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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
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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).
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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.
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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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