Demystifying the Black-Scholes formula

The Black-Scholes formula is an option valuation model developed by two academics, Fischer Black and Myron Scholes, who first described it in a 1973 article. The article appeared in the same year that the Chicago Board Options Exchange (CBOE) was founded, and the model effectively democratized the use of options. Previously, the use of options had been limited to institutions with the necessary technological resources.

The Black-Scholes formula was revolutionary because it could easily be programmed into the calculators available at the time. This meant that anyone could assess the value of an option. The formula is easy to use, even though the underlying mathematics are not that simple. In this article, I will try to demystify the model, which at first glance may appear quite daunting, but I will not discuss the underlying mathematics.

First, here follows the famous Black-Scholes formula for a European style call option that does not pay dividends.* Remember that European style options may only be exercised at expiration, while American style options may be exercised at any time.
C0 = S0 N(d1) – Ke-rT N(d2)

Where,
C0 = the price of a European style call option that does not pay dividends
S0 = the price of the underlying stock at the time of valuation
d1 = (ln(S0/K) + (r + σ2/2)T)/(σ√T)
N(d1) = a statistical measure (normal distribution) corresponding to the call option’s delta
d2 = d1 – (σ√T)
N(d2) = a statistical measure (normal distribution) corresponding to the probability that the call option will be exercised at expiration
Ke-rt = the present value of the strike price
r = the risk-free interest rate
T = the time remaining to expiry, in years
σ = the volatility of the price of the underlying stock
As you can see, the Black-Scholes formula contains the six factors that influence the value of options and that we have discussed previously: the stock price (S0), the strike price (K), the time remaining to expiration (T), the risk-free interest rate (r), the volatility of the stock price (σ) and the dividend, which in this example is equal to zero.

If we ignore for the moment the terms N(d1) and N(d2), we can see that the Black-Scholes formula is simply based on the expression “S0 – Ke-rT.” Does this remind you of anything? In fact, the basis for the Black-Scholes formula is simply the current intrinsic value of the call option. So when the difference between these two terms increases, the intrinsic value of the call option increases, and when the difference between the terms decreases, the intrinsic value of the call option decreases.

But what happens if the variable S0 is less than Ke-rT? Since the call option’s price (C0) cannot be negative, variables N(d1) and N(d2) come to the rescue to give it a positive value, preventing the intrinsic value from falling below zero. N(d1) and N(d2) are statistical variables representing probabilities, with their values falling in a range from 0 to 1. As a result, the greater the amount by which S0 is less than KerT, the more that variables N(d1) and N(d2) approach zero. And when N(d1) and N(d2) are exactly zero, then the value of C0 is also nil. By taking this brilliant approach, Black and Scholes eliminated the possibility that the price, C0, could have a negative value.

Taking a closer look, we see that the expression S0 N(d1) is the amount that will likely be received on selling the stock at expiration, while the expression Ke-rT N(d2) is the payment that will likely be made to purchase the stock when the call option is exercised at expiration. So the value of the call option depends on the difference between these two expressions.
For example, consider a European call option on the stock of Shopify Inc. This security does not pay dividends and is trading at $114.92 on June 16, 2017. We will choose a strike price of $110 with an expiration on January 19, 2018.

We enter the following values in the Black-Scholes formula:
S0 = $114.92, K = $110, r = 0.57%, σ = 43.82%, T = 0.59 (217 days)
We are looking for:
C0 = S0 N(d1) – Ke-rT N(d2)
The following results are obtained:
Ke-rT = $109.63
d1 = 0.3083
d2 = -0.0283
N(d1) = 0.6211
N(d2) = 0.4887
C0 = $114.92 x 0.6211 – $109.63 x 0.4887
C0 = $71.37 – $53.58
C0 = $17.80

As you can see, the value of the call option is the difference between $71.37, which represents the funds likely to be received upon selling the stock at expiration, and $53.58, which represents the amount likely to be paid to buy the stock when the call option is exercised at expiration, for a total value of $17.80.

In closing, even though the Black-Scholes formula was developed using complex mathematical concepts, it is not that difficult and can be used by the average person. All things considered, the Black-Scholes formula essentially calculates the current intrinsic value, adjusted for the probability that the security will be worth more than the strike price at expiration. This probability is taken into account by the variables N(d1) and N(d2).

* The formula for an European put option that does not pay dividends is P0 = Ke-rT N(-d2) – S0 N(-d1)

Good luck with your trading, and have a good week!

The strategies presented in this blog are for information and training purposes only, and should not be interpreted as recommendations to buy or sell any security. As always, you should ensure that you are comfortable with the proposed scenarios and ready to assume all the risks before implementing an option strategy.