APPENDIX 10A: Exposure ad swaptio aalogy. Sorese ad Bollier (1994), effectively calculate the CVA of a swap positio ad show this ca be writte as: CVA swap = LGD V swaptio (t; t i, T) PD(t i 1, t i ). i=1 Whilst more details o CVA computatio are give i Chapter 12 ad Appedix 12, we ca ote that i the above formula the expected exposure (EE) is represeted by V swaptio (t; t i, T) which is the value today of a Europea swaptio o the uderlyig swap. The swaptio has a exercise date of t i, which is the potetial default time i the discretised CVA formula ad the uderlyig swap is defied by the period [t i, T]. 1 The above shows that the EE for the purpose of calculatig the CVA of the swap ca be represeted as a series of swaptio values. The ituitio is that the couterparty has the optio to default at ay poit i the future ad therefore effectively cacel the swap. Not oly is this formula useful for aalytical calculatios, it is also quite ituitive for explaiig CVA. A iterest rate swaptio ca be priced i a modified Black-Scholes framework via the formula: [FΦ(d 1 ) XΦ(d 2 )]D(t, T) [ FΦ( d 1 ) + XΦ( d 2 )]D(t, T) (payer swaptio) (receiver swaptio) d 1 = l (F X ) + 0.5σ S 2 (t t) σ S t t = d 2 + σ S t t Where F is the forward rate of the swap, X is the strike (the fixed swap of the uderlyig swap), σ S is the swap rate volatility, t is the maturity of the swaptio (the time horizo of iterest). The fuctio D(t, T) represets the uderlyig swap duratio (aity value) for which the maturity is (T t ). The exposure of the swap will be defied by the iteractio betwee two factors: the swaptio payoff, e.g. FΦ(d 1 ) XΦ(d 2 ), ad the duratio. These quatities respectively icrease ad decrease mootoically over time. The overall swaptio value therefore peaks somewhere i-betwee as illustrated i Figure 10.1 ad Spreadsheet 10.1. 1 There could be a questio of whether the swaptio is cash or physically settled. This relates to the close out discussio i Chapter 14. Copyright 2015 Jo Gregory 1
APPENDIX 10B: Commets o exposure models by asset class i) Iterest rates I the Hull ad White model, the short rate (short-term iterest rate) is assumed to follow the followig process: dr t = [θ(t) ar t ] + σ r dw t, which is a Browia motio with mea reversio. Mea reversio dictates that whe the rate is above some mea level, it is pulled back towards that level with a certai force accordig to the size of parameter a. The mea reversio level θ(t) is timedepedet which is what allows this model to be fitted to the iitial yield curve. The mea reversio has the effect of dampig the stadard deviatio of discout factors, B(t, T), writte as: 1 exp ( a(t t)) σ(b(t, T)) = σ r [ ]. a(t t) Although the yield curve is ot modellig directly, it ca be recostructed at ay poit, give kowledge of the above parameters ad the curret short rate. Hece, usig such a approach i a Mote Carlo simulatio is relatively straightforward. Usig historical data (real world calibratio), oe ca estimate the stadard deviatio (volatility) of zero-coupo bod prices of various maturities, the it is possible to estimate values for σ r ad a. For risk-eutral pricig, these parameters would be calibrated to the prices of iterest-rate swaptios with a time depedet volatility fuctio σ r (. ). Spreadsheet 10.2 uses this model for a flat yield curve (Vasicek model). ii) FX FX is typically modelled via a geometric Browia motio (GBM): dx t X t = k[θ l (X t )] + σ FX (t)dw t, where k is the rate of mea reversio to a log-term mea level θ. For real world calibratio the the volatility fuctio will be flat ad mea reversio may be relevat to avoid the FX rate explodig which is particularly importat for log time horizos. For risk-eutral calibratio the the mea reversio is less importat sice the volatility fuctio, σ FX (t), will be calibrated directly. It is geerally ecessary to make assumptios about log-dated volatility (e.g. above 5-years) i this respect. iii) Equity A stadard geometric Browia motio (GBM) is defied by: Copyright 2015 Jo Gregory 2
ds t S t = μ(t)dt + σ E (t)dw t. where S t represets the value of the equity i questio at time t, μ(t) is the drift, σ E (t) is the volatility ad dw t is a stadard Browia motio. It is geerally preferable to simulate sigle stocks via their relatioship (beta) to idices. iv) Commodities A simple ad popular model (see Gema 2005) is: l(s t ) = f t + Z t. dz t = [α βz t ]dt + σ C (t)dw t. Where f is a determiistic fuctio, which may be expressed usig si or cos trigoometry fuctios to give the relevat periodicity ad the parameters α ad β are mea reversio parameters. v) Credit A typical model for credit icludig jumps could be: dλ t = θ[η λ t ]dt + σ λ λ t dw t + jdn, where λ t is the itesity (or hazard rate) of default 2 ad θ ad η are mea reversio parameters. This model (depedig o the calibrated parameters) ca prevet egative hazard rates as required. Additioally, dn represets a Poisso jump with jump size j. This jump size ca itself be radom such as followig a expoetial distributio. 2 This meas that the default probability i a period dt coditioal o o default before time t is λ t dt. The hazard rate is related to the credit spread as is explaied i Chapter 12. Copyright 2015 Jo Gregory 3
