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Consistent Re-Calibration of the Discrete-Time Multifactor Vasiček Model^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Discrete-Time Multifactor Vasiček Model and Hull–White Extension

#### 2.1. Setup and Notation

#### 2.2. Discrete-Time Multifactor Vasiček Model

**Assumption 1.**

**Remark.**

**Remark.**

**Theorem 2.**

#### 2.3. Hull–White Extended Discrete-Time Multifactor Vasiček Model

^{(k)}denotes that time point and corresponds to the time shift we apply to the Hull–White extension θ in Model (5). The factor process ${\mathit{X}}^{\left(k\right)}$ generates the spot rate process and the bank account process as in (1).

**Remark.**

^{(k)}in the notation is important since the conditional distribution depends explicitly on the lag $m-k$.

**Theorem 3.**

**Remark.**

#### 2.4. Calibration of the Hull–White Extended Model

**Theorem 4.**

**θ**fulfills:

## 3. Consistent Re-Calibration

#### 3.1. Consistent Re-Calibration Algorithm

#### 3.1.1. Initialization $k=0$

#### 3.1.2. Increments of the Factor Process from $k\to k+1$

#### 3.1.3. Parameter Update and Re-Calibration at $k+1$

**Remark.**

#### 3.2. Heath–Jarrow–Morton Representation

**Theorem 5.**

**Key observation.**

**Further remarks.**

- CRC of the multifactor Vasiček spot rate model can be defined directly in the HJM framework assuming a stochastic dynamics for the parameters. However, solely from the HJM representation, one cannot see that the yield curve dynamics is obtained, in our case, by combining well-understood Hull–White extended multifactor Vasiček spot rate models using the CRC algorithm of Section 3; that is, the Hull–White extended multifactor Vasiček model gives an explicit functional form to the HJM representation.
- The CRC algorithm of Section 3 does not rely directly on ${\left({\mathit{\epsilon}}^{*}\left(t\right)\right)}_{t\in \mathbb{N}}$ having independent and Gaussian components. The CRC algorithm is feasible as long as explicit formulas for ZCB prices in the Hull–White extended model are available. Therefore, one may replace the Gaussian innovations by other distributional assumptions, such as normal variance mixtures. This replacement is possible provided that conditional exponential moments can be calculated under the new innovation assumption. Under non-Gaussian innovations, it will no longer be the case that the HJM representation does not depend on the Hull–White extension ${\theta}^{\left(k\right)}\in {\mathbb{R}}^{\mathbb{N}}$.
- Interpretation of the parameter processes will be given in Section 5, below.

## 4. Real World Dynamics and Market Price of Risk

**Corollary 6.**

## 5. Choice of Parameter Process

#### 5.1. Interpretation of Parameters

#### 5.1.1. Level and Speed of Mean Reversion

#### 5.1.2. Instantaneous Variance

#### 5.2. State Space Modeling Approach

#### 5.2.1. Transition System

**Remark.**

#### 5.2.2. Measurement System

#### 5.2.3. Anchoring

#### 5.2.4. Forecasting the Measurement System

#### 5.2.5. Bayesian Inference in the Transition System

#### 5.2.6. Forecasting the Transition System

#### 5.2.7. Likelihood Function

**Θ**, given the data. As in the EM (expectation maximization) algorithm, maximization of the likelihood function is alternated with Kalman filtering until convergence of the estimated parameters ${\widehat{\mathbf{\Theta}}}^{\mathrm{MLE}}$ is achieved.

