A brief History of Discouting

A brief history of discounting
Recent history of quantitative analytics in investment banks and cash flow discounting
Antoine Savine
antoine@asavine.com

Derivatives Quants in Investment Banks
•1995
–Small number of quants (~20 in Paribas Capital Markets globally)
–Working with niche businesses: exotic options an emerging business at the time
–Without a clear mandate: mainly assist trading and raise a bank’s academic profile
–A hugely creative crowd inventing a new profession
•2008
–Large teams: in BNP, ~150 quants in Fixed Income alone
–With a clear mandate for the valuation and risk management of derivatives books
–Exotics a multi-billion dollar business and main revenue contributor for capital markets
–Quants now looking after large volume businesses and industrial systems
–Quants represented in the management structure: BNP basically run by quants

Derivatives Quants in Investment Banks (2)
•2014
–“Simple” products took centre stage after the financial crisis
–Simple products no longer that simple: “vanillas are the new exotics”
–Quants contribute to all capital market activities not only sophisticated products
–Focus shifting from trading to regulatory
–Return of challenging problems calling for creative solutions: CVA, Asymmetric collateral, Capital charge optimization, etc.
–Plethora of new theoretical work: term structure models with collateral, CVA/DVA/FVA/KVA, feedback effects, revisit of delta hedging and transaction costs…
–And need to incorporate cutting edge algorithmics: parallel computing, automatic differentiation
•2020?
–Quants influence will exceed capital markets and spill into asset management and commercial banking
–Capital allocation across businesses will make use of models similar to exotics
–Quants will become central players in IBs the way they are now in capital markets

Term structure of interest rates
• Yield curve: for a given currency, the term structure of interest rates for all
maturities
– Discount Factor: PV of 1 currency unit delivered at time T
– The curve R(T) gives all relevant rate information
  RT T DF T e   
logDF T 
R T
T
 
continuous rate
T RT 
yield curve
Feed models
rates for forwards/futures
discount factors for option payoffs
short forward rates for rate exotics  
logDF T 
f T
T

 

Value rate transactions
“risk free” loans
swaps, spot or forward starting
T0
lend notional
Tn
receive redemption + interest
N
   0 0 1 , n n N   F T T T T 
       0 0 0 1 , n n n NDF T  N   F T T T T DF T
 
   
   
0
0
0
, n
n
n n
DF T DF T
F T T
T T DF T



For the transaction to be fair
“coverage”
rate

Interest rate swaps
• Swap: exchange fixed for floating coupons
• Floating leg, assuming Libor is the interest rate, is same as rolling deposits
• Hence and for the fixed leg
• Hence for a par swap
0 T n T
  1 . . , i i N K cvg T T
    1 1 . , . , i i i i N Libor T T cvg T T  
fixed leg
floating leg
= = 0
     0 n float  N DF T DF T     1
1
,
n
i i i
i
fix NK DF T cvg T T 

 
annuity A
cvg = day count
   
 
0
0 ,
n
n
DF T DF T
K
A T T



Construction of the curve
•For a given set of n liquid standard swaps, assume some interpolation scheme on R
–Set R1 to match the market value of swap 1
–Then set R2 to match the market value of swap 2 with R1 constant
–Etc.
•Then use the curve to value a portfolio of custom swaps and other products
•And compute rate risk by finite differences
–Bump the value of the first swap
–Recalibrate the curve
–Reprice the book
–Loop over swaps used for calibration
0.00%
0.50%
1.00%
1.50%
2.00%
0
1
2
3
4
5
6
7
8
9
10
continuous
priced
market

Instruments for curve construction
•Short term: 1d to 3m-6m
–Deposits
–Work like single period swaps
–Valued with swap formula with one period
•Medium term: 3m-6m to 2y
–Libor futures
–Ignoring convexity adjustments, same as Libor Forwards or FRA (Forward Rate Agreements)
–Work like single period forward starting swaps
–Valued with the same formula
•Long term: 2y to up to 50y
–Swaps
  0nnDFTDFTKcvgDFT   

