Can you explain the assumptions behind Black Scholes?

Answer 1

1. The option is European and can only be exercised at expiration.
2. No dividends are paid out during the life of the option.
3. Markets are efficient (i.e., market movements cannot be predicted).
4. There are no transaction costs in buying the option.
5. The risk-free rate and volatility of the underlying are known and constant.
6. The returns on the underlying are normally distributed.
7. It is assumed you can borrow and lend at the risk free rate. Funding purchase of the underlying at this rate and earning this rate with the proceed of shorting the asset.. if this is down

The key point is if returns are normally distributed and the volatility of the returns is known and constant then if you buy (or sell) an at the money option would be a fair bet. We look at European options so we are looking at the expected value of a single random cash flow at a certain date in the future. I will try to give a simple example

Suppose we have an option struck at 100 maturing some time in the future.. We can determine the probability it will be worth something quite easily. How much we expect it to be worth is another story. We know the distribution of possible outcomes and the probability that it will be worth MORE than 200 is .04. and we know that the probability it will be more than 195 . So it has a .02 probability of being worth 97.5 (on average).. Suppose the probability of being higher than 190 is .07. The probability of being between 190 and 195 is .03. so there is a probability of .03 that the option will be worth about 92.5. We can do this incrementally moving towards the strike. If we are at or below strike the option will be worthless at expiration.The probability the option will be worthless is 1- the probability it will be worth something. If we add up these probability weighted values we will have an approximate future value of this contingent claim. the smaller we make these increments the more precise the answer will be. Note that I said future value. We must take this expected future value and discount it back to present value to come up with the premium or "fair bet price."

Several other things come out of this. If you buy(sell) an option and constant rebalance an offsetting delta hedge you will make (lose) money if the realized volatility is greater than the implied volatility used to price it you will make(lose) money. Just the opposite will happen if realized volatility is less than implied.

Please don't hesitate to reach out for any option questions. I ran very large option and derivative books for the big banks on wall street and managed proprietary option portfolios at several hedge funds