#### 5.3. Estimation Motivated by Continuous Time Modeling

#### 5.3.1. Rescaling the Time Grid

**Ψ**of the dynamics on the refined time grid with size Δ from

**μ**, γ and Γ.

**Ψ**from γ and Γ as follows.

**Ψ**, we observe that the ${\mathcal{F}}_{t-1}$-conditional mean of ${\mathcal{D}}_{t}\mathit{Z}$:

#### 5.3.2. Longitudinal Realized Covariations of Yields

- It does not depend on the unobservable factors $\mathit{X}$.
- It allows for direct cross-sectional estimation of β and Σ. That is, β and Σ can directly be estimated from market observations without knowing the market-price of risk.
- It is helpful to determine the number of factors needed to fit the model to market yield curve increments. This can be analyzed by principal component analysis.
- It can also be interpreted as a small-noise approximation for noisy measurement systems of the form (18).

#### 5.3.3. Cross-Sectional Estimation of β and Σ

#### 5.4. Inference on Market Price of Risk

**λ**and Λ of the change of measure specified in Section 4. For this purpose, we combine MLE estimation (Section 5.2) with estimation from realized covariations of yields (Section 5.3). First, we estimate β and Σ by ${\widehat{\beta}}^{\mathrm{RCov}}$ and ${\widehat{\Sigma}}^{\mathrm{RCov}}$ as in Section 5.3. Second, we estimate $\mathit{a}$, $\mathit{b}$ and α by maximizing the log-likelihood:

**λ**by:

## 6. Numerical Example for Swiss Interest Rates

#### 6.1. Description and Selection of Data

- Short times to maturity. The SAR is an ongoing volume-weighted average rate calculated by the Swiss National Bank (SNB) based on repo transactions between financial institutions. It is used for short times to maturity of at most three months. For SAR, we have the Over-Night SARONthat corresponds to a time to maturity of Δ (one business day) and the SAR Tomorrow-Next (SARTN) for time to maturity $2\Delta $ (two business days). The latter is not completely correct, because SARON is a collateral over-night rate and tomorrow-next is a call money rate for receiving money tomorrow, which has to be paid back the next business day. Moreover, we have the SAR for times to maturity of one week (SAR1W), two weeks (SAR2W), one month (SAR1M) and three months (SAR3M); see also [10].
- Short to medium times to maturity. The LIBOR reflects times to maturity, which correspond to one month (LIBOR1M), three months (LIBOR3M), six months (LIBOR6M) and 12 months (LIBOR12M) in the London interbank market.
- Medium to long times to maturity. The SWCNB is based on Swiss government bonds, and it is used for times to maturity, which correspond to two years (SWCNB2Y), three years (SWCNB3Y), four years (SWCNB4Y), five years (SWCNB5Y), seven years (SWCNB7Y), 10 years (SWCNB10Y), 20 years (SWCNB20Y) and 30 years (SWCNB30Y).

#### 6.2. Model Selection

#### 6.2.1. Discussion of Identification Assumptions

#### 6.2.2. Determination of the Number of Factors

#### 6.2.3. Determination of Vasiček Parameters

#### 6.2.4. Selection of a Model for the Vasiček Parameters

**φ**, ϕ and Φ. We rewrite the equation for the evolution of the volatility as:

**φ**, ϕ and Φ. From the regression residuals, we estimate the correlations between $\mathit{\epsilon}\left(t\right)$ and $\tilde{\mathit{\epsilon}}\left(t\right)$. Figure 16, Figure 17 and Figure 18 show the estimates of

