Calculating the Funding Valuation Adjustment

(FVA) of Value-at-Risk (VAR) based Initial

Margin∗

Andrew Green†and Chris Kenyon‡

August 6, 2019

Version 1.0

Abstract

Central counterparties (CCPs) require initial margin (IM) to be postedfor derivative portfolios cleared through them. Additionally, the BaselCommittee on Banking Supervision has proposed (BCBS-261 2013) thatall significant OTC derivatives trading must also post IM by 2019. IMis typically calculated using Value-at-Risk (VAR) or Conditional Value-at-Risk (CVAR, aka Expected Shortfall), based on historical simulation.As previously noted (Green 2013b), (Green 2013a) IM requirements giverise to a need for unsecured funding similar to FVA on unsecured deriva-tives. The IM cost to the derivatives originator requires an integral of thefunding cost over the funding profile which depends on VAR- or CVAR-based calculation. VAR, here, involves running a historical simulationMonte Carlo inside a risk-neutral Monte Carlo simulation. Brute forcecalculation is computationally unfeasible. This paper presents a compu-tationally efficient method of calculating IM costs for any derivative port-folio: Longstaff-Schwartz Augmented Compression (LSAC). Essentially,Longstaff-Schwartz is used with an augmented state space to retain accu-racy for VAR-relevant changes to the state variables. This method allowsrapid calculation of IM costs both for portfolios, and on an incrementalbasis. LSAC can be applied wherever historic simulation VAR is requiredsuch as lifetime cost of market risk regulatory capital using internal mod-els. We present example costs for IM under (BCBS-261 2013) for interestrate swap portfolios of up to 10000 swaps and 30 year maturity showingsignificant IM FVA costs and two orders of magnitude speedup comparedto direct calculation.

∗The views expressed are those of the author(s) only, no other representationshould be attributed.†Contact: andrew.green2@lloydsbanking.com‡Contact: chris.kenyon@lloydsbanking.com

1

arX

iv:1

405.

0508

v1 [

q-fi

n.PR

] 2

May

201

4

1 Introduction

1.1 Initial Margin and Funding Costs

Historical simulation VAR-based IM calculations will become almost universalwith (BCBS-261 2013), but there is no efficient way to calculate their lifetimeportfolio costs as required for (incremental) trade pricing. This paper introducesa Longstaff-Schwartz-based method, Longstaff-Schwartz Augmented Compres-sion (LSAC) for efficient lifetime portfolio computation of any VAR- or CVAR-based costs (funding, capital, etc.). We also provide numerical results for IMFVA for interest rate swap portfolios of from 100 to 10000 swaps and 30-yearmaturities showing constant calculation times for LSAC that are up to 100 timesfaster than brute-force calculation.

Central counterparties (CCPs) require their members to post collateral throughseveral mechanisms including initial margin, variation margin, volatility or bid-offer costs and contributions to the default fund. The recent proposal from theBasel Commmittee on Banking Supervision (BCBS-261 2013) would require allfinancial firms and systemically important non-financial institutions that tradederivatives to post initial margin in support of OTC derivatives between them.This reflects the fact that most CSA agreements in place that support OTCderivatives trading do not include initial margin requirements and the desireof regulators to incentivise the use of central clearing and disincentivise OTCderivatives between financial firms.

The methodology used to calculate the size of initial margin for a givenportfolio is often based on a historical Monte Carlo simulation approach tocalculating Value-at-Risk (VAR) or variants of it such as Conditional Value-at-Risk (CVAR)1. For example, LCH.Clearnet SwapClear uses a proprietary modelcalled PAIRS based on CVAR (LCH 2013). Historical simulation parametersvary between CCPs with differences in the length of look-back period, confidenceinterval and close-out period (Cameron 2011), (Rennison 2013). The Baselcommittee recommendations on the Initial Margin to be based on the potentialincrease in value of the derivative consistent with a one-tailed 99% confidenceinterval with a 10-day time horizon and calibrated to a period that includessignificant market stress. The period can be at most five years, and should beappropriate for each broad asset class.