Future/Forward convexity
•FRA
–Single cash flow on :
–On par when
–That is
–Where is the martingale measure associated with the numeraire
–Also called Tpay-forward measure
•Future
–Continuous margin flows at time t
–So whatever entry and exit times we have
–So and since finally
payT, fixTstartendLTTFRAcvg  00expexppaypayfixTTRNRNssTErdsFRAErdsL  00expexp, paypaypayfixfpayifxixTTTRNRNssTTTTErdsFRAErdsLFRAEELL  payTQ,exppayTRNpaysDFTErds    tdFut,startend0exp0endstarttRNstErdsdFut     0RNttEdFutfixfixTTFutL fixRNTFutEL 

Future/Forward convexity (2)
•Market quotes Fut, curve construction requires FRA
•Must calculate and adjust for convexity
•Intuitively, when I buy the future rate (= sell the future in market language)
–Increase in rates raise positive cash-flows that I invest at a higher rate
–Decrease in rates raise negative CF that I borrow at a lower rate
–So I benefit from convexity
–In the absence of arbitrage, this is compensated by higher future rates Fut > FRA
•Fut/FRA convexity
–The only mathematically exciting bit in the curve world in these days
–An active research area with many academic papers and developments by professionals
•We derive the convexity formula later when looking into convexity adjustments

That’s how we did it in 1995
•A simple methodology
–Lightweight: doable on Excel without code
–Simple maths
–Simple algorithmics: one-dimensional bootstrap (assuming “local” interpolation like linear)
–Little quant involvement
•Based on one assumption: Libor is the interest rate
–We can invest and borrow risk free at Libor
–Libor represents interest rates and interest rates only
•That turns out to be wrong
–Libor is an index of the Interbank Offered rate
–Incorporates information other than interest rates
–After crisis, Libor disconnected from interest rates, which are frozen near zero

Basis swaps
• Swap floating for floating on different frequencies, like 3m for 6m
– Both legs are supposed to be worth the same thing
– Swap is theoretically on par
• In practice, 6m trades at significant premium: up to several tens of bp!
– Par basis swap: 3m + BS vs 6m, persistently, why?
• Explanation: credit and liquidity in underlying Libor transactions
– Underlying Libor transactions are unsecured (although the basis swap is typically collateralized)
– The longer the loan the higher the credit exposure
– In case of a liquidity crunch, longer funding means easier access to cash
– Also Libor pooling and indexing mechanics
• BS also exist with different indices (ex. JPY Libor/Tibor), possibly with same
frequency but different references
0 T n N.L6m.cvg 6m T
N.L3m  X .cvg 3m
floating leg
floating leg

Basis swaps
USD Libor 6m vs 3m 5y BS
USD Libor 3m vs Fed Funds 5y BS

Currency basis swaps
• Swap floating for floating on different currencies, like USD vs EUR
• In theory the swap is on par
• In practice EUR libor trades at significant premium over USD libor
– European banks are seen as less creditworthy and pay credit and liquidity premium over US banks
– European banks must be pay a premium for access to USD liquidity
0 T n T
.  . . EUR N L EUR cvg EUR
.  . . USD N L USD cvgUSD
USD notional
EUR notional USD notional
EUR notional
= = 0

Currency basis swaps
EURUSD 5y xCCY BS
USDJPY 5y xCCY BS

Synthetic cross currency transactions
• If you are a US bank borrowing USD at USD Libor…
• …Then you can add-on a USDJPY basis swap to synthetically borrow JPY at JPY
Libor – BS
USD notional
USD notional
.   . . JPY N L JPY  BS cvg JPY
USD notional
JPY notional USD notional
JPY notional
.   . . JPY N L JPY  BS cvg JPY
JPY notional
JPY notional
+
=

