Discounting · 2011. 2. 12. · Derivatives valuation should not use an assumption for the...

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Discounting

Jeroen Kerkhof

22 September 2010

Overview

❖ Overview

Introduction to InterestRates

Linear Interest RateDerivatives

Uncollateralised trades

Collateralised trades

Non-linear Interest RateDerivatives

Introduction to Interest Rates

Linear Interest Rate Derivatives

Non-linear Interest Rate Derivatives

Introduction to Interest Rates

❖ Overview

❖ Time Value of Money

Time Value of Money

❖ Overview

● We all know that a euro today is worth more than a eurotomorrow, but how much...

● This depends on

✦ Type of investment opportunities, e.g.

● In order to uniquely define the value of a euro in the future abenchmark investment is needed.

✦ Finance textbooks introduce the so-called risk free rateon a bank account

✦ Not that clearly defined in practice...

Time Value of Money

❖ Overview

● Not everyone has the same investment opportunities...

● For derivatives valuation ”risk-free” is not clearly defined

● Used to assumed to be Libor

● Nowadays Overnight Index more relevant

● or maybe repo?

● Derivatives valuation should not use an assumption for therisk-free rate

● It should use the terms in the bilateral contracts (CSA)

Linear Interest Rate Derivatives

❖ Overview

❖ Interest-rate Swaps

❖ The Old Days

❖ Valuation Interest-rateSwaps

❖ Curve ConstructionRecipe

Interest-rate Swaps

❖ Overview

❖ The Old Days

Standard Fixed-Floating interest rate swap:

03/06/11 03/06/12 03/06/13 03/06/14 03/06/15

6M Libor

K K K K K

Leg Day count basis Date rule FrequencyFixed 30/360 Mod. Following Annual

Floating Act/360 Mod. Following Semi-Annual

The Old Days

❖ Overview

❖ The Old Days

● Libor fixings (up to 1 year)

● Futures (1 year up to 3 years)

● Fixed-Floating (6M) Swaps (the long end of the curve)

● Derive the swap curve (bootstrapping or solver) See, forinstance, your favorite edition of Hull: Options, Futures andOther Derivatives

● The curve will provide D(t, T ) for all T (up to end of curve) t =today, T = maturity date and D(t, T ) represents todays valueof one euro receivable at T

Valuation Interest-rate Swaps

❖ Overview

❖ The Old Days

● Value other linear interest rate derivatives using this curve

● For instance, suppose we need to value the following swap

03/06/11 03/06/12 03/06/13 03/06/14 03/06/15

K K K K K

3M Libor

● Use the swap curve to find

● F3M (t, Ti) =1δi

D(t,Ti)D(t,Ti+1)

− 1)

✦ Ti+1 = Ti + 3M

● Determine K

Valuation Interest-rate Swaps

❖ Overview

❖ The Old Days

● The value of a Libor payment equals

δiD(t, Ti + 3M)× F3M (t, Ti) =

δiD(t, Ti+3M)× 1δi

D(t,Ti)D(t,Ti+3M) − 1

= D(t, Ti)−D(t, Ti+3M)

● Iteration gives the value of the floating leg equals

● For the 6m index we had something similar

● As Tn = T2k the values of the 2 floating legs are the same

● Hence fixed rate should be the same

Old days are gone...

❖ Overview

❖ The Old Days

● Let’s look at the market for 3M-6M tenor basis swaps

History Libor 3M - OIS spread: Euro

❖ Overview

❖ The Old Days

History Libor 3M - OIS spread: US

❖ Overview

❖ The Old Days

Market Data

❖ Overview

❖ The Old Days

These days the market trades

● OIS vs fixed / floating

● Tenor basis swaps (e.g. 3M vs 6M)

● Cross-currency basis swaps

● OIS swaps on ECB dates (Euro), MPC dates (UK)

● fixed-floating swaps

● We should use this information to build our interest ratecurve...