To see how funding costs arise when initial margin is required, consider thefollowing examples. Consider first a derivative transacted between a bank anda corporate client under a CSA. The bank hedges the market risk on this tradeby trading the opposite transaction through the interbank market, also undera CSA agreement. For simplicity assume that both CSA agreements allowexactly the same assets to be posted and that there is no threshold, that theMinimum Transfer Amount is infinitesimally small and that collateral transferoccurs instantly.2 The trade could be any derivative but it is easier to see theargument if it can have a positive or negative value so assume it is an interestrate swap. Under this CSA no initial margin is required but variation marginis of course required and we assume that both CSAs allow full rehypothecationof the collateral. The example is illustrated in figure 1.

1Also known as Expected Shortfall2That is both CSAs are perfect and there is no loss on the default of any party.

2

Bank Corporate Market Interest Rate Swap A Interest Rate Swap B

Case I Trade A –ve Value

to Corp

Trade A +ve Value to Bank

Trade B -ve Value

to Bank

Trade B +ve Value

to Mkt

Corporate Gives Collateral to Bank

Bank Rehypothecates Collateral to Market

Variation Margin

Case II Trade A

+ve Value to Corp

Trade A -ve Value to Bank

Trade B +ve Value

to Bank

Trade B -ve Value

to Mkt

Bank Rehypothecates Collateral to Corporate

Market gives Collateral to Bank

Variation Margin

Figure 1: The collateral flows associated with a trade between a bank and acorporate and the associated hedge with the interbank OTC derivative market.Both trade and hedge are supported by perfect CSA agreements and rehypoth-ecation is allowed. No funding costs or FVA occur in this case.

When the trade has a negative value to the corporate they must post collat-eral to the bank under the terms of the CSA agreement. The bank receives thecollateral to mitigate the credit risk from the counterparty. The bank’s hedgetrade has a negative value to it and it must post collateral to the hedge counter-party under the second CSA. Given rehypothecation is allowed the bank simplypasses the collateral it received from the corporate to the hedge counterparty.If the corporate trade has a positive value to it, the collateral flows are exactlyreversed. Given there is no shortage of collateral there is no funding valuationadjustment.

In the second example, the bank trades with a corporate under a CSA agree-ment as before but chooses to hedge with an interbank counterparty but clearthe derivative through a CCP. For simplicity we assume that the variation mar-gin required by the CCP is exactly equivalent to the terms of the bank-corporateCSA. The CCP, however, also required initial margin to be posted. This exam-ple is illustrated in figure 2.

In this second case the variation margin is again provided by rehypothecationwith the direction of the collateral flow determined by the sign of the value ofthe corporate trade. The initial margin, however, must be provided irrespectiveof the sign of the trade value and the size is determined by by a percentile of thedistribution of possible changes in trade value. There is no natural source forthis collateral and the bank will be forced to borrow unsecured in the market

3

Bank Corporate CCP Interest Rate Swap A Interest Rate Swap B

Case I Trade A –ve Value

to Corp

Trade A +ve Value to Bank

Trade B -ve Value

to Bank

Trade B +ve Value

to CCP

Corporate Gives Collateral to Bank

Bank Rehypothecates Collateral to Market

Variation Margin

Case II Trade A

+ve Value to Corp

Trade A -ve Value to Bank

Trade B +ve Value

to Bank

Trade B -ve Value

to CCP

Bank Rehypothecates Collateral to Corporate

Market gives Collateral to Bank

Variation Margin

No Collateral Source – Bank Borrows Unsecured

Initial Margin

No Collateral Source – Bank Borrows Unsecured

Initial Margin

Figure 2: The collateral flows associated with a trade between a bank and acorporate and the associated hedge cleared through a CCP. The variation marginis met with rehypothecation of collateral, however, the initial margin remainsirrespective of the variation margin position. Given there is no rehypothecablesource of collateral the bank must borrow unsecured to fund the initial marginand this unsecured borrowing gives rise to FVA.

unless it has spare unused assets on its balance sheet that can be used to postas collateral. Hence the initial margin gives rise to a funding cost and hence afunding valuation adjustment on the corporate trade. Of course if the corporateand bank did not have a CSA in place then an FVA would also arise on thevariation margin requirement.

