Clarification on Option Proceeds

I was going through the WSO LBO model and was having trouble understanding option proceeds.

Let's say our buy in sponsor equity amount is $100 and assume a management option pool of 10%

If we exit at $200, I would calculate proceeds as the following:

Assume $1 per share (arbitrarily) so sponsor has 100 shares, management has 11.1111 shares -

Exit Equity (pre-options): $200 Cash from options: $11.11

Total Diluted Equity: $211.11

Cash to Management: $21.111 (Overall proceeds being $10) Cash to Sponsor: $190.

However, going through he course, it seems that instead of calculating proceeds to management as the following:

Exit Equity (pre-options): $200 Cash from options: $11.11

Total Diluted Equity: $211.11

Cash to Management: $20 (Overall Proceeds being 8.9) Cash to Sponsor: $191.11

The calculation in the course was by taking the implied price per share after the dilution of the shares (lower than pre-options), subtracting the strike price (original price per share) and multiplying by 10%

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