Black Scholes Option Pricing Formula Explained: Options, Futures, Derivatives & Commodity Trading

Black Scholes Option Pricing Formula Explained

Continunig further from our part I of this article, Black–Scholes Model Explained with Example for Options Pricing, we discussed the following three points:

1. the probability that Microsoft stock will actually move higher than $25
2. the probability that if it moves higher, how high will it go?
3. the time value of money for the one year timeframe (time to option expiry)

Let's see the second part here, for the third point of time value of money.

If I am willing to give away 100 bucks for an option which will lock my money in for a year, I would also like to factor in the loss of interest that I will incur if I pay 100 bucks to the seller for buying this option.
Black Scholes Option pricing model takes this into account by assuming a constant interest rate over the period of option duration and assumes it to be continously compounded. (hence the term "e" raised to - r*T where r is the interest rate and T is time to expiry)

Black–Scholes Option Pricing Formula Explained

So now, we attempt to build the Black Scholes model with all possibilities and time value of money factored in:
c = S*N(d1) - K*N(d2)* e(raised to -r*T)

S indicates the current stock price
N(d1) indicates the probability of current stock price movement
So the way to explain first part of the formula is as follows:
If we exercise this call option today (if it is in the money), the we will get one share of underlying. That will happen on the day we exercise the call option and will be represented by the expected value (current stock price * probability) and is represented by S*N(d1)

For the second part, "K*N(d2)* e(raised to -r*T)"
K represents strike price
N(d2) indicates probability that the stock price is ABOVE the strike price
So KN(d2) indicates the expected value of the stock price being ABOVE strike price

Also, this is for future, hence it needs to be discounted for time value of money, hence the multiplication by "e(raised to -r*T)"

Hope the above explanation makes it clear to understand the underlying concept behind Black-Scholes Model for Options Pricing.
Here is a brief summary of it again - calculate the volatility of the underlying stock price to identify the probability of the stock price reaching a partuicular level, and if so by how much. Then factor in the current market price of the stock and the time value of money to get to the price.
In case it is still not clear, leave your questions in the "Post Your Comments" section below and I'll be happy to clarify your doubts.

A word of caution about Black Scholes model for option pricing, that model is full of assumptions and those assumptions may not be applicable practically in real world:

1. No dividends are paid out on the underlying stock during the option life:

This usually never happens - atleast not for sure and there is uncertainity on whether a particular company will pay dividends or not. Even for index based options, the methodology used in index constitution may or may not adjust for the dividends paid by the constituent companies.

2. The option can only be exercised at expiry (European characteristics):
Usually, all stock based options are American type i.e. they can be exercised any time during the life of option.

3. Efficient markets (Market movements cannot be predicted)
It assumes a perfectly lognormal stock price movement i.e. if there is 5% probability of 10% upmove, then there is same 5% probability of 10% downmove. However, this is not true in reality and volatility skews do come in due to crashophobia effects.

4. Commissions are non-existent
Brokerages and Stock commissions eat away into lot of profits for option trading. But they are not considered in Black Scholes model for option pricing.

5. Interest rates do not change over the life of the option (and are known)
Again a unrealistic assumption, as interet rates keep changing

6. Stock returns follow a lognormal distribution
Same as three above.
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