Why Vasicek Simulation Tracks Short Rates and Integrated Discount Factors
Summary
The note distinguishes a zero-coupon bond price from the pathwise discount factor used to discount cash flows. In the Vasicek model, the bond price has an analytic expression conditional on the current short rate. The discount factor along a simulated path instead depends on the accumulated short rate between two dates, represented by the integral of the rate over time.
A numerical path can approximate this integral with a Riemann sum, but discretization can introduce noise. The note explains that the short rate and its time integral are jointly Gaussian in this model, so they can be simulated jointly and precisely. This supports more accurate pathwise discounting in Monte Carlo calculations. The discussion is a conceptual answer rather than a worked simulation or empirical comparison, and its joint Gaussian result is specific to the model assumptions being discussed.
Key ideas
- A bond price conditional on the current short rate is distinct from a realized pathwise discount factor.
- The pathwise discount factor depends on the integral of short rates over the cash flow period.
- Approximating that integral from discrete rate observations can introduce discretization error.
- In the Vasicek model, the short rate and its integrated value have a joint Gaussian distribution that can be simulated directly.
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# Vasicek model: joint simulation with discount factor
# Vasicek model: joint simulation with discount factor
In Vasicek model, we have the following relation to get Discount factors given the value of short rate: $$P(t\,,T)={{e}^{A(t,T)\,-\,B(t,T){{r}_{t}}\,}}$$
So, Discount factors are known as soon as we know the short rate. But then in some references like Glasserman (pg. 115) there is a whole subsection on "Joint Simulation [of short rate] with the Discount Factor" where he talks about simulating the pair $$({r}_{t},\int_{0}^t{r(u)}du)$$.
Piterbarg's book has something similar too. So my question is - why do we need to simulate Discount factor if we have an exact analytical result.
## Answer by Aguelmame (score 4, accepted)
https://quant.stackexchange.com/a/48681
Although it's been a long time this question has been asked, I'd like to propose an answer in case someone was looking for the same thing.
First, I think there's a confusion between $P(t,T)$ and $DF(t,T)$. The former is the $t-$price of a contract paying $1$ unit of currency at date $T$ while the later is the (stochastic) discount factor at $t$ for flows occuring at $T$. The two are linked through the relationship $$ P(t,T)=\mathbb{E}^Q[DF(t,T) | \mathcal{F}_t]$$
If $r_t$ is the instantaneous short rate, then $DF(t,T)$ is given by $$ DF(t,T)=e^{-\int_t^T r_s ds}$$ and is a random variable.
Now, the argument of Glasserman is about computing $\int_t^T r_s ds$. In theory, since one has $r_t$ up to maturity on a given path, this is just a matter of doing a Riemann sum. However, this may be very "noisy" because of discretization errors. It turns ou, as AXH mentionned, that $(r_t, \int_t^T r_s ds)$ are jointly gaussian and can be simulated precisely.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.