1.2 Funding Valuation Adjustment

During and after the financial crisis bank unsecured funding costs rose signif-icantly. One measure of bank funding costs, the Libor-OIS spread, peaked ataround 350bp during early 2009 (Thornton 2009). Banks could no longer ig-nore the cost of unsecured funding on their derivative portfolios and began tolook at ways of including this effect. Anecdotal evidence suggests that the firstsuch models involved simple discounting adjustments for unsecured derivatives.Later more mathematically consistent models were proposed by Antonov et al.(2013), Burgard and Kjaer (2011b) (2011a) (2012), Crepey (2012a) (2012b),Fries (2010) (2011), Kenyon and Stamm (2012), Morini (2011) and Pallaviciniet al. (2012).

In 2012 and 2013 John Hull and Alan White wrote a series of papers which

4

argued that FVA did not exist (Hull and White 2012), (Hull and White 2013).These papers prompted a strong practitioner response, arguing that FVA wasa real cost and should be priced into derivatives as illustrated by the debate inthe pages of Risk magazine, (Carver 2012) and (Laughton and Vaisbrot 2012),and elsewhere in (Castagna 2012), (Morini 2012) and (Kenyon 2013). Industrydiscussion about accounting policy in relation to FVA is underway (Cameron2013), with eight banks reporting FVA reserves or adjustments by end Q1 2014,and anecdotal evidence suggesting all major banks include FVA in pricing tosome degree. One of the big-four accounting firms has also recently (December2013) published recommendations including the use of FVA (Sommer, Todd,Peter, and Carstens 2013).

1.3 Longstaff-Schwartz and CVA

In 2001 Longstaff and Schwartz proposed a regression based model to allow alower bound on the value of derivatives with American or Bermudan option ex-ercise. This approach was quickly accepted by practitioners and became widelyused in the valuation of structured derivative products such as Bermudan swap-tions. It has also been used for hedging (Wang and Caflish 2009). In its originalform the regression method was effectively used to identify the exercise bound-ary for the Bermudan or American call by regressing for the continuation value.However, structured products often had embedded optionality in both the non-exercise and exercised state requiring the regression approach to be used forboth continuation and exercise value.

Later it was realised that the same method could be applied in the contextof CVA (Cesari, Aquilina, Charpillon, Filipovic, Lee, and Manda 2010). Inthis context the Longstaff-Schwartz approach can be through of as providinga polynomial approximation function to the value of a derivative using statevariables that are evolved inside a Monte Carlo simulation. This approximationis then used to provide the value of the exotic derivative rather than call acomputationally expensive valuation routine that itself may involve Monte Carlosimulation. The polynomial approximation is, of course, very fast and herewe propose making use of this same approximation to quickly compute thedistribution of values under the VAR model.

2 The FVA Model

We propose that the Funding Valuation Adjustment for initial margin is simplythe integral over the Initial Margin requirement at a given time t with thefunding density. The appropriate funding rate, sF (θ), is simply the averagecost of providing the cheapest to deliver collateral to the CCP. This could, forexample, be the cost of providing GBP cash for the initial margin on a book ofGBP interest rate swaps. We do not propose to discuss further in this paperhow this rate is obtained. We assume for the purposes of this paper that theCCP itself cannot default. With these conditions the FVA on the initial marginis given simply by

VFVA-IM = −∫ T

0

sF (θ)e−∫ θ0sF (s)dse−

∫ θ0r(s)dsE[IM(θ)]dθ. (1)

5

3 Longstaff-Schwartz for Expected Exposure

The Longstaff-Schwartz algorithm for pricing American or Bermudan style isdescribed in numerous publications including (Longstaff and Schwartz 2001),(Andersen 1999) and (Piterbarg 2003). In the context of CVA the aim of theLongstaff-Schwartz approach is to produce a polynomial approximation to thevalue of a derivative or a portfolio of derivatives. Note that if a portfolio isregressed this approach can also be thought of as a type of portfolio compressionalgorithm. To this we adopt the following procedure:

1. Choose a series of explanatory variables

O1(ω, tk), . . . , OnO (ω, tk)

on which the value and cash flows of the derivative or portfolio depend.Note that empirically the choice of explanatory variables has a significantimpact on the quality of the approximation and so it is very important tochoose these carefully.

