collateral consistent derivatives pricing... collateral consistent derivatives pricing fric...
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Collateral consistent derivatives pricing
FRIC Practitioner Seminar, CBS
Martin D. Linderstrøm, [email protected] Counterparty Credit & Funding Risk
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Agenda
• The intuition behind collateral consistent pricing. − A benchmark case: A multi-currency calibration under EUR cash collateral.
• The complexities of a multi CSA book − Which collateral assumptions hold for calibration instruments?
− The ISDA Standardized CSA approach
− Market fragmentation between CCP cleared and bilateral trades?
• Case studies in curve calibration − What are reasonable bounds for forward curves?
− Arbitrages in fragmented markets?
• Pricing and hedging discounting risks under different CSA regimes? − The collateral valuation adjustment.
− The cheapest-to-deliver optionality in CSAs
− Hedge ratios with and without optionality?
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Swaps in the old way
• In the “old” days (until Aug’07) many market participants had just one swap curve for each currency.
− Forward rates – irrespective of tenor – were calculated on this.
− Discount factors were also derived from this curve.
• This implicitly assumes:
− No money market basis (e.g. 3s6s basis is zero).
− No cross currency basis (e.g. EUR/USD basis is (close to) zero).
− Traders can fund themselves at xIBOR.
− Note that on a single curve, a Floating Rate Note trades at par at fixing time.
• These assumptions are no longer valid.
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5Y 3M-6M Basis 5Y EONIA-3M Basis
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5Y EUR/USD X-CCY
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Swaps in the new way
• Need for multiple projection curves for each currency.
− We cannot compute 3M xIBOR and 6M xIBOR forwards on the same curve.
• Need for a single discounting curve for each currency.
− This should reflect CCS spreads.
− But what should be my anchor in terms of currency and credit premium?
− If your trade is collateralised, you should discount with the collateral rate.
− What is your collateral rate?
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10M
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Forward LIBOR Surface
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1.2 Discount Factors
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The institutional setting
• The ISDA Master agreement
− The legal umbrella underpinning netting.
− Default and early termination provisions.
• ISDA Definitions
− Sets standards for methodologies such as settlement of options, application of floating rates etc.
• ISDA Credit Support Annex (Credit Support Deed)
− Defines the terms for collateralisation.
− Sets Thresholds, Independent Amounts, Mininmum Transfer Amounts and valuation frequency.
− Eligible collateral and specifies interest earned.
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The intuition behind collateral consistent pricing
Flow analysis: EUR Derivative - EUR Cash collateral
Counter Party Trading Desk
Collateral Management
Cash Desk
T0: Contract (PV=-100 EUR)
T0: Payment (=+100 EUR)
T0: Payment (=+100 EUR)
T0: Payment (=+100 EUR) T0: Payment (=+100 EUR)
T1: Interest= REUR/OIS*(1/360)*100 EUR T1: Interest= RIntern*(1/360)*100 EUR
T1: Interest= RIntern*(1/360)*100 EUR
•Cash desk is passing through the liquidity – no haircuts or disagreement on valuation. •Internal loop can be ”closed” if rIntern=rOIS
•See Piterbarg (2010).
•For the setup to be arbitrage free, the trader needs to be discounted at the rate his cash position earns, i.e. RDisc = ROIS. •He could in principle hedge his cash exposure via an EONIA swap.
Bank
T1: Contract (PV=-100*(1+ RDisc/360) EUR)
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A benchmark case: A multi-currency calibration under EUR
cash collateral
• Stylised market:
− Only IRSs against 3M xIBOR.
− 3M xIBOR-OIS basis swaps.
− X-CCY basis swaps against 3M XIBOR.
− Only swap instruments 1-30Y.
• Setup
− Separate forward and discounting curves.
− Single collateral assumption – all products are EUR cash collateralised.
− Want a CCS consistent valuation setup.
• Approach
− Calibrate jointly EUR3M and EUROIS=EURDISC curves.
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0 60 120 180 240 300 360
Forward start (months)
EUR: 3M forward rates
EUROIS EUR3M
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Benchmark case – cont’d
• Approach cont’d:
− Calibrate jointly USD3M, USDOIS and USDDISC curves...
