the discount factor that takes a payout at time t back to time s. The price for the swap at timetis

Π(t) = N X i=mink∈{1,...,N}:t≤tk KL(ti−1, ti)δif loatD(t, ti)− M X j=mink∈{1,...,M}:t≤tj Krf ixδ_if ixD(t, tj) (28)

4.3.1.1 A Note on Discounting in the Presence of CVA

In the pre-crisis times, before counterparty credit risk was consistently incorporated in the pricing of OTC derivatives, interest rate and foreign exchange products were valued and discounted using swap curves, which were derived from the LIBOR market. The LIBOR rate is the reference rate for unsecured transactions given by an average of deposit rates that a number of member banks offer each other and is quoted for terms ranging from one day to one year. The LIBOR rate was generally seen as a good proxy of the risk-free interest rate, which was a result of the credit and liquidity risk in the interbank market being considered low and therefore ignored to a large extent as Bianchetti and Carlicchi (2012) explain. Alongside LIBOR the OIS rate is another reference rate for unsecured transactions based on overnight indexing. This rate is the average of the rates at which overnight transactions have been executed among a number of member banks. Because of the short-term horizon this rate is considered as the best proxy of the risk-free rate, especially in the post-crisis era. In the past the spreads between LIBOR

4.3 Implementation of a CVA Tool for Interest Rate Swaps Credit Value Adjustment

and OIS rates have been small but from 2007 and onwards the spreads have become larger, which can be seen in Figure 13, adopted from Bianchetti and Carlicchi (2012). The figure depicts the historical time series of the EURIBOR 6M rate, the EONIA 6M rate and the spread between them. According to the authors the spread has its origin from the different credit and liquidity risk that the rates represent.

Figure 13: Historical time series of the EURIBOR 6M rate, EONIA 6M rate and the spread between them. Adopted from Bianchetti and Carlicchi (2012).

As a result of the yield spread between LIBOR and OIS the LIBOR rate is no longer considered as a good proxy of the risk-free rate and the principles behind the valuations of interest rate and foreign exchange contracts have changed. In the case of OTC deriva- tives with cash collateral where margining is done daily it is market consensus to use the OIS curve for discounting because this is exactly the rate that the collateral will earn. Clarke (2010) provides a justification of this based on an arbitrage argument. In the case of uncollateralised OTC derivatives different opinions have been uttered regard- ing discounting. Bianchetti and Carlicchi (2012) suggest that a counterparty-specific

funding curve should be used, whereas Hull and White (2013) claim that the correct way is to derive discount factors from the OIS curve because credit risk is incorporated through CVA and using LIBOR rates for discounting would result in credit risk being accounted for twice. In this thesis the swap curve is simulated through time and used for determining future PVs and these values are subsequently discounted using the OIS curve23.

23

Since the T-forward measure is used for simulation the discount factors can be derived from the empirical OIS curve at valuation date with the advantage that the spread between the swap curve and the OIS curve does not need to be incorporated in the market model.

4.3 Implementation of a CVA Tool for Interest Rate Swaps Credit Value Adjustment