Intuitive Black‑Scholes Options Pricing Explained Simply

Introduction

Financial options — the right to purchase (call) or sell (put) stock or other assets at a fixed price on a future date — have been around for a long time. They are attractive to the option buyer because the buyer’s total risk is limited to the premium (price) paid for the option, unlike futures contracts, which create obligations to buy or sell at a future date.

Until recently, pricing these contracts presented great difficulty and controversy: how do you put a fair price on an asset deliverable at a future date, sometimes years ahead? In 1973 Drs. F. Black, M. Scholes, and R. C. Merton published a model for options pricing based essentially on Gaussian (random-walk) probability.

Black‑Scholes equation

The following is the famed Black‑Scholes equation for determining the “fair” price of a call or put option.

Derivation excerpt (some math)

Some of the involved mathematics may be gleaned from the following excerpt of its derivation:

Black‑Scholes closed form

And finally the formula itself:

where c is the cost of a call option and p is the cost of a put option. I won’t discuss the remaining parameters since we won’t be using this model directly; instead I present a simpler, intuitive approach that yields the same quantitative results.

An intuitive alternative: overview

As a purely theoretical inquiry, what mathematics lies behind the Black‑Scholes formula? I have long wondered and marveled at its complexity. After all, it is supposedly based on Gaussian probability (random walk), yet the closed form does not lend itself to immediate intuition. Is there a simpler, more intuitive approach that yields the same quantitative results and deeper insight into the rationale for options pricing? I propose one here.

I assume only a basic familiarity with put and call options: you need the spot price, strike price, duration, and (annualized) volatility. By “volatility” a financier specifically means the standard deviation of the spread of underlying stock prices over one year. (Interest rate is usually another input and will be touched on later.)

Note: the only option price that must be determined is the call price. The put price for the same inputs follows immediately from put–call parity: (call price) – (put price) = (spot price) – (strike price). I therefore limit the discussion to call options.

This approach applies, strictly speaking, to European options; however, price differences between European and American options are usually very small (well below 1%). An online Black‑Scholes options price calculator illustrates this. Another easy-to-use Black‑Scholes calculator for call options is available at mystockoptions.com.

Example 1 — an in‑the‑money call

Let us go by example. Assume a spot price of $20 and a strike price of $18. Suppose the volatility is $1 on a one-month basis (1/20 = 5% on that month). The monthly standard deviation is therefore $1; the annualized volatility relates to this by a factor of $\frac 1 {\sqrt12}$. The mean is $20 and the standard deviation for one month of data is $1. See Figure 1.

We now develop the data to price a 3‑month call. At three months the distribution spreads by a factor of $\sqrt 3$, so the strike price represents less than 2 standard deviations. The 3‑month distribution looks like Figure 2.

Notably, the $18 strike now corresponds to -2/$\sqrt 3$ = -1.15σ. The area under the probability density function to the right of that value is the probability that the price will exceed the strike after three months. But we also need the expected payoff conditioned on being in the money — i.e., the average of the x-axis to the right of the strike price after shifting x so that x = -1.15 corresponds to zero payoff.

Probability calculation

Let f(x) be the probability density function and F(x) the cumulative distribution (area from -$\infty$ to x). The average x (with the strike shifted to -1.15) is

$$ \bar x = \int_{-1.15}^{\infty} (x – (-1.15)) f(x) \, dx$$

$$ = 1.15\int_{-1.15}^{\infty} f(x) \, dx + \int_{-1.15}^{ \infty} x f(x) \, dx.$$

Using the relation x f(x) = – f'(x) we obtain

$$ \bar x = 1.15\int_{-1.15}^{\infty} f(x) \, dx ~ – \int_{-1.15}^{\infty} f'(x) \, dx$$

or

$$ \bar x = 1.15F(1.15) – [(f(\infty) – f(1.15)] $$

= 1.15F(1.15) + f(1.15)

Numerically, $\bar x$ = 0.875 + 0.206 = 1.212.

Finally multiply by the dollar scale factor corresponding to $\sqrt 3$: c = ($1.73)(1.212) = $2.09.

Comparing with the Black‑Scholes calculator (with the substitution $\sigma_{1 yr} = \sqrt{(12/3)}\sigma_{3 mos.}$ to annualize volatility) yields the same result: $2.09.

Example 2 — an out‑of‑the‑money call

Now consider an out-of-the-money case with these inputs:

• spot = $20
• strike = $25
• 1 month volatility = $2 = 1σ
• duration = 3 months

Figure 3 shows the distribution for this case.

The σ-equivalent of the strike is ($25 − $20) / (2.5 × 1.73) = 1.44. The expected payoff (shifted to start at 1.44) is

$$\int_{1.44}^{\infty} (x – 1.44) f(x) \, dx$$

= f(1.44) – 1.44(1 – F(1.44)) = 0.142 – 1.44(0.075) = 0.034. Multiply by ($2 × 1.73) = $0.11. The Black‑Scholes result is $0.19; small differences here are likely due to rounding and the particular volatility scaling used.

General formula

In general, the price of any call option in this framework is

$$ c = \left( \frac $ \sigma \right) \int_{\sigma_{strike}}^{\infty} (x – \sigma_{strike}) f(x) \, dx $$

with $\sigma = \sqrt{n/\sigma_v}$, where n is the duration in years and $\sigma_v$ is the annualized volatility. (The original derivation in the Black‑Scholes literature formalizes this scaling.)

I further tested the model with spot = strike = $20 and obtained $0.69, which agreed exactly with the Black‑Scholes calculation.

Interest rates and practical considerations

The prevailing interest rate should be included by discounting the expected payout to present value. The same treatment applies for inflation: account for it as appropriate in the present-value calculation.

Should this simplified model replace Black‑Scholes in practice? No. Black‑Scholes has been refined and validated over decades and can accommodate additional inputs. Online calculators and trading systems implement it efficiently. The point of this presentation is pedagogical: to provide an alternative, intuitive basis for evaluating options that is still rooted in Gaussian variation but is easier to understand.

Finally, how valuable is any model based on random theory? Historic failures such as the collapse of the hedge fund Long‑Term Capital Management in 1999 (in which Dr. Scholes participated) and the 2008 financial crisis illustrate the limits of stochastic models. Nassim Taleb’s The Black Swan argues that rare, severe events (fat tails) often break Gaussian assumptions.

Resources

• (1) Copy of rude man call option calculator
• Additional (duplicate) PowerPoint slides for the three embedded figures (better quality): blk-sch.ppt