A Shared Term-Structure Framework for Rates and Volatility

Summary

The document identifies a modeling connection between interest-rate options and volatility options: both markets have term structures. It points to a variance-curve model in which variance across maturities evolves as a martingale driven by Brownian shocks, a structure described as broadly analogous to the Heath-Jarrow-Morton framework for interest rates.

The cited approach derives conditions under which such a stochastic process can represent a variance curve and how to calculate functionals that remain consistent with it. This offers a mathematical framework for comparing rate and volatility modeling, rather than a trading strategy or pricing recipe. The evidence is a reference to the model’s equations and derivation, not empirical results. The answer also gives a practical limitation: implementing the machinery for volatility-exotic trading was found difficult, so theoretical similarity does not guarantee straightforward production use.

Key ideas

• Interest rates and implied volatility both have term structures across maturities.
• A variance curve can be modeled as a martingale driven by Brownian factors.
• The variance-curve framework has a structure analogous to HJM interest-rate modeling.
• Consistency conditions are needed for the model to represent variance and calculate related functionals.
• The framework can be difficult to implement in a volatility-exotics trading system.

Tags

# Common point between IR and Vol option pricing models?


# Common point between IR and Vol option pricing models?












What is the common point between pricing models on options on Interest Rates and options on Volatility?

## Answer by Brian B (score 0, accepted)

https://quant.stackexchange.com/a/12825

One common point is that both implied volatility and interest rates come with term structures. This is exploited by H Buehler in this paper (and a few of his others). In particular, in Equation 2.10 he defines a variance curve model $v$ as a martingale represented as

$$ dv_t(T) = \sum_{j=1}^d \beta^j_t(T) \, dW^j_t $$

which is more or less the same as the HJM interest rate model. He then goes on to derive when this can describe a variance curve and how to compute consistent functionals with it, namely that for a process $z$ one has to satisfy a variance-establishing equation like 3.1

$$ \partial_x G = \sum_{i=1}^m \mu_i \partial_{z_i} G +\frac12 \sum_{i,k=1}^m \left( \sum_{j=1}^d \sigma_i^{(j)} \sigma_k^{(j)} \right) \partial_{z_i,z_k} G $$

I really like all this machinery, though in practice I found it extremely difficult to implement for a volatility exotics trading system.

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.