Interest rates after the credit crunch crisis: SINGLE versus

MULTIPLE curve approach

Oleksandr DmytriievYining Geng

Cem Sinan Ozturk

Barcelona Graduate School of Economics

July 1, 2014

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Outline

Introduction

SINGLE versus MULTIPLE curve approach

Generalization of the lattice approach for the multiple curveframework

Our approach: Black- Derman-Toy Model

Conclusions

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Brief introduction and motivation

2007 crisis is a turning point for interest rate derivative pricing;

Prior to the crisis: market interest rates were consistent and singleyield curve was used for both forwarding and discounting;

After the crisis: the inconsistencies in the market interest rates anddevelopment of the Multi-Curve Framework;

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Brief introduction and motivation

Why are interest rate derivatives important?The interest rate derivatives market is the largest derivatives marketin the world.

80% of the world’s top 500 companies (as of April 2003) usedinterest rate derivatives to control their cashflows (InternationalSwaps and Derivatives Association).

The notional amount in June 2012: US $494 trillion of OTC interestrate contracts, and US $342 trillion of OTC interest rate swaps (theSemiannual OTC derivative statistics of the Bank for InternationalSettlements).

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Money market rates before and after thefinancial crisis

Tenor basis spread: longer tenors are riskier; forwards related tolonger tenors should be priced higher;Separation of forward curve and discounting curve: a uniquediscounting curve for all tenor forward curves is used;the Overnight Indexed Swap (OIS) curve is now commonly used todiscount for collateralized derivative deals.

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Main steps of multi-curve framework

• Build the discounting curve using a bootstrapping technique.The typical instruments are OIS swaps.

• Select vanilla instruments linked to LIBOR/EURIBOR for eachtenor curve homogeneous in the underlying rate (typically with 3M,6M, 12M tenors).The typical instruments are FRA contracts, Futures, Swaps andBasis swaps.

• Build the forward curves using the selected instruments by meansof bootstrapping technique; use these forward rates to findcorresponding cashflows. Portfolio of interest rate derivatives withdifferent underlying tenors requires separate forward curves.

• A unique discounting curve for all tenor forward curves is used todiscount the cashflows and calculate their present value.

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Outline

Introduction

SINGLE versus MULTIPLE curve approach

Generalization of the lattice approach for the multiple curveframework

Our approach: Black- Derman-Toy Model

Conclusions

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A little bit of common knowledge

Interest rate swap: contract in which two counter-parties agree toexchange interest payments of differing character based on anunderlying notional principle amount that is not exchanged.

• Coupon swaps: exchange of fixed rate for floating rateinstruments in the same currency;

• Basis swaps: exchange of floating rate for floating rateinstruments in the same currency;

• Cross currency interest rate swaps: exchange of fixed rateinstruments in one currency for floating rate in another

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Coupon interest rate swap

• Notional principle: N=100 million;

• Maturity: 5 years;

• Payment frequency: both fixed and floating rate payments aremade semiannually (6M tenor);

• Coupon dates of the swap: T0 < T1 < ... < T10

T0 = 0;T1 = 0.5 years ; ...; T10 = 5 years

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Swap valuation (General idea)

• Present value (PV) of the interest payments on the fixed legs

PVfixed = N10∑

j=1

δ (Tj−1,Tj ) · K · P (0,Tj )

• PV of the interest payments on the floating legs:

PVfloating = N10∑

j=1

δ (Tj−1,Tj ) · F (Tj−1,Tj ) · P (0,Tj )

• Day-count fraction: δ(Tj−1,Tj ) = 0.5;

• Fixed and forward rates: K, F (Tj−1,Tj );

• Discount factor (price of zero coupon bonds): P(0,Tj )

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Swap fixed rate

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Data

The Macro Financial Analysis Division of the Bank of Englandestimates three kinds of continuously compounded yield curves for theUK on a daily basis:

• based on yields on UK government bonds (gilts);

• based on sterling interbank rates (LIBOR) and on yields oninstruments linked to LIBOR;

• based on sterling overnight index swap (OIS) rates, which areinstruments that settle on overnight unsecured interest rates (theSONIA rate in the UK).

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Money Market rates

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Money Market rates

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Result: Swap fixed rate

Fixed rate for a swap with the maturity of 5 years and both fixed andfloating rate ( LIBOR) payments are made semiannually (6M tenor):

• Single curve approach (LIBOR is used for both discounting andforwarding): 1.97%

• Multi-curve approach (OIS rate is used for discounting and LIBORis used for forwarding): 1.98%

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Outline

Introduction

SINGLE versus MULTIPLE curve approach

Generalization of the lattice approach for the multiple curveframework

Our approach: Black- Derman-Toy Model

Conclusions

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More complicated case:Option on a swap (Swaption)

• We need a model! Black-Scholes-like approach for a forwardswap rate, which depends on all forward and discount curves(Mercurio (2008), Bianchetti, Carlicchi (2012))

• Problems with specification and justification of SDE for theforward swap rate.

• Our approach: Separate description of forward and discountingcurves using short rate models→ approximation of continuous time models using lattice approach→ generalization of lattice approach for multiple curve framework.This approach is general and works for any interest rate derivative!

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Black-Scholes-like approachfor forward swap rate

• An European swaption gives buyer the right to enter at timeT F

a = TKa an interest rate swap (IRS) with floating payments at

time {T Fa+1, ...,T

Fn } and fixed payments at time {TK

c+1, ...,TKm }.

