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## FTX Move Option as Forward Option Butterfly

The FTX BTC move contract payoff is $|S_T - S_t|$ and correspond to the $t$ forward starting butterfly contract $c+p$. The option prices are consistent with FTX BTC move contracts for 1 wk and 1 quarter. We compare the implied volatility to median and average breakeven price:

## Implied Median Vol and Implied Breakeven Vol

I compute the median vol and the implied vol corresponding to the average move payoff:

The weekly move implied volatility contract on FTX is trading at 80% implied volatility (around 9%). The quarterly move contract trade at 93% implied volatility around 35%.

## Rubinstein (1990) formula for forward start option

We consider a forward strike set option where $\alpha S_t$ is the $t$ forward strike setting date and $T$ the maturity. $d$ is the dividend yield and $r$ the risk neutral rate. Formula from Espen Haug p.36 consistent with numerical application:

\begin{eqnarray} b &=& r -d \\ d_1 &=& \frac{\ln(1/\alpha) + (b + \frac{1}{2} \sigma^2)(T-t)}{\sigma \sqrt{T-t}} \\ d_2 &=& d_1 - \sigma \sqrt{T-t} \\ c &=& S \exp((b-r)t) (\exp((b-r)(T-t)) N(d_1) - \alpha \exp(-r(T-t)) N(d_2)) \\ p &=& S \exp((b-r)t) (-\exp((b-r)(T-t)) N(-d_1) + \alpha \exp(-r(T-t)) N(-d_2)) \end{eqnarray}

Until fixing date $t$, the moneyness terms $d_1$ and $d_2$ depend only on a constant $\alpha$ and volatility and not on spot. This leads to option price being linear w.r. to spot $S$. Unlike standard option, no gamma and no vanishing of option time value when far in or out of the money.