When Time-Changed Models Can Be Expressed as Stochastic Volatility

Summary

The document asks whether a price model driven by Brownian motion evaluated at a random time can be represented as a stochastic volatility model. It proposes identifying the random clock with integrated variance, so that instantaneous volatility accumulates to the same random time, and notes that the drift may need adjustment under the risk-neutral measure to preserve the martingale condition.

The key qualification is that the proposed equivalence is easiest to consider when the random clock is independent of the Brownian motion. The author asks how to construct an equivalent volatility process when the clock and Brownian motion are dependent, including a correlation parameter. No derivation, answer, or empirical evidence is provided, so the document frames an open modeling question rather than establishing a general equivalence. Any representation would depend on assumptions about the joint dynamics and the resulting risk-neutral drift.

Key ideas

• A random time change can be compared with accumulated variance in a stochastic volatility model.
• The proposed identification equates the random clock with the time integral of instantaneous variance.
• The author raises the martingale condition as a reason the drift may require adjustment.
• Dependence between the random clock and Brownian motion complicates the proposed equivalence.
• The document poses the correlated case as an open question and gives no solution.

Tags

# Equivalence time-changed models and stochastic volatility models


# Equivalence time-changed models and stochastic volatility models












I suspect that time changed models can be written as a stochastic volatility model (and vice versa) if the random time is independent of the Brownian motion. Specifically, suppose under the risk-neutral measure $$ S_T = S_0 e^{\mu T + \beta \tau_T + \nu W_{\tau_T}} $$ where $\tau_T$ is the random time and is independent of the standard Brownian $W$, then we could say there is some stoch instantaneous vol $\sigma_u$ such that $$ \tau_T = \int_0^T \sigma_t^2 \, dt $$ and I think we can fiddle around with $\mu$ such that $S$ is a martingale.

But if $\tau_T$ is not independent of $W$ does anybody know who to write the process for $S$ as a stoch vol process with an `equivalent' stochastic instantaneous vol process $\sigma_u$ with correlation parameter $\rho$?

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