Black-Scholes Derivation

The Step-by-Step Mathematical Journey

1

Start: Option Price Function

We begin with the option price as a function of time and stock price:

$$C = C(t, S_t)$$

Where:

2

Find dC Using ItΓ΄'s Lemma

Since $S_t$ is stochastic, we use ItΓ΄'s lemma to find $dC$:

$$dC = \frac{\partial C}{\partial t}dt + \frac{\partial C}{\partial S}dS_t + \frac{1}{2}\frac{\partial^2 C}{\partial S^2}(dS_t)^2$$
Key insight: The extra $(dS_t)^2$ term appears because we're dealing with stochastic processes. In regular calculus, $(dx)^2 = 0$, but in stochastic calculus, $(dW_t)^2 = dt$!
3

Model Stock Price Dynamics (dS)

Under the risk-neutral measure, stock prices follow:

$$dS_t = rS_t dt + \sigma S_t dW_t$$

Where:

And importantly: $(dS_t)^2 = \sigma^2 S_t^2 dt$

Critical Assumption: We're treating $\sigma$ as a constant! If volatility were stochastic (i.e., $\sigma = \sigma(t, S_t)$), we'd need to include $d\sigma$ terms, which opens a whole Pandora's box of complexity. This constant volatility assumption is one of the key limitations of the basic Black-Scholes model.
4

Substitute dS into dC

Now we substitute our stock dynamics into the $dC$ expression:

$$dC = \frac{\partial C}{\partial t}dt + \frac{\partial C}{\partial S}(rS_t dt + \sigma S_t dW_t) + \frac{1}{2}\frac{\partial^2 C}{\partial S^2}(\sigma^2 S_t^2 dt)$$

Collecting terms:

$$dC = \left(\frac{\partial C}{\partial t} + rS_t\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S_t^2\frac{\partial^2 C}{\partial S^2}\right)dt + \sigma S_t\frac{\partial C}{\partial S}dW_t$$
5

Apply Martingale Property

Under the risk-neutral measure, the discounted option price $e^{-rt}C(t,S_t)$ must be a martingale.

What this means: The expected return on the option must equal the risk-free rate. This eliminates arbitrage opportunities!

For a martingale, the drift term must be zero when we discount:

$$\frac{\partial C}{\partial t} + rS_t\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S_t^2\frac{\partial^2 C}{\partial S^2} - rC = 0$$

πŸŽ‰ THE BLACK-SCHOLES PDE! πŸŽ‰

$\frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 C}{\partial S^2} = rC$

This is the famous Black-Scholes partial differential equation that governs option pricing!

6

Set Up Terminal & Boundary Conditions

Now we need to solve this PDE! We know the option value at expiry and the boundaries:

For a European Call Option:
  • At expiry (t = T): $C(T, S) = \max(S - K, 0)$ - the "hockey stick" payoff
  • At S = 0: $C(t, 0) = 0$ - worthless if stock is worthless
  • At S β†’ ∞: $C(t, S) \approx S - Ke^{-r(T-t)}$ - behaves like stock minus discounted strike
For a European Put Option:
  • At expiry (t = T): $P(T, S) = \max(K - S, 0)$
  • At S = 0: $P(t, 0) = Ke^{-r(T-t)}$ - maximum value when stock worthless
  • At S β†’ ∞: $P(t, S) = 0$ - worthless when stock very expensive
7

The Mathematical Magic: Analytical Solution

Here's where Black, Scholes, and Merton performed mathematical wizardry. Using transform methods and the special structure of the PDE, they found an exact solution!

European Call Option:
$C = S_0 N(d_1) - Ke^{-rT}N(d_2)$
European Put Option:
$P = Ke^{-rT}N(-d_2) - S_0 N(-d_1)$

Where:

$d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$ $d_2 = d_1 - \sigma\sqrt{T}$

And $N(\cdot)$ is the cumulative standard normal distribution function.

πŸš€

The Revolutionary Impact

This changed everything!

Before Black-Scholes: "How much is this option worth?" β†’ Educated guesswork and intuition
After Black-Scholes: "How much is this option worth?" β†’ Plug numbers into formula, get exact answer!

Why it was revolutionary:

The Nobel Prize (1997): Myron Scholes and Robert Merton won for developing "a new method to determine the value of derivatives" - both the PDE derivation AND the analytical solution!
⚠️

When the Magic Breaks

The closed-form solution only works under specific conditions:

Reality Check: Once you relax these assumptions (American options, stochastic volatility, barriers), you lose the analytical solution and must use numerical methods like finite differences, Monte Carlo, or other computational techniques.

But the foundation remains: Even complex models like Heston and SABR build on the Black-Scholes framework - they just require more computational firepower!

βœ“

The Journey Complete

What we did:

  1. Started with $C = C(t, S_t)$
  2. Applied ItΓ΄'s lemma to get $dC$
  3. Defined stock dynamics $dS$
  4. Substituted $dS$ into $dC$
  5. Used martingale property to eliminate drift
  6. Arrived at the Black-Scholes PDE!