Consistent yield curve modelling Philipp Harms joint work with - - PowerPoint PPT Presentation

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 + βr(t)
• dt + σdW (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 (%) time to maturity (τ) Y(0, τ)

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 0.6 1.2 1.8 2.4 3

Market Model (b, β, σ) Model (b, β, σ)

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 (%) time to maturity (τ) b(τ)

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 −0.6 −0.2 0.2 0.6 1

Least squares b = b Exact calibration b

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 dft = ∂ ∂τ ft + µHJM

β,σ

• dt + σHJM

β,σ dWt,

where each ft 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 (%) time (t)

05 06 07 08 09 10 11 12 13 14 0.75 1.5 2.25 3

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 time

05 06 07 08 09 10 11 12 13 14 0.5 1.5 2.5 3.5 4.5 5.5 6.5

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 dft = ∂ ∂τ ft + µHJM

βt,σt

• dt + σHJM

βt,σt dWt.

• Here, µHJM

β,σ , σHJM β,σ

denote the HJM drift and volatility of the Vasiˇ cek model with parameters β, σ.

• βt, σt are stochastic processes.

Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 13 / 19

Simulation

Zero−coupon bond yields (%) time to maturity (τ) Y(0, τ)

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 1 2 3 4

Market Vasicek for (r0, b0, β0, σ0) HWE Vasicek for (r0, b0, β0, σ0)

• Assume that (βt, σt) is piecewise constant.
• Fix r0 and parameters

b0, β0, σ0 calibrated to the market.

• Simulate r1 starting from r0 using these parameters.

Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 14 / 19

Simulation

Zero−coupon bond yields (%) time to maturity (τ) Y(δ, δ + τ)

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 1 2 3 4

HWE Vasicek for (r1, b0, β0, σ0) HWE Vasicek for (r1, b0, β1, σ1) HWE Vasicek for (r1, b1, β1, σ1)

• Choose new parameters β1, σ1.
• Calibrate

b1 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 (βt, σt) is Markovian.
• Then the simulation scheme with piecewise constant parameter

process (βt, σt) is an exponential Euler splitting schemes for the joint evolution of (ft, βt, σt).

Convergence of the simulation scheme

• By semigroup methods, one obtains convergence to solutions of

dft = ∂ ∂τ ft + µHJM

βt,σt

• dt + σHJM

βt,σt dWt.

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 (βt, σt), 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 τi, τj ∈ {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¨

• rsek 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 (%) time (t) Y(t, t + τ)

05 06 07 08 09 10 11 12 13 14 1 2 3 4 5 6

τ = 0.25 τ = 0.5 τ = 1 τ = 2 τ = 5 τ = 10 τ = 15 τ = 20 τ = 30

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 (%) time to maturity (τ) Y(0, τ)

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 1 2 3 4 5

2005−01−03 2006−01−02 2007−01−02 2008−01−02 2009−01−02 2010−01−04 2011−01−03 2012−01−02 2013−01−02 2014−01−02

Term structure of yields on a fixed day

Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19

Short-rate along time series

Short rate (%) time (t) r(t)

05 06 07 08 09 10 11 12 13 14 1 2 3 4 5

The short rate, as obtained by approximation by 3-month yields.

Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19

Instantaneous volatility σ of the short rate

Volatility parameter (%) time ai

05 06 07 08 09 10 11 12 13 14 3.8 7.6 11.4 15.2 19

Vasicek a CIR a

The instantaneous volatility σ = √a of the short rate, obtained by path-wise estimation.

Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19

Vasiˇ cek mean reversion parameter β

Speed of mean−reversion parameter time

|βi|

05 06 07 08 09 10 11 12 13 14 1 2 3 4 5 6

Vasicek |β| CIR |β|

The mean reversion parameter β < 0, estimated from the covariation of yields.

Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19

Vasiˇ cek mean reversion parameter β

Speed of mean reversion |β| time

05 06 07 08 09 10 11 12 13 14 1 2 3 4 5 6

τ1 = τ2 = 0.5 τ1 = τ2 = 0.75 τ1 = τ2 = 1 τ1 = τ2 = 2 τ1 = τ2 = 3 τ1 = τ2 = 5 τ1 = τ2 = 10 τ1 = τ2 = 15 τ1 = τ2 = 20 τ1 = τ2 = 30

The estimated β depends on the times to maturity of the yields used in the estimation.

Philipp Harms (ETH Z¨ urich) Consistent yield curve modelling November 29, 2014 19 / 19