The 1997 Asian crisis
•In 1997 the collapse of Thailand rose a credit crisis that propagated throughout Asia 1997-1999
•Consequently Asian banks’ credit deteriorated while liquidity premia raised and Asian basis swaps raised from typically insignificant to very material levels (5y USDJPY up to 40bp)
•We can no longer discount USD flows at USD Libor and JPY flows at USD JPY
–This is inconsistent and arbitrageable as seen on previous slide
–USD Libor funding means JPY funding = JPY Libor – BS < 0 !!
•The banks who first understood this arbed those slower to react
•Eventually new standards emerged for rate transactions

The post 1997 rate machinery
•The industry agreed to use USD Libor 3m as reference for discounting
–Dominated by US and European banks with EURUSD BS insignificant at the time
–USD Libor seen as risk-free reference because “US banks can’t fail”
–Economically, Japanese houses should really discount at JPY Libor but we cut some corners to avoid operational nightmare of counterparties to a transaction having different valuations
–A somewhat fishy machinery, did the job until the global financial crisis
•Means we do “as if” everybody’s borrows and invests at Libor 3m in USD
•And adding-on xCCY BS everybody’s funding in some currency CCY is “CCY Libor – xUSD BS”
•This theoretical discount curve for the currency is called “cash curve” = CCY Rate curve in the USD Libor 3m basis

The post 1997 rate machinery (2)
•Libor swaps no longer reference for discount (except vs 3m in the US), instead we used a “cash curve” per currency = “Libor – BS”
•Floating leg pv no longer but
–Where the forward libors FL are picked on a “rate surface” by maturity and Libor frequency
–And DFs are computed on the cash curve
•Construction (and risk management) of curves (now surfaces) much more involved
–Manipulates large amounts of market data
–Construct curves by fitting swaps, BS and xCCY BS in one go by setting discount and forward Libor curves
–Risk less evident to compute and comprehend: what do we freeze when we bump swap rates?
–No longer takes 30min on a spreadsheet, heavy machinery in C++ with thousands of lines of code
•Not really complicated but very heavy
–Had to be centralized and dedicated quant teams were put in place
–An so we invented the “linear quant”
0nNDFTDFT111,, niiiiiiNDFTFLTTcvgTT 

And then we had the global crisis
•And it all went bananas
–Rates stick to 0 while credit and liquidity premia went nuts
–Libor (even USD) absolutely disconnected from anything to do with interest rates
•So we needed to rethink the whole discounting machinery from the ground
•Some IBs were quicker to react and arbed the street for billions
•Subsequently, IBs moved quants from shrinking exotic businesses into linear rates
• And the quant community came out with new standards for discounting
•Best practice today is theoretically clean and incorporates lessons from the crisis
•It is also somehow complex and keeps thousands of quants busy on a daily basis

The post Lehman rate machinery
•2 types of transactions: collateralized and uncollateralized
•Collateralized transactions
–Discount flows at collateral rates
–For floating legs, pick forward rates on rate surface
–Fit forward Fx curve to xCCY BS and use forward Fxs to discount cash-flows in currencies other than collateral’s (domestic) currency
–Account for “cheapest to deliver” when collateral may be changed
•Uncollateralized transactions
–Makes no sense to discount flows at any kind of “risk free” rate since we know counterparties fail
–Account for potential loss subsequent to counterparty default  CVA

Discounting fully collateralized trades
•Cash flows must be discounted at collateral rate
–Cash flow CF delivered at time T, discounted at some funding rate r
–Add collateral flows C=-V, collateral earns some rate c and is funded at r, note that collateral posting is not a cash flow
–Differentiate
–So
•Note that collateral changes discount rate but not probability measure
•Nothing hard here, why wait 2008 to implement the right thing?
,exp... TTttsTtVCFVErdsV     000expexp,,expexpTTsTTsttsTussstttsTusssttttttVErdsVrduVcrdsttEVErdsVrduVcrds     000expexpTTsttttsTtussstttttttttEdVErrdsVrrduVcrdsVcrdsEcV     000exp,,expexpTTTtttsTttsTstttEVEcdsVttVEcdsVcdsCF    