− ...requires EUR model as input since X-CCY legs have initial PV...
− ...USDDISC curve is not dependent on USDOIS.
• Pricing implication
− This creates a X-CCY dependence for the pricing of every USD cashflow.
− Hedging tool for USD net liquidity is to trade USD fixed-EONIA float CCS…
− …this delivers the required EONIA floater to collateral mgmt.
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Forward start (months)
USD: 3M forward rates
USDOIS
USD3M
USDDISC
EUR/USD X-CCY 3M (rhs.)
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The intuition behind collateral consistent pricing (cont.)
Flow analysis: EUR Derivative - USD Cash collateral
Counter Party Trading Desk
Collateral Management
Cash Desk
T0: Contract (PV=-100 EUR)
T0: Payment (=+100 EUR)
T0: Payment (=+100 EUR)
T0: Payment (=+126 USD) T0: Payment (=+126USD)
T1: Interest= RUSD/OIS*(1/360)*126 USD T1: Interest= RIntern*(1/360)*126 USD
T1: Interest= RIntern*(1/360)*100 EUR
•To produce the collateral posting in USD an Eonia/Fed-Funds CCS is entered. •Notice that there is a spread s on the EUR leg! •See Piterbarg (2012).
•The discount rate needs to reflect the spread in the CCS. •In reality there may be multiple currencies, and hence a cheapest-to-deliver option for the collateral poster!
Bank
T1: Contract (PV=-100*(1+ RDisc/360) EUR)
CCS Counter Party
T0: 100 EUR
T0: 126 USD
T1: Interest= RUSD/OIS*(1/360)*126 USD
T1: Interest= (REUR/OIS+s)(1/360)*100 EUR
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Calibration instrument assumptions
• Fundamentals − What do we mean by calibration instruments?
− Our model tells where to price one product reletive to others…
− …so we should calibrate it market prices at which we can execute hedges.
• ”The Market” − How is ”The Market” collateralised?
− No single answer…
− …CSAs are bilateral agreements – and they vary substantially.
− CCP collateralisation rules are however very clear.
• My calibration should depend carefully on the collateral assumptions that I will face once I start using the calibration instruments for hedging. − Each market segment offers one source of risk – but can be collateralised differently:
− On several CCPs EUR trades are EONIA collateralised, USD trades are FF collateralised…
− …the same goes for the ISDA Standardised CSA.
− But what holds true for FX products?
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Changing the assumptions
• Back to USD:
− Let us instead calibrate by using Fed Funds discounting…
− …most of ”The Market” for USD swaps clears via LCH…
− We are using the same market quotes for spot instruments…
− …but see slight changes in the 3M Fwd curve for 3M USD LIBOR.
• Intuition:
− A par-swap rate is a weighted average of xIBOR forward rates.
− Changing the discounting assumption alters the weighting of the individual fwd xIBOR rates.
− A typical swap market calibration has many degrees of freedom.
• Conclusion:
− Depending on your assumptions, you can easily misprice forward starting swaps with 1.0-1.5 bps.
− This is huge in a market that trades with bid-offer spreads in the 0.25-1 bps range.
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Forward start (months)
USD3M Fwd: EONIA vs. Fed Funds based
EONIA based calibration
Fed Funds based calibration
Diff (bps), rhs
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Changing the assumptions – cont’d
• Cross currency swaps:
− The same effect holds true for CCSs.
− In most markets, the fwd curves for the CCS breaks are less steep than xIBOR fwd curves…
− …this means that the discounting effect is smaller.
• ISDA Standardised CSA:
− Is promoting USD cash collateral for FX products incl. CCS…
− …so Fed Funds discounting must be right…
− …but what about the fwd curves needed to price up this product?
This introduces a multi-step calibration requirement…
…need to calibrate ”silo” models first and subsequently introduce a new discounting curve.
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Forward start (months)
CCS 3M fwd breaks: EONIA vs. Fed Funds based
EONIA based calibration
Fed Funds based calibration
Diff (bps), rhs
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Changing the assumptions – cont’d
• An aside on CCSs:
− The basic building block for CCSs is in itself tricky…
− …MtM FX resets or constant notionals?
