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How does my broker determine how much option margin they can or will allow me? For example if I have $1,000,000 in cash or stocks how do they determine how much option margin I may use?

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Many brokers will apply the exchange margin calculation rule. For example if you buy a "SPY" option listed on the CBOE, the rules are detailed at the bottom of this page:

Margin: Uncovered writers must deposit 100% of the options proceeds plus 15% or 20% of the aggregate contract value (current ETP price multiplied by $100) minus the amount by which the option is out-of-the-money, if any. Minimum margin is 100% of the option proceeds plus 10% of the aggregate contract value. Long puts or calls must be paid in full.

However some brokers may require more than that minimum margin amount and some may require less if they take hedging positions into account. The exact policy being broker dependent it is generally detailed on the broker's website or contract. For example interactive broker's policy is an example of the latter and as you can see the calculation can be fairly complex.

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