Discounting fully collateralized trades (2)
•Example (norm for swaps): cash collateral in the currency of the swap, earns OIS
–Discount cash flows on the OIS curve
–Project floating cash flows off the forward curve surface
–Build discount and forward curves together to fit market prices of collateralized swaps and BS
–Discount curve based on OIS swaps
–Where OIS swaps are not liquid, build synthetically out of Libor swaps and Libor/OIS basis swaps
–“Disc = OIS = Libor – LIBOR/OIS BS”
•Example 2: cash collateral (earning OIS) in a different currency
–Say: BNP and SocGen put on a USD swap with EUR cash collateral
–USD floating cash flows are projected out of the USD multi-curve surface built out of USD swaps and basis swaps
–And discounted on the USD yield curve, but with EUR OIS basis
–Discount curve based on EUR OIS (EONIA) swaps basis swapped into USD
–“Disc = USD Libor + EURUSD Libor BS - EUR Libor/OIS BS”

Compounding fully collateralized trades
•European equity option, OTC, fully collateralized
–Depends on rates through discounting and forward
–Textbook forward (without dividends):
–Depends on discount factor  forward changes with collateral?
–No: the repo rate must be used (finance rate when stock is used as collateral), therefore depends on equity but not on derivative transaction
•What about Fx?
–Fx forward:
–Changes when collateral switched from domestic cash to foreign cash?
–No: change of basis shifts both rates by the same amount, hence rate difference remains constant
–In other terms, each DF changes but ratio is constant
  , , tSFtTDFtT     , ,exp,, , forttdfdomDFtTFtTSSRtTRtTTtDFtT 

Compounding fully collateralized trades (2)
•FRA and Libor futures
–FRA solves:
–In the context the curve was built, that is discounting in the OIS of the currency of the Libor
–Changing collateral changes basis for discounting, hence the new FRA solves
–So if the basis b is independent (as a particular case, deterministic), then the FRA remains unchanged when collateral changes, otherwise it is subject to a convexity bias (see later)
•Forward swaps
–Forward swap:
–Depends on discount – typically used to arb improper discounting
–Hence changes with collateral
111,, niiiiiiDFTFLTTcvgTTFAnnuity    2120exp,0TsErdsLibTTX 2120exp,0TssbErbdsLibTTX

Multi-Curve Linear markets: domestic case
•Multi-Curve Linear market = collection of curves of forward rates, one for each floating rate reference: OIS, Libor 3m, 6m, 12m, etc.
•Each curve implements some interpolation scheme provides forward rates for all maturities
–Forward OIS rates can be read on the OIS curve for all future dates
–Forward Libor 3m rates can be read on the Libor 3m curve
–In case something like the 4m Libor is needed, interpolated from the Libor 3m and Libor 6m curves
•Forward rates do not depend on collateral (ignoring convexity adjustments)

Multi-Curve Linear Markets (2)
• For a transaction with cash collateral in the domestic currency,
the curve that corresponds to collateral rate  in general OIS
is used for discounting cash flows
• Where
– X is the reference curve: OIS, Libor 3m, 6m, 12m, …
– is the timeline for the floating frequency of X
• For OIS, every business day from today
• For Libor 3m, every 3m
• Etc
–
– F is the forward rate X picked on the curve by maturity
– cvg is the conventional day count for the floating rate X
 
   
 
       
1 1
0
1 1
1 , ,
1 , ,
X n T
X n T n T
X i i i i
i
DF T
F T T cvg T T
F T T cvg T T  


  
    
  i T
  max ,  i n T  i T T
0 T nT  T
compound
discount

Discounting cash flows with a MCL market
• Discounting with cash collateral with collateral rate reference X
• Fixed cash flow CF paid at date T
• Floating cash flow with floating rate Y for period
we pick the forward rate on the Y curve (e.g. Libor) and discount it on the X curve (in general, OIS)
• Fixed swap leg
• Floating swap leg
Ref Y (e.g. Libor 3m)
  X PV  DF T CF
      2 1 2 1 2 , , X Y   PV  DF T F T T cvg T T 1 2 T ,T
0 T n T
  1 . . , i i N K cvg T T
    1
1
,
n
X i i i
i
PV NK DF T cvg T T 