2. Choose a set of functions of the explanatory variables that will form abasis for a linear least squares regression for the value function,

f1(Oi(ω, tk)), . . . , fnf (Oi(ω, tk)).

3. Run an appropriate Monte Carlo simulation in the risk-neutral measureincluding the dates upon which the derivative or portfolio value is required.Additional dates of importance to the derivative or portfolio such as optionexercise dates may be required to obtain accurate results.

4. Using backward induction from the maturity of the derivative or portfoliotK = T ,

(a) For each path ω at observation date tk, calculate the value of thederivative or portfolio, F (ω; tk), as a discounted sum of cash flowsC(ω, tj) for tj > tk including any future exercise decisions, giving

F (ω; tk) =

K∑j=k+1

e−∫ tjtk

r(ω,s)dsC(ω, tj). (2)

(b) At each observation date tk perform a linear least squares regressionof all the continuation values, F (ω; tk) on the basis functions. hencewe find the coefficients αm such that,

min

∑ω

‖nf∑

m=1

αmfm(Oi(ω, tk))− F (ω; tk)‖2, αm ∈ R

. (3)

(c) The functions,

F (αm, Oi(ω, tk), tk) =

nf∑m=1

αmfm(Oi(ω, tk)), (4)

provide a fast approximation to the value of the derivative or portfolioat each observation time, tk.

6

5. Run a second simulation to calculate the expected positive exposure wherethe value of each derivative or the whole portfolio is replace by a call tothe relevant function, F . The input explanatory variables are calculatedfrom the Monte Carlo simulation along each path and at each observationdate.

4 Calculating VAR efficiently inside a MonteCarlo Simulation

The initial margin, IM(t) requirement, is normally determined by a VAR orCVAR type model using historical simulation. In general, to calculate the Ex-pected Initial Margin Profile, E[IM(t)], we would need to run a Monte Carlosimulation to calculate the expectation and run a second historical Monte Carlomodel on each path, and at each observation point, to estimate the initial mar-gin required in that state. This inner Monte Carlo, if done by brute force, wouldrequire the full revaluation of the portfolio hundreds to thousands of times (e.g.around 1294 for the 5-year period in (BCBS-261 2013)). Hence we need anappropriate and efficient approximation.

In BCBS-261 the historical period must include a period of market stress.CCPs, by typically using long periods, also include periods of market stress.Furthermore VAR, and CVAR, are used at high percentiles, 99% one-sided gen-erally in Basel III and BCBS-261 (VAR), and 97.5% in BCBS-265 (ES/CVA).Thus we have not just one stress (the high percentile), but two: high per-centile and market stress. Hence we must ask just how big do we expect theVAR-relevant changes to be? The changes we consider here are relative 10-dayshocks (as BCBS-261, i.e. IM), and leave detailed discussion of alternatives toa forthcoming paper (Kenyon and Green 2014). Jumping ahead to Figure 5 wecan see that VAR and CVAR relevant relative shocks for USD zero yields rangefrom -30% to +35%. In general we expect the relevant shock range to be large,suggesting that local approximations (e.g. delta-gamma) may be sub-optimal.

The large range of shocks required for VAR means that we must augment thestate space. Consider with a simple example where our model is driven by anOrstein-Uhlenbeck process (i.e. mean reverting), dx = η(µ−x)dt+σdW, x(0) =x0, where W is the driving Weiner process. Figure 3 shows the analysis of thestate space with 1024 paths. It is clear that the 1024 paths shown do not coverthe state space required by VAR calculation when that uses shocks of up to30% because these start from the paths. The range of the 1% to 99% confidenceinterval (between red lines) is also insufficient. Hence we not only need anefficient approximation technique, but it must be appropriate for large changesnot covered by a standard simulation. This is what our Longstaff-SchwartzAugmented Compression technique provides.

While a delta-gamma approach might seem feasible computationally, it wouldstill require the estimation of delta and gamma, itself a potentially expensivecomputation. Furthermore, using a technique based on derivatives at a point(delta and gamma) for changes of the order of 30% to 40% appears problematic.We do not expect the large changes to be limited to the current example, it aconsequence of the double stress: high percentile and market stress period.

