Interest rates after the credit crunch crisis: SINGLE versus
MULTIPLE curve approach
Oleksandr Dmytriiev
Yining Geng
Cem S...
Outline
Introduction
SINGLE versus MULTIPLE curve approach
Generalization of the lattice approach for the multiple curve
f...
Brief introduction and motivation
2007 crisis is a turning point for interest rate derivative pricing;
Prior to the crisis...
Brief introduction and motivation
Why are interest rate derivatives important?
The interest rate derivatives market is the...
Money market rates before and after the
financial crisis
Tenor basis spread: longer tenors are riskier; forwards related to...
Main steps of multi-curve framework
• Build the discounting curve using a bootstrapping technique.
The typical instruments...
Outline
Introduction
SINGLE versus MULTIPLE curve approach
Generalization of the lattice approach for the multiple curve
f...
A little bit of common knowledge
Interest rate swap: contract in which two counter-parties agree to
exchange interest paym...
Coupon interest rate swap
• Notional principle: N=100 million;
• Maturity: 5 years;
• Payment frequency: both fixed and floa...
Swap valuation (General idea)
• Present value (PV) of the interest payments on the fixed legs
PVfixed = N
10
j=1
δ (Tj−1, Tj...
Swap fixed rate
11 of 34
Data
The Macro Financial Analysis Division of the Bank of England
estimates three kinds of continuously compounded yield c...
Money Market rates
13 of 34
Money Market rates
14 of 34
Result: Swap fixed rate
Fixed rate for a swap with the maturity of 5 years and both fixed and
floating rate ( LIBOR) payments...
Outline
Introduction
SINGLE versus MULTIPLE curve approach
Generalization of the lattice approach for the multiple curve
f...
More complicated case:
Option on a swap (Swaption)
• We need a model! Black-Scholes-like approach for a forward
swap rate,...
Black-Scholes-like approach
for forward swap rate
• An European swaption gives buyer the right to enter at time
TF
a = TK
...
Black-Scholes-like approach
for forward swap rate
• The payoff can be priced under the risk neutral swap measure
Qc,m whose...
Outline
Introduction
SINGLE versus MULTIPLE curve approach
Generalization of the lattice approach for the multiple curve
f...
Our approach
Step1: Separation of forward and discounting curves using
general short rate model:
The most general form of ...
Our approach
Under the framework of factor short rate models, there are a dozen of
models:
e.g. Ho-Lee Model, Hull-White M...
Our approach
Step2: Approximation of continuous time models using the
lattice approach
Clewlow and Strickland (1998) show ...
Our approach
Step 3: Generalization of lattice approach for multiple curve
framework
• Using observed market term-structur...
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, i...
LIBOR/OIS binomial trees
26 of 34
Cash flow binomial trees:
MULTIPLE curve framework
Under risk-neutral probability, interest rate can develop into one of
tw...
Cash flow binomial trees:
SINGLE v.s. MULTIPLE curve framework
28 of 34
• The estimated price of the swaption at t = 0 is the sum of all
possible, proper discounted and proper weighted future ca...
Outline
Introduction
SINGLE versus MULTIPLE curve approach
Generalization of the lattice approach for the multiple curve
f...
Conclusions
• We studied the influence of the modern, after crisis multi-curve
framework on the pricing of interest rate de...
Limitations and possible future research
• We used Historical volatility. Alternatively, we could use implied
volatility.
...
Acknowledge
To
our scientific supervisor Prof. Eulalia Nualart
&
all our professors
33 of 34
Thanks.
34 of 34
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Interest rates after the credit crunch crisis: SINGLE versus MULTIPLE curve approach (Slides)

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Barcelona GSE Master Project by Oleksandr Dmytriiev, Yining Geng, and Cem Sinan Ozturk

Master Program: Finance

About Barcelona GSE master programs: http://j.mp/MastersBarcelonaGSE

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

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 11. Swap fixed rate 11 of 34
  12. 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. 13. Money Market rates 13 of 34
  14. 14. Money Market rates 14 of 34
  15. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26. LIBOR/OIS binomial trees 26 of 34
  27. 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. 28. Cash flow binomial trees: SINGLE v.s. MULTIPLE curve framework 28 of 34
  29. 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. 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. 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. 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. 33. Acknowledge To our scientific supervisor Prof. Eulalia Nualart & all our professors 33 of 34
  34. 34. Thanks. 34 of 34

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