**φ**, ϕ and Φ.

#### 6.3. Simulation and Back-Testing

#### 6.3.1. Simulation

#### 6.3.2. Back-Testing

#### 6.3.3. Regulatory Framework

## 7. Conclusions

- Flexibility and tractability. Consistent re-calibration of the multifactor Vasiček model provides a tractable extension that allows parameters to follow stochastic processes. The additional flexibility can lead to better fits of yield curve dynamics and return distributions, as we demonstrated in our numerical example. Nevertheless, the model remains tractable. In particular, yield curves can be simulated efficiently using Theorem 5 and Corollary 6. This allows one to efficiently calculate model quantities of interest in risk management, forecasting and pricing.
- Model selection. CRC models are selected from the data in accordance with the robust calibration principle of [4]. First, historical parameters, market prices of risk and Hull–White extensions are inferred using a combination of volatility estimation, MLE and calibration to the prevailing yield curve via Formulas (23–26, 10). The only choices in this inference procedure are the number of factors of the Vasiček model and the window length K. Then, as a second step, the time series of estimated historical parameters are used to select a model for the parameter evolution. This results in a complete specification of the CRC model under the real world and the pricing measure.
- Application to modeling of Swiss interest rates. We fitted a three-factor Vasiček CRC model with stochastic volatility to Swiss interest rate data. The model achieves a reasonably good fit in most time periods. The tractability of CRC allowed us to compute several model quantities by simulation. We looked at the historical performance of a representative buy and hold portfolio of Swiss bonds and concluded that a multifactor Vasiček model is unable to describe the returns of this portfolio accurately. In contrast, the CRC version of the model provides the necessary flexibility for a good fit.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CRC | Consistent re-calibration |

HJM | Heath–Jarrow–Morton |

ZCB | zero-coupon bond |

## Appendix A Proofs

**Proof of Theorem 2.**

**Proof of Theorem 3.**

**Proof of Theorem 4.**

**θ**, such that the condition is satisfied, can be calculated recursively in the following way.

- First component ${\theta}_{1}$. We have ${A}^{\left(k\right)}(k+1,k+2)=0$, $\mathit{B}(k+1,k+2)=\mathbf{1}\Delta $ and:$${A}^{\left(k\right)}(k,k+2)=-{\mathbf{1}}^{\top}\mathit{b}\Delta -{\theta}_{1}\Delta +\frac{1}{2}{\mathbf{1}}^{\top}\Sigma \mathbf{1}{\Delta}^{2},$$$${\theta}_{1}=\frac{1}{2}{\mathbf{1}}^{\top}\Sigma \mathbf{1}\Delta -{\mathbf{1}}^{\top}\mathit{b}-{A}^{\left(k\right)}(k,k+2){\Delta}^{-1}.$$$${A}^{\left(k\right)}(k,k+2)={\mathbf{1}}^{\top}\left(\right)open="("\; close=")">\mathbb{1}-{\beta}^{2}\mathit{x}\Delta -2{y}_{2}\Delta .$$$${\theta}_{1}=\frac{1}{2}{\mathbf{1}}^{\top}\Sigma \mathbf{1}\Delta -{\mathbf{1}}^{\top}\mathit{b}-{\mathbf{1}}^{\top}\left(\right)open="("\; close=")">\mathbb{1}-{\beta}^{2}\mathit{x}+2{y}_{2}.$$
- Recursion $i\to i+1$. Assume we have determined ${\theta}_{1},\dots ,{\theta}_{i}$ for $i=1,\dots ,M-2$. We want to determine ${\theta}_{i+1}$. We have ${A}^{\left(k\right)}(k+i+1,k+i+2)=0$, and iteration of the recursive formula for ${A}^{\left(k\right)}$ in Theorem 3 implies:$${A}^{\left(k\right)}(k,k+i+2)=-\sum _{s=k+1}^{k+i+1}\mathit{B}{(s,k+i+2)}^{\top}\left(\right)open="("\; close=")">\mathit{b}+\theta (s-k){\mathit{e}}_{1}$$$$\begin{array}{cc}\hfill {\theta}_{i+1}=& -\frac{1}{\Delta}{A}^{\left(k\right)}(k,k+i+2)-\frac{1}{\Delta}\sum _{s=k+1}^{k+i}\mathit{B}{(s,k+i+2)}^{\top}\left(\right)open="("\; close=")">\mathit{b}+\theta (s-k){\mathit{e}}_{1}-{\mathbf{1}}^{\top}\mathit{b}\hfill \end{array}$$$${A}^{\left(k\right)}(k,k+i+2)={\mathbf{1}}^{\top}\left(\right)open="("\; close=")">\mathbb{1}-{\beta}^{i+2}\mathit{x}\Delta -(i+2){y}_{i+2}\Delta .$$$$\begin{array}{cc}\hfill {\theta}_{i+1}& =(i+2){y}_{i+2}-{\mathbf{1}}^{\top}\left(\right)open="("\; close=")">\mathbb{1}-{\beta}^{i+2}{\left(\right)}^{\mathbb{1}}-1\hfill & \mathit{x}-\frac{1}{\Delta}\sum _{s=k+1}^{k+i}\mathit{B}{(s,k+i+2)}^{\top}\left(\right)open="("\; close=")">\mathit{b}+{\theta}_{s-k}{\mathit{e}}_{1}\end{array}$$