Overview

❖ Overview

❖ The Old Days

Eonia Swap data

❖ Overview

❖ The Old Days

Eonia Swap data

❖ Overview

❖ The Old Days

Eonia Swap data

❖ Overview

❖ The Old Days

Fixed-Floating Swaps

❖ Overview

❖ The Old Days

Tenor basis swaps

❖ Overview

❖ The Old Days

Interest-rate Swaps

❖ Overview

❖ The Old Days

● 3M vs fixed

Interest-rate Swap

❖ Overview

❖ The Old Days

● Rather than having just D(t,T) for all T we will have

✦ D(t, T ) for all T

✦ D1M (t, T ) for all T

● The value of e.g. the 12M index will be

F12m(t, Ti) = δi

D12m(t,Ti)D12m(t,Ti+1)

− 1)

● Its value today is given by D(t, Ti + 12m)× F12(t, Ti) =

D(t, Ti + 12m)× δi

D12m(t,Ti)D12m(t,Ti+1)

− 1)

What is appropriate for D(t, T )

Curve Construction Recipe

❖ Overview

❖ The Old Days

In order to construct discount curves we follow the following steps:

● Select relevant instruments

● Get the relevant market data (e.g. ICAE)

● Select an interpolating scheme forD(t, T ), D1M (t, T ), D3M (t, T ), D6M (t, T ), D12M (t, T )

● Use a solver forD(t, T ), D1M (t, T ), D3M (t, T ), D6M (t, T ), D12M (t, T )

❖ Overview

❖ The Old Days

● Suppose we have fixed rate OIS swaps out to end of thecurve

● Then we can iteratively determine D(t, T ) for all T on whichthere is a fixed cash flow.

● Using the interpolating scheme we have modelled D(t, T ) forall T

● Using these discount factors we can determine value of thefixed legs of fixed-floating (6M Euribor) swaps. By iterationand the fact that swaps are 0-NPV at inception we find thevalue of the D6M (t, T ) for all T on which there is a cash flow.

● Using the interpolating scheme we have modelled D6M (t, T )for all T

❖ Overview

❖ The Old Days

● Armed with D(t, T ) and D6M (t, T ) we value the 6M floatingindex leg of 3M-6M tenor basis swaps. We can bootstrapD3M (t, T ) using an iterative procedure. Again interpolationdoes the rest.

● Continue with 1M-3M basis swaps and 6M-12M basis swaps

● This finally givesD(t, T ), D1M (t, T ), D3M (t, T ), D6M (t, T ), D12M (t, T ) for all T

● Unfortunately, the market does not quote fixed OIS swapsbeyond 2y, but OIS-3M Euribor swaps

● As such there is a circular dependency between OIS, 3M, and6M index

● Therefore, a solver is to be preferred

❖ Overview

❖ The Old Days

● We know how to build a curves in line with market instruments

● How do we discount client cash-flows?

● Uncollateralised

● Collateralised (different types)

❖ Overview

Uncollateralised trades❖ CVA: Credit ValuationAdjustment

❖ DVA: Debt ValuationAdjustment

❖ Funding

❖ Wrong-way risk

❖ Netting

CVA: Credit Valuation Adjustment

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

For quite some time (some) banks realized that

● Not all counterparties are of ”Libor”-credit quality

● CVA adjustments are needed to reflect credit quality of thecounterparty

Typically,

● CVA adjustments were unilateral (only default risk of the clientwas considered)

● CVA desks (if existent) were not very sophisticated

CVA: Credit Valuation Adjustment

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

Nowadays,

● Every (self-respecting) bank has a CVA desk

● however, sophistication levels vary widely

● most banks are looking at DVA as well

What is DVA?

DVA: Debt Valuation Adjustment

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

● Not only the client can default

● Own default probability should be a factor in pricing

● This leads to so-called bilateral CVA

● What happens if your own credit rating deteriorates?

● Which banks can be most aggressive in pricing?

DVA: How to monetize

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

My credit rating deteriorates, I am making money!

● Really?

● How to lock-in this ”profit”?

● Sell protection on yourself?

● Ideally, yes, but practically no (more later)

Funding

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

● Often ignored (for valuation purposes)

● Critical component of derivatives business

● What does it cost to run a derivatives business?

