Modeling Time-Varying Drift and Volatility Relationships
Summary
This discussion asks how to model an asset whose drift and volatility vary over time and appear to move together. It distinguishes a deterministic relationship between time-varying drift and volatility from stochastic dependence: if both are deterministic functions of time, their shapes can be specified as term structures without requiring a special model that links them. A proposed linear relation is one possible parameterization, but the answers do not develop or validate it.
For volatility that varies with the asset price as well as time, the response points to local volatility models, which use a volatility surface to match vanilla option markets and may be extended with jumps. Another response suggests stochastic volatility as a broader framework when the quantities are random, while noting that the specific drift-volatility linkage may require a custom extension. The discussion is conceptual and provides no estimation procedure or empirical evidence that one approach is best; the appropriate setup depends on whether the observed relationship is deterministic or stochastic.
Key ideas
- Deterministic time variation in drift and volatility can be represented with separate term structures.
- A relationship between deterministic functions does not by itself require a specialized stochastic model.
- Local volatility models allow volatility to depend on the asset price and time.
- Stochastic volatility is a broader framework when volatility itself is random, but the proposed drift linkage may need custom modeling.
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Full text
# how to extend lognormal model so that $\sigma$ is correlated to $\mu$?
# 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?
## Answer by Brian B (score 3, accepted)
https://quant.stackexchange.com/a/8805
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).
## Answer by athos (score 0)
https://quant.stackexchange.com/a/10948
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.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.