[Paper Review] Localizing Volatilities

Marc Atlan|arXiv (Cornell University)|Apr 13, 2006

Market Dynamics and Volatility4 citations

TL;DR

This paper establishes a rigorous link between stochastic volatility models with stochastic interest rates and local volatility models with deterministic rates using Gyöngy's (1986) mimicking diffusion theorem. It derives explicit formulas for local volatility in Bessel-based stochastic volatility models and extends Dupire's local volatility framework to hybrid interest rate-volatility settings, showing that the local volatility surface can be constructed stably from a stochastic volatility model rather than market option prices.

ABSTRACT

We propose two main applications of Gyöngy (1986)'s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous Itô processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based stochastic volatility models in which this relation is used to compute analytical formulas for the local volatility. Secondly, we use these mimicking techniques to extend the well-known local volatility results to a stochastic interest rates framework.

Motivation & Objective

• To extend Dupire's (1994) and Derman and Kani's (1994) local volatility framework to models with stochastic interest rates.
• To demonstrate that local volatility surfaces can be derived analytically from Bessel-based stochastic volatility models using scaling and conditioning properties.
• To establish a stable, model-based alternative to market-implied local volatility surfaces, which are known to be numerically unstable.
• To formalize the relationship between local volatility and stochastic volatility in hybrid interest rate-volatility models under the risk-neutral measure.
• To investigate the impact of stock-volatility correlation and hybrid correlation risk on the local volatility surface.

Proposed method

• Applies Gyöngy's (1986) theorem to construct an inhomogeneous Markovian SDE that mimics the one-dimensional marginal distributions of a continuous Itô process.
• Uses Bessel processes and their scaling properties to derive explicit expressions for the joint law of integrated volatility and stock price in stochastic volatility models.
• Derives the local volatility function as the conditional expectation of the instantaneous variance under the T-forward measure: $\sigma^2(t,x) = \mathbb{E}^T[V_t + 2\rho\sqrt{V_t}\sigma_B(t,T) + \sigma_B^2(t,T) \mid S_t = xB(t,T)e^{\int_t^T f(0,s)ds}]$.
• Applies the theorem to forward rate dynamics under the T-forward measure, showing that the local volatility of the forward price matches the conditional expectation of the instantaneous volatility in the stochastic model.
• Uses the Lévy area formula and Laplace transforms of Bessel processes to compute characteristic functions of integrated volatility and stock price.
• Derives a general framework where the volatility diffusion is a deterministic time and space transformation of a Bessel process, enabling analytical tractability.

Experimental results

Research questions

• RQ1 Can local volatility models with deterministic interest rates be constructed to replicate the joint law of stock price and volatility in a stochastic volatility model with stochastic interest rates?
• RQ2 How does the correlation between the stock and its stochastic volatility affect the shape of the local volatility surface?
• RQ3 Can analytical formulas for local volatility be derived in Bessel-based stochastic volatility models using the mimicking diffusion theorem?
• RQ4 What is the role of the hybrid correlation risk (between interest rates and volatility) in calibrating local volatility surfaces in a stochastic interest rate setting?
• RQ5 Is it possible to construct a stable local volatility surface from a stochastic volatility model rather than from market option prices, and what are the advantages?

Key findings

• The local volatility surface in a deterministic interest rate model can be derived as the conditional expectation of the instantaneous variance in a stochastic volatility model with stochastic interest rates, under the T-forward measure.
• The formula $\sigma^2(t,x) = \mathbb{E}^T[V_t + 2\rho\sqrt{V_t}\sigma_B(t,T) + \sigma_B^2(t,T) \mid S_t = xB(t,T)e^{\int_t^T f(0,s)ds}]$ provides a closed-form expression for local volatility in hybrid models.
• The local volatility surface constructed from a stochastic volatility model is smoother and more stable than the one obtained via the Dupire PDE from market option prices.
• Bessel processes provide a tractable class of stochastic volatility models where analytical computations of local volatility are feasible due to their scaling and conditioning properties.
• The framework allows for both independent and correlated stock-volatility dynamics, with the correlation parameter $\rho$ directly influencing the skew of the local volatility surface.
• The paper establishes that the (HC)-Hypothesis (hybrid correlation) must be accounted for in the local volatility surface when interest rates are stochastic, implying a market premium for hybrid risk.