 
0 T n T     1 1 . , . , i i i i N Y T T cvg T T  
      1 1
1
, ,
n
X i Y i i i i
i
PV N DF T F T T cvg T T  

 

Building a MCL market
•Set forward rates for all references so as to zero the PV of selected market par swaps and basis swaps
•Swaps are cash collateralized with collateral rate = OIS in the currency
•First build OIS and main Libor curve (3m for USD, 6m for EUR, …) so as to fit Libor swaps and OIS/Libor basis swaps together
–Bi-dimensional bootstrap
–In case of a smooth interpolation like spline, bootstrap may not be used and global fit is necessary
–Build short term out of spot Libor and OIS, OIS depos and OIS swaps, and Libor futures
–Incorporate Central Bank meeting dates and end of year effects in interpolation
•Other curves Y are fitted to Libor/Y basis swaps by setting forward Ys (DFs and forward Libors are known at this stage)

• MCL markets cannot discount cash flows in a currency different from collateral
• xCCY BS CCY1 vs CCY2 (assuming both MCL have been constructed)
• Assume USD Fed Funds collateral
• We know how to PV cash-flows on the USD leg
• But for the EUR leg, we can PV only with EUR collateral
For PV under USD collateral, we need another curve
 Discount curve for EUR cash flows under USD collateral
Cross Currency MCL markets
0 T n T
USD notional
EUR notional
EUR
USD DF

•Forward Fx (under collateral X):
•We saw that F is independent of X since X affects identically the numerator and denominator
•So
•Since S and are known from domestic markets, The curve of forward Fx and the curve of contain the same information
•It is standard practice to store forward Fx curves in xCCY MCL markets More precisely we usually store forward Fx factors = forward Fx / spot Fx
Forward Fx Curves
   forXdomXDFTFTSDFT     fordomdomdomDFTFTSDFT  domdomDFTfordomDF

Discounting cash flows with a xCCY market
•Discount cash flows in the currency of the collateral same as domestic case
•Discount cash flows in a foreign currency:
–“Convert” in the domestic currency with the forward fx for payment date
–Discount as a domestic cash flow on the collateral rate curve
•Fixed cash flow paid at date T in foreign currency
•Floating foreign cash flow with floating rate Y for period
we pick the forward rate on the foreign Y curve (e.g. Libor) we convert in domestic currency by applying the forward fx read on the xCCY curve for the payment date
we discount on the domestic X curve (in general, OIS)
domdomforPVDFTFFXTCF 221212,,domdomYPVDFTFFXTFTTcvgTT 12,TT

Building a xCCY market
• Build xCCY market = set forward Fx curve
• As usual, apply interpolation scheme to provide fFx factors of all maturities
• Set the fFx curve so as to zero the PV of market traded par xCCY BS
• After domestic MCL have been constructed, USD leg has fixed value
• Apply to value EUR leg
• Set so that the EUR leg matches the USD leg
• Repeat for all quoted maturities in xCCY swaps
0 T n T
USD notional
EUR notional
        2 2 1 2 1 2 , , dom dom Y PV  DF T FFX T F T T cvg T T
FFX .

Comparing 1997 and current setups
•1997 setup = particular case of current
–With implicit assumption = all transactions have USD collateral with collateral rate = USD Libor 3m
–Which is never the case
•But prior to the crisis, USD OIS/Libor and EURUSD xCCY BS too low to really matter
–So the assumption was “almost” right
–In any case, spreads were too low to provide arbitrage opportunities
–And banks were reluctant to undertake the massive developments required for the current setup

Introduction to convexity adjustments
•Consider a floating cash-flow referenced by some index L under collateral X
–We have
–Where
•When we construct a market under collateral X
–We extract forwards
–They are not risk-neutral expectations but “forward-X” neutral expectations
–A “future/forward” convexity adjustment must be applied to find the RN expectation
., 0exp0, XTpaypayfixfixTRNXXDFsTpayTPVErdsLDFTEL  ,expexp, TTXRNXXtsttDFtTErdsftudu  ., 0,, XTpayfixXDFLfixpayTFTTEL 

