Variance Swap Strike Formula and Its Earliest Attribution

Summary

The document asks about the origin of a formula expressing a variance swap strike as an integral of squared implied volatility across Black–Scholes log-moneyness, weighted by the standard normal density. It defines the relevant Black–Scholes quantity and notes an early reference to a 1999 Goldman Sachs risk modelling paper by Morokoff and coauthors.

The post does not derive the formula or establish who first published it. It raises an attribution question based on the earliest source its author located, so the citation is a lead rather than proof of priority. Readers seeking to verify the result would need to consult the cited paper and compare earlier literature; the document itself offers no derivation, supporting evidence, or discussion of assumptions and limitations.

Key ideas

• The variance swap strike is presented as a weighted integral of squared implied volatility.
• The weighting uses the standard normal density of a Black–Scholes variable.
• A 1999 Goldman Sachs paper is identified as the earliest reference found by the author.
• The post asks whether that reference establishes priority but does not answer the question.

Tags

# Origin of the formula Varswap strike $= \int_{\mathbb R} I^2(d_2) n(d_2)\, \mathrm d d_2$


# Origin of the formula Varswap strike $= \int_{\mathbb R} I^2(d_2) n(d_2)\, \mathrm d d_2$












As stated in the title, who first derived the formula $$ \text{Varswap strike} = \int_{\mathbb R} I^2(d_2) n(d_2) \, \mathrm d d_2 $$ where $d_2$ is the Black-Scholes quantity $$ d_2 = \frac{ \log S_t/K}{I\sqrt{T-t}} - \frac{I\sqrt{T-t}}{2} $$ and $n(d_2)$ is the standard normal density function?

The earliest reference I could find is to a GS paper by Morokoff et al., Risk management of volatility and variance swaps, Firmwide Risk Quantititave Modelling Notes, Goldman Sachs, 1999.

Is this then the source, i.e. the formula can be attributed to Morokoff?

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