**θ**. Note that Equation (28) can be written as:

**Proof of Theorem 5.**

## References

- O. Vasiček. “An equilibrium characterization of the term structure.” J. Financ. Econ. 5 (1997): 177–188. [Google Scholar] [CrossRef]
- J.C. Cox, J.E. Ingersoll, and S.A. Ross. “A theory of the term structure of interest rates.” Econometrica 53 (1985): 385–407. [Google Scholar] [CrossRef]
- D. Heath, R. Jarrow, and A. Morton. “Bond pricing and the term structure of interest rates: A new methodology for contingent claim valuation.” Econometrica 60 (1992): 77–105. [Google Scholar] [CrossRef]
- P. Harms, D. Stefanovits, J. Teichmann, and M.V. Wüthrich. “Consistent re-calibration of yield curve models.” 2015. Available online: arxiv.org/abs/1502.02926 (accessed on 1 May 2016).
- Bank for International Settlements (BIS), and Basel Committee on Banking Supervision (BCBS). “Basel II: International convergence of capital measurement and capital standards. A revised framework—Comprehensive version.” 2006. Available online: http://www.bis.org/publ/bcbs128.htm (accessed on 1 May 2016).
- P.J. Brockwell, and R.A. Davis. Time Series: Theory and Methods. Berlin/Heidelberg, Germany: Springer, 1991. [Google Scholar]
- J. Hull, and A. White. “Branching out.” Risk 7 (1994): 34–37. [Google Scholar]
- L.E. Svensson. Estimating and Interpreting forward Interest Rates: Sweden 1992–1994. Cambridge, MA, USA: National Bureau of Economic Research, 1994. [Google Scholar]
- M.V. Wüthrich, and M. Merz. Financial Modeling, Actuarial Valuation and Solvency in Insurance. Berlin/Heidelberg, Germany: Springer, 2013. [Google Scholar]
- T.J. Jordan. “SARON—An innovation for the financial markets.” Launch event for Swiss Reference Rates, Zurich, 25 August 2009. Available online: https://www.snb.ch/en/mmr/speeches/id/ref_20090825_tjn_1 (accessed on 1 May 2016).
- S.L. Heston. “A closed-form solution for options with stochastic volatility with applications to bond and currency options.” Rev. Financ. Stud. 6 (1993): 327–343. [Google Scholar] [CrossRef]

**Figure 1.**Yield rates (lhs): Swiss Average Rate (SAR) and (rhs) London InterBank Offered Rate (LIBOR) from 8 December 1999, until 15 September 2014.

**Figure 2.**Yield rates: (lhs) Swiss Confederation Bond (SWCNB) and (rhs) a selection of SAR, LIBOR and Swiss Confederation Bond (SWCNB) from 8 December 1999, until 15 September 2014. Note that LIBOR looks rather differently from SAR and SWCNB after the financial crisis of 2008.

**Figure 3.**SAR realized volatility $\widehat{\mathrm{RCov}}{(t,\tau ,\tau )}^{\frac{1}{2}}$ for $\tau =1,2,5,10,21,63$, window length $K=21$ (lhs) and $K=126$ (rhs).

**Figure 4.**LIBOR realized volatility $\widehat{\mathrm{RCov}}{(t,\tau ,\tau )}^{\frac{1}{2}}$ for $\tau =21,63,126,252$, window length $K=21$ (lhs) and $K=126$ (rhs).