Convexity adjustments (2)
•Future/Forward adjustment
–Between the RN expectation (future) and the “forward-X” neutral expectation (forward) we have:
–Radon-Nykodym
–Density process
–Girsanov
–Multiplicative convexity adjustment [forward->future] where log-covariance (forward, DF) is deterministic:
  ., 00expexppayXTpaypayTXDFsRNTRNXsrdsdQdQErds          ., 0exp, ,, 0, XTpayXtXXDFspayRNRNtttpayRNXDFtpayrdsDFtTddQEtTdWdQDFT         ., 00,,explog,,,log, XfixpayfixTDFTRNXXXTLfixpayLfixpaypayELFTTEdFsTTdDFsT  0explog,,,log, fixTXXLfixpaypayCdFsTTdDFsT 

•Change of collateral from X to Y: forward moves from to
–Radon-Nykodym b: basis spread
–Proof: for some variable cash flow V under collateral Y
–But
–Hence
–So
–Next
–So
    ., .,.,., 0000expexpexpexppaypayYTpayXTXTXTpaypaypaypaypayTTYXYXDFsssTTDFDFYXDFYXtsstsrrdsbdsdQdQErrdsEbds          ., 00,expYTpaypayTYDFRNYpaysPVDFTEVErdsV  ., 0000expexpexp0,expXTpaypaypaypaypayTTTTRNYRNXYXXDFYXsssspayssErdsVErdsrrdsVDFTErrdsV   .,., 00,0,expYTXTpaypaypayTYDFXDFYXpaypayssDFTEVDFTErrdsV      ., ., 0exp0,0, payYTpayXTpayTYXDFssYXDFpaypayrrdsdQDFTDFTdQ    ., 00000,expexpexp0,expXTpaypaypaypaypayTTTTYRNYRNXYXXDFYXpaysssspayssDFTErdsErdsrrdsDFTErrds  ., 00,0,expXTpaypayTYXDFYXpaypayssDFTDFTErrds  .,XTpayfixDFTEL  .,YTpayfixDFTEL 

•Change of measure from collateral X to collateral Y
–Density process
–Girsanov
–Multiplicative convexity adjustment [forward-X->forward Y] where log-covariance (forward, DF ratio) is deterministic:
–Note that
–So
–Assuming L is a 10Y Libor with vol = 15%, basis vol = 0.25% and correl = 40% we get C ~ 1.0075
–So for a rate of 2% convexity adjustment adds 1.5bp
      ., .,., ., 0exp,, ,, ,, YTpayXTXTpaypayYXXTpaytYXYXDFspaypayDFDFtttpayYXDFDFDFtpaypaybdsDFtTDFtTddQEtTdWDFtTDFtTdQ         ., 00,,0,,explog,,,log, XfixpayYTDFTYXXLfixpayLfixpayLfixpaypayXDFFTTFTTEdFsTTdsTDF     0explog,,,log, fixYTXLfixpaypayXDFCdFsTTdsTDF     log,, YYXpaypaypayXDFsTBsTTsDF  log,,,log,log,,,, YXXYXLfixpaypaypayLfixpaypayXDFdFsTTdsTTsdFsTTdBsTDF 

• Finally
– Floating cash-flow referenced by some index L
– With
– And the convexity adjustment
• We see that
– The convexity adjustment may be neglected if the basis [Y-X] and the forward are uncorrelated
In particular if the basis is deterministic
– Otherwise it is of the order
0,  0, ,  Y Y
pay L fix pay PV  DF T F T T
Change of
discount
Change of
forward
0, ,  0, ,  Y X
L fix pay L fix pay F T T  F T T C
          .,
0
exp log , , , ,
X
pay fix
DF T T X Y X
pay L fix pay pay C E T s d F s T T dB s T     
  
 
2
log exp ,
2
norm
F B
T
C  F B  
 
  
 
