Collateral consistent derivatives pricing

Collateral consistent derivatives pricing FRIC Practitioner Seminar, CBS Martin D. Linderstrøm, marlin@danskebank.dk Counterparty Credit & Funding Risk

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? 2

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 These assumptions are no longer valid. 70 60 50 40 30 20 10 5Y 3M-6M Basis 0 2005 2006 2007 2008 2009 2010 2011 2012 5Y EUR/USD X-CCY 5Y EONIA-3M Basis 10 0-10 -20-30 -40-50 -60-70 -80 2005 2006 2007 2008 2009 2010 2011 2012 3

23-Aug-10 29-Nov-10 7-Mar-11 15-Jun-11 21-Sep-11 29-Dec-11 5-Apr-12 17-Jul-12 23-Oct-12 1-Feb-13 15-May-13 21-Aug-13 Swaps in the new way Need for multiple projection curves for each currency. Forward LIBOR Surface 2.5% 2.0% We cannot compute 3M xibor and 6M xibor forwards on the same curve. Need for a single discounting curve for each currency. 10M 5M 1B 1.5% 1.0% 0.5% 0.0% 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? 1.2 Discount Factors 1.0 0.8 0.6 0.4 0.2 0.0 4

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. 5

The intuition behind collateral consistent pricing Flow analysis: EUR Derivative - EUR Cash collateral T 0 : Payment (=+100 EUR) Bank T 0 : Payment (=+100 EUR) Counter Party T 0 : Contract (PV=-100 EUR) Trading Desk T 1 : Interest= R Intern *(1/360)*100 EUR T 1 : Contract (PV=-100*(1+ R Disc /360) EUR) Cash Desk T 1 : Interest= R EUR/OIS *(1/360)*100 EUR T 1 : Interest= R Intern *(1/360)*100 EUR T 0 : Payment (=+100 EUR) Collateral Management T 0 : Payment (=+100 EUR) Cash desk is passing through the liquidity no haircuts or disagreement on valuation. Internal loop can be closed if r Intern =r OIS 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. R Disc = R OIS. He could in principle hedge his cash exposure via an EONIA swap. 6

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. 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% EUR: 3M forward rates EUROIS EUR3M 0 60 120 180 240 300 360 Forward start (months) 7

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. 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% USD: 3M forward rates USDOIS USD3M USDDISC EUR/USD X-CCY 3M (rhs.) 0.0% -0.1% -0.2% -0.3% -0.4% -0.5% -0.6% 0 60 120 180 240 300 360 Forward start (months) 8

The intuition behind collateral consistent pricing (cont.) Flow analysis: EUR Derivative - USD Cash collateral T 0 : Payment (=+100 EUR) Bank T 0 : Payment (=+100 EUR) Counter Party T 0 : Contract (PV=-100 EUR) T 1 : Contract (PV=-100*(1+ R Disc /360) EUR) T 1 : Interest= R USD/OIS *(1/360)*126 USD Trading Desk T 1 : Interest= R Intern *(1/360)*100 EUR Cash Desk T 1 : Interest= R Intern *(1/360)*126 USD T 1 : Interest= (R EUR/OIS +s)(1/360)*100 EUR T 0 : 100 EUR T 0 : 126 USD CCS Counter Party T 0 : Payment (=+126USD) Collateral Management T 0 : Payment (=+126 USD) T 1 : Interest= R USD/OIS *(1/360)*126 USD 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! 9

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? 10

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. 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% USD3M Fwd: EONIA vs. Fed Funds based EONIA based calibration Fed Funds based calibration Diff (bps), rhs 0.25 0.00-0.25-0.50-0.75-1.00-1.25-1.50-1.75-2.00 0 60 120 180 240 300 360 Forward start (months) 11

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. 0.0% -0.1% -0.2% -0.3% -0.4% -0.5% -0.6% CCS 3M fwd breaks: EONIA vs. Fed Funds based EONIA based calibration Fed Funds based calibration Diff (bps), rhs 0.10 0.05 0.00-0.05-0.10-0.15-0.20-0.25-0.30-0.35-0.40 0 60 120 180 240 300 360 Forward start (months) 12

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 silobased 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. 0.25 0.00-0.25-0.50 CCS 3M fwd breaks: Difference to Fed Funds (bps) EONIA based calibration Silo based calibration 0 60 120 180 240 300 360 Forward start (months) 13

