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The short rate under the Vasicek one factor model under the real-world measure $\mathbb{P} $ follows : $$ dr(t)=(a\theta - (a+\lambda \sigma)r(t))dt + \sigma dW(t),$$ $$ r(0)=r_0 $$

where $ \lambda $ is the market price of risk .ie. risk premium

Now, under the Vasicek model, $\lambda$ is usually chosen to be constant .ie. $ \lambda(t)=\lambda$

But what if this was not the case, and instead, the risk premium was time-dependent .ie. $ \lambda(t) $? In that case:

  1. What would be my mathematical starting point if I wanted to determine the value of $ \lambda(t) $?
  2. Would the traditional formula of $ \lambda = (\mu - r_f)/\sigma $ work in this case?

Is this even a valid point to think about? Apologies if this doesn't make sense and thank you in advance.

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    $\begingroup$ What is the purpose of specifying the dynamics under $\mathbb{P}$? Usually the Vasicek model is used for pricing and is specified directly under the risk-neutral measure $\mathbb{Q}$ by calibrating the model to fixed income instruments. Chapter 23 in Thomas Björk - Arbitrage Theory in Continuous Time (3rd edition) has a nice discussion about $\mathbb{Q}$ vs. $\mathbb{P}$ dynamics of the short rate. $\endgroup$
    – mmencke
    May 29 at 13:15
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    $\begingroup$ I understand that. It's definitely desirable to have tractable dynamics under $\mathbb{Q}$. The chapter from Bjork is great. But my question was more from a how-about standpoint .ie. if I were to just think about the evolution of the short rate in the Vasicek model, how do I factor in the risk-premium? How is that estimated? Can it even be estimated? $\endgroup$ May 29 at 13:47
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    $\begingroup$ One motivation to specify a real-world rate process is if someone wanted to project realistic future values of the short rate for pricing beyond T0. $\endgroup$ May 29 at 14:05
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    $\begingroup$ Most term structure models other than Vasicek assume a stochastic market price of risk. For the purpose of creating realistic real world yield curve dynamics, you typically estimate the parameters by fitting the model implied yield curve to data on zero coupon yields. $\endgroup$
    – fesman
    May 29 at 14:48
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I recommend two papers that should help you with this exercise.

The first is "Kalman Filtering of Generalized Vasicek Term Structure Models." This paper provides a general framework for calibrating term structure models using the Kalman filtering technique. The method is capable of disentangling the parameters under the risk neutral and physical measures, with the difference attributed to risk premium. The paper is for the generalized Vasicek models, to which the single-factor Vasicek model belongs.

When applying the methodology from the paper above, we generally find that the risk premium estimates can be very unstable and potentially nonsensical. A second paper, Term Structure Estimation with Survey Data on Interest Rate Forecasts (and a sister paper, An Arbitrage-Free Three-Factor Term Structure Model and the Recent Behavior of Long-Term Yields and Distant-Horizon Forward Rates), provides a simple extension that allows you to incorporate survey-based interest rate forecasts (i.e., observed rate expectations) into the estimation process, allowing you to pin down the parameters in the physical measure with more confidence.

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With only a short rate $r_t$ (not a tradable asset) given, we have a context where there is no risky asset, but there is at least one Brownian motion driver (incomplete market model). The only traded asset is the bank account given by:

$$ d\beta_t = r_t \beta_t dt, \; \beta_0 = 1, $$

which is a locally riskless asset. Zero-coupon bonds would be viewed as interest rate derivatives with $r$ as underlying. The existence of an equivalent martingale measure is assured by the fact that

$$ e^{-\int_0^t r_u du} \beta_t =1, $$

so any measure $Q$ equivalent to $P$ is an equivalent martingale measure in this context.

Also, the drift condition for risky assets (relationship between their $\mu$, $\sigma$ and $r$, $\lambda$) is automatically met as there is no risky asset. So, any process whose local exponential martingale is $P$-martingale can be chosen as market price of risk. See, for example, Market price of risk specifications for affine models: Theory and evidence for such specifications in affine model class (Vasicek as special case).

This suggests that one can, in practice, specify the drift of $r$ SDE (linear function of $r$ in the Vasicek case) and the market price of risk directly under a $Q$ measure (diffusion coefficient of $r$ SDE stays unchanged under Girsanov measure change). Choosing $Q$ is then equivalent to choosing the drift of $r$ SDE (via calibration to, say, traded current zero-coupon bond term structure and other interest rate derivatives, as selected $Q$ is already risk-neutral.)

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