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- 1. Interest rates after the credit crunch crisis: SINGLE versus MULTIPLE curve approach Oleksandr Dmytriiev Yining Geng Cem Sinan Ozturk Barcelona Graduate School of Economics July 1, 2014 1 of 34
- 2. Outline Introduction SINGLE versus MULTIPLE curve approach Generalization of the lattice approach for the multiple curve framework Our approach: Black- Derman-Toy Model Conclusions 2 of 34
- 3. 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 single yield curve was used for both forwarding and discounting; After the crisis: the inconsistencies in the market interest rates and development of the Multi-Curve Framework; 3 of 34
- 4. Brief introduction and motivation Why are interest rate derivatives important? The interest rate derivatives market is the largest derivatives market in the world. 80% of the world’s top 500 companies (as of April 2003) used interest rate derivatives to control their cashﬂows (International Swaps and Derivatives Association). The notional amount in June 2012: US $494 trillion of OTC interest rate contracts, and US $342 trillion of OTC interest rate swaps (the Semiannual OTC derivative statistics of the Bank for International Settlements). 4 of 34
- 5. Money market rates before and after the ﬁnancial crisis Tenor basis spread: longer tenors are riskier; forwards related to longer tenors should be priced higher; Separation of forward curve and discounting curve: a unique discounting curve for all tenor forward curves is used; the Overnight Indexed Swap (OIS) curve is now commonly used to discount for collateralized derivative deals. 5 of 34
- 6. 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 each tenor curve homogeneous in the underlying rate (typically with 3M, 6M, 12M tenors). The typical instruments are FRA contracts, Futures, Swaps and Basis swaps. • Build the forward curves using the selected instruments by means of bootstrapping technique; use these forward rates to ﬁnd corresponding cashﬂows. Portfolio of interest rate derivatives with diﬀerent underlying tenors requires separate forward curves. • A unique discounting curve for all tenor forward curves is used to discount the cashﬂows and calculate their present value. 6 of 34
- 7. Outline Introduction SINGLE versus MULTIPLE curve approach Generalization of the lattice approach for the multiple curve framework Our approach: Black- Derman-Toy Model Conclusions 7 of 34
- 8. A little bit of common knowledge Interest rate swap: contract in which two counter-parties agree to exchange interest payments of diﬀering character based on an underlying notional principle amount that is not exchanged. • Coupon swaps: exchange of ﬁxed rate for ﬂoating rate instruments in the same currency; • Basis swaps: exchange of ﬂoating rate for ﬂoating rate instruments in the same currency; • Cross currency interest rate swaps: exchange of ﬁxed rate instruments in one currency for ﬂoating rate in another 8 of 34
- 9. Coupon interest rate swap • Notional principle: N=100 million; • Maturity: 5 years; • Payment frequency: both ﬁxed and ﬂoating rate payments are made semiannually (6M tenor); • Coupon dates of the swap: T0 < T1 < ... < T10 T0 = 0; T1 = 0.5 years ; ...; T10 = 5 years 9 of 34
- 10. Swap valuation (General idea) • Present value (PV) of the interest payments on the ﬁxed legs PVﬁxed = N 10 j=1 δ (Tj−1, Tj ) · K · P (0, Tj ) • PV of the interest payments on the ﬂoating legs: PVﬂoating = N 10 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 ) 10 of 34
- 11. Swap ﬁxed rate 11 of 34
- 12. Data The Macro Financial Analysis Division of the Bank of England estimates three kinds of continuously compounded yield curves for the UK on a daily basis: • based on yields on UK government bonds (gilts); • based on sterling interbank rates (LIBOR) and on yields on instruments linked to LIBOR; • based on sterling overnight index swap (OIS) rates, which are instruments that settle on overnight unsecured interest rates (the SONIA rate in the UK). 12 of 34
- 13. Money Market rates 13 of 34
- 14. Money Market rates 14 of 34
- 15. Result: Swap ﬁxed rate Fixed rate for a swap with the maturity of 5 years and both ﬁxed and ﬂoating rate ( LIBOR) payments are made semiannually (6M tenor): • Single curve approach (LIBOR is used for both discounting and forwarding): 1.97% • Multi-curve approach (OIS rate is used for discounting and LIBOR is used for forwarding): 1.98% 15 of 34
- 16. Outline Introduction SINGLE versus MULTIPLE curve approach Generalization of the lattice approach for the multiple curve framework Our approach: Black- Derman-Toy Model Conclusions 16 of 34
- 17. More complicated case: Option on a swap (Swaption) • We need a model! Black-Scholes-like approach for a forward swap rate, which depends on all forward and discount curves (Mercurio (2008), Bianchetti, Carlicchi (2012)) • Problems with speciﬁcation and justiﬁcation of SDE for the forward swap rate. • Our approach: Separate description of forward and discounting curves 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! 17 of 34
- 18. Black-Scholes-like approach for forward swap rate • An European swaption gives buyer the right to enter at time TF a = TK a an interest rate swap (IRS) with ﬂoating payments at time {TF a+1, ..., TF n } and ﬁxed payments at time {TK c+1, ..., TK m }. Note that TF n = TK m , ﬁxed rate is K. • In a multi-curve framework, the swaption payoﬀ at the time TF a = TK c is 18 of 34