**Figure 5.**SWCNB realized volatility $\widehat{\mathrm{RCov}}{(t,\tau ,\tau )}^{\frac{1}{2}}$ for $\tau /252=2,3,4,5,7,10,20,30$, window length $K=21$ (lhs) and $K=126$ (rhs).

**Figure 6.**A selection of SAR, LIBOR and SWCNB realized volatility $\widehat{\mathrm{RCov}}{(t,\tau ,\tau )}^{\frac{1}{2}}$ for $\tau =$ 1, 63, 252, 504, window length $K=21$ (lhs) and $K=126$ (rhs). Note that LIBOR looks rather differently from SAR and SWCNB after the financial crisis of 2008.

**Figure 7.**SAR realized volatility $\widehat{\mathrm{RCov}}{(t,\tau ,\tau )}^{\frac{1}{2}}$ for $K=126$, $\tau =1,2,5,10,21,63$ and three observation dates compared to the realized volatility of the two- (lhs) and three-factor (rhs) Vasiček model fitted by optimization (23) for $M=6$, ${\tau}_{1}=1$, ${\tau}_{2}=2$, ${\tau}_{3}=5$, ${\tau}_{4}=10$, ${\tau}_{5}=21$, ${\tau}_{6}=63$ and ${w}_{ij}={1}_{\{i=j\}}$. The three-factor model achieves an accurate fit.

**Figure 8.**SAR realized volatility $\widehat{\mathrm{RCov}}{(t,\tau ,\tau )}^{\frac{1}{2}}$ for $K=126$, $\tau =1$ (lhs), $\tau =2$ (rhs) and all observation dates compared to the realized volatility of the two- and three-factor Vasiček models fitted by optimization (23) for $M=6$, ${\tau}_{1}=1$, ${\tau}_{2}=2$, ${\tau}_{3}=5$, ${\tau}_{4}=10$, ${\tau}_{5}=21$, ${\tau}_{6}=63$ and ${w}_{ij}={1}_{\{i=j\}}$.

**Figure 9.**SAR realized volatility $\widehat{\mathrm{RCov}}{(t,\tau ,\tau )}^{\frac{1}{2}}$ for $K=126$, $\tau =5$ (lhs), $\tau =10$ (rhs) and all observation dates compared to the realized volatility of the two- and three-factor Vasiček models fitted by optimization (23) for $M=6$, ${\tau}_{1}=1$, ${\tau}_{2}=2$, ${\tau}_{3}=5$, ${\tau}_{4}=10$, ${\tau}_{5}=21$, ${\tau}_{6}=63$ and ${w}_{ij}={1}_{\{i=j\}}$.

**Figure 10.**SAR realized volatility $\widehat{\mathrm{RCov}}{(t,\tau ,\tau )}^{\frac{1}{2}}$ for $K=126$, $\tau =21$ (lhs), $\tau =63$ (rhs) and all observation dates compared to the realized volatility of the two- and three-factor Vasiček models fitted by optimization (23) for $M=6$, ${\tau}_{1}=1$, ${\tau}_{2}=2$, ${\tau}_{3}=5$, ${\tau}_{4}=10$, ${\tau}_{5}=21$, ${\tau}_{6}=63$ and ${w}_{ij}={1}_{\{i=j\}}$.

