Variance Additivity for Equity Volatility with Stochastic Rates

Summary

The question concerns how to infer an equity’s local volatility when its drift is the short rate and that rate is stochastic. The author proposes adjusting market implied volatility on a forward for short-rate volatility, then asks whether cumulative variance can be subtracted across two horizons to recover local volatility over the intervening period, or whether only forward variance is additive.

The document supplies no answer, derivation, model specification, or empirical evidence. It therefore identifies a modeling issue rather than establishing a method. Any variance subtraction would depend on consistent definitions of the volatility measures, the numeraire and the treatment of the equity-rate relationship; the prompt does not resolve those assumptions. It is useful as a focused question about volatility and rates, but readers need additional theory to determine what quantity is additive and how the proposed correction should be calculated.

Key ideas

• The question models an equity whose drift follows a stochastic short rate.
• It proposes adjusting forward implied volatility for the volatility of the short rate.
• It asks whether total variance can be differenced to infer volatility over an intermediate interval.
• The document provides no derivation or answer, so the additivity claim remains unresolved.

Tags

# Total Variance of an asset in case of stochastic rates


# Total Variance of an asset in case of stochastic rates












Let's suppose the underlying S follows a BS dynamic with the drift being the short rate that follows a short dynamic model.

the "local volatility" of the equity should be the implied volatility from the market (of the forward) corrected from the volatility of the short rate.

Now let's suppose that for [0,T1] and [0,T2] I have deduced the "local volatility" like described above.

Can I use the fact that the variance is additive to the local volatility to deduce the local volatility in [T1,T2].

Or is it only the variance of the forward that is additive?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.