# 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 $$f^{''}(S_T)=\frac{2}{T}{\frac{1}{S_T^2}}$$

• I think he is simply defining the log contract as the above expression you wrote. Others might say the log contract payoff is just $\ln S_T/S_t$. It's almost a matter of taste. In Rouah's definition I think the cash bond holding and the dynamic hedge in the stock is included in the log contract definition. In any case, for variance swaps it really doesn't matter as you'll end up with the same price and replicating portfolio. – ilovevolatility May 17 '19 at 9:22