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I have got a question regarding the Vasicek Model and the corresponding Bond Pricing Equation (BPE).

Starting with a short-rate process (under measure $P$ or real world drift $u(r,t)$) of the form:

$dr = u(r,t)dt + w(r,t)dX$ with $dX$ being a GBM

Applying the steps to derive the BPE for the Bond Price $V(r,t)$:

  1. Set up hedged portfolio: $\Pi$ $= V_1 - \Delta$$V_2$
  2. Apply no arbitrage condition: $d\Pi$ $= dV_1 - \Delta$$dV_2 = r\Pi$$dt$

Using Ito and removing the risk by defining $\Delta =$ $\frac{\frac{V_1}{\delta r}}{\frac{V_2}{\delta r}}$ we end up having to define the universal contsant $a(r,t)$ which allows us to drop the subscripts $_1$ and $_2$ making the Bond price independent of its maturity $T_1$ and $T_2$:

$a(r,t) = \lambda (r,t) w(r,t) - u(r,t)$ with $\lambda (r,t)$ being the Market price of risk. Applying all these steps and definitions we end with the parabolic partial differential heat equation for the Bond price:

$ \frac{\delta V}{\delta t}+\frac{1}{2}w^2\frac{\delta ^2V}{\delta r^2}+(u-\lambda w)\frac{\delta V}{\delta r} - rV = 0$

which displays a risk-neutral form since $\lambda $ can be seen as a Sharpe ratio defining the excess return for each unit of taken risk $w$.

Ok, this was kind of a long way to finally derive at my question :-). When using the Vasicek Model for the short-rate process defined as:

$dr = (\eta - \gamma r)dt + \beta ^\frac{1}{2}dX$

the BPE in all textbooks I have seen is given by:

$ \frac{\delta V}{\delta t}+\frac{1}{2}\beta \frac{\delta ^2V}{\delta r^2}+(\eta - \gamma r)\frac{\delta V}{\delta r} - rV = 0$

Hence we define $a = \gamma r - \eta = \lambda w - u = \lambda \beta ^\frac{1}{2} - (\eta - \gamma r)$

I struggle to understand this. Do we assume that in the Vasicek model the market price of risk $\lambda = 0$?

Thank you very much in advance for any helpful comments!

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  • $\begingroup$ Are you sure about all the interest rate dynamics stated? Particularly, beware of which probability measure you are using. Referring to quant.stackexchange.com/questions/17657/…, you should have that $u^*(r,t) = u(r,t) - \lambda w(r,t)$ with $u(r,t)$ (resp. $u^*(r,t)$) being the drift under P (resp. under Q, the risk neutral probability). $\endgroup$ – JejeBelfort May 25 '17 at 10:21
  • $\begingroup$ yes, that's correct. $\endgroup$ – friend1 May 25 '17 at 10:25
  • $\begingroup$ risk neutral drift (under measure Q): $u(r,t) - \lambda w(r,t)$; real world drift (under measure P): $u(r,t)$. The universal constant I defined is not the risk-neutral drift. It seems that Vasicek itself is a risk-neutral model. However, I miss one puzzle piece to understand this (I hope it is only one piece :-)) $\endgroup$ – friend1 May 25 '17 at 10:27

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