Premium-Adjusted Greeks for Inverse Crypto Options Hedged with Futures

# Inverted Crypto Options - How to Derive the Greeks and Delta-Hedge?


# Inverted Crypto Options - How to Derive the Greeks and Delta-Hedge?












Most of my question is based off this paper by Lucic and Sepp (2024). An inverted cyptocurrency option that is traded on the Deribit platform. It is a cash-settled option that settles in crypto and is also priced/traded in crypto. E.g. An ETH call option with a $T=1-week$, $K=2,500$, is priced as $C=0.04ETH$ and if $S_T=2,600$, then the call option owner is given $\frac{2600-2500}{2600}=0.03846 ETH$ at expiry.

I am trying to derive the greeks, specifically delta and gamma wrt to the future, $F$, with the same maturity as the option. (The future is also cash-settled in ETH at maturity.

Greek Derivation:

The paper derives $\Delta$ for the spot, $F$, by:

$$V(t,F) = F_t \tilde{V}(t,F),$$ where $V$ is the price of the option quoted in USD, and $\tilde{V}$ is the price of the option quoted in ETH.

By taking the partial wrt the spot, we get:

$$\partial_F V(t,F) = \tilde V(t,F) + F\partial_F \tilde{V}(t,F)$$

$$\partial_F \tilde{V}(t,F) = \frac{1}{F}\left[\partial_F V(t,F) - \tilde{V}(t,F)\right] $$

$$\therefore \tilde{\Delta} (t,F) = \Delta(t,F) - \frac{V(t,F)}{F},$$

where $\tilde{\Delta}$ is the delta of future with the premium-adjusted. This is known as "NDelta" on the platform. From what I understand thus far, NDelta is similar to the premium-adjusted delta for an FX option (I am not too familiar with FX options), and we are essentially removing $\frac{V(t,F)}{F}$ from the $\Delta$ since we already received some of the underlying when we sold the premium.

In the paper, it uses $S$ instead of $F$, but still says $S$ can be the future. (Equation 12, $\tilde{\Delta} (t,S) = \Delta(t,S) - \frac{V(t,S)}{S}$. This confuses me because in reality, if you were to sell the option, you are receiving physical ETH at the inception of the trade, which has some spot price, $S$. So I would think intuitively, the last equation in the derivation should be:

$$\tilde{\Delta} (t,F) = \Delta(t,F) - \frac{V(t,F)}{S},$$

1: Why am I wrong here? It relates the premium-adjustment to how many futures units we need to buy to hedge (which will be less since we collected spot for the premium), but we received SPOT, not futures. So I am confused here.

Further, going from $\tilde{\Delta} (t,F) = \Delta(t,F) - \frac{V(t,F)}{F}$, then is $\Gamma$ simply:

$$\tilde{\Gamma}(t,F) = \Gamma (t,F) - \frac{F\Delta(t,F) - V}{F^2}$$

2: Or does NGamma not make sense in this framework since our position direction evolves by Delta and NDelta is just an extra subtraction of the premium we received?

Delta-Hedging:

Using a practical example, how would the delta-hedge with a future for this inverted option be in reality? If we sell 1 ETH call option for 0.03 ETH, the maturity is 1-week, $K=2,650$, current spot is $S_t = 2,550$ and the futures is $F=2,600$.

Since we received 0.03 ETH, which has a value of $0.03 \cdot 2550$ at spot, the implied volatility $\sigma(T,K,F) \approx 66.642\%$, with 365 trading days and using the numerical solution to the Black76 equation.

Since the USD denominated $\Delta=N(d_1)$, with $d_1 = \frac{ln(F/K) + (\sigma^2 / 2) T}{\sigma\sqrt{T}}$, the NDelta becomes:

$$\tilde{\Delta} = N\left(\frac{ln(F/K) + (\sigma^2 / 2) T}{\sigma\sqrt{T}}\right) - \frac{V}{F}$$ $$\tilde{\Delta} = 0.43926 - \frac{0.03 \cdot 2550}{2600}$$ $$\tilde{\Delta} = 0.40984$$

3: Is this what my long position should be on the 1-week future until the delta changes if I want to be delta-neutral? (Ignoring basis and roll costs if I were to exit the position on the option before the future expires)