Solana Options Standard

An open source software/program development kit (SDK). Inside of the SDK, there are a few things that will be built and optimized over a course of time. More detailed documentation and descriptions written in docs.epicentrallabs.com.

Black Scholes Model Formula:

Reference/Citation(s) - Wikipedia: Black Scholes Model

Call Option

$C(S,t) = N(d_+)S_t - N(d_-)Ke^{-r(T-t)}$

Put Option

$P(S,t) = N(-d_+)Ke^{-r(T-t)} - N(-d_-)S_t$

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Where:

$d_1 = \frac{1}{\sigma\sqrt{T-t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T-t)\right]$

$d_2 = d_+ - \sigma\sqrt{T-t}$

Above are the two standardized normal variables used in the Black-Scholes formula. They are crucial for the Black-Scholes model, as they are used to determine the option's price and its sensitivity to various factors.

d1 and d2 Calculations

The d1 and d2 parameters are integral components of the Black-Scholes option pricing model. These parameters are used to determine the theoretical price of options and their sensitivities to various market factors.

d1 Calculation

The d1 parameter is calculated using the following formula:

$d_1 = \frac{1}{\sigma\sqrt{T-t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T-t)\right]$

• Parameters:

  • spot_price (( S_t )): The current market price of the underlying asset.
  • strike_price (( K )): The price at which the option can be exercised.
  • risk_free_rate (( r )): The annualized risk-free interest rate.
  • volatility (( \sigma )): The volatility of the underlying asset.
  • time_to_expiry (( T-t )): The time remaining until the option's expiration, expressed in years.

• Purpose: The d1 parameter is a standardized measure that helps determine the probability of option exercise, adjusted for the time value of money and volatility risk premium.

d2 Calculation

The d2 parameter is calculated using the formula:

$d_2 = d_1 - \sigma\sqrt{T-t}$

• Parameters:

  • d1: The previously calculated d1 parameter.
  • volatility (( \sigma )): The volatility of the underlying asset.
  • time_to_expiry (( T-t )): The time remaining until the option's expiration, expressed in years.

• Purpose: The d2 parameter is used alongside d1 in the Black-Scholes formula to calculate the option's price. It represents the adjusted probability of the option expiring in-the-money.

$C(S,t) = N(d_+)S_t - N(d_-)Ke^{-r(T-t)}$

Where:

$d_+ = \frac{1}{\sigma\sqrt{T-t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T-t)\right]$

$d_- = d_+ - \sigma\sqrt{T-t}$

Notation:

Market Related:

$S$ = Current price of the underlying token/asset.

$σ$ = Volatility of the underlying token/asset.

$t$ = is a time in years; with $t = 0$ generally representing the present year.

$r$ = is the annualized "risk-free" interest rate, continuously compounded (APY).

Option Related:

$V(S,t)$ = is the current option price based on the asset price and time.

$C(S,t)$ = is the call option price and $P(S,t)$ is the put option price.

$T$ = is when the option expires.

$\tau$ = is time left until expiry ($T - t$).

$K$ = is the strike price (agreed price to buy/sell).

$e$ = Euler's number (≈ 2.718281823) a mathematical constant and is the base of the natural logarithm.

$ln$ = is the natural logarithm.

$N(x)$ = denotes the standard normal cumulative distribution function (CDF):

$t$ is a time in years; with $t = 0$ generally representing the present year.

$r$ is the annualized "risk-free" interest rate, continuously compounded (APY).

Option Related:

$V(S,t)$ is the current option price based on the asset price and time.

$C(S,t)$ is the call option price and $P(S,t)$ is the put option price.

$T$ is when the option expires.

$\tau$ is time left until expiry ($T - t$).

$K$ is the strike price (agreed price to buy/sell).

$N(x)$ denotes the standard normal cumulative distribution function (CDF):

$N(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-z^2/2} dz.$

Greeks

Reference/Citation(s) - Wikipedia: Option Greeks

The Greeks serve as essential metrics for managing risk in options trading. They help traders understand how their portfolio value changes when specific market factors fluctuate. By analyzing each Greek independently, traders can assess individual risk components and adjust their positions to maintain their target risk profile.

Delta

Delta shows how much the option value changes when the underlying asset price changes. It represents the number of tokens needed to hedge the option.

$\Delta = \frac{\partial V}{\partial S}$

Theta

Theta shows how much value an option loses as time passes. It represents the daily decay in option value as it gets closer to expiration.

$\Theta = -\frac{\partial V}{\partial \tau}$

Vega

Vega shows how much the option value changes when volatility changes. It measures the impact of a 1% change in volatility.

$\nu = \frac{\partial V}{\partial \sigma}$

Gamma

Gamma shows how fast delta changes when the asset price moves. It helps measure how stable an option position is.

$\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2}$\

Rho

Rho shows how much the option value changes when interest rates change. It measures the impact of a 1% change in rates.

$\rho = \frac{\partial V}{\partial r}$