**Figure 11.**Estimation of ${\beta}_{11}$, ${\beta}_{22}$ and ${\beta}_{33}$ (lhs) and ${({\Sigma}^{\frac{1}{2}}\Lambda )}_{11}={\beta}_{11}-{\alpha}_{11}$, ${({\Sigma}^{\frac{1}{2}}\Lambda )}_{22}={\beta}_{22}-{\alpha}_{22}$ and ${({\Sigma}^{\frac{1}{2}}\Lambda )}_{33}={\beta}_{33}-{\alpha}_{33}$ (rhs) by optimizations (23) and (24) in the three-factor model for $K=126$, $M=6$, ${\tau}_{1}=1$, ${\tau}_{2}=2$, ${\tau}_{3}=5$, ${\tau}_{4}=10$, ${\tau}_{5}=21$, ${\tau}_{6}=63$, ${w}_{ij}={1}_{\{i=j\}}$ and $S={10}^{-5}\xb7\mathbb{1}$. The values determine the speed of mean reversion of the factors. Since we are considering a daily time grid, values close to one (slow mean reversion) are reasonable. We observe that the difference in the speed of mean-reversion under the risk-neutral and real-world measures is negligible.

**Figure 12.**Estimation of ${\Sigma}_{11}$, ${\Sigma}_{22}$ and ${\Sigma}_{33}$ (lhs) and correlations ${\rho}_{21}$, ${\rho}_{31}$ and ${\rho}_{32}$ (rhs) by optimization (23) in the three-factor model for $K=126$, $M=6$, ${\tau}_{1}=1$, ${\tau}_{2}=2$, ${\tau}_{3}=5$, ${\tau}_{4}=10$, ${\tau}_{5}=21$, ${\tau}_{6}=63$ and ${w}_{ij}={1}_{\{i=j\}}$. We observe large spikes in the volatilities and strong correlations among the factors during the European sovereign debt crisis and after the SNB intervention in 2011.

**Figure 13.**Estimation of ${b}_{1}$, ${b}_{2}$ and ${b}_{3}$ (lhs) and ${({\Sigma}^{\frac{1}{2}}\mathit{\lambda})}_{1}={b}_{1}-{a}_{1}$, ${({\Sigma}^{\frac{1}{2}}\mathit{\lambda})}_{2}={b}_{2}-{a}_{2}$ and ${({\Sigma}^{\frac{1}{2}}\mathit{\lambda})}_{3}={b}_{3}-{a}_{3}$ (rhs) by optimizations (23) and (24) in the three-factor model for $K=126$, $M=6$, ${\tau}_{1}=1$, ${\tau}_{2}=2$, ${\tau}_{3}=5$, ${\tau}_{4}=10$, ${\tau}_{5}=21$, ${\tau}_{6}=63$, ${w}_{ij}={1}_{\{i=j\}}$ and $S={10}^{-5}\xb7\mathbb{1}$. The difference between $\mathit{b}$ and $\mathit{a}$ is considerable in 2000–2002 and 2006–2009.

**Figure 14.**Objective function $log{\mathcal{L}}_{t}$ (lhs) and ${({\Sigma}^{\frac{1}{2}}\mathit{\lambda})}_{1}={b}_{1}-{a}_{1}$, ${({\Sigma}^{\frac{1}{2}}\mathit{\lambda})}_{2}={b}_{2}-{a}_{2}$ and ${({\Sigma}^{\frac{1}{2}}\mathit{\lambda})}_{3}={b}_{3}-{a}_{3}$ (rhs) given by optimization (24) in the three-factor model for $K=126$, $M=6$, ${\tau}_{1}=1$, ${\tau}_{2}=2$, ${\tau}_{3}=5$, ${\tau}_{4}=10$, ${\tau}_{5}=21$, ${\tau}_{6}=63$, ${w}_{ij}={1}_{\{i=j\}}$ and $S={10}^{-5}\xb7\mathbb{1}$. We compare the value of the objective function for $(\mathit{b},\beta ,\Sigma ,\mathit{a},\alpha )=(\mathbf{0},{\beta}^{\mathrm{RCov}},{\Sigma}^{\mathrm{RCov}},\mathbf{0},{\alpha}^{\mathrm{RCov}})$ and the numerical solution of the optimization. The configuration $(\mathbf{0},{\beta}^{\mathrm{RCov}},{\Sigma}^{\mathrm{RCov}},\mathbf{0},{\alpha}^{\mathrm{RCov}})$ is almost optimal in low interest rate times.

