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Co-Authored by Rohan Douglas and Peter Decrem (Quantifi)

• A new generation of interest rate modelling based on dual curve pricing and integrated CVA is evolving• This new framework requires a rethink of derivative modelling from first principles and presents significant challenges for existing valuation, risk management, and margining systems

WHITE PAPEROIS AND CSA DISCOUNTING

About the Authors

Rohan DouglasRohan has over 25 years experience in the global financial industry. Prior to founding Quantifi in 2002, he was a Director of Research at Salomon Brothers and Citigroup, where he worked for ten years. He has extensive experience working in credit, interest rate derivatives, emerging markets and global fixed income. Mr.Douglas teaches as an adjunct professor in the graduate Financial Engineering program at NYU Poly in New York and the Macquarie University Applied Finance Centre in Australia and Singapore and is the editor of the book Credit Derivative Strategies by Bloomberg Press.

Peter DecremPeter heads the Rates Group at Quantifi. As Director, Peter is responsible for managing the product development process of all Rates Solutions within the Quantifi product suite. Peter started in Research and Technology at Bear Stearns and Deutsche Bank. He traded fixed income derivatives, government bonds and agencies for Lehman Brothers and Salomon Brothers. He was responsible for fixed income derivatives trading desk for a number of European banks. Most recently he refocused on technology and specifically concentrated on machine learning and high frequency trading on parallel systems prior to joining Quantifi in 2009.

OIS and CSA discounting

IntroductionPrior to the credit crisis, interest rate modelling was generally well understood. The underlying fundamental principles had existed for over thirty years with steady evolutions in areas that were most relevant to options and complex products. Credit and liquidity were ignored as their effects were minimal. Pricing a single currency interest rate swap was straightforward. A single interest rate curve was calibrated to liquid market products and future cash flows were estimated and discounted using this single curve. There was little variation between implementations and results across the market were consistent.

Following the credit crisis, interest modelling has undergone nothing short of a revolution. During the credit crisis, credit and liquidity issues drove apart previously closely related rates. For example, Euribor basis swap spreads dramatically increased and the spreads between Euribor and Eonia OIS swaps diverged. In addition, the effect of counterparty credit on valuation and risk management dramatically increased. Existing modelling and infrastructure no longer worked and a rethink from first principles has taken place.

Today a new interest rate modelling framework is evolving based on OIS discounting and integrated CVA. Pricing a single currency interest rate swap now takes into account the difference between projected rates such as Euribor that include credit risk and the rates appropriate for discounting cash flows that are risk free or based on funding cost. This approach is referred to as dual curve, OIS discounting, or CSA discounting and forces a re-derivation of derivatives valuation from first principles. In addition, the counterparty credit risk of (uncollateralised) OTC transactions are measured as a credit valuation adjustment (CVA) which takes into account the likelihood the counterparty will default, along with expected exposures, volatility of these expected exposures, and wrong way risk. In this paper we will focus on OIS discounting as part of this new interest rate modelling framework.

Interest Rate modelling prior to the credit crisisPrior to the credit crisis, interest rate derivatives were valued with models that focused on the dynamics and term structure of interest rates but generally ignored other elements including:

• Credit risk

• Liquidity risk

• Collateral agreements

• Funding costs

Since the introduction of Black-Scholes in 1973, interest rate modelling has evolved steadily. There have been several key milestones with the most recent evolutions prior to the credit crisis relating to volatility skew modelling.

Valuing a single currency vanilla interest rate swap involved calculating forward rates and discounting expected cash flows from a single interest rate curve based on no-arbitrage assumptions. These curves were calibrated from liquid interest rate products including money market securities, Eurodollar futures, FRAs, and interest rate swaps. A common reference rate for Euro denominated swaps is the Euro Interbank Offer Rate (Euribor). Euribor rates are published 11am Central European Time each day and are calculated as the average (excluding the top and bottom 15%) offer rates from over 40 contributing banks for Euro Interbank deposits. 15 different maturities are published. These deposits are unsecured but prior to the credit crisis were considered a proxy for a risk-free rate. Similar interbank deposit rates are calculated for other currencies.

Figure 1: Key milestones in the history of interest rate modelling

The key modelling components are curve construction, pricing, and risk management.

Curve construction

1. Select a set of liquid interest rate securities. For example – EUR deposits, Euribor futures and Euribor swaps2. Make selections for the handling of overlapping securities, interpolation methods and seasonality effects3. Fit a single yield curve using bootstrapping, tension splines, or other fitting methods

Figure 2 shows a typical calibration screen where funding rates, money market rates, FRAs and Swap rates are combined to calibrate a discount curve.

