Consistent Re-Calibration in Yield CurveModeling: an Example

Mario V. Wuthrich

RiskLab, ETH Zurich

joint work with

Philipp Harms, David Stefanovits, Josef Teichmann

January 6-8, 2016International Conference of the Thailand Econometric Society

Chiang Mai, Thailand

Aim of this presentation

Calibrate term structure models to continuously changing

market conditions in a consistent way (free of arbitrage).

Choose maturity date m > k + 1.

Y (k,m; Θ(k)) = yield rate at time k under market conditions Θ(k),

Y (k + 1,m; Θ(k+1)) = yield rate at time k + 1 under market conditions Θ(k+1).

� Naıve re-calibration from Θ(k) to Θ(k+1) induces arbitrage in the model.

� The theory of consistent re-calibration (CRC) gives a natural split of the drift

term into (a) no-arbitrage part and (b) market-price of risk part.

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Hull-White extended Vasicek model

� Harms et al. (2015): general theory for affine term structure models.

� Here: Hull-White extended discrete time Vasicek model (as simplest example).

� This simple example is sufficient to explain the basic idea behind CRC.

� As a result we obtain a Heath-Jarrow-Morton (HJM) representation.

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Discrete time Vasicek model

� Spot rate process (rt)t∈N0 in the discrete time Vasicek model is

rt = b+ βrt−1 + σε∗t ,

for parameters b, β, σ ∈ R, and ε∗t i.i.d. standard Gaussian under EMM Q.

� For given parameter Θ = (b, β, σ) we have affine term structure

Y (k,m; Θ) = − 1

m− k

[A(k,m; Θ)− rk B(k,m; Θ)

].

� This model is calibrated at time point k which provides parameter

Θ(k) = (b(k), β(k), σ(k)).

� Often, market yield ymkt(k,m) differs from model yield Y (k,m; Θ(k)).

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Hull-White extended Vasicek model

� Spot rate process (rt)t∈N0 in the Hull-White (HW) extended Vasicek model is

rt = bt + βrt−1 + σε∗t ,

for parameters bt, β, σ ∈ R, and ε∗t i.i.d. standard Gaussian under EMM Q.

� b = (bt)t∈N is called HW extension.

� Calibrate HW extension at time k such that for Θ(k) = (b(k), β(k), σ(k))

Y (k,m; Θ(k)) = ymkt(k,m).

Theorem. There exists a matrix C(β) and a vector z(β, σ, ymkt(k,m)) such that

b(k) = C−1(β) z(β, σ, ymkt(k,m)

).

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Consistent re-calibration CRC algorithm

(a) Initialization k = 0. Initialize β(0) and σ(0) and choose HW extension

b(0) = C−1(β(0)) z(β(0), σ(0), ymkt(0,m)

).

(b) Spot rate dynamics k → k + 1. For given Θ(k) = (b(k), β(k), σ(k))

rk+1 = b(k)k+1 + β(k)rk + σ(k)ε∗k+1.

(c) Re-calibrate at time k + 1. Calibrate β(k+1) and σ(k+1) to actual marketconditions at time k + 1 and choose

b(k+1) = C−1(β(k+1)) z(β(k+1), σ(k+1), Y (k + 1,m; Θ(k))

).

(d) Iterate (a)-(c).

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Market-price of risk and real-world measure

� Derivations above under pricing measure Q (EMM).

� Observations under real-world measure P (historical measure).

� Introduce the following change of measure for final time horizon T

dPdQ

= exp

{−1

2

T∑s=1

(λ0s−1 + λ1s−1rs−1

)2+

T∑s=1

(λ0s−1 + λ1s−1rs−1

)ε∗s

},

for market price of risk process λt = (λ0t , λ1t ), t ∈ N0.

� This model has a Heath-Jarrow-Morton (HJM) representation under Q and P.

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Calibration to real-world observations

� Real-world dynamic parameters

a(k)k+1 = b

(k)k+1 − σ

(k)λ0k, α(k) = β(k) − σ(k)λ1k and σ(k),

are calibrated from real-world short rate observations, e.g., with MLE.

� Mean reversion rate β(k) is calibrated from historical realized co-variations.

� HW extension b(k) is obtained as above from the CRC algorithm (no arbitrage).

� Estimate of market-price of risk λk is naturally obtained.

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Swiss currency CHF example

8

Real-world parameter calibration α(k) and a(k)k+1

MLE of α(k) and a(k)k+1 for different rolling window lengths (which provides different

viscosity to the parameter processes).

9

Volatility calibration σ(k)

(lhs) MLE and (rhs) realized volatility calibration of σ(k).

10

Mean reversion rate β(k)

One-factor Vasicek model cannot cope with the situation after 2008.

11

Market-price of risk (1/2)

(lhs) resulting λ1k = (β(k) − α(k))/σ(k);

(rhs) estimates b(k)k+1 and a

(k)k+1.

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Market-price of risk (2/2)

(lhs) resulting λ0k = (b(k)k+1 − a

(k)k+1)/σ

(k);

(rhs) resulting drift from market-price of risk λk(rk) = λ0k + λ1krk.

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Conclusions

� Model parameters Θ(k) are considered to be (stochastic) processes.

� Hull-White extension is used to make model re-calibration free of arbitrage.

� We obtain a natural split of the no-arbitrage drift.

� CRC provides a functional form to the Heath-Jarrow-Morton framework.

� Can be generalized to various affine term structure models, see Harms et al. (2015).

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References

[1] Harms, P., Stefanovits, D., Teichmann, J., Wuthrich, M.V. (2015). Consistentrecalibration of yield curve models. arXiv:1502.02926.

[2] Harms, P., Stefanovits, D., Teichmann, J., Wuthrich, M.V. (2015). Consistentre-calibration of the discrete time multifactor Vasicek model. Working paper.

[3] Wuthrich, M.V. (2015). Consistent re-calibration in yield curve modeling: anexample. To appear in conference volume.

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