− Should FX-Basis correlation be included?
− Does the market standard CCS product rather warrant a full hybrid model?
• Conclusion:
− The full sequential calibration of the silo-based model matters in certain curve segments.
− Is obviously dependent on interpolation settings…
− …but for plausible choices, the difference in a 5Y5Y EUR/USD CCS can be up 0.25 bps.
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Forward start (months)
CCS 3M fwd breaks: Difference to Fed
Funds (bps)
EONIA based calibration
Silo based calibration
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The intuition behind collateral consistent pricing
Flow analysis: EUR Derivative - EUR security collateral
Counter Party Trading Desk
Collateral Management
Cash Desk
T0: Contract (PV=-100 EUR)
T0: Payment (=+100 EUR)
T0: Payment (=+100 EUR)
Bond Bond
T1: Interest= RIntern*(1/360)*100 EUR
•Security collateral can be financed at their respective repo rate. •Note the role of haircuts: Cash desk potentially receives one, but collateral management will have to provide one in the CSA. Only differences in haircuts matter – and then becomes a question of unsecured funding rates.
Bank
T1: Contract (PV=-100*(1+ RDisc/360) EUR)
Repo Cpty.
T1: Interest= RRepo *(1/360)*100 EUR
Bond
Cash
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Case study: Potential for market fragmentation in SEK
• CCP vs. Bilateral:
− Clearing is not standard in all markets – yet.
− In SEK, a large share of the IRS market is cleared but much is still bilateral.
− Among the market makers security collateral is allegedly common place…
− …and some of this is closer to STIBOR funded.
• CCP valuation vs. cash accrual
− LCH.SwapClear uses STIBOR discounting for VM calculation…
− …but still pays T/N rate on SEK cash.
− First order (accrual rate) vs. second order (accrual balance) effect.
• Conclusion:
− If there is still only one broker price, there should be fragmentation in the forward swap market.
− Screen prices should be different. -1.0
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Forward start (months)
SEK3M: Fwd curve diffs (bps)
STINA-Fed Funds
STINA-EONIA
STINA-STIBOR
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Collateral valuation adjustments
• CSA optionality:
− Many (older) CSAs contain long lists of eligible collateral.
− If collateral can be freely substituted, this creates a cheapest-to-deliver option for the posting party.
− This creates a need for an effective discount curve – created from more than one curve.
• Example:
− Can choose between placing EUR cash earning EONIA and USD cash earning Fed Funds.
− This is effectively a series of call options on the EONIA-FF CCS spread.
• Intrinsic value of CSA option:
− Find the upper convolution of the EONIA disc curve and the Fed Funds adjusted curve (in fwd terms).
− Use these forward rates to generate effective discount curve.
− In the specific example, it is expected to be cheapest to deliver EUR for all 30Y years...
− …but there is a risk that USD will be cheaper.
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Forward start (months)
3M forward EURDISC rates
Fed Funds based calibration EONIA based calibration
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The intuition behind collateral consistent pricing (cont.)
Expected collateral flow – 100M EUR 20Y IRS Payer
• Net Flow
− Take forward Euribor rates and par fixed rate as given, assume EUR OIS discounting.
− Forward curve is upward sloping
− We pay out net the first 5 years, and receive net the last 15 years.
• Future Value as expected collateral balance.
− Starts and ends at zero for the ATM trade.
− Increses since we are owed more and more.
− Decreases when we start to receive.
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Flow Rec
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FV
FV Rec
FV Pay
FV Net
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The intuition behind collateral consistent pricing (cont.)
Forward Cross Currency Basis Spreads – 1Y Forward CCS next 20Y
• ColVA
− Consider the Collateral Valuation Adjustment if collateral should be posted in USD Cash rather than EUR Cash.
− User the FV Net as the CCS notional profile, compute the value of paying the spread.
− The spread is determined through the CCS with the Fed Funds rate flat on the one leg and Eonia plus a spread on the other.