7

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

Time HYears L

Sta

teV

alu

e

Figure 3: Analysis of state space for an Orstein-Uhlenbeck process with pa-rameters: x0 = 1; µ = 1; η = 1/4; σ = 0.3. There are 1024 thin lines eachrepresenting one path. Thick red lines show 1% and 99% confidence intervalsfor each date separately. Thick blue lines show those confidence intervals witha relative shift of 30% to represent the range of VAR pricing assuming that usesup to 30% relative shifts. Clearly the original (blue) confidence intervals areinsufficient for the state space required for VAR calculation on every path andtimestep.

4.1 Longstaff-Schwartz Augmented Compression (LSAC)

To make VAR and CVAR calculations efficient the revaluation of the portfolioneeds to be very fast. The Longstaff-Schwartz regression functions provide ameans to do this as each is simply a polynomial in the explanatory variablesOi(ω, tk). Hence we can approximate the value of the portfolio in each of thescenarios by using the regression functions with the explanatory variables cal-culated using the shocked rates,

V IMq ≈ F (αm, Oi(y

aq , tk), tk). (5)

It is important to note that we apply the Longstaff-Schwartz approach to allderivative products from vanilla linear products to more complex exotic struc-tures. We also use the resulting regressions as a compression technique, thatis, we seek one set of functions, F , as an approximation of the whole portfoliovalue. Hence not only does the Longstaff-Schwartz approach replace each tradevaluation with a polynomial but it replaces the need to value each trade indi-vidually thus giving further significant performance benefits. Hence the use ofLongstaff-Schwartz Augmented Compression provides a portfolio valuation costthat is constant and independent of portfolio size.

Longstaff-Schwartz, in its original form, requires augmentation for the VARand CVAR calculations because:

• At t = 0 portfolio NPV has exactly 1 value, so regression is impossible.

• For t > 0 the state region explored by the state factor dynamics is muchsmaller than the region explored by VAR shocks (which are additional tothe state factor dynamics).

8

• If the payoff does not contain a risk factor how can we obtain any regressionagainst it?

We leave the third point to a forthcoming paper dealing with Longstaff-SchwartzAugmented Compression Composition (LSACC) as it is beyond what we requirehere.

There are two augmentation methods we can apply depending on whether,or not, there are any American-callable or Bermudan-callable options (calledAB-Callable here) in the portfolio.

• With AB-Callable Use an Early Start for the simulation, i.e. start as farback as required to span the state space (calibrate and start at t−s wheres is the shift. An early start has been suggested previously for hedging(Wang and Caflish 2009), but our early start criterion is different in thatit must provide a sufficiently large state space for VAR at all times t ≥ 0.

Augmentation = Early-Start.

– Required to preserve path-continuity for comparison with continua-tion values from regression.

– Potential loss of accuracy due to non-stationarity of processes leadingto different regression functions.

• No AB-Callable At each stopping date and on each path shock the statespace of the t = 0-calibrated simulation.

Augmentation = Shocked-State

– No continuation value comparison required so we do not need path-continuity. No loss of accuracy.

– Many choices as to how to apply the shocks. We used one shock, ran-domly chosen, per path. In fact any method that covers the requiredstate space is sufficient. For high-dimensional state spaces systematicexploration is not possible so generally a statistical exploration willbe required.

Our example here uses an interest rate swap portfolio with a central coun-terparty, these are all non-American-callable and non-Bermudan-callable (cur-rently). Hence we use the Shocked-State approach.

The algorithms were implemented in C++/CUDA and run on a GPU (NVIDIAK40c) for efficiency.

5 Numerical Results

We calculate FVA for IM based on (BCBS-261 2013) for portfolios of US Dollarinterest rate swaps (IRS) using LSAC. We calculate FVA for IM, and comparewith FVA for exposure, i.e. collateral requirements. (BCBS-261 2013) specifies:

• 99% one-sided VAR;

• 10-day overlapping moves;

• 5-year window including a period of significant stress. Our portfolio con-sists of IRS so a suitable period starts January 2007.

9

Test portfolios have n swaps with maturities ranging up to 30 years, and eachswap has the following properties,

• n swaps with maturity i× 30n where i = 1, . . . , n

• notional = USD100M ×(0.5 + x) where x ∼U(0,1)

• strike = K ×(y + x) where x ∼U(0,1), K = 2.5%. y = 1 usually, ory = 1.455 for the special case where we balance positive and negativeexposures.

