# how to extend lognormal model so that $\sigma$ is correlated to $\mu$?

Consider a log-normal model, $dx / x = \mu dt + \sigma dW$, where $W(t)$ is a Wiener process.

Let's say $\mu$ and $\sigma$ change with time, slowly, so we note them by $\mu(t)$ and $\sigma(t)$.

Consider $dx / x$, where the drift rate is $\mu$, and the volatility is $\sigma \sqrt{dt}$. Here, $\mu(t)$ and $\sigma(t)$ is not correlated.

Now, if in some cases the data shows a strong correlation, such as when $\mu(t)$ goes up, $\sigma(t)$ would also go up -- the 2 are almost in a linear relationship, something like $\sigma(t) = \sigma_0 + k \mu(t)$ -- how could I set a model for that?

Of course, I could just put it as $$dx/x = \mu(t) dt + (\sigma_0 + k \mu(t)) dW$$

But I wonder, is there some already established model/methods for such situation? for example for general stochastic model there are HJM, for mean-reverse there are Hull-White, for stock price there is the log-normal.

Is there some model already researched or even better, used in industry, that extend lognormal model $dx / x = \mu dt + \sigma dW$, so that $\mu(t)$ and $\sigma(t)$ would be correlated?

• What do you mean saying mu(t) is correlated to sigma(t)? They are determenistic functions, there can only be case when sigma = f(mu(t)), something you have already mentioned. – Rustam Aug 16 '13 at 9:25
• sigma = f(mu(t)), is what i observed in saving account, i'm wondering if there are also such case in, say, stock, interest rate, and if so, are there already some established models? – athos Aug 16 '13 at 13:48
• @athos: I guess what you will end up with in the end will something like the "market price of risk" (the linear relationship you mentioned) and the notion of a risk-neutral measure. Please do not mix up correlation and simply a functional relationship. The correlation of deterministic quantities is always $0$. If you really want to pour time into this: This seems like a dead end to me. – vanguard2k Apr 15 '14 at 8:48

As @Rustam notes, "correlation" of deterministic functions in the sense you describe is a special case of allowing $\mu$ and $\sigma$ to have a term structure of arbitrary shape. Since the latter is easy to treat, no one bothers with restricted forms of it.
Now, there quite a few people who deal with models that let $\sigma$ change with $S$. I am thinking in particular of local volatility models, which have an explicit surface $\sigma(S,t)$ to match vanilla option markets. These are used on exotics desks (and the models sometimes have jump terms also).
to answer my own question, there's no popular model for the question, that $dS/S=\mu(t)dt+\sigma(t)dW$, and $\sigma(t)$ is correlated with $\mu(t)$. the general framework should be stochastic volatility model, but need do the extension on my own.