Vasicek Bond Pricing and the Market Price of Short-Rate Risk
Summary
The document asks how the market price of risk enters the bond-pricing equation for a Vasicek short-rate model. It outlines a hedged-portfolio argument for deriving a bond pricing PDE, then compares a general short-rate process with the Vasicek specification, whose textbook PDE uses the physical drift parameters and diffusion variance. The central question is whether this form assumes a zero market price of risk.
The discussion highlights a distinction between physical-measure dynamics and risk-neutral pricing dynamics: a change of measure adjusts the drift, while the bond price is valued under risk-neutral dynamics. However, the text is a question rather than a full explanation or derivation, and it does not establish that the market price of risk must be zero. The relationship between the Vasicek parameters and the risk premium depends on which measure defines the model parameters; the notation in the question also appears to mix variance and volatility conventions.
Key ideas
- A hedged-portfolio argument leads to a partial differential equation for bond prices.
- Risk-neutral pricing changes the short-rate drift through the market price of risk.
- The Vasicek pricing equation is presented with mean-reverting drift and constant diffusion variance.
- The document asks whether the stated form implies zero market price of risk but does not answer it.
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Full text
# Basic Interest Rate Modelling Ques
# Basic Interest Rate Modelling Ques
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 $V(r,t)$:
- Set up hedged portfolio: $\Pi$ $= V_1 - \Delta$$V_2$
- Apply no arbitrage condition: $d\Pi$ $= dV_1 - \Delta$$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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.