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Short-Rate-Dependent Volatility Models

Tim Leung, Matthew Lorig

TL;DR

The paper addresses European option pricing when the instantaneous volatility is driven by the short-rate process, specifying $R_t=r(Y_t)$ and $dX_t=(r(Y_t)-\tfrac{1}{2}c^2(Y_t))dt+c(Y_t)(\rho dW_t+\bar{\rho} dB_t)$. It develops a Fourier pricing framework where the price is $V_t=\mathbb{E}_t[ e^{-\\int_t^T r(Y_s) ds}\phi(X_T)]$ and can be computed via a characteristic function $G$ under a tilted measure, reducing pricing to evaluating $G$. The authors provide two analytically tractable instances—CIR- and Jacobi-driven short-rate processes—with explicit $G$ and zero-coupon-bond factors, yielding closed-form integral expressions for European calls and computable implied-volatility surfaces. They show that short-rate–dependent volatility can generate asymmetric smiles even when forward–price co-movements vanish in certain regimes, highlighting monetary-policy transmission to equity volatility. The framework is extensible to multi-factor rate models and exotic payoffs, offering a bridge between reduced-form stochastic volatility and term-structure modelling with practical pricing and calibration applications.

Abstract

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.

Short-Rate-Dependent Volatility Models

TL;DR

The paper addresses European option pricing when the instantaneous volatility is driven by the short-rate process, specifying and . It develops a Fourier pricing framework where the price is and can be computed via a characteristic function under a tilted measure, reducing pricing to evaluating . The authors provide two analytically tractable instances—CIR- and Jacobi-driven short-rate processes—with explicit and zero-coupon-bond factors, yielding closed-form integral expressions for European calls and computable implied-volatility surfaces. They show that short-rate–dependent volatility can generate asymmetric smiles even when forward–price co-movements vanish in certain regimes, highlighting monetary-policy transmission to equity volatility. The framework is extensible to multi-factor rate models and exotic payoffs, offering a bridge between reduced-form stochastic volatility and term-structure modelling with practical pricing and calibration applications.

Abstract

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.
Paper Structure (9 sections, 2 theorems, 50 equations, 1 figure)

This paper contains 9 sections, 2 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A general class of short-rate-dependent volatility models
  3. Pricing of European-style claims
  4. Call options and Implied Volatility
  5. Example: CIR-driven short rate
  6. Example: Jacobi-driven short rate
  7. Conclusion
  8. Dynamics of $F^T$ under the $T$-forward measure in the CIR setting
  9. Wolfram Mathematica © code

Key Result

Lemma 2

Fix $\omega \in \mathds{C}$ and suppose Consider the following (complex-valued) change of measure Then the dynamics of $Y$ under $\widetilde{\mathds{P}}$ are given by where the process $\widetilde{W} = (\widetilde{W}_t)_{0 \leq t \leq T}$ is a $(\widetilde{\mathds{P}},\mathds{F})$-Brownian motion and where the function $G$ is defined by with $\widetilde{\mathds{E}}$ denoting expectation under

Figures (1)

  • Figure 1: The implied volatility $\sigma_0(T,L)$ is plotted as a function of log-moneyness $L$ for two different maturities, $T=0.25$ and $T=1.00$, and three correlation coefficients, $\rho = 0.00$ (solid), $\rho = 0.5$ (dashed) and $\rho = 1.00$ (dotted). Other parameters are fixed at the following values: $\kappa = 0.5$, $\theta = 0.05$, $\delta = 0.95 \sqrt{2 \kappa \theta}$, $\gamma = 0.2 \sqrt{\theta}$, $t=0$, $X_0 = \log 100$, and $Y_0 = \theta$.

Theorems & Definitions (8)

  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7