- 19. Black-Scholes-like approach for forward swap rate • The payoﬀ can be priced under the risk neutral swap measure Qc,m whose associated numeraire is the annuity m j=c+1 δ(TK j−1, TK j ) · P(t, TK j ) • In the multi-curve approach, the forward swap rate Ka,n,c,m(TF a ) depends on all yield curves and correspondingly has very complicated dynamics. If we assume that we know its volatility function and it evolves under 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. 19 of 34
- 20. Outline Introduction SINGLE versus MULTIPLE curve approach Generalization of the lattice approach for the multiple curve framework Our approach: Black- Derman-Toy Model Conclusions 20 of 34
- 21. Our approach Step1: Separation of forward and discounting curves using general short rate model: The most general form of SDE for one factor short rate model is the following: df [r(t)] = {θ(t) + ρ(t)g[r(t)]}dt + σ[r(t), t]dWt, where f 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), which can be chosen by the user of the model or dictated bythe market; σ is the local volatility of the short rate. 21 of 34
- 22. Our approach Under the framework of factor short rate models, there are a dozen of models: e.g. Ho-Lee Model, Hull-White Model, Kalotay-Williams- Fabozzi Model, 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. 22 of 34
- 23. Our approach Step2: Approximation of continuous time models using the lattice approach Clewlow and Strickland (1998) show that for each step i in the binomial tree, SDE can be approximated in the lattice as: ri,j = ai eσi j √ ∆t where j represents diﬀerent possible states for every step i. ai are found from the calibration to the observed term- structure of corresponding market spot rates. σi is the volatility of the forward rate with tree-period tenor and expiration at time period i. Two types of volatility: historical volatility and implied volatility. 23 of 34
- 24. Our approach Step 3: Generalization of lattice approach for multiple curve framework • Using observed market term-structure, calibrate the model to ﬁnd the parameters and construct the binomial/trinomial trees for discounting and forwarding interest rates separately • Using separate trees, get the valuation of the interest rate derivative by calculating the present value of the cash ﬂows. 24 of 34
- 25. 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 an 8-period swap with semi-annual ﬁxed and ﬂoating payments • Floating payment are based on the prevailing LIBOR rate of the previous months. • The annual ﬁxed rate is set at 2%, which is what we found previously. 25 of 34
- 26. LIBOR/OIS binomial trees 26 of 34
- 27. Cash ﬂow binomial trees: MULTIPLE curve framework Under risk-neutral probability, interest rate can develop into one of two possible (binomial) states with equal 1/2 probability in next period. • We use LIBOR interest rate tree to compare ﬁxed rate with the ﬂoating rate for every period; • We discount cash ﬂow at each state by OIS interest rate tree; 27 of 34
- 28. Cash ﬂow binomial trees: SINGLE v.s. MULTIPLE curve framework 28 of 34
- 29. • The estimated price of the swaption at t = 0 is the sum of all possible, proper discounted and proper weighted future cash ﬂows. • For a notional amount of one unit, the swaption price in a multiple curve framework is 0.0027; In the single curve framework (we used only LIBOR tree for both discounting and forwarding), it is 0.0025. • The diﬀerence can be explained as OIS rate is lower comparing to LIBOR, so it leads to a lower discounting. • If we consider a swaption with notional amount of $100 million, we obtain $20,000 diﬀerence in price between two approaches. 29 of 34
- 30. Outline Introduction SINGLE versus MULTIPLE curve approach Generalization of the lattice approach for the multiple curve framework Our approach: Black- Derman-Toy Model Conclusions 30 of 34
- 31. Conclusions • We studied the inﬂuence of the modern, after crisis multi-curve framework on the pricing of interest rate derivatives. • We calculated and compared the price of a interest rate swap in both multi-curve and single curve frameworks. • We suggested the generalization of the lattice approach with short interest rate models for multi- curve framework. This technique can be used for pricing any interest rate instruments. This is a novel result, which have not been developed in the scientic literature. • As an example, we showed how to use the Black-Derman-Toy interest rate model on binomial lattice in multi-curve framework and calculated the price of the 2-8 period swaption in a single (LIBOR) curve and two- curve (OIS+LIBOR) frameworks. 31 of 34
- 32. Limitations and possible future research • We used Historical volatility. Alternatively, we could use implied volatility. • We applied one-factor short interest rate model: Black- Derman-Toy Model. In future, we could extend our generalized approach to multiple-factor interest rate models or the LIBOR and Swap Market Models (LFM and LSM) • We built the binomial tree for derivatives pricing. In future, we could use trinomial or more sophiscated tools. 32 of 34
- 33. Acknowledge To our scientiﬁc supervisor Prof. Eulalia Nualart & all our professors 33 of 34
- 34. Thanks. 34 of 34

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