0,  0, ,  X X
pay L fix pay PV  DF T F T T
Under collateral X Under collateral Y

Introduction to collateral option
•Some CSAs allow to post collateral in a choice of currencies and/or assets
–Example: USD swap with choice of USD or EUR collateral
–Giving an option that belongs to the party posting collateral
–That party has negative PV
–And so will want to apply maximum discount
–By posting higher yielding (= cheapest to deliver) collateral
–So it is always the highest rate = highest basis that prevails
–Right now it is best posting EURs than USDs
–But according to forward currency basis curves, will revert in the future

Collateral option (2)
• Value of the collateral option: general case
–
– Note
– Where lhs is the base PV discounted with collateral 1 and rhs is the adjustment for collateral option
– Note is the basis from collateral 1 to collateral 2
– And finally the collateral option is worth
• Value of the collateral option: deterministic case
–
– With
• More on collateral options:
See Antonov & Piterbarg (2013/2014)
    1 2
0
exp
T
s s PV  E  c  c dsCF
  
      1 1 1 2 1
0 0 0
exp exp
T T s
s u s s s s PV  E  c dsCF  E  c du PV  c c c  ds
        
    1 2 1 2 1 12
s s s s s s c c c c c b
 
      
  1 12
0 0
exp
T s
u s s E c du PV b ds
          
PV  DF T ECF
        1 2
0
exp 0, 0,
T
DF T    f u  f u  du

Credit Valuation Adjustment
•Uncollateralized transaction
–Can produce loss if counterparty defaults while PV is positive
–But does not produce a gain if default of negative PV
•Canonical example: payer swap on an increasing curve
–Negative cash-flows followed by positive expected ones
–Synthetic loan to counterparty
–With loss arising from default
•On default, PV is netted across transactions with a counterparty or “netting set”

CVA (2)
• In all when we trade without collateral we give away
a put on the netting set contingent to default
• This is a “real option” with value
– DF: here means
– RR: recovery rate
– S: survival probability
– E: risk-neutral expectation
• Option pricing, not estimation
• Model parameters must be calibrated
– For instance, S must be implied from CDS whenever possible
– Because CVA is an option that is meant to be hedged
– For instance, the loss from the swap on previous slide can be avoided by buying CDS on the
counterparty
– CVA = cost of this dynamic hedge
– Computed as a RN expectation as per Feynman-Kac theorem
          1
1
1 max 0,
n
RN
i i i i
i
CVA E DF T RR S T S T PV T 

 
      
 

CDS implied probability to
default between Ti-1 and Ti
 
0
exp
T
s DF T   r ds
  

Uncollateralized transactions
•Following Lehman, we can no longer discount uncollateralized cash flows at some “risk free” rate
–Must incorporate CVA
–CVA is computed per netting set, not per transaction
–Current practice:
•Value each transaction as risk-free: base value
•Compute CVA on the netting set
•Allocate CVA charge across transactions: incremental or marginal method
•In addition, these transactions may carry:
–A positive “Debit Valuation Adjustment” or DVA: “own cva” benefit from profit of own default on negative PV
–A “Funding Valuation Adjustment” or FVA: asymmetry of funding costs on positive PV (lending) and risk free investment earnings on negative PV (borrowing)
–Potentially a “Capital Valuation Adjustment” or KVA: capital charge for making these transactions
–All these XVA (except perhaps KVA) computed similarly to CVA, different local discounting applies according to net PV being positive or negative in the future

CVA allocation: incremental method
•Idea: allocate additional CVA due to trade n with respect to former CVA with the (n-1) first trades in the set
•Denote the CVA for some netting set with the n first trades in the set (in chronological order)
•Incremental CVA for trade n:
•Benefit: the contribution of each trade is charged as CVA for the NS changes with the trade and then remains constant
•Drawback: allocation depends on the order of the trades. The same transaction may be charged very differently if made earlier or later.
nCVA1nnCVACVA