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FV Net
FV ColVA
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Flow ColVA
CCS Spread
Flow ColVA
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The intuition behind collateral consistent pricing (cont.)
Discount Curve Risk wrt 1Y Forward CCS Spreads - 100M EUR 20Y IRS Payer
• Compute Discount Curve Risk wrt. 1Y Fwd swaps to derive 1st order ColVA impact estimate from shifting collateral type.
• Example continued:
− ATM, ITM (ATM-100bp), OTM(ATM+100bp)
− Positive FV implies negative Fwd Disk Risk.
− ITM/OTM have the extra disk risk from an annuity.
− Result:
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Disc Risk
ATM Disc Risk OTM Disc Risk
ITM Disc Risk CCS Spread
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-50,000
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150,000
1st Order ColVA: Disc Risk * CCS Spread
ATM Impact OTM Impact ITM Impact
ATM OTM ITM
Impact Impact Impact
203k EUR -356k EUR 761k EUR
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Option adjusted collateral consistent pricing
• Realised volatility on CCS spreads: − Spot (normal) volatility is in the 20-50 bps
range on an annualised basis.
− Forward spreads are however less volatile.
• How to include volatility? − Simple model, can only EUR or USD cash.
− Assume Gaussian model.
− Collateral poster is long a series of caplets on CCS breaks, struck at 0 bps.
• Conclusion − Given the shape of the CCS fwd break
curve, the short expiries are deep OTM…
− …little effect on effective discounting curve.
− But significant increases for long dated expiries (closer to ATM and higher vega).
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Feb-10 Aug-10 Feb-11 Aug-11 Feb-12 Aug-12
EONIA-FF CCS:
Rolling 30D realised volatility (annualised)
5Y 10Y
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Forward start (months)
3M forward EURDISC rates
Intrinsic 20 bps vol 50 bps vol
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Option adjusted collateral consistent pricing – cont’d
• Theory: − Fujii & Takahashi (2011) and Piterbarg (2012)
• Example:
− 30Y EUR Payer, 100m 250 bps OTM.
• Risk:
− Using the intrinsic approach, not CCS hedge is required (EUR trade, EUR cash is CTD with certainty).
− But this will change as basis spreads increase Risk will ”jump”.
− Stability in hedges is an important argument for developing CTD models…
− …especially in ”naive” bump-and-re-run” mode.
Model PV Initial Difference
Intrinsic CTD -46.67m -
Option adj. CTD, 20 bps -45,80m 878k
Option adj. CTD, 50 bps -43.92m 2.756k -15,000
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30,000
1Y 2Y 3Y 4Y 5Y 7Y 10Y 12Y 15Y 20Y 25Y 30Y
Intrinsic (0 bps): CCS Dv01 (EUR)
Base +10 bps +20 bps
Note, this is a typical pension fund trade – a difference of 6% of the PV of derivatives can mean insolvency.
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Option adjusted collateral consistent pricing – cont’d
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Option adj (20 bps): CCS risk (EUR)
Base case +10 bps +20 bps
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Option adj (50bps): CCS risk (EUR)
Base case +10 bps +20 bps
• Option adjusted discount deltas:
− Results in stable hedges.
− Intuition fits well against USD cash-only benchmark case.
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1Y 2Y 3Y 4Y 5Y 7Y 10Y 12Y 15Y 20Y 25Y 30Y
USD cash only: CCS risk (EUR)
Base case +10 bps +20 bps
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Conclusion
• There is a direct link between collateral terms and discount factors.
• This is important – it is not just for market makers in derivatives.
• It is not trivial to construct collateral consistent swap curves – and arbitrages are sometimes not far away.
• The ”poor man’s” collateral consistent approach can bring most market participants far.
• While the value of CTD options embedded in CSAs is debatable – the risk implications are clear.
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References
• Piterbarg, V. (2010), ”Funding beyond discounting: Collateral agreements and derivatives pricing”, Risk Magazine February, pp.97-102
• Fujii, M. & Takahashi, A. (2011), ”Choice of collateral currency”, Risk Magazine January, pp. 120-125
• Piterbarg, V. (2012), ”Cooking with collateral”, Risk Magazine, pp. 58-63
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