Does sigma in Black-Scholes attempt to take into account future events?

Question

I am learning basic information about Black-Scholes wrt options pricing.

I see that the standard deviation of the stock price is taken into account. This seems to only rely on existing (historical) data. The formula does not seem to take into account unusual events that are expected to occur soon (e.g. legislation), or unusual events that have very recently occurred (and do not yet affect the standard deviation of the stock price much).

Do options prices ignore future / very recent events (being priced on Black-Scholes, a seemingly imperfect model)? Or is Black Scholes just a rough guide? I believe I know the answer to this, but I'd like to get an advanced perspective and explanation.

Answer 1

Black-Scholes is a classic case of "all models are wrong but some are useful".

Black-Scholes is a formula that tells you what the price of an option would be if:

1. The stock's returns are log-normally distributed
2. You know the stock's volatility
3. Dividends are continuous
4. Trading is continuous and free of transaction costs
5. Borrow is always available if shorting is required to hedge

All four of these assumptions are untrue. 4 is pretty ugly because (as far as I know) you cannot trade 24/7/365.

Black-Scholes does not require the historical volatility. It requires future volatility.

Despite these flaws, Black-Scholes is still useful for some purposes.

Using Black-Scholes, options prices can be used to back into volatility to allow for useful comparisons. If you have option prices for two different stocks with very different share prices, the Black-Scholes implied volatility can give you a sense of which stock options traders expect to be more volatile. Similarly, if a stock has an important event tomorrow, an option with one week to expiration might have a higher implied volatility than an one with one month to expiration.

In the opposite direction, over the counter traders might agree to execute an options trade at a price determined using Black-Scholes, an implied volatility they negotiate, and the underling's price at the time the negotiations end. This makes quotes from different brokers more comparable. If broker A tells a trader the indicative price of a call option is $5 and later broker B says it is $5.05, it would be difficult for the trader to decide who to go to for execution if the stock price had gone up between the time when the two quotes were received. If implied volatility is more stable than the stock price, receiving quotes in terms of implied volatility can help resolve this issue.

Answer 2

There are 6 variables used in the Black Scholes formula: Stock price, strike price, time remaining, volatility, carry cost and dividend (if any). Input them and you get the theoretical price of the put and the call. Do any of those variables sound like legislation or earnings or any other pending unusual event?

Options price do not ignore future events. They are reflected by increased premium/increased implied volatility.