• gearing = (0.5 + x) where x ∼U(0,1)

• P[payer] = {90%, 50%, 10%}

All swaps have standard market conventions for the USD market. For n = 1000the expected exposure profile of the portfolio is illustrated in Figure 4. We usen = {50, 100, 1000, 10000} in our examples. For the example VAR calcula-

- 2,000 4,000 6,000 8,000

10,000 12,000 14,000

0 10 20 30

Exp

osu

re (

MU

SD)

Years

n=1,000

EPE

ENE

Figure 4: The expected positive exposure (EPE, blue) and expected negativeexposure (ENE, red) when n = 1000 and P[payer] = 90%.

tion five years of shock values gives 1294 shocks. The 1% and 99% confidenceintervals for each tenor separately are shown in Figure 5. The regression isperformed using a linear combination of 2m+ 1 basis functions including a con-stant, m swaps, and m annuities, with length i

m × 30, i = 1, . . . ,m. Note thatwe choose the basis functions once and use them for all stopping dates. Thischoice was motivated by the fact that we can construct any swap of the samematurity but different fixed rates, from linear combination of a swap and anannuity and that we can construct forward-starting swaps from two swaps ofdifferent maturities. We use 1024 paths in all our simulations with a horizon of30 years and 6-monthly stopping dates. The interest rate simulation model wascalibrated to 10th March 2014.

Regression accuracy on portfolio price is illustrated in Figure 6. With a smallnumber of basis functions high accuracy is achieved. Fewer basis functions areneeded at later time points as the portfolio ages, this is handled automaticallyas we keep the same basis functions for all stopping dates. As time progresses

10

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30

1%

an

d 9

9%

10

-Day

R

ela

tive

Ch

ange

, 2

00

7+5

Y, U

SD Y

ield

s

Maturity (years)

Figure 5: 1% and 99% confidence intervals for each tenor separately of the1294 shock examples from January 2007 to December 2011.

some of the basis functions mature, as does some of the portfolio. The accuracyof the VAR, and IM funding, is shown in Figure 7.

Note that most of the regression is happening over time rather than overstate because our basis functions are very well adapted to the portfolio. Thatis, we are using just a few swap and annuity maturities to price swaps of manydifferent maturities.

0.0%0.5%1.0%1.5%2.0%2.5%3.0%

9 21 41 81

Erro

rs in

Re

gre

ssio

n

Po

rtfo

lio P

rice

s

Basis Size (#instruments)

n=1,000

mean

max

Figure 6: The regression accuracy as a function of basis size for a portfolio ofn = 1000 swaps. Mean and max refer to errors in regressions across the rangeof stopping dates in the simulation between zero and 30Y.

The performance of the LSAC approach is illustrated in Figure 8 and iscompared with a brute force calculation. Under the brute force approach theES and VAR calculations take most of the time, particularly for larger portfolios.The regression approach, by contrast, is a constant-time algorithm. The relativespeedup increases with larger problem size, reaching x100 for medium-sized swapportfolio (10,000 swaps).

11

-30

-20

-10

0

10

50 100 1000 10000

Ave

rage

Err

or

in U

se

%/1

00

(b

asis

po

ints

)

Portfolio Size (#swaps)

#basis=41

Value

VAR

Funding

Figure 7: Accuracy portfolio as a function of portfolio size using 41 basisfunctions (2×20 + 1), for the expected portfolio value, the 99% one-sided VAR,and the IM funding cost.

Portfolio Exposure FVA (exposure) FVA (IM)P[payer] % notional bps of notional bps of notional

90% 26.4 259 11850% 2.6 0 410% 8.9 -255 123

Table 1: FVA costs relative to portfolio notional. Note that FVA from IM isalways a cost, where as FVA from exposure can be a net cost or a net benefitdepending on portfolio direction.

Figure 9 shows the relative cost of funding the initial margin versus the costof funding the net exposure. Table 1 augments this picture by showing costsas percentages (or bps upfront) of portfolio notional (caveat that the portfoliohas roughly linear notional rolloff). Obviously FVA from IM is always a cost,where as FVA from exposure can be a net cost or a net benefit depending onportfolio direction. FVA from IM is the same order of magnitude for one-wayportfolios. For exposure-flat portfolios where the exposure-FVA is zero there isstill IM FVA, this is infinitely larger (division by zero).

