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Black-Scholes From The Ground Up

#Edcuation #Finance

About

The Black-Scholes model provides a theoretical framework for pricing European-style options.

European-style options are only exercisable at expiration

The central principle is that an option can be perfectly replicated by dynamically buying and selling the underlying asset and a risk-free bond in just the right proportions. This strategy, called continuously revised delta hedging, eliminates risk. Since arbitrage is not allowed, the option’s fair value must equal the cost of building this replicating portfolio.

The core idea

Why it matters

Goal

We want the fair price today of a European option.

At expiry $T$, payoff is known:

Question: What’s the correct price at time $t<T$?

Setup and definitions

Let

Stock price model

Brownian motion

Brownian Motion describes a stochastic process as a function of time where every step is random. Where a random variable is a single draw from a distribution, Brownian motion describes uncertainty over time.

A standard Brownian motion $W_t$ is

  1. $W_0 = 0$
  2. Increments are independent: $W_{t+\delta} - W_t \sim \mathcal{N}(0, \Delta t) $
  3. Paths are continuous but nowhere differentiable
  4. Variance grows linearly with time, $\mathbb{E}[W_t] = 0$, $\text{Var}[W_t] = t$

Why assume Brownian motion for stocks?

The intuition behind Brownian motion in finance is that stock prices result from the cumulative effect of many small, random buying and selling decisions. Each trade is like a small random step. In the limit of many rapid, independent steps, this process converges to Brownian motion, a continuous random process with normally distributed increments.

GBM setup

Assume geometric Brownian motion (GBM)

\[dS_t = \mu S_t\, dt + \sigma S_t\, dW_t\]

Interpretation:

A tiny change in the stock price over a small interval is:

Is GBM a good assumption?

Evidence for

Evidence against

Extensions

Ito’s Lemma (stochastic chain rule)

Ordinary calculus uses the chain rule. Because Brownian motion has variance that scales with $dt$, in stochastic calculus, we need an extra correction term.

If $f(S,t)$ is a smooth function of stock price and time and follows $dS_t = \mu S_t dt + \sigma S_t dW_t$, then Ito’s Lemma says:

\[df = \frac{\partial f}{\partial t}\,dt + \frac{\partial f}{\partial S}\,dS + \tfrac{1}{2} \frac{\partial^2 f}{\partial S^2} (dS)^2\]

With $(dW_t)^2 = dt$.

This “extra” second-derivative term is the core of stochastic calculus.

Apply Ito’s Lemma to option price

Let the option price be $C(S,t)$. Applying Ito’s Lemma:

\[dC = \Big( \frac{\partial C}{\partial t} + \mu S \frac{\partial C}{\partial S} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \Big) dt + \sigma S \frac{\partial C}{\partial S} dW_t\]

Interpretation:

Construct a hedged portfolio

We now form a portfolio that combines an option and the stock in such a way that the random part of its value change disappears.

\[\Pi = C - \Delta S\]

This corresponds to being long one option and short $\Delta$ shares of stock. The reason for the minus sign is that a call option already behaves like a partial long stock position (its Delta). To cancel that exposure, we must take the opposite side in the stock.

If instead we went long both the option and the stock, their risks would reinforce each other and stock rises increase the value of both, so the randomness would be additive rather than hedged. By choosing $\Delta$, we can cancel the random parts out, leaving the portfolio riskless.

Change in portfolio:

\[d\Pi = dC - \Delta\, dS\]

Substitute expressions for $dC$ and $dS$:

\[d\Pi = \Big( \frac{\partial C}{\partial t} + \mu S \frac{\partial C}{\partial S} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} - \Delta \mu S \Big) dt + \Big( \sigma S \frac{\partial C}{\partial S} - \Delta \sigma S \Big) dW_t\]

Eliminate randomness (Delta Hedging)

If we choose

\[\Delta = \frac{\partial C}{\partial S}\]

Then the $dW_t$ term vanishes, so the portfolio becomes riskless and has no randomness left in it

This step shows that derivative pricing can be anchored to no-arbitrage and hedging arguments, not investor psychology. That’s why Black–Scholes was revolutionary: it turned option pricing into physics-like mechanics instead of subjective forecasting.

Interpretation:

Implication

Now:

\[d\Pi = \Big( \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} \Big) dt\]

where

No-arbitrage condition

A riskless portfolio must grow at the risk-free rate $r$.

Portfolio value:

\[\Pi = C - S \frac{\partial C}{\partial S}\]

So:

\[d\Pi = r\Big(C - S \frac{\partial C}{\partial S}\Big) dt\]

Equating both expressions

We now equate the two expressions for $d\Pi$. We are explicity linking the changes in portfolio value with the risk-free return.

\[\frac{\partial C}{\partial t} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} = rC - rS \frac{\partial C}{\partial S}\]

Rearrange into the Black–Scholes PDE:

\[\frac{\partial C}{\partial t} + rS \frac{\partial C}{\partial S} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} - rC = 0\]

This is the central equation of the Black–Scholes model.

Solve PDE with boundary condition

Boundary condition at maturity:

\[C(S,T) = \max(S-K,0)\]

Solving the PDE gives the closed-form solution:

\[C(S,t) = S_t \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2)\]

with

\[d_1 = \frac{\ln(S_t/K) + (r + \tfrac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}, \quad d_2 = d_1 - \sigