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Log Contract Payoff and Its First Two Derivatives

Article Quant Q&A · Author: TryingtobeQuant
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Summary

The note presents a variance-related log contract payoff as a function of the terminal asset price, initial price, and time horizon. It also states the payoff’s first and second derivatives with respect to the terminal price. The first derivative depends on the difference between the reciprocal initial and terminal prices, while the second derivative is positive for positive terminal price and horizon, indicating convexity in terminal price.

The document is framed as a request for help locating or understanding the formula, rather than a derivation. It gives no explanation of how the payoff is constructed, what assumptions connect it to variance, or how it might be replicated or used in pricing. The formula therefore serves as a compact starting point for studying log contracts, but the reader must consult other material to understand its financial interpretation and conditions of use.

Key ideas

  • The stated log contract payoff depends on the terminal price relative to the initial price and the time horizon.
  • The first derivative is the payoff’s sensitivity to the terminal asset price.
  • The second derivative is positive when terminal price and horizon are positive, so the payoff is convex in terminal price.
  • The note gives formulas but does not derive them or explain the link to variance swaps.

Tags

OptionsDerivatives pricingVolatility
Full text
# Log Contract payoff function


# Log Contract payoff function












I can’t get where Dr. Rouah gets payoff function of log contract. Could you please take a look at that?

https://frouah.com/finance%20notes/Variance%20Swap.pdf

It’s on page 2, section 3. I couldn’t find same function in anywhere else:

The log contract has the payoff function

$f(S_T) = \frac{2}{T}\left(\ln \frac{S_0}{S_T}+\frac{S_T}{S_0}-1\right)$

Note that $f^{'}(S_T)=\frac{2}{T}(\frac{1}{S_0}-\frac{1}{S_T})$ and $f^{''}(S_T)=\frac{2}{T}{\frac{1}{S_T^2}}$

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.