# Options Hedge Ratio
The Pascal Protocol employs a risk-based margining methodology that aligns the margin requirements of options positions with corresponding futures contracts. This method is grounded on the concept of continuously revised delta hedging methodology derived from the Black-Scholes Model, which quantifies the rate of change in the option's premium price per unit change in the underlying asset's price (and thus futures price).
Specifically, the delta hedge ratio determines the quantity of futures contracts needed to offset the directional price risk in an options position. Margin requirements for options are thus effectively represented as those of a futures contract, with the quantity equating to the number of options contracts multiplied by the delta hedge ratio. For call and put options, respectively, delta hedge ratio can be derived from BSM equation:
$$
\Delta\_{call} = \Phi(d\_1)
$$
$$
\Delta\_{put}=\Phi(d\_1) - 1
$$
For a call option, delta ranges from $$0$$ to $$1$$, while for a put option, delta ranges from $$-1$$ to $$0$$, with at-the-money strikes producing deltas around $$0.5$$ and $$-0.5$$.
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**Example**
If a trader holds 10 out-of-the-money call option contracts with a delta of 0.25, the delta hedge ratio would be 2.5 futures contracts. This means 2.5 futures contracts would offset the directional risk of the 10 calls.
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In addition to directional price risk captured by delta, Pascal Protocol also incorporates into the options margin requirements a gamma risk, the rate of change of delta, component. By including this gamma component in the margin calculations, the model accounts for the nonlinear risks associated with options as they move further in-the-money (for calls) or out-of-the-money (for puts).
Gamma is the second partial derivative of the option's price with respect to the underlying price:
$$
\Gamma = \frac{d(\Delta)}{dF}=\frac{1}{F\sqrt{2\pi}} \exp\left(-\frac{d\_1^2}{2}\right) \bigg/ \left(F \sigma\sqrt{t}\right)
$$
Gamma-adjusted delta hedge ratio calculated from $$\Delta$$, $$\Gamma$$, and $$R$$, expected magnitude of movement in the price of the underlying asset over the margin period of risk, provided by Risk Oracle:
$$
Hedge Ratio = \Delta +\frac 1 2 \Gamma R^2
$$
Incorporating gamma adjustment in the hedge ratio calculation allows Pascal Protocol to provide better margin coverage of deep in-the-money (ITM) and out-of-the-money (OTM) options – scenarios frequently encountered in volatile crypto markets.
Pascal Protocol dynamically calculates the gamma-adjusted delta hedge ratio for options contracts, with each Futures Mark Price and Options Implied Volatility update, and uses this ratio to determine margin requirements for options positions. The combined margin requirements for the options and futures positions reflect the real-time net risk exposure for traders portfolio. Consequently, this alignment streamlines the margining process, while still ensuring adequate collateralization across both derivatives markets.
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