12

0

2,000

4,000

6,000

8,000

50

10

0

10

00

10

00

0

Co

mp

ute

Tim

e

(se

c)

Portfolio Size (#swaps)

#basis=41

Direct

Regression

Figure 8: The performance of the LSAC approach (red) for the calculationof Initial Margin compared to the use of a brute force Monte Carlo withinMonte Carlo (blue). The units of the vertical scale are not very importantbecause they change with updates to pricing algorithms. However, we expectthe relative difference to at least remain at current levels.

-

500

1,000

1,500

-3,000 -1,000 1,000 3,000

IM F

VA

(M

USD

)

Exposure FVA (net cost or benefit, MUSD)

Figure 9: FVA costs for exposure and IM. We consider four portfolios, all thesame size (n=1000). The first two have the exposure profile of Figure 4 fromeither side. The center two points are from adjusting the fixed rates on theswaps so that the exposure FVA was zero. Even in this case the IM FVA willbe non-zero as it is VAR-driven. The lower center point is just from settingP[payer]=50%, adjusting the fixed swaps rates, the upper point adjusts thenotionals to preserve the net exposure to be the same as the one-way portfolios(i.e. those with P[payer]=10% or 90%). See also Table 1 for FVA costs aspercentage of notional.

13

6 Conclusions

Historical simulation VAR-based IM calculations will become almost universalwith (BCBS-261 2013), and VAR-based calculations are already required underBasel regulations. Central Counterparties (CCPs) too base their IM require-ments on VAR or CVAR. This paper has shown that funding costs or FVAshould also be applied to any initial margin posted, as well as variation marginor collateral. Given the use of VAR-based methodologies these IM FVA compu-tations for CCPs and in (BCBS-261 2013) are very expensive if done directly.We have introduced an efficient Longstaff-Schwartz-based method, Longstaff-Schwartz Augmented Compression (LSAC) for lifetime portfolio computation ofany VAR- or CVAR-based costs (funding, capital, etc.). This is a constant-timeapproach because it compresses any portfolio, or trade, into a set of regressionpolynomials. The cost of the initial margin has been shown to be significant, soderivative dealers will need to consider this effect in pricing.

The approach used here can also be applied in the context of regulatorycapital cost calculations or KVA. Green, Kenyon and Dennis (2014) proposes afully integrated model for XVA including CVA, FVA and KVA terms. Marketrisk capital under the Internal Model Method (IMM) uses VAR and SVAR, whileunder the Fundamental Review of the Trading Book (BCBS-219 2012; BCBS-265 2013) expected shortfall and VAR will be used. Hence the calculation ofKVA for market risk capital under IMM resolves to exactly the same approachdescribed here.

Acknowledgments

The authors would like to acknowledge the feedback from participants and re-viewers at the GPU Technology Conference 2014, and Quant Europe 2014,where early results were presented.

References

Andersen, L. (1999). A Simple Approach to the Pricing of Bermudan Swap-tions in the Multi-Factor Libor Market Model. SSRN . Available at SSRN:http://ssrn.com/abstract=155208 or DOI: 10.2139/ssrn.155208.

Antonov, A., M. Bianchetti, and I. Mihai (2013). Funding value adjustmentfor general financial instruments: Theory and practice. SSRN eLibrary .

BCBS-219 (2012). Fundamental review of the trading book — consultativedocument. Basel Committee for Bank Supervision.

BCBS-261 (2013). Margin requirements for non-centrally cleared derivatives.Basel Committee for Bank Supervision.

BCBS-265 (2013). Fundamental review of the trading book - second consul-tative document. Basel Committee for Bank Supervision.

Burgard, C. and M. Kjaer (2011a). In the balance. Risk 24 (11).

Burgard, C. and M. Kjaer (2011b). Partial differential equation representa-tions of derivatives with bilateral counterparty risk and funding costs. TheJournal of Credit Risk 7, 75–93.

14

Burgard, C. and M. Kjaer (2012). Generalised CVA with Funding and Col-lateral via Semi-Replication. SSRN .