Figure 2: Calibrating an interest rate curve in a single curve valuation framework (source Quantifi)

Pricing

1. Calculate expected future floating coupon payments of the floating leg of the swap by calculating the forward rate

2. Calculate the expected value of the floating leg of the swap by present valuing each of the expected cashflows

Figure 3: Pricing an off-market interest rate swap in a single curve valuation framework (source Quantifi)

Figure 3 shows a typical pricing screen for an off-market interest rate swap. The net interest rate sensitivity shown as Net IR01 is the change in the mark-to-market value of the swap if interest rates increase by one basis point. Note that under this pricing framework, a floater will price to par.

Risk Management

1. Calculate the sensitivity of expected value of the swap to changes in the yield curve

2. Calculate the sensitivity of the chosen liquid hedge securities to changes in the yield curve to determine the correct hedge ratios

3. Hedge the swap with with the chosen hedge securities. For example, an off market Euribor swap may be hedged with a combination of liquid at-market Euribor swaps

Figure 4 shows the interest rate sensitivity by tenor for an off market interest rate swap of roughly 8 ½ years maturity. The appropriate hedge is a combination of liquid at-market 8 year and 9 year interest rate swaps.

Figure 4: Hedging analysis for an off-market interest rate swap in a single curve valuation framework (source Quantifi)

Impact of the Credit Crisis on the Rates MarketAs the credit crisis unfolded, there were significant impacts on the structure and dynamics of the rates market. Credit and liquidity drove segmentation and rates that were previously closely related diverged, causing a rethink of how these rates should be modelled.

Basis swap spreads increased dramaticallyA basis swap is an interest rate swap where the parties exchange floating rate payments of different tenors. For example, a swap where one party pays Euribor 3M + a spread and the other party pays Euribor 6M. The swaps are quoted in terms of the spread.

Reflecting the different credit risk and market segmentation between different Euribor rate tenors, basis swap spreads blew out during the crisis from being fractions of a basis point (where they had been quoted for decades) to double digits in a matter of months. In Figure 6 we see a graph of the Basis Swap Spreads for 3M Vs 6M Euribor of 1Y maturity. The 3M Vs 6M Euribor Basis swap spread went from under a basis point to peak at over 44bp around October 2008, after the Lehman default.

This divergence is again a reflection of different credit and liquidity risks between these indices. The longer term deposits (6M) carry more credit risk than the shorter (3M) deposits.

Figure 5: Euribor 3M Vs 6M Basis Swap

Figure 6: 3M Vs 6M Euribor 1 year basis swap spreads (source Bloomberg)

OIS and LIBOR swap rates divergedOvernight Index Swaps are swaps with a fixed leg versus a floating rate leg that is indexed to an overnight rate. For Euros, the OIS swap is indexed to the Eonia rate. The Eonia rate is a weighted (by volume) average of all overnight unsecured lending transactions in the European interbank market. The contributing banks are the same as those contributing to the Euribor index and the index is calculated and published by the European Central Bank (ECB). Other currencies have similar reference rates – for example the Fed Funds rate in the US. These rates are typically targeted by the respective central banks as a part of monetary policy. In Europe, the Eonia trades between the Deposit Facility rate and the Margin Lending Facility rates set by the ECB.

The daily tenor of the Eonia transactions means that these rates carry negligible credit and liquidity risk. The spread between Euribor Deposits and the Eonia OIS rates blew out through the credit crisis. Figure 7 shows a graph of the spread between the Euribor 6M deposit rate and the 6M Eonia OIS rate.

The basis widened dramatically from under 10 basis points to peak at 222.5 basis points around Lehman’s default in October 2008. The Euribor/Eonia spread has persisted even after the credit crisis and reflects the revised view of the different credit and liquidity characteristics between these two rates. The Eonia rate has now become the market standard proxy for a EUR market risk-free rate.

Figure 7: Euribor 6M Deposit rate Vs 6M Eonia OIS rate (source Bloomberg)

The relationship between forward rates of different tenors diverged.Prior to the credit crisis there were small but generally negligible differences between forward rates implied from interest rate products of different tenors. No-arbitrage arguments held and a 6 month rate implied from a 3 month rate and a 3x6 month forward would match. As the credit crisis continued, the market segmented and this previously arbitrage-free relationship broke down.

Figure 8 shows two examples of the divergence between implied rates from different tenors. This segmentation meant that you could no longer price swaps paying a 3M floating rate coupon using curves calibrated from swaps paying 6M floating rate coupons.

Increased use of Collateral AgreementsAnother effect of the credit crisis has been a dramatic increase in the use of collateral agreements as a method of managing counterparty risk. The 2010 ISDA Margin Survey reports that 70 percent of OTC derivatives net credit exposure worldwide is covered by collateral, compared with 29 percent in 2003.

Collateral agreements provide a margining facility whereby parties in an OTC transaction post or receive margin on a regular basis to mitigate counterparty risk. The Credit Support Annex (CSA) is a part of the standard International Swaps and Dealers Association (ISDA) master agreement that provides the legal framework for OTC derivative trades between two parties. The margining facility is similar to that used by a clearing house and provides for posting of collateral based on a calculated mark-to-market valuation for a transaction. The collateral can be cash or assets of the same value. The holder of the collateral pays an interest rate called the ‘collateral rate’. Under default, the collateral is available to the holder to cover the net market value owed. The collateral agreement typically allows for netting across all trades covered by that ISDA master agreement. Typical collateral agreements provide for daily collateral calls and a collateral rate such as Eonia, Fed Funds, Sterling overnight index average (Sonia) or Jibar.

Figure 8: 3x6M forward rates implied by 3M and 6M Euribor Vs 3x6M FRA and Euribor 6M deposit Vs 6M rate implied by Euribor 3M and 0x3M FRA. (Source Banca IMI)

The New Interest Rate Modelling ParadigmClearly the credit crisis had a significant impact on the interest rates market. A large part of this related to the increased importance of credit and liquidity risk along with structural changes such as an increased use of collateral agreements. These changes have driven a profound shift in the way all OTC products are valued and risk managed. The result has been an abandonment of the classic derivatives pricing framework based on single interest rate curves and the introduction of a new approach that takes into account current interest rate dynamics and market segmentation using multiple curves.

Dual curve/OIS discountingThe old-style no-arbitrage, single-curve derivatives valuation framework where Euribor was a reasonable proxy for a risk-neutral discount rate has been permanently changed by the credit crisis. An understanding of the credit risk embedded in Euribor and similar rates and an increased importance in the modelling of funding have driven a separation between the index rates used for the floating legs of the swap (the projection rates) and the appropriate rates used for present value (the discount rates). The market-standard rate to discount future cash flows is now OIS rates.

The method of projecting rates using Euribor and discounting rates using Eonia changes the fundamental framework for existing derivative modelling. It has required a rethink from first principles that continues to be discussed and refined. Pricing and risk managing even a vanilla single currency swap has become significantly more complex. Curve construction, pricing and hedging now involve multiple instruments and additional basis risks. These complexities compound for interest rate products such as cross currency swaps, Caps/Floors and Swaptions.

Funding costThe debate about what are appropriate discount rates is still in progress. The role of funding and funding costs is a complex one. The impact of different market participants funding costs, the uncertainty in some institutions about measuring funding costs, and the impact of LVA are the subject of current academic and market debate.

Counterparty risk and CVAA broader and evolving understanding of valuing and managing counterparty credit risk was well underway before the credit crisis. Many of the larger global banks had been actively measuring and managing counterparty credit risk many years prior to the crisis. The crisis, however, dramatically increased the focus for market participants as well as regulators and accelerated the impact on the broader OTC markets. The measurement and management of counterparty risk is now something that impacts all market participants. Accurate valuation of OTC products now requires accurate valuation of the credit component of each transaction. In addition, regulatory initiatives such as Basel III and Solvency II, along with accounting rules such as ASC 820 (FAS 157) and IAS 39 have mandated more accurate counterparty risk valuation and risk management.

The larger banks have led the evolution of valuing and managing counterparty credit risk. Over time they have converged to generally consistent methods and processes. The concept of a Credit Value Adjustment (CVA) is now widely accepted and consistently calculated across the markets. OTC transactions that carry counterparty exposure executed by all the larger institutions now have a CVA component as part of the valuation. An accurate CVA calculation takes into account all transactions in the portfolio with that counterparty as well as any netting agreements, CSAs and collateral.

Dual Curve OIS Discounting Curve ConstructionFollowing the credit crisis, interest rate derivatives are now valued with models that reflect the observed market segmentation, counterparty risk, and interest rate dynamics. Valuing a single currency vanilla interest rate swap involves calculating forward rates based on Euribor rate curves and discounting expected cash flows using Eonia rates. As in the single-curve case, these curves are calibrated from liquid interest rate products. For the EUR curves this includes money market securities, futures, FRAs, IONA swaps, basis swaps and interest rate swaps. The process is complicated, however, by changes to the modelling principles around calculating the expected forward rates. These forward rates must be conditional on the Eonia rates used for discounting.

