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I understood there were 3 alternative methods of dealing with dividends in BS: 1) using a continuous dividend yield as an input; or 2) setting dividends to zero and subtracting the PV of divs from the spot; or 3) setting dividends to zero and adding the FV of divs to the strike.
When I use a high dividend stock these diverge widely. Why? Which is more accurate?
I suspect that using the continuous dividend yield is providing the most accurate answer but would like to understand why.
If you assume that dividends are discrete but proportional to the pre dividend date stock price then the BS formula is exact provided you correctly compute the expiry date stock forward price, hence the continuous dividend yield case calibrated to the correct forward will give the correct result (this is because with proportional dividends, discrete or continuous, the stock terminal price is always log normal).
If you assume on the other hand that dividends are discrete and have fixed amounts then there is no closed form formula. Common approximations either move the dividends to the valuation date by subtracting their PV from the spot or move the dividends to the expiry date by adding their FV to the strike, or to use a combination of both. There are however more accurate approximations that use the log normal closed form formula around the stock forward price with an adjustment on volatility see for instance Bos R., Gairat A., Shepeleva A. (2003) Dealing with discrete dividends, Risk Magazine January 2003.
As an alternative you can forgo closed form formulas and solve for the exact model with discrete fixed dividends (or a combination of discrete fixed and proportional amounts) using a numerical method such as Monte Carlo or finite differences.
You cannot say "the continuous dividend yield is providing the most accurate answer" because you haven't really defined the problem. Accurate in what sense? What we are doing here is modelling a real-world situation. In a real world situation for equity options, dividends are always discrete. So a numerical solution that handles dividends properly (say finite differences, applying a jump-condition and interpolating at the known dividend date) would be the most "accurate" approach. Now, when as you say the dividends are large, you have another problem you need to define: what happens when at the ex-dividend date the dividend amount is greater the the stock price?! Again, we are just modeling here, so one needs to ask what would actually happen in practice, what dividend policy does the company have? A good discussion of all this can be found in Back to Basics: a new approach to the discrete dividend problem by Haug, Haug and Lewis.
Check out this great paper about BS and dividend estimation: https://optionmetrics.com/wp-content/uploads/2015/02/DividendForecasting_WhitePaper.pdf
In short, the difference can be attributed to risk-free vs risk-specific assumptions and the non-deterministic nature of future payouts in high-dividend stocks. The magnitude of change in future dividend payout is greater than in low dividend stocks.
