Pricing with Black-Sholes Model

European Option Pricing

There are several mathematical models available to price options, including the Black-Scholes Model and the Binomial Model. In our case, we have chosen to use the Black-Scholes Model, which is widely used in the financial industry.

Below are the formulas for pricing a call option $(C)$ and a put option $(P)$ using the Black-Scholes Model.

$$C(t) = N(d_1)S_t-N(d_2)Ke^{-rt}$$

$$P(t) = -N(-d_1)S_t+N(-d_2)Ke^{-rt}$$

Where:

$$d_1 = \frac{ln\frac{S}{K}+(r+\frac{\sigma^2}{2})}{\sigma\sqrt t}$$

$$d_2 = d_1 - \sigma \sqrt t$$

And:

$N$is the cumulative distribution function (CDF) of the normal distribution.

$S$ is the spot price of the underlying asset in USD.

$K$ is the strike price of the option in USD.

$r$ is the risk-free rate, set to 0 because it doesn't exist in DeFi.

$t$ is the time to maturity in years.

$\sigma$ is the implied volatility of the underlying asset.

Eonian Option Pricing

Eonian Options are perpetual derivatives that utilize a rolling cost, and have the potential to solve issues related to option portfolio rolling and liquidity fragmentation. Unlike other derivatives, the rolling cost charged daily for Eonian Options does not aim to incentivize price convergence to the spot price. Instead, it is intended to cover the rolling cost, which is the difference between the value of the trader's portfolio at time $t$ and the value of the same portfolio at time $t + 1$ due to the theoretical repurchase of new options on a daily basis.

The rolling cost for an Eonian Options is determined by the formula $(MARK - PAYOFF)$, which holds a unique significance - it represents the time value of the option.

The formula for the payoff of a call option is:

$$PAYOFF = max(S-K, 0)$$

The formula for the payoff of a put option is:

$$PAYOFF = max(K-S, 0)$$

$MARK$ represents the quoted price of the option provided by the pool.

The payment frequency of the rolling cost is denoted by $F$.

The pricing formula of the Eonian Option, based on the pricing formula of the everlasting option can be expressed as follows [6]:

$$E = \frac{1}{F} \times \sum_{i=1}^{\infty} (\frac{1}{F+1})^i \times P_{t_{i}}$$

Where $E$is the price of the Eonian Option and $P_{t_i}$ is the price of a European option (given by the Black Sholes Model) which expires in $t_i$ years.

As this is more convenient in DeFi, since the rolling cost is paid every block, we tend the payment frequency $F$ towards $\infty$. Therefore, the summation converges to the following integral.

$$E = \int_{0}^{\infty} \frac{1}{T} e^{\frac{-t}{T}} P_t \, \mathrm{d}t \$$

Thus, for an Eonian Call option the pricing formula is given by:

$$E_{call} = \int_{0}^{+\infty} \frac{1}{T} e^{\frac{-t}{T}} C(t) \, \mathrm{d}t \$$

And for an Eonian Put option the pricing formula is given by:

$$E_{put} = \int_{0}^{+\infty} \frac{1}{T} e^{\frac{-t}{T}} P(t) \, \mathrm{d}t \$$

Where $T$ is the funding period in days, which, in our case, is set to 1.

Next, we will examine how the pool determines the variables involved in these pricing formulas.