Cameron, M. (2011). Margin models converge as CCPs battle for dealer sup-port. Risk (online) October.

Cameron, M. (2013). Quant Congress Europe: No consensus on FVA account-ing. Risk June.

Carver, L. (2012). Traders close ranks against FVA critics. Risk 25 (9).

Castagna, A. (2012). The impossibility of DVA replication. Risk 10.

Cesari, G., J. Aquilina, N. Charpillon, Z. Filipovic, G. Lee, and I. Manda(2010). Modelling, Pricing, and Hedging Counterparty Credit Exposure:A Technical Guide . London: Springer Finance.

Crepey, S. (2012a). Bilateral Counterparty Risk under Funding ConstraintsPart I: Pricing. Mathematical Finance. doi: 10.1111/mafi.12004.

Crepey, S. (2012b). Bilateral Counterparty Risk under Funding ConstraintsPart II: CVA. Mathematical Finance. doi: 10.1111/mafi.12005.

Fries, C. (2010). Valuations under funding costs, counterpartyrisk and collateralization. Technical report. Available at SSRN:http://ssrn.com/abstract=1609587.

Fries, C. (2011). Funded replication: Valuing with stochastic funding. SSRNeLibrary .

Green, A. (2013a). FVA and Collateral. Infoline: FVA & DVA: PracticalImplementation, London, July.

Green, A. (2013b). Trade Economics In Derivative Pricing. Global Deriva-tives, Amsterdam, April.

Green, A., C. Kenyon, and C. R. Dennis (2014). KVA: Capital Valuation Ad-justment. SSRN . Available at SSRN: http://ssrn.com/abstract=2400324.

Hull, J. and A. White (2012). Is FVA a Cost for Derivatives Desks?Risk 25 (9).

Hull, J. and A. White (2013). Valuing Derivatives: Funding ValueAdjustments and Fair Value. Financial Analysts Journal . Forth-coming, and online at: http://www-2.rotman.utoronto.ca/~hull/

DownloadablePublications/FVAandFairValue.pdf.

Kenyon, C. (2013). Short Rate Models with Stochastic Basis and Smile. InM. Morini and M. Bianchetti (Eds.), Interest Rate Modelling after theFinancial Crisis. London: Risk Books.

Kenyon, C. and A. Green (2014). VAR and ES/CVAR Dependence on Data-Cleaning and Data-Models, Analysis and Resolution. Forthcoming .

Kenyon, C. and R. Stamm (2012). Discounting, Libor, CVA and Funding:Interest Rate and Credit Pricing. Palgrave Macmillan.

Laughton, S. and A. Vaisbrot (2012). In Defence of FVA. Risk 25 (9).

LCH (2013). Clearing House Procedures Section 2C.

Longstaff, F. and E. Schwartz (2001). Valuing american options by simulation:A simple least-squares approach. The Review of Financial Studies 14 (1),113–147.

15

Morini, M. (2012). Model risk in todays approaches to funding and collateral.8th Fixed Income Conference, Vienna, October.

Morini, M. and A. Prampolini (2011). Risky Funding: A Unified Frameworkfor Counterparty and Liquidity Charges. Risk 24 (3).

Pallavicini, A., D. Perini, and D. Brigo (2012). Funding, Collateral and Hedg-ing: uncovering the mechanics and subtleties of funding valuation adjust-ments. SSRN . http://ssrn.com/=2161528.

Piterbarg, V. (2003). A Stochastic Volatility Forward Libor Model with aTerm Structure of Volatility Smiles. Published: Available at SSRN:http://ssrn.com/abstract=472061.

Rennison, J. (2013). CMS vs LCH.Clearnet: Clients may face CCP-specificpricing, warn FCMs. Risk (online) July.

Sommer, D., D. Todd, M. Peter, and H. Carstens (2013). FVA — PuttingFunding into the Equation (December 2013 Update ed.). London: KPMG.

Thornton, D. L. (2009). What the Libor-OIS Spread says. Economic Syn-opses (24).

Wang, Y. and R. Caflish (2009). Pricing and Hedging American-Style Op-tions: A Simple Simulation-Based Approach. Journal of ComputationalFinance 13(4), 95–125.

16