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A central limit theorem and its application to the limiting distribution of volatility target index

Xuan Liu, Michel Gauthier

TL;DR

The paper addresses the limiting distribution of volatility target indices as the discretisation time step vanishes. It develops a LLN and a CLT for a dependent sequence to characterize the exact limiting diffusion, with drift and diffusion coefficients scaled by multipliers $U(\lambda)$ and $V(\lambda)$. The results yield an explicit limiting Itô diffusion for the volatility target index and a rho-vega conversion formula linking sensitivities to the underlying diffusion, with bounds and asymptotics for the multipliers. The analysis extends from a constant-coefficient case to a general deterministic setting via reduction arguments and continuity, and is complemented by numerical tests validating the theoretical findings and illustrating practical implications for pricing and hedging. The work provides a rigorous framework for understanding the stochastic behavior of volatility-targeted strategies and their sensitivity properties in the vanishing-discretisation limit.

Abstract

We study the limiting distribution of a volatility target index as the discretisation time step converges to zero. Two limit theorems (a strong law of large numbers and a central limit theorem) are established, and as an application, the exact limiting distribution is derived. We demonstrate that the volatility of the limiting distribution is consistently larger than the target volatility, and converges to the target volatility as the observation-window parameter $λ$ in the definition of the realised variance converges to $1$. Besides the exact formula for the drift and the volatility of the limiting distribution, their upper and lower bounds are derived. As a corollary of the exact limiting distribution, we obtain a vega conversion formula which converts the rho sensitivity of a financial derivative on the limiting diffusion to the vega sensitivity of the same financial derivative on the underlying of the volatility target index.

A central limit theorem and its application to the limiting distribution of volatility target index

TL;DR

The paper addresses the limiting distribution of volatility target indices as the discretisation time step vanishes. It develops a LLN and a CLT for a dependent sequence to characterize the exact limiting diffusion, with drift and diffusion coefficients scaled by multipliers and . The results yield an explicit limiting Itô diffusion for the volatility target index and a rho-vega conversion formula linking sensitivities to the underlying diffusion, with bounds and asymptotics for the multipliers. The analysis extends from a constant-coefficient case to a general deterministic setting via reduction arguments and continuity, and is complemented by numerical tests validating the theoretical findings and illustrating practical implications for pricing and hedging. The work provides a rigorous framework for understanding the stochastic behavior of volatility-targeted strategies and their sensitivity properties in the vanishing-discretisation limit.

Abstract

We study the limiting distribution of a volatility target index as the discretisation time step converges to zero. Two limit theorems (a strong law of large numbers and a central limit theorem) are established, and as an application, the exact limiting distribution is derived. We demonstrate that the volatility of the limiting distribution is consistently larger than the target volatility, and converges to the target volatility as the observation-window parameter in the definition of the realised variance converges to . Besides the exact formula for the drift and the volatility of the limiting distribution, their upper and lower bounds are derived. As a corollary of the exact limiting distribution, we obtain a vega conversion formula which converts the rho sensitivity of a financial derivative on the limiting diffusion to the vega sensitivity of the same financial derivative on the underlying of the volatility target index.

Paper Structure

This paper contains 9 sections, 17 theorems, 215 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Proof of The Main Results
  3. Proof of Theorem \ref{['thm:-4']} and Theorem \ref{['thm:-5']}
  4. Reduction to An Equivalent Convergence
  5. Proof of Theorem \ref{['thm:-3']}: Constant Coefficient Case
  6. Proof of Theorem \ref{['thm:-3']}: General Case
  7. Asymptotic Properties of The Functions $U(\lambda)$ and $V(\lambda)$
  8. Numerical Tests
  9. Appendix

Key Result

Theorem 1.4

(i) As $\Delta t\to0$, the process $\{I_{t}^{(N)}\}_{t\ge0}$ converges in law to the Itô diffusion where and (ii) The multipliers $U(\lambda)$ and $V(\lambda)$ satisfy that $U(\lambda)>1$, $V(\lambda)>1$ and $\lim_{\lambda\to1-}U(\lambda)=\lim_{\lambda\to1-}V(\lambda)=1$. Moreover, for $\lambda\in(0.7,1)$, the following bounds for $U(\lambda)$ and bounds for $V(\lambda)$ hold.

Figures (6)

  • Figure 4.1: Approximation of $U(\lambda)$.
  • Figure 4.2: Approximation of $V(\lambda)$.
  • Figure 4.3: Approximation of sample density. Parameters used are $T=1.0$, $\lambda=0.9$, $\sigma=0.5$, $v_{0}=0.02$, $r=0.05$, $\rho=0.03$, $S_{0}=I_{0}=1$, and $\bar{\sigma}=0.2$
  • Figure 4.4: Convergence of volatility for different $\lambda$ ($x$-axis). Parameters used are $T=1.0$, $\sigma=0.5$, $v_{0}=0.02$, $r=0.05$, $\rho=0.03$, $S_{0}=I_{0}=1$, and $\bar{\sigma}=0.2$
  • Figure 4.5: Convergence of European call option price for different $\lambda$ ($x$-axis). Parameters used are $T=1.0$, $\sigma=0.5$, $v_{0}=0.02$, $r=0.05$, $\rho=0.03$, $S_{0}=I_{0}=1$, and $\bar{\sigma}=0.2$
  • ...and 1 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 32 more