APPENDIX 10C: Computatio of icremetal ad margial exposure i) Icremetal exposure I order to calculate the icremetal exposure we simply eed to add the simulated values for a ew trade (i) to those for the rest of the ettig set. Workig from equatio (10.1), we ca write K V NS+i (s, t) = V(k, s, t) + V(i, s, t) = V NS (s, t) + V(i, s, t), k=1 givig the future value of the ettig set, icludig the ew trade i each simulatio (s) ad at each time poit (t). From this, it is easy to calculate the ew EE, which ca be compared with the existig EE as required by equatio (10.2). What is helpful here is that we eed oly kow the future value of the ettig set, ot the costituet trades. From a systems poit of view this reduces the storage requiremets from a cube of dimesio K S T (which could be extremely costly) to a matrix of dimesio S T. Typically, systems hadle the computatio of icremetal exposure by calculatig ad storig the ettig set iformatio V NS (s, t) (ofte i a overight batch) ad the geeratig the simulatios for a ew trade, V(i, s, t) o-the-fly as ad whe required. The reaggregatio is straightforward ad recalculatio of measures such as EE is the a quick calculatio. ii) Margial exposure Suppose we have calculated a etted exposure for a set of trades uder a sigle ettig agreemet. We would like to be able write the total EE as a liear combiatio of EEs for each trade, i.e.: EE total = EE i. If there is o ettig the we kow that the total EE will ideed be the sum of the idividual compoets ad hece the margial EE will equal the EE (EE i = EE i ). However, sice the beefit of ettig is to reduce the overall EE, we expect i the evet of ettig that EE i < EE i. The aim is to fid allocatios of EE that reflect a trade s cotributio to the overall risk ad sum up to the total couterparty level EE (EE total ). As described i Chapter 10, this type of problem has bee studied for other metrics such as value-at-risk (VAR). I the absece of a collateral agreemet, EE (like VAR) is homogeous of degree oe which meas that scalig the size of the uderlyig positios by a costat will have the same impact of the EE. This is writte as: i=1 αee(x) = EE(αx), Copyright 2015 Jo Gregory 4
where α = (α 1, α 2,, α ) is a vector of weights. By Euler s theorem we ca the defie the margial EE as: EE i = EE total(α) α i. Oe way to compute the above partial derivative is to chage the size of a trasactio by a small value ad calculate the margial EE usig a fiite differece. This does ot require ay additioal simulatio but just a rescalig of the future values of oe trade by a amout (1 + ε) followed by a recalculatio of the EE for the ettig set 3. The margial EE of the trade i questio is the give by the chage i the EE divided by. The sum of the margial EEs will sum to the total EE 4. Alteratively, as show by Rose ad Pykhti (2010), it ca be also computed via a coditioal expectatio: EE i = E[max(V i, 0) V NS > 0] = S 1 max(v i,s, 0) I(V NS > 0) where V i,s represets the future value for the trasactio i i simulatio s (igorig the time suffix) ad V NS = i=1 V i is the future value for the relevat ettig set. The fuctio I(. ) is the idicator fuctio that takes the value uity if the statemet is true ad zero otherwise. Such calculatios are illustrated i Spreadsheet 10.6. More detail, icludig discussio o how to deal with collateralised exposures ca be foud i Rose ad Pykhti (2010). The ituitio behid the above formula is that the future values of the trade i questio are added oly if the ettig set has positive value at the equivalet poit. A trade that has a favourable iteractio with the overall ettig set may the have a egative margial EE sice its future value will be more likely to be egative whe the ettig set has a positive value. Whilst margial EE is easy to calculate as defied above, it does require full storage of all the future values at the trade level. From a systems poit of view, margial EE could be calculated durig the overight batch with little additioal effort whereupo the trade-level future values ca be discarded. However, for aalysig the chage i margial EE uder the ifluece of a ew trade(s) the, ulike icremetal EE, all trade-level values must be retaied. S k=1 3 ε is a small umber such as 0.001. 4 At least i the curret case where o collateral is assumed as discussed below. Copyright 2015 Jo Gregory 5