Short-Rate-Dependent Volatility Models

We price European options in a class of models in which the volatility of the underlying risky asset depends on the short rate of interest. Our study results in an explicit pricing formula that depends on knowledge of a characteristic function. We provide examples of models in which the characteristic function can be computed analytically and, thus, the value of European options is explicit. Numerical implementation to produce the implied volatility is also presented.

💡 Research Summary

The paper introduces a flexible class of short‑rate‑dependent volatility models in which the diffusion coefficient of a risky asset’s log‑price is a deterministic function of the short‑rate driver. The short rate Rₜ is modeled as a non‑negative function r(Yₜ) of an auxiliary diffusion Yₜ, which follows dYₜ = b(Yₜ)dt + a(Yₜ)dWₜ. The log‑price Xₜ = log Sₜ then satisfies

dXₜ =