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- The Ornstein-Uhlenbeck Process
The Ornstein-Uhlenbeck Process
Sometimes, an Ornstein-Uhlenbeck process is used to make volatility stochastic.
The process is: 
where 
is a mean reversion rate, 
is a mean reversion level and v is a vol-of-vol parameter.
It is a normal process (as opposed to lognormal since there is no xt multiplying the dWt), and therefore xt can be negative.
For this reason, when using the process to model a stochastic volatility, we do not use xt directly, but instead some positive function of it.
Typically, this function will be an exponential so that we set 
and the SDEs for the model become:
This is called the exponential Ornstein-Uhlenbeck model.
The reason the Ornstein-Uhlenbeck process is interesting for us is that it has a simple solution. This is not a solution for the full model in the 3 equations above, but just for the xt that drives the spot volatility. Nevertheless, it will give us some useful understanding of the interplay between mean reversion and vol-of-vol.
To solve for xt, we introduce a new variable ut with: 
Applying Ito's lemma, we obtain:
which is a simple normal model with drift.
We can integrate to obtain a solution:
So that for xT we have:
This solution confirms our intuition about mean reversion.
In the limit of large time T we obtain: 
so that xT is normally distributed about the mean reversion level, even though it started at x0 at time t = 0, which might be far from .
What is particularly interesting is that the variance becomes 
at large times.
In contrast to normal or lognormal models without mean reversion, the variance of instantaneous spot volatilty does not increase with time. The spot volatility remains under control over long time intervals when there is mean reversion. When there is no mean reversion, the variance grows with time so that the spot volatility is likely to become very large or very small.
If we think of the IV smile at a particular expiry date, we saw that it is the variance of the total realised volatility up to this date that gives the smile its convexity. So it is the variance of the quantity 
that controls the convexity of the IV smile at expiry T.
Adjusting the parameters in our stochastic volatility model gives us control over the behaviour of the instantaneous volatility 
, and therefore over this total variance. Then we can play off mean reversion versus vol-of-vol in order to achieve the required convexity of smile.
When the time scale is not too large, the factor 
appearing in the variance is important. As it depends on time T, it allows us to adjust the variance at two times T1 et T2, by adjusting the parameter mean reversion rate rate and vol-of-vol. In this way, we can use the mean reversion to control the convexity of smile at two expiries without needing to introduce time-dependent parameters into the model.