Download - Interest rates after the credit crunch crisis: SINGLE versus MULTIPLE curve approach (Slides)
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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