# Two Types of Inverse Options

## [Deribit Inverse Options](https://support.deribit.com/hc/en-us/articles/31424939096093-Inverse-Options)

Inverse options on Deribit are settled in the underlying cryptocurrency (e.g., BTC for Bitcoin options). The strike price is quoted in USD, but the premium and settlement are in BTC.

The payoff for a call option is calculated as:

$$\text{Payoff}\_{\text{call}} = \frac{\max(S\_T - K, 0)}{F\_0}$$

For a put option, the payoff is:

$$\text{Payoff}\_{\text{put}} = \frac{\max(K - S\_T, 0)}{F\_0}$$

Where:

* $$S\_T$$ is the price of the underlying asset at expiration (in USD).
* $$K$$ is the strike price of the option (in USD).

Note this is the definition of the payoff of the Inverse Option in Deribit.

This structure means that the payoff is inversely proportional to the price of the underlying asset at expiration. As the price of the underlying asset increases, the payoff in BTC decreases, and vice versa.

Based on the definition above, the option price is calculated as:

$$C = \frac{S\_0N(d\_1) - Ke^{-rT}N(d\_2)}{S\_0} = N(d\_1) - \frac{KN(d\_2)}{F} \quad P = \frac{Ke^{-rT}N(-d\_2) - S\_0N(-d\_1)}{S\_0} = \frac{KN(-d\_2)}{F} - N(-d\_1)$$

Where:

* $$F = S\_0e^{rT}$$ is the forward price of the underlying asset.
* $$d\_1 = \frac{\ln{\frac{F}{K}} + \frac{\sigma^2T}{2}}{\sigma\sqrt{T}}$$
* $$d\_2 = d\_1 - \sigma\sqrt{T}$$

For more information, refer to [Deribit Inverse Contracts – Calculating Profit in BTC and USD](https://insights.deribit.com/education/deribit-inverse-contracts-calculating-profit-in-btc-and-usd/).

## [OKX Dual Investment](https://www.okx.com/earn/dual)

The essence of OKX Dual Investment is the sale of an inverse option. In this case, the investor sells an inverse option and receives a premium upfront. The settlement is also in the underlying cryptocurrency. But the payoff structure of the defined product is different.

For a Sell High DCD, it is actually selling a call option with a higher strike than current spot price, the payoff of the call option is:

$$\text{Payoff}\_{\text{call}} = \max(\frac{S\_T - K}{S\_T}, 0)$$

For a Buy Low DCD, it is equivalent to selling a put option with lower strike compared with current spot price, and the payoff of the put option is:

$$\text{Payoff}\_{\text{put}} = \max(\frac{K - S\_T}{S\_T}, 0)$$

And the option price is calculated as:

Since $$S\_T$$ is lognormally distributed, and the mean of $$\ln{S\_T}$$ is $$m$$ and the standard deviation of $$\ln{S\_T}$$ is $$w$$.

Define $$g(S\_T)$$ as the probability density function of $$S\_T$$, and given

$$S\_T=e^{m+\frac{1}{2}w^2}$$

Given

$$Q = f(S\_T) = \frac{\ln{S\_T} - m}{w}$$ $$S\_T = e^{m+wQ}$$ $$h(Q) = \frac{1}{\sqrt{2\pi}}e^{-Q^2/2}$$

So that

$$dS\_T = we^{m+wQ}dQ$$ $$g(S\_T) = \frac{1}{S\_Tw\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{\ln{S\_T} - m}{w})^2} = \frac{1}{S\_Tw}h(Q)$$

Alternatively, we can derive the transformation: the pdf of Q is $$h(Q)$$, so by change of variables:

$$g(S\_T)dS\_T = h(Q)dQ => g(S\_T) = h(Q)|\frac{dQ}{dS\_T}| = h(Q)\frac{1}{wS\_T}$$ $$dS\_T = we^{m+wQ}dQ$$

We can get the price of Sell High DCD as:

$$
\begin{aligned}
C = E\left \[ \max(\frac{S\_T-K}{S\_T}, 0) \right ]
&= \int\_{K}^{+\infty}\frac{S\_T-K}{S\_T}g(S\_T)dS\_T \\
&= \int\_{(\ln{K}-m)/w}^{+\infty}\frac{e^{Qw+m}-K}{e^{Qw+m}}h(Q)\frac{1}{wS\_T}(we^{m+wQ})dQ \\
&= \int\_{(\ln{K}-m)/w}^{+\infty}\frac{e^{Qw+m}-K}{e^{Qw+m}}h(Q)\frac{1}{wS\_T}(wS\_T)dQ \\
&= \int\_{(\ln{K}-m)/w}^{+\infty}(1-\frac{K}{e^{Qw+m}})h(Q)dQ \\
&= \left \[1-N(\frac{\ln{K}-m}{w})\right]-K\int\_{(\ln{K}-m)/w}^{+\infty}{e^{-(Qw+m)}}h(Q)dQ \\
&= N(\frac{m-\ln{K}}{w}) -K\int\_{(\ln{K}-m)/w+w}^{+\infty}{e^{-(Qw+m)}}h(Q+w)d(Q+w) \\
&= N(\frac{m-\ln{K}}{w}) -Ke^{-m+w^2/2}\left\[ 1 - N(\frac{\ln{K}-m}{w}+w) \right] \\
&= N(\frac{m-\ln{K}}{w}) -Ke^{-\ln{E(S\_T)}+w^2}N(-\frac{\ln{K}-m}{w}-w) \\
&= N(\frac{\ln{\frac{E(S\_T)}{K}}-w^2/2}{w}) -\frac{Ke^{w^2}}{E(S\_T)}N(\frac{\ln{\frac{E(S\_T)}{K}}-\frac{3w^2}{2}}{w}) \\
&=N(d\_2) -\frac{Ke^{w^2}}{E(S\_T)}N(d\_2^\*)
\end{aligned}
$$

where

$$
d\_2 = \frac{\ln(E(S\_T)/K) - w^2/2}{w}, \quad d\_2^\* = \frac{\ln(E(S\_T)/K) - 3w^2/2}{w}.
$$

Based on the derivation provided, the formula for the **Buy Low DCD** (which corresponds to the payoff = $$\max\left(\frac{K - S\_T}{S\_T}, 0\right)$$) is:

$$
P = \frac{K e^{w^2}}{E(S\_T)} N(-d\_2^\*) - N(-d\_2),
$$

with

$$
d\_2 = \frac{\ln(E(S\_T)/K) - w^2/2}{w}, \quad d\_2^\* = \frac{\ln(E(S\_T)/K) - 3w^2/2}{w}.
$$


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