Mathematics

Contrast itself, through its PvP architecture, does not participate in the pricing of digital options listed on the platform. The following sections will cover protocol equations and assumptions made.

Modeling Payoffs

First, we provide the expected profit (loss) a user would see on the front-end UI for a single-sided position, assuming that only one deposit was made:

$$r = \begin{cases} {\frac{l_u \times (1-a)}{p_1} \times (p_2-f)}, \,\, &&&\text{Outcome = True} \\ \\ l_u\times0,\, &&&\text{Outcome = False} \end{cases}$$

where,

$$r = \text{return}\\ l_u = \text{Liquidity a user has deposited in true proposition pool} \\p_1 = \text{Liquidity of all users in true proposition pool}\\ p_2 = \text{Liquidity of all users in false proposition pool}\\ f = \text{protocol fee on contract expiration}\\ a = \text{protocol fee on withdrawal, if applicable}\\$$

Probability Delta

Because single-sided deposits ultimately disrupt the ratio of liquidity at a given strike, thus effecting the expected payout for all other users in a pool, a user can deposit liquidity on both sides in a way that has no impact on the current implied market probability. First, we define the probability of outcome x in terms of y:

$$\text{Market Implied Probability of Outcome} = \frac{p_1}{p_1 + p_2}\\$$

where,

$$\\p_1 = \text{Liquidity of all users for x outcome}\\ p_2 = \text{Liquidity of all users for y outcome}\\$$

Thus, in order for the payoff at a given strike to be the same pre-deposit and post-deposit the following condition must be true when additional liquidity is added for outcome x (variable a) and outcome y (variable b):

$$\frac{p_1}{p_2}=\frac{p_1+a}{p_2+b}$$

Assuming a trader has capital, c, to deploy and wishes to benefit from outcome x (variable a), the optimal liquidity (deferring fees) to add to both outcome x and y is given:

$$\frac{p_1\times (1+\frac{c}{p_1+p_2+c})-p_1}{p_2\times(1+\frac{c}{p_1+p_2+c})-p_2}$$