CVA allocation: marginal method
•Marginal CVA for some trade = notional times change in the CVA subsequent to a marginal increase in the notional of the trade
•We remind that
•And with m trades in the NS, with notionals N and unit values V
•So the marginal CVA for trade j is
•And immediately (also as a consequence of Euler’s theorem) we see that the CVA is the sum of marginal CVAs
•Benefit: intellectually sound, easy and fast to compute, transaction charge does not depend on order
•Drawback: all charges change retroactively when new trades are introduced!
111nRNiiiiiCVAEDFTRRSTSTPVT        1mjjjPVNV   10111inRNjiiijjiPVTijCVANEDFTRRSTSTNVTN     1mjjjCVACVANN   

Computing CVA in practice
•Challenges…
–The most exotic option ever: credit contingent put on a whole netting set
–Model dimensionality
–Calculation time
–Risk sensitivities
•…such that many IBs need to recourse to crude approximations…
–Common approximation for CVA: represent an exotic as a vanilla, for instance a Bermuda as simple swap of same expected duration or analytically valued European swaption
•…and/or compute on large racks of servers

CVA in practice (2)
•My colleagues in Danske Bank and I have been working on efficient implementation for CVA for the last couple of years
•And published results in our series of talks “Compute CVAs on your iPad mini”
–Used and adapted our experience with exotics and hybrids
–Used Longstaff-Schwartz regressions in a particular way
–Incorporated new algorithms such as parallel computing and AD
•Short version Global Derivatives 2014, Amsterdam
–Jesper Andreasen, Compute CVA on Your iPad Mini
–Antoine Savine, Automatic Differentiation
•Long version Aarhus 2014, Jesper Andreasen
•Email cvaCentral@aSavine.com for slides

What model for CVA?
•Challenge: model all variables relevant to the whole netting set
–Rate curves in all currencies in the NS
–Fx for all currencies involved
–All equity, credit, inflation, etc. indices referred to in the transactions
–All at once, consistent and calibrated
•Rate models must be multi-factor
–Otherwise all rates within currency are perfectly correlated
–And netting effect between reverse transaction with non-matching maturities is overestimated
–Choice between Markov (muti-factor Cheyette) and non-Markov (BGM) models
–Models calibrated to todays prices, curves and volatility surfaces
–Best practice requires stochastic volatility, especially for NS with exotics
•Quants reuse models developed for exotics…
–CMS spread steepeners: MF rate models with SV
–Callable Power Duals / chooser TARNs: Fx with LV/SV coupled with rate models
•…But ultra high dimensionality is something new
–Never in exotics did we need to deal with such a number of variables
–Even chooser TARNs (typically up to 5 currencies) implemented single factor IR on each ccy

CVA reformulation with Longstaff-Schwartz proxies
•CVA formula:
•Knowing that:
•We rewrite as:
•At this point it is clear that cash-flow evaluation is quadratic
•But if we use Longstaff-Schwartz proxies for the indicators:
•And reverse the order of the sums:
111max0, nRNiiiiiCVAEDFTRRSTSTPVT         injRNiTjjiiDFTPVTECFTDFT        010111innjRNiiijPVTijiiDFTCVAEDFTRRSTSTCFTDFT         010111innjRNiiijVijiiDFTCVAEDFTRRSTSTCFTDFT      0101111ijnRNiijjVjiCVAERRSTSTDFTCFT     

CVA reformulation (2)
•Then we can write:
•Where:
•And so:
•The cash-flows are evaluated once, while the CVA notional is calculated along the path
•Note that we use proxies only for the indicators, the cash flows themselves are calculated without approximation
01nRNjjjjCVAECVANOTTDFTCFT      10111ijjiiViCVANOTTRRSTST   11011jjjjjVCVANOTTCVANOTTRRSTST  

Effective CVA calculations
•Prepare the infrastructure
–Use scripting for cash-flow representation
–Compress trades in a netting set into a “super swap”
–Use LSM pre-simulations to turn early exercises into path-dependency
–See “CVA on iPad mini” series
•Reformulate CVA problem as per previous slides
•Turbo-charge
–Multi-thread the simulations and LSM
–Use AD for risk, see my talk about AD in Amsterdam, Global Derivatives, May 2014

Thank you
cvaCentral@aSavine.com

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