An Implied-Volatility Approximation for Variance Option Bounds

Summary

The note proposes an approximate lower bound for a call option on average realized variance. It starts with a stochastic-volatility price process and conditions the option payoff on the terminal asset price. Jensen’s inequality moves the positive-part function outside the conditional expectation, producing a lower bound involving conditional expected average variance. The proposal then approximates that conditional expectation with the square of the implied volatility for the corresponding strike and maturity, weighted by the terminal-price density.

The document poses the expression as a question and provides no numerical tests, proof of the implied-volatility substitution, or comparison with simulated option values. The inequality applies to the conditional expectation step; the final expression’s quality depends on the most-likely-path approximation and model assumptions. It therefore presents a research idea rather than a validated pricing method, and gives no evidence that the approximation is accurate or useful across volatility models or market conditions.

Key ideas

• Conditioning a variance option payoff on the terminal asset price allows integration over terminal prices.
• Jensen’s inequality gives a lower bound by applying the positive-part function to conditional expected variance.
• The proposal approximates conditional expected average variance using squared implied volatility at each strike and maturity.
• The note supplies no empirical or simulation results validating the most-likely-path approximation.

Tags

# An approximate lower bound for options on variance


# An approximate lower bound for options on variance












Question: I am wondering if anybody has looked at the following lower bound, based on the most-likely path approximation, and/or tested it?

Let $dS_t = \sigma_t S_t dW_t$, where $\sigma_t$ is a stochastic spot vol.

Options on variance lower bound:

\begin{align} E \left[ \left( \frac1T \int_0^T \sigma^2_t dt - l \right)_+ \right] &= \int_0^\infty E \left[ \left.\left( \frac1T \int_0^T \sigma^2_t dt - l \right)_+ \right| S_T=K \right] p(K) dK \\ &\geq \int_0^\infty \left( E \left[ \left. \frac1T \int_0^T \sigma^2_t dt \right| S_T=K \right] - l \right)_+ \; p(K) dK \\ &\approx \int_0^\infty \left( I^2(K,T) - l \right)_+ \; p(K) dK. \\ \end{align} where the last step uses the MLP approximation.

I'm really curious if this gives a reasonable lower bound. As I cannot find any papers on this and this is quite an easy approx to derive, my guess is that it doesn't work that well. But just double checking here at QFSE.

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