Black-Scholes Stock Option Valuation Calculator

The tables below calculate estimates of the economic fair value of a stock option using different variations of the Black-Scholes formula. You can see how the variations compare against each other and against the plain vanilla method. For puts, it calculates an alternative valuation based on a conversion (synthetic put) formula. 

Instructions:

1. Fill in the blanks. (Stock name and symbol are optional.) Numeric fields should contain no characters except .0123456789. All others will be converted to spaces. Date fields must contain valid dates. 
2. Click Calculate.

References

• A very technical discussion of the Black-Scholes equation is at Wikipedia.  
• How To Make Money In Stock Options, by Norman Saint-Peter, Prentice-Hall, 1984.
• Programmable TI-58/59 calculator Solid State Software module documentation: Master Library, Securities Analysis, and Applied Statistics modules, Texas Instruments, 1977-78.
• Options As A Strategic Investment, by Lawrence McMillan.

Notes

1) The basic formula does not take dividends into account. This is unrealistic because dividends do affect option prices. On the morning a stock goes ex-dividend, the stock price is reduced by the amount of the dividend. This permanently affects the price projections into the future. It makes the stock less likely to be above call strike prices and more likely to be below put strike prices.

2) This variation (which is used by the TI-59 calculator routines and appears to be common elsewhere) subtracts the present value of each dividend from the current stock price before calculating the option value. 

3) This variation is my invention. It subtracts the total value of all dividends from the current stock price before calculating the option value, without discounting to present value. The reasoning: As option holders, we are neither paying nor earning any monetary amount. What matters is that the stock price will drop, not by the dividend's present value but by its full amount. The stock price, if it is to regain its position prior to the drop, must recoup the entire amount, not the present value, and the interest rate is irrelevant to whether it will succeed in doing that. The stock price doesn't "grow" based on the interest rate. It must regain the ground it lost through volatility-induced price fluctuations alone. I've found no support for this reasoning online. It is possible that the reason dividends should be discounted can be found in the principle of Put-Call Parity or in the existence of synthetic calls whose value must always be the same as real calls and which do involve the purchase of the stock and therefore do result in the holder receiving dividends in the future, which should rightly be discounted.

4) The Hedge Ratios are based on the valuations in the Using Present Value Divs columns since that seems to be the most commonly used calculation method.

5) A synthetic put consists of shorting the stock and buying a call, a position with the same characteristics as buying a put but purportedly less expensive for floor traders. When the conversion put value is below the calculated Black-Scholes value, arbitrage trading of synthetic puts against real puts can drive real put prices to their conversion value. In this situation, puts can appear to be undervalued even though they are unlikely ever to reach their Black-Scholes valuations. The conversion put formula used here is based on information from the Norman Saint-Peter book referenced above.

6) For American-style options, handling dividends by adjusting the current stock price before calculating the option value fails to give the option "credit" for the time it spends in its higher trading range (pre-dividend), during which it has a higher probability of reaching a call strike price, for a while, than it will after the dividend. A more realistic handling of dividends for an American option might be to calculate its value from now to the first dividend, then calculate a new value (based on the lower stock price) to the time of the next dividend, and so on. Then calculate a weighted average of the multiple valuations such that each valuation's weight is in proportion to the time period it spanned. That is, if you buy a 6-month option on a stock that is currently at $60, but it pays a $5 dividend next Friday, its current $60 price isn't worth much because it won't be there much longer, and that should be reflected in the option's valuation. On the other hand, if the dividend will be 3 months from now, the stock will spend a long time fluctuating around its current $60 price, so a call in this situation should be more valuable. 

7) The program does not catch or warn you about all possible errors in the input data. Make sure your inputs match the formats of the examples shown. 

8) Black-Scholes is only one method of valuing stock options, which many people use as a guide, but there are other valuation methods. 

9) Use the rate of a Treasury with a maturity about the same as your anticipated holding period. 

10) Although Black-Scholes inherently takes into account the stock's probable price range, it is possible to do those probability calculations independently of the option valuation.

11) Questions, suggestions, comments are welcome in the discussion forum.

12) Most options traders lose money. If you use these valuation estimates, you will probably lose money trading options. If you don't use these valuation estimates, you will probably lose money trading options. You will probably lose money trading options no matter what calculations you do or what methods you try.