Note that T Fn = TK

m , fixed rate is K.

• In a multi-curve framework, the swaption payoff at the timeT F

a = TKc is

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Black-Scholes-like approachfor forward swap rate

• The payoff can be priced under the risk neutral swap measureQc,m whose associated numeraire is the annuity∑m

j=c+1 δ(TKj−1,T

Kj ) · P(t,TK

j )

• In the multi-curve approach, the forward swap rate Ka,n,c,m(T Fa )

depends on all yield curves and correspondingly has verycomplicated dynamics.If we assume that we know its volatility function and it evolvesunder Qc,m according to a driftless geometric Brownian motion:

dKa,n,c,m(t) = σa,n,c,mKa,n,c,m(t)dWt

• The price for the swaption is defined by the generalized Black-Scholes formula.

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Outline

Introduction

SINGLE versus MULTIPLE curve approach

Generalization of the lattice approach for the multiple curveframework

Our approach: Black- Derman-Toy Model

Conclusions

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Our approach

Step1: Separation of forward and discounting curves usinggeneral short rate model:The most general form of SDE for one factor short rate model is thefollowing:

df [r(t)] = {θ(t) + ρ(t)g [r(t)]}dt + σ[r(t), t]dWt ,

wheref and g are suitably chosen functions;θ is the drift of the short rate, is determined by the market;ρ is the tendency to anequilibrium short rate (mean reversal), whichcan be chosen by the user of the model or dictated bythe market;σ is the local volatility of the short rate.

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Our approach

Under the framework of factor short rate models, there are a dozen ofmodels:e.g. Ho-Lee Model, Hull-White Model, Kalotay-Williams- FabozziModel, Black-Karasinski Model, Black- Derman-Toy Model.

We use Black- Derman-Toy Model:

d ln r(t) = {θ(t) +σ′(t)

σ(t)ln r(t)}dt + σ(t)dWt .

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Our approach

Step2: Approximation of continuous time models using thelattice approachClewlow and Strickland (1998) show that for each step i in thebinomial tree, SDE can be approximated in the lattice as:

ri ,j = aieσi j√

∆t

wherej represents different possible states for every step i.ai are found from the calibration to the observed term- structure ofcorresponding market spot rates.σi is the volatility of the forward rate with tree-period tenor andexpiration at time period i. Two types of volatility: historical volatilityand implied volatility.

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Our approach

Step 3: Generalization of lattice approach for multiple curveframework

• Using observed market term-structure, calibrate the model to findthe parameters and construct the binomial/trinomial trees fordiscounting and forwarding interest rates separately

• Using separate trees, get the valuation of the interest ratederivative by calculating the present value of the cash flows.

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Example: 2-8 Swaption

• It is an option with expiration of 2 periods (1 year in our case).In the end of first 2 periods, investors have option to enter an8-period swap with semi-annual fixed and floating payments

• Floating payment are based on the prevailing LIBOR rate of theprevious months.

• The annual fixed rate is set at 2%, which is what we foundpreviously.

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LIBOR/OIS binomial trees

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Cash flow binomial trees:MULTIPLE curve framework

Under risk-neutral probability, interest rate can develop into one oftwo possible (binomial) states with equal 1/2 probability in nextperiod.

• We use LIBOR interest rate tree to compare fixed rate with thefloating rate for every period;

• We discount cash flow at each state by OIS interest rate tree;

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Cash flow binomial trees:SINGLE v.s. MULTIPLE curve framework

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• The estimated price of the swaption at t = 0 is the sum of allpossible, proper discounted and proper weighted future cash flows.

• For a notional amount of one unit, the swaption price in a multiplecurve framework is 0.0027;In the single curve framework (we used only LIBOR tree for bothdiscounting and forwarding), it is 0.0025.

• The difference can be explained as OIS rate is lower comparing toLIBOR, so it leads to a lower discounting.

• If we consider a swaption with notional amount of $100 million, weobtain $20,000 difference in price between two approaches.

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Outline

Introduction

SINGLE versus MULTIPLE curve approach

Generalization of the lattice approach for the multiple curveframework

Our approach: Black- Derman-Toy Model

Conclusions

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Conclusions

• We studied the influence of the modern, after crisis multi-curveframework on the pricing of interest rate derivatives.

• We calculated and compared the price of a interest rate swap inboth multi-curve and single curve frameworks.

• We suggested the generalization of the lattice approach with shortinterest rate models for multi- curve framework. This techniquecan be used for pricing any interest rate instruments. This is anovel result, which have not been developed in the scienticliterature.

• As an example, we showed how to use the Black-Derman-Toyinterest rate model on binomial lattice in multi-curve frameworkand calculated the price of the 2-8 period swaption in a single(LIBOR) curve and two- curve (OIS+LIBOR) frameworks.

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Limitations and possible future research

• We used Historical volatility. Alternatively, we could use impliedvolatility.

• We applied one-factor short interest rate model: Black-Derman-Toy Model. In future, we could extend our generalizedapproach to multiple-factor interest rate models or the LIBOR andSwap Market Models (LFM and LSM)

• We built the binomial tree for derivatives pricing. In future, wecould use trinomial or more sophiscated tools.

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Acknowledge

Toour scientific supervisor Prof. Eulalia Nualart&all our professors

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Thanks.

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