# Premium Mark Price

The final step in the Pascal Protocol involves calculating the **Premium Mark Price** for each strike of call or put options, $$C$$ and $$P$$ respectively, that plays the same role as Mark Price for futures or perpetual contracts. For each strike, the protocol selects an implied volatility value from the interpolated volatility surface, based on strike price to futures price ratio. It then calculates the Premium Mark Price using the Futures Mark Price, strike price, chosen implied volatility, time to maturity, and risk-free interest rate, based on solving of Black Scholes equation:

$$
C = e^{-rt} \[F\Phi(d\_1) - K\Phi(d\_2)]
$$

$$
P = e^{-rt} \[K\Phi(-d\_2) - F\Phi(-d\_1)]
$$

Where $$d\_1$$and $$d\_2$$ parameters calculated as:

$$
d\_1 = \frac{1}{\sigma\sqrt{T}} \left\[ \ln \left(\frac{F}{K}\right) + \left(r + \frac{\sigma^2}{2}\right) T \right]
$$

$$
d\_2 = d\_1 - \sigma\sqrt{T}
$$

Here, $$K$$ is strike price of the option, and $$\Phi$$ represents the cumulative distribution function (CDF) of the standard normal distribution:

$$
\Phi(x) = \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{x} e^{-\frac{t^2}{2}} dt
$$

<img src="/files/nh3v10yZGzGk0YN9aZ2Y" alt="" class="gitbook-drawing">

This methodology ensures that the Premium Mark Price remains sensitive to actual market fluctuations, capturing real-time market dynamics and trends via connection to the current Futures Mark Price. Simultaneously, it is safeguarded against price manipulations, as the reliance on an interpolated volatility surface provides a buffer against sporadic price anomalies.


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