Choosing the Measure for Vasicek Bond Price Simulations
Summary
The document asks which probability measure to use when simulating short rates to plot the future price of a zero-coupon bond in the Vasicek model. It distinguishes the real-world process, with its objective long-run rate, from the risk-neutral process used to derive the bond pricing formula. The bond price is expressed as a deterministic factor multiplied by an exponential function of the simulated short rate, with pricing parameters specified under the risk-neutral measure.
The author leans toward simulating under the real-world measure but notes that some explanations simulate under the risk-neutral measure without discussing real-world dynamics. The document presents this as an unresolved question and gives no answer, derivation, empirical evidence, or simulation comparison. Its main value is identifying that the choice depends on what the trajectory is intended to represent: the text does not establish how to interpret either simulated path or explain how to calibrate the two parameter sets.
Key ideas
- The Vasicek setup distinguishes real-world and risk-neutral short-rate dynamics.
- The zero-coupon bond pricing expression uses risk-neutral parameters.
- The question concerns which measure describes simulated future bond price paths.
- The document raises the distinction but does not resolve it.
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# 35013
# Vasicek Model - Should I simulate short-rate under the real-world or risk-neutral measure if I am interested in simulating future bond prices
In the classic Vasicek model, the market's short rate process $(r_t)_{t \geq 0 }$ is given through the SDEs:
$$ dr_t=\alpha \left( \bar{\mu} - r_t\right) dt+\sigma d W^{\mathbb{P}}(t), $$ $$ dr_t=\alpha \left( \mu - r_t\right) dt+\sigma d W^{\mathbb{Q}}(t), $$
where $W^{\mathbb{P}}$ is a Wiener process under the objective, real-world probability measure $\mathbb{P}$, and $W^{\mathbb{Q}}$ is a Wiener process under the risk-neutral measure $\mathbb{Q}$ (measure equivalent to $\mathbb{P}$).
A standard result of this set-up is that the time $t$ price of a T-maturity zero-coupon bond is equal to:
$$P(t,T)=A(t,T)e^{-r(t)B(t,T)},$$
where $A(t,T)$ and $B(t,T)$ are deterministic functions of the risk-neutral parameters $\alpha, \mu$ and $\sigma$.
Let's say that the current time is $t=0$ and we want to plot one trajectory of $P(t,T)$, i.e. $t \mapsto P(t,T)$ for $t \in \left[0,T\right]$. Thus, we want to calculate $P(\Delta_k , T)$, where $\Delta_k =k \times T/n$, and $k=0, 1, \dots, n$.
We have that $$ P( \Delta_k , T)= A(\Delta_k ,T)e^{-r( \Delta_k )B(\Delta_k ,T)}.$$
My question is: Should we simulate $r(\Delta_k )$ under the risk-neutral or under the real-world measure. That is, should we simulate it as:
$$r( \Delta_k ) \sim r(\Delta_{k-1} )+\alpha (\bar{\mu}-r(\Delta_{k-1} ))\Delta_1 + \sigma \Delta_1 \mathcal{N}(0,1), $$ or as $$r( \Delta_k ) \sim r(\Delta_{k-1} )+\alpha (\mu-r(\Delta_{k-1} ))\Delta_1 + \sigma \Delta_1 \mathcal{N}(0,1). $$
From my understanding, we should do it under the real-world measure, but I have some resources that go on talking about the trajectory of $P(t,T)$ without even touching the real-world dynamics of the short-rate, so I don't quite understand if I am right, what are they even getting when they only use the risk-neutral measure to simulate the short-rate.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.