**Figure 15.**Three-factor Hull–White extended Vasiček yield curve (lhs) and Hull–White extension θ (rhs) as of 29 September 2006. The parameters are estimated as in Figure 11, Figure 12 and Figure 13. The initial factors are obtained from the Kalman filter for the estimated parameters. The calibration of the Hull–White extension requires yields on a time to maturity grid of size Δ. These are interpolated from SAR and SWCNB using cubic splines.

**Figure 16.**Estimation of ${\phi}_{1}$, ${\phi}_{2}$ and ${\phi}_{3}$ by least square regression (two different scales). We use a time window of 252 observations for the regression.

**Figure 17.**Estimation of ${\varphi}_{11}$, ${\varphi}_{22}$ and ${\varphi}_{33}$ (lhs) and ${\Phi}_{11}$, ${\Phi}_{22}$ and ${\Phi}_{33}$ (rhs) by least square regression. We use a time window of 252 observations for the regression.

**Figure 18.**Estimation of correlations ${\tilde{\rho}}_{21}$, ${\tilde{\rho}}_{31}$ and ${\tilde{\rho}}_{32}$ (lhs) and correlations $\mathrm{Cor}\left(\right)open="["\; close="]">\mathit{\epsilon}\left(t\right),\tilde{\mathit{\epsilon}}\left(t\right)\mid \mathcal{F}(t-1)$ (rhs). We use a time window of 252 observation for the regression. The residuals ε are calculated using the parameter estimates of Figure 11, Figure 12 and Figure 13.

**Figure 19.**Logarithmic monthly returns of a buy and hold portfolio investing in equal wealth proportions in the zero-coupon bonds with times to maturity of 2, 3, 4, 5, 6 and 9 months and 1, 2, 3, 5, 7 and 10 years. For each monthly period, we calculate the logarithmic return of this portfolio assuming that at the beginning of each period, we are invested in the bonds with these times to maturity in equal proportions of wealth.

**Figure 20.**Confidence intervals computed from ${10}^{4}$ simulations of the test portfolio returns in the Vasiček model and its CRC counterpart with stochastic volatility. For each monthly period, we check if the market return lies in the confidence interval. This is more often the case for the CRC than for the standard Vasiček model. A one-sided binomial test assuming the independence of monthly periods shows that the difference is statistically significant ($p=0.0013$ for the $25\%$ and $p=$ 0.00017 for the $5\%$ quantiles). The result remains significant if every second month is discarded to account for dependencies ($p\approx 0.01$). This suggests that the CRC Vasicěk model is able to match the return distribution better than its counterpart with constant parameters.

**Table 1.**Statistics computed from simulations of the test portfolio returns for some of the monthly periods in the Vasiček model. For each monthly period, we simulate ${10}^{4}$ realizations.

**Table 2.**Statistics computed from the simulations of the test portfolio returns for some of the monthly periods in the consistent re-calibration (CRC) counterpart of the Vasiček model with stochastic volatility. For each monthly period, we simulate ${10}^{4}$ realizations.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Harms, P.; Stefanovits, D.; Teichmann, J.; Wüthrich, M.V.
Consistent Re-Calibration of the Discrete-Time Multifactor Vasiček Model. *Risks* **2016**, *4*, 18.
https://doi.org/10.3390/risks4030018

**AMA Style**

Harms P, Stefanovits D, Teichmann J, Wüthrich MV.
Consistent Re-Calibration of the Discrete-Time Multifactor Vasiček Model. *Risks*. 2016; 4(3):18.
https://doi.org/10.3390/risks4030018

**Chicago/Turabian Style**

Harms, Philipp, David Stefanovits, Josef Teichmann, and Mario V. Wüthrich.
2016. "Consistent Re-Calibration of the Discrete-Time Multifactor Vasiček Model" *Risks* 4, no. 3: 18.
https://doi.org/10.3390/risks4030018