● How should this be reflected in derivatives prices

CVA/DVA/Funding: Example

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

How does it work? Assume party A (that’s us) extends a loan toparty B payable at time T with a value X

● party B can default. How to account for this?

● use party B’s CDS spread

Funding costs

● loan needs to be funded. At which rate?

● use party A’s unsecured bond spread

Note: party A’s unsecured bond spread can be different from it’sCDS’s spread

CVA/DVA/Funding: Example

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

D(t, T ) = e−(r+rcds,b−rubs,a)(T−t) (1)

● r is the reference discount rate

● rcds,b denotes the cds spread for company B

● rubs,a denotes the unsecured bond spread for company A

What with derivatives?

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

What do loans have to do with derivatives?

● Trade in-the-money = loan to counterparty

● Most trades start ATM.

● Need a model for future values of derivative

Typical CVA formula (no wrong-way risk)

CV A =

D(0, t)s(t)EPE(t)dt (2)

● D(0, t) denotes the relevant risk-free discount factor for t

● s(t) denotes the relevant credit spread at t

● EPE(t) denotes the expected positive exposure of the tradeat time t

Wrong-way risk

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

Can we separate derivatives valuation and credit charge? I.e.

● Compute exposures

● Multiply by survival probabilities

● Multiply by discount rate

Unfortunately, not

● payoff/exposure can be (typically is) correlated to survivalprobability

● Possible big jumps in case of default

● liquidity issues

Netting

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

A way to mitigate counterparty risk is netting over all trades with aparticular counterpartyThis introduces

● Some non-linearities

● Hence, simple CVA formula no longer holds

DVA: How to monetize?

❖ Overview

❖ Funding

❖ Wrong-way risk

❖ Netting

Back to DVA. We can’t sell protection on ourselves, so...

● Sell protection on (group of) competitors

● Obviously, less than perfect correlation

● Who is going to buy my protection?

● If my competitor goes under, I am likely in trouble as well

❖ Overview

❖ CSA Agreement

❖ Reference Rates inEurope: Eoniar and

Euriborr

❖ Funding CollateralisedDerivatives

OTC Derivatives

❖ Overview

❖ CSA Agreement

Euriborr

● Derivative contracts are typically privately negotiatedcontracts (e.g. no exchange in between) between 2counterparties.

● In order to help standardize these contracts ISDA(International Swaps and Derivatives Association) has set-upstandard master agreements.

● Before two counterparties enter into OTC transactions theytypically set-up a so-called CSA (Credit Support Annex)

● The CSA serves to mitigate counterparty credit exposure

● It specifies the collateral posting procedure in case the trademoves away from the money

OTC Derivatives in General

❖ Overview

❖ CSA Agreement

Euriborr

● Big picture:

● CP that is out-of-the money (OTM) posts collateral equal tothe value of the swap. CP that is in-the-money (ITM) paysinterest on the received collateral.

● (Some) details:

✦ Which collateral is posted?

■ Cash (which currency?)■ Bonds (which bonds?)■ CDOs etc

✦ How frequent?

✦ Which interest rate?

✦ Minimum adjustment of position

CSA Agreement

❖ Overview

❖ CSA Agreement

Euriborr

● What is standard?

● Practically all banks will trade with one another via theLondon Clearing House (LCH, www.lchclearnet.com).

● The LCH CSA says the counterparties need to post cash inthe local currency and will receive the overnight index (e.g.EONIA for Euro and SONIA for Sterling)

● Is this the CSA that Banks have with their clients as well?

● Typically no

Reference Rates in Europe: Eoniar andEuriborr

❖ Overview

❖ CSA Agreement

Euriborr

● Reference rates within the Eurozone

● www.euribor.org

● Euriborr (Euro Interbank Offered Rate) is the rate at whicheuro interbank term deposits within the euro zone are offeredby one prime bank to another prime bank.

● Eoniar (Euro OverNight Index Average) is an effectiveovernight rate computed as a weighted average of allovernight unsecured lending transactions in the interbankmarket, initiated within the euro area by the contributing panelbanks.