The intuition behind collateral consistent pricing Flow analysis: EUR Derivative - EUR security collateral T 0 : Payment (=+100 EUR) Bank T 0 : Payment (=+100 EUR) Counter Party T 0 : Contract (PV=-100 EUR) Trading Desk T 1 : Interest= R Intern *(1/360)*100 EUR Cash T 1 : Contract (PV=-100*(1+ R Disc /360) EUR) Cash Desk Bond Repo Cpty. Bond Collateral Management Bond T 1 : Interest= R Repo *(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. 14

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. 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0-0.5-1.0 SEK3M: Fwd curve diffs (bps) STINA-Fed Funds STINA-EONIA STINA-STIBOR 0 60 120 180 240 300 360 Forward start (months) 15

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. 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% -0.5% -1.0% 3M forward EURDISC rates Fed Funds based calibration EONIA based calibration 0 60 120 180 240 300 360 Forward start (months) 16

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. 1,000,000 800,000 600,000 400,000 200,000 0-200,000-400,000-600,000 40,000,000 35,000,000 30,000,000 25,000,000 20,000,000 15,000,000 10,000,000 5,000,000 0 Flow 0Y 5Y 10Y 15Y FV 0Y 5Y 10Y 15Y Flow Rec Flow Pay Flow Net FV Rec FV Pay FV Net 17

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. 0.000% -0.100% -0.200% -0.300% -0.400% -0.500% 8,000,000 7,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,000,000 1,000,000 0 Flow ColVA -0.600% 0Y 5Y 10Y 15Y FV ColVA 0Y 5Y 10Y 15Y 10,000 8,000 6,000 4,000 2,000 0 250,000 200,000 150,000 100,000 50,000 0 CCS Spread Flow ColVA FV Net FV ColVA 18

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. 2,000 1,000 0-1,000-2,000-3,000 Disc Risk 0.00% -0.20% -0.40% -0.60% Example continued: ATM, ITM (ATM-100bp), OTM(ATM+100bp) ATM Disc Risk ITM Disc Risk OTM Disc Risk CCS Spread Positive FV implies negative Fwd Disk Risk. ITM/OTM have the extra disk risk from an annuity. Result: ATM OTM ITM 150,000 100,000 50,000 0-50,000-100,000 1st Order ColVA: Disc Risk * CCS Spread Impact Impact Impact 203k EUR -356k EUR 761k EUR ATM Impact OTM Impact ITM Impact 19

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). 0.80% 0.60% 0.40% 0.20% 0.00% Feb-10 Aug-10 Feb-11 Aug-11 Feb-12 Aug-12 4.0% 3.0% 2.0% 1.0% 0.0% EONIA-FF CCS: Rolling 30D realised volatility (annualised) 5Y 10Y 3M forward EURDISC rates Intrinsic 20 bps vol 50 bps vol 0 60 120 180 240 300 360 Forward start (months) 20

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. 30,000 25,000 20,000 15,000 10,000 5,000 Model PV Initial Difference -5,000 Intrinsic CTD -46.67m - -10,000 Option adj. CTD, 20 bps -45,80m 878k Option adj. CTD, 50 bps -43.92m 2.756k -15,000 Note, this is a typical pension fund trade a difference of 6% of the PV of derivatives can mean insolvency. 0 Intrinsic (0 bps): CCS Dv01 (EUR) Base +10 bps +20 bps 1Y 2Y 3Y 4Y 5Y 7Y 10Y 12Y 15Y 20Y 25Y 30Y 21

Option adjusted collateral consistent pricing cont d Option adj (20 bps): CCS risk (EUR) Option adj (50bps): CCS risk (EUR) Base case +10 bps +20 bps Base case +10 bps +20 bps 15,000 15,000 10,000 10,000 5,000 5,000 0 0-5,000 1Y 2Y 3Y 4Y 5Y 7Y 10Y 12Y 15Y 20Y 25Y 30Y -5,000 1Y 2Y 3Y 4Y 5Y 7Y 10Y 12Y 15Y 20Y 25Y 30Y Option adjusted discount deltas: Results in stable hedges. Intuition fits well against USD cashonly benchmark case. 25,000 20,000 15,000 10,000 5,000 0 USD cash only: CCS risk (EUR) Base case +10 bps +20 bps 1Y 2Y 3Y 4Y 5Y 7Y 10Y 12Y 15Y 20Y 25Y 30Y 22

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. 23

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 24

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