Consistent yield curve modelling Philipp Harms joint work with David Stefanovits, Josef Teichmann, and Mario W¨ uthrich ETH Z¨ urich, Department of Mathematics November 29, 2014 Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 1 / 19
Interest rate models Challenge: • Consistent recalibration of model parameters. Classical approach: affine factor models for the short rate • Main example: Vasiˇ cek model � � dr ( t ) = b + β r ( t ) dt + σ dW ( t ) . • More generally, affine multi-factor models, possibly with jumps. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 2 / 19
Calibration to initial yield curves Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 3 / 19
Calibration to initial yield curves Problem • Homogeneous models: No exact fit to market yield curves. • Therefore, inconsistency between model and market. Solution • Use Hull-White extensions. • Obtain reformulations as HJM models. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 4 / 19
Calibration to initial yield curves Zero − coupon yields (%) 3 Market Model ( b , β , σ ) 2.4 Model ( b , β , σ ) 1.8 Y ( 0 , τ ) 1.2 0.6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 time to maturity ( τ ) Calibration of Hull-White extensions to initial yield curves. (Vasiˇ cek model) Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 5 / 19
Hull-White extensions Time − dependent drift (%) 1 0.6 0.2 b ( τ ) − 0.2 − 0.6 Least squares b = b Exact calibration b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 time to maturity ( τ ) Constant drift b versus time-dependent drift � b ( t ). Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 6 / 19
Factor models as HJM models HJM equation • In Vasiˇ cek models with fixed parameters ( β, σ ), forward rates satisfy � ∂ � ∂τ f t + µ HJM dt + σ HJM df t = β,σ dW t , β,σ where each f t is a curve of forward rates indexed by τ . Properties • Finite-dimensional realisation of the HJM equation. • Easy to simulate. • Calibration reduces to an estimation problem because of the analytical formulas for bond prices. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 7 / 19
Time-varying parameters Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 8 / 19
Iterative recalibration of factor models Volatility parameter (%) 3 2.25 1.5 0.75 05 06 07 08 09 10 11 12 13 14 time ( t ) Time series of calibrated Vasiˇ cek volatilities σ (AAA rated Euro area government bonds) Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 9 / 19
Iterative recalibration of factor models Drift parameter 6.5 5.5 4.5 3.5 2.5 1.5 0.5 05 06 07 08 09 10 11 12 13 14 time Time series of calibrated Vasiˇ cek speeds of mean reversion − β (AAA rated Euro area government bonds) Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 10 / 19
Time-varying parameters Motivation • Iterative recalibration results in time series of model parameters. • The model should anticipate that parameters are subject to change. Problem • Introducing stochastic parameters in affine factor models destroys their good properties. Solution (Consistent Recalibration Models) • Make HJM parameters stochastic, but stick to HJM volatilities coming from affine factor models. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 11 / 19
Consistent Recalibration Models Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 12 / 19
Consistent recalibration models Definition • In consistently recalibrated Vasiˇ cek models, forward rates satisfy � ∂ � ∂τ f t + µ HJM dt + σ HJM df t = β t ,σ t dW t . β t ,σ t • Here, µ HJM β,σ , σ HJM denote the HJM drift and volatility of the Vasiˇ cek β,σ model with parameters β, σ . • β t , σ t are stochastic processes. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 13 / 19
Simulation Zero−coupon bond yields (%) 4 3 Y ( 0 , τ ) 2 1 HWE Vasicek for ( r 0 , b 0 , β 0 , σ 0 ) Market Vasicek for ( r 0 , b 0 , β 0 , σ 0 ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 time to maturity ( τ ) • Assume that ( β t , σ t ) is piecewise constant. • Fix r 0 and parameters � b 0 , β 0 , σ 0 calibrated to the market. • Simulate r 1 starting from r 0 using these parameters. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 14 / 19
Simulation Zero−coupon bond yields (%) 4 3 Y ( δ, δ + τ ) 2 1 HWE Vasicek for ( r 1 , b 0 , β 0 , σ 0 ) HWE Vasicek for ( r 1 , b 1 , β 1 , σ 1 ) HWE Vasicek for ( r 1 , b 0 , β 1 , σ 1 ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 time to maturity ( τ ) • Choose new parameters β 1 , σ 1 . • Calibrate � b 1 to the yield curve at t = 1 of the model with old parameters and repeat. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 15 / 19
A semigroup perspective The simulation algorithm as a splitting scheme • Assume that the parameter process ( β t , σ t ) is Markovian. • Then the simulation scheme with piecewise constant parameter process ( β t , σ t ) is an exponential Euler splitting schemes for the joint evolution of ( f t , β t , σ t ). Convergence of the simulation scheme • By semigroup methods, one obtains convergence to solutions of � ∂ � ∂τ f t + µ HJM dt + σ HJM df t = β t ,σ t dW t . β t ,σ t Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 16 / 19
A geometric perspective Foliations of the space of forward rate curves • Each choice of ( β, σ ) corresponds to a foliation of the space of forward rate curves. • In factor models, forward rate curves evolve on single leaves of the foliation. • In CRC models, forward rate evolutions are tangent to the foliation corresponding to ( β t , σ t ), at all t . Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 17 / 19
An empirical perspective Realised covariations • Consider the 10 × 10 matrix of realised covariations (on time-windows [ t , t + 1] of one year) between yields of maturities τ i , τ j ∈ { 1 , . . . , 10 } . • On the Euro-area government bond market, this matrix had ranks 7–10 over the years 2005–2013. • In the Vasiˇ cek and CIR model, this matrix has rank 1. • In Vasiˇ cek CRC models, where β is updated stochastically every week, this matrix has rank 7–9, in our simulations. • In the continuous-time limit of the model, the matrix has full rank. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 18 / 19
Conclusion Advantages of HJM models. . . • Exact fits to initial yield curves can be achieved. • Dynamics can be specified independently of the initial fit. • Time-dependent parameters pose no problem. • No arbitrage. . . . combined with advantages of factor models • Simulation is easy. • Analytical bond pricing formulas hold. • Calibration reduces to an estimation problem. Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19
Thank you Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19
Literature [1] Anja Richter and Josef Teichmann. “Discrete Time Term Structure Theory and Consistent Recalibration Models”. In: arXiv preprint arXiv:1409.1830 (2014). [2] Philipp D¨ orsek and Josef Teichmann. “Efficient simulation and calibration of general HJM models by splitting schemes”. In: SIAM Journal on Financial Mathematics 4.1 (2013), pp. 575–598. [3] Jan Kallsen and Paul Kr¨ uhner. “On a Heath-Jarrow-Morton approach for stock options”. In: arXiv preprint arXiv:1305.5621 (2013). Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19
Term structure of interest rates Zero−coupon bond yields (%) 6 5 4 Y ( t , t + τ ) 3 2 τ = 0.25 τ = 2 τ = 15 1 τ = 0.5 τ = 5 τ = 20 τ = 1 τ = 10 τ = 30 05 06 07 08 09 10 11 12 13 14 time ( t ) Stochastic evolution of yields with fixed maturities Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19
Term structure of interest rates Zero−coupon bond yields (%) 5 4 3 Y ( 0 , τ ) 2 2005−01−03 2009−01−02 2013−01−02 2006−01−02 2010−01−04 2014−01−02 1 2007−01−02 2011−01−03 2008−01−02 2012−01−02 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 time to maturity ( τ ) Term structure of yields on a fixed day Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19
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