● http://www.euribor.org/html/content/panelbanks.html

Eoniar

❖ Overview

❖ CSA Agreement

Euriborr

● Daycount convention: Act / 360

● 3 decimals

● ’Overnight’ means from one TARGET day (i.e. day on whichthe Trans-European Automated Real-Time Gross-SettlementExpress Transfer system is open) to the next TARGET day

● total volume of all unsecured overnight lending transactionsthat day and the weighted average lending rate for thesetransactions

● Eonia be published between 6.45 p.m. and 7.00 p.m. (CET)on the same evening the Eonia will be computed as aweighted average of all (without exceptions) overnightunsecured lending transactions in the interbank market

Euriborr

❖ Overview

❖ CSA Agreement

Euriborr

● Euriborr (Euro Interbank Offered Rate) is the rate at whicheuro interbank term deposits are offered by one prime bank toanother prime bank and is published at 11.00 a.m. CET forspot value (T+2).

● Euriborr is quoted for spot value (T+2) and on an Act/360day-count convention. It is displayed to three decimal places.

● Panel Banks contribute for one, two and three weeks and fortwelve maturities from one to twelve months Reuters shall, foreach maturity, eliminate the highest and lowest 15% of all thequotes collected. The remaining rates will be averaged androunded to three decimal places.

Multi-currency CSA

❖ Overview

❖ CSA Agreement

Euriborr

● What if the counterparty is allowed to post other currencies?

● Say $ on Euro swaps

● In order to properly value the euro swap we need to know thecost of it in dollars

● As such we need to translate the cost of Euro cash flows intodollars

● Use cross-currency basis swaps

Small CSA Test

❖ Overview

❖ CSA Agreement

Euriborr

● Lets do some qualitative tests on CSA conditions

● You have a euro swap which is massively in the money

● Currently you have a LCH type CSA with the bank

● What happens to the value of your position if we change

✦ The frequency to 1m and the index to 1M Euribor?

✦ The minimum increments from 0 to 1mln?

✦ Allow the option that either EUR or GBP cash is posted?

✦ Allow the option that subprime CLOs are posted?

Valuation Collateralised Derivatives

❖ Overview

❖ CSA Agreement

Euriborr

Value of collateral during the trade

C(t) = [V (t)− Tcpty]II(V (t)− Tc −MTc)

− [−V (t)− Tb]II(V (t)− Tb −MTb)

= (V (t)− Tc −MTc)+ +MTcII(V (t)− Tc −MTc)

− (−V (t)− Tb −MTb)+ +MTbII(−V (t)− Tb −MTb),(3)

● V (t) denotes the value of the derivative

● MTx denotes the Minimum Transfer amount for bank,counterparty

● Tx denotes the Threshold for bank, counterparty

Valuation Collateralised Derivatives

❖ Overview

❖ CSA Agreement

Euriborr

The value of a collateralised derivatives therefore depends on

● dynamics of V (t)

● MTx denotes the Minimum Transfer amount for bank,counterparty

● Tx denotes the Threshold for bank, counterparty

Funding Collateralised Derivatives

❖ Overview

❖ CSA Agreement

Euriborr

If your credit rating is low, collateral is expensive

● asset in-the-money you receive collateral

● asset out-of-the-money you pay collateral and receive OISreward

● however, it costs (much) more to fund the collateral payment!

Hence, not a symmetric trade

● volatile assets require higher credit charge

Linear Discounting

❖ Overview

❖ CSA Agreement

Euriborr

Proper discounting requires serious modelling

● time of bootstrapping and rootsolving is over

● proper valuation requires dynamics modelling and treatmentof non-linearities

❖ Overview

Clearly, this has an effect on Non-linear derivatives as wellNowadays, swaption premiums are quoted without discount factore.g.

D(t, T )= IET [(S(T )−K)+g(S(T ))] (4)

where g(S(T )) denotes the IRR-settled PV01 at maturity and S(T )the swap rate at maturity.

Discounting · 2011. 2. 12. · Derivatives valuation should not use an assumption for the...

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