# 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?

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