Curve construction

1. Select a set of liquid interest rate securities for each curve.

2. The Eonia interest rate curve is fitted to Eonia rates including deposits and OIS swap spreads. As with single-curve interest rate curve construction, key issues include

– Selection of curve instruments

– Convexity adjustments for Future prices

– Overlapping securities

– Seasonal effects

– Interpolation methods (fit Vs smoothness)

3. In addition there are specific complexities relating to bootstrapping the Eonia curve including

– Modelling the step nature and key policy dates for central bank target rates

– Dealing with less liquid tenors

– Bootstrapping longer dated parts of the curve from swap spreads

4. Fit the Euribor interest rate curve conditional on the Eonia interest rate curve. This calibration needs to be carefully done to provide robust and accurate matching of Euribor swap rates. For example, the resulting curve should reprice a standard Euribor interest rate swap at par.

Pricing

1. Calculate expected future floating coupon payments of the floating leg of the swap by calculating the forward rate

2. Calculated the expected value of the floating leg of the swap by present valuing each of the expected cash flows

Figure 9 shows a typical pricing screen for an off-market interest rate swap. In addition to the Euribor rate sensitivity (Net IR 01), there is an additional basis risk Vs Eonia rates (Net Spread 01) introduced. Note that under this pricing framework, an at-market floater will not price to par.

Figure 9: Single currency interest rate swap pricing under dual curve (source Quantifi)

Risk Management

1. Calculate the sensitivity of expected value of the swap to changes in the forward rate curve

2. Calculate the sensitivity of expected value of the swap to changes in the discount rate curve

3. Calculate the sensitivity of the chosen hedge securities to changes in the interest curves to determine the correct hedge ratios.

4. Hedge the swap with the chosen hedge securities. For example an off market Euribor swap may be hedged by a combination of liquid at-market Euribor swaps and a combination of at-market Eonia OIS swaps.

Posting collateral in multiple currenciesIn many cases, CSA agreements provide for posting of collateral in one of several currencies. This adds a level of complexity to the selecting of the appropriate discount curve to use and adds a degree of optionality that is difficult to model. Often these effects are managed on an ad-hoc basis.

Figure 10: Interest rate risk for a single currency swap priced using OIS discounting (source Quantifi)

Dual currency pricing of Cross Currency SwapsHistorically, cross currency swaps were priced based on a single interest rate curve for each currency, a cross currency basis spread, and the current spot FX quote. Depending on the currency pair, different methods of calibration would be used to take advantage of the most liquid markets and quotes (for example, foreign swap rates Vs FX forwards). Dual curve pricing of cross currency swaps adds challenges in terms of the relationship between the appropriate discount curves, projection curves, and cross currency basis. One method for calibrating the interest rate curves for a fixed/floating cross currency swap is:

• Select a set of liquid interest rate securities for each curve

• Fit the OIS discount curves for each currency

• Fit the LIBOR projection curves for the floating rate currency conditional on the OIS discount curves

• Adjust the floating rate projection curve based on cross currency basis swap rates

Figure 11: Basis swap risk for a single currency interest rate swap using OIS Discounting (source Quantifi)

Figure 12 shows the process flow for calibrating interest rate curves for valuing a USD/EUR fixed/floating cross currency swap. The EUR leg is discounted using the Eonia curve. The USD leg would be projected using the cross-currency basis adjusted LIBOR interest rate curve and discounted using the Fed Funds curve.

The calibration process shown here is simplified. The calibration and pricing processes have significant implications for the hedging and risk management of these products and need to be carefully thought through.

Figure 12: Cross currency curve construction in a dual curve valuation framework

Moving to integrated dual curve OIS discounting and CVAThere is clear evidence that the market has moved to valuing interest rate swaps using OIS discounting and integrated CVA. The move towards central clearing and standardised products has accelerated this trend and the recent press release from the London Clearing House (LCH.Clearnet) is a very clear testimony to this wider adoption.

“LCH.Clearnet Ltd (LCH.Clearnet), which operates the world’s leading interest rate swap (IRS) clearing service, SwapClear, is to begin using the overnight index swap (OIS) rate curves to discount its $218 trillion IRS portfolio.

Previously, in line with market practice, the portfolio was discounted using LIBOR. However, an increasing proportion of trades are now priced using OIS discounting. After extensive consultation with market participants, LCH.Clearnet has decided to move to OIS to ensure the most accurate valuation of its portfolio for risk management purposes.”

LCH’s decision to move to OIS discounting reflected the fact that most swaps cleared through their system were subject to standard CSAs with daily collateral calls and a collateral rate based on OIS rates. The large market-making banks are now pricing using OIS discounting and their margining and collateral systems are being converted to also reflect this practice.

ConclusionA new generation of interest rate modelling is evolving. An approach based on dual curve pricing and integrated CVA has become the market consensus. There is compelling evidence that the market for interest rate products has moved to pricing on this basis, but not all market participants are at the stage were existing legacy valuation and risk management systems are up to date. The changes required for existing systems are significant and present many challenges in an environment where efficient use of capital at the business line level is becoming increasingly important.

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