I consider the Vasicek model under the risk-neutral measure $\mathbb{Q}$: $$ dr_t=\kappa(\theta−r_t) dt+\sigma dW^{\mathbb{Q}}_t.$$ I have already determined $$\mathbb{E}^{\mathbb{Q}}\left[e^{−\int\limits_0^T r_u \textrm{d}u}\right] = \exp\left(-\left(r_0 \frac{1-e^{-\kappa T}}{\kappa} + \theta \left(T - \frac{1- e^{-\kappa T}}{\kappa}\right)\right)+\frac{1}{2} \left(\frac{\sigma^2}{2 \kappa^3} ( 2 \kappa T - 3 + 4 e^{-\kappa T}-e^{-2\kappa T}\right)\right)$$, but now I have to calculate the same expression under the real-world measure $\mathbb{P}$, i.e. $\mathbb{E}^{\mathbb{P}}\left[e^{−\int\limits_0^T r_u \textrm{d}u}\right]$. How can I do this? Do I have to apply Girsanov's Theorem?


1 Answer 1


Vasnicek by itself does not specify what form the change of measure should be and how you should parameterise the market price of risk.

A very natural parameterisation is affine in the factor, i.e., $$dW^* = dW + (\lambda_0+\lambda_1 r) dt$$ where $W$ is the Wiener process under $\mathbb{Q}$ and $W^*$ for $\mathbb{P}$.

Effectively, under $\mathbb{P}$ you will have a different set of parameters $\kappa, \theta$, but the process will still be an OU process.

There is nothing in Vasicek which requires an affine change of measure. If you want to know more about how to use models in both the $\mathbb{P}$ and $\mathbb{Q}$ measures simultaneously, a decent starting place is Singleton-Dai (I can't find the original citation, but this one works as an overview):

  • Dai, Qiang, and Kenneth Singleton. "Term structure dynamics in theory and reality." The Review of financial studies 16.3 (2003): 631-678.


  • Ang, Andrew, and Monika Piazzesi. "A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables." Journal of Monetary economics 50.4 (2003): 745-787

In each paper, they have affine models in the physical measure and affine in the risk-neutral. Affine in the physical measure is basically just a vector auto-regression (VAR) (they do not consider stochastic vol, CIR type models). Due to the pricing formulas in the risk-neutral measure, the VAR has nonlinear constraints on its parameters, unfortunately making the solution of fitting to timeseries of bond data challenging.

The industry of macro-affine models lead to hundreds of papers in this area, which were ultimately not terribly successful, as the paper by Gregory Duffee, in my opinion, put an end to the endeavour:

  • Duffee, Gregory R. Forecasting with the term structure: The role of no-arbitrage restrictions. No. 576. Working papers//the Johns Hopkins University, Department of Economics, 2011.

Basically, Duffee shows that, for yield curve forecasting, constraints on a VAR are good, because it reduces the overall number of parameters (this is a classic case of bias-variance tradeoff), and increases forecast accuracy. But he also shows that no-arbitrage constraints are not that helpful. You might as well use another easier-to-use constraint (like a PCA or other dimension-reduction, or some regularization method).

Irrespective, it helps to start with Dai-Singleton and with Ang-Piazzessi, and look at some of the work of Rudebusch-Wu and others before realising th limits of trying to force a no-arbitrage model into the realm of actually forecasting yield curve dynamics.

And, while people still publish papers in this area, there are many who instead look at Nelson-Siegel or Nelson-Siegel-Svensson and use it, instead for forecasting. (e.g., Diebold and his coauthors) and there is some work on extending the Factor-Augmented VAR (FAVAR) models that Bernanke et al liked so much, instead using NS/NSS with macroeconomic factors.

  • $\begingroup$ Thanks for your detailed answer! For my thesis I have to implement the two expressions in R. Therefore, I have numbers for $\kappa, \theta, r_0$ and $\sigma$, so under the risk-neutral measure the implementation works. But how do I have to choose $\lambda_0$ and $\lambda_1$? $\endgroup$
    – Stephanie
    Commented Aug 21, 2018 at 23:07
  • $\begingroup$ @Stephanie please look at Ang and Piazzessi. Much better explanation than anything I could write. They get formulas for bond prices in terms of the parameters much as you do but then fit them using panel data as a VAR. In their case there are hopelessly large numbers of parameters so it doesn’t work well. In your case far fewer so more tractable. (They have some 300+). I think Dai and Singleton also complicated. Rudebusch-Wu are much more judicious. They all use the same method (which you will use too!). Good luck! $\endgroup$
    – NBF
    Commented Aug 29, 2018 at 6:53

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