Using Interest Rate Models to Shape Implied Volatility Term Structures

# Applying interest rate models for volaility rate


# Applying interest rate models for volaility rate












To what extent may the interest rate models be applied for modeling implied volatity?

The story: I was checking different stochastic option pricing models for being able to replicate implied volatility term strucure (namely its hump shape). While doing that, it came to my mind that interest rate term structure is roughthly the same thing:

Interest rate TS: $\frac{1}{h} E^P \int\limits_t^{t+h}r_t dt$

Volatility TS: $\frac{1}{h} E^Q \int_t\limits^{t+h}\sigma_t^2 dt$

Where P is a physical measure, Q - risk-neutral measure and volatility is derived from some option pricing model, for instance, Heston (under risk-neutral measure):

$dS_t = rS_tdt + \sigma_t S_t dW_t^r$

$d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \zeta \sigma_t dW_t^\sigma$

While conceptually interest rate and volatility of asset price are different things, they apper to be just the same thing in analytical sence.

The question: So the question arises: to what extent can we use interest rate models for modeling implied volatity?

Techically what I'm up to is:

We need a model for underlying asset which will perform hump shape in implied volatility term-structure?

Just take the interest rate model which allows for a hump in a yield curve, write $\sigma^2$ instead of $r$ and we're done.

Motivation: I thought that checking various option pricing models for being able to generate humps in TS was a smart research idea for my master thesis (it appeared to that there is a lack of literature on this topic). But if the results for interest rate models may be easily applied for volatility what I'm doing is futile since a bunch of literature exists on replicating all kinds of yield curves (hump, tilt etc).

## Answer by Brian B (score 2, accepted)

https://quant.stackexchange.com/a/21356

Hans Buehler investigated this in some detail, including in his doctoral thesis.

When I tried it out some years ago, back when volatility exotics were more liquid, I found the models nearly impossible to calibrate to my satisfaction, even for the SP500 complex.

I think the mathematical analogy is fair, and enjoyed Buehler's work, but in practice it won't work because the way we approach interest rate modeling tends to depend on liquidity in the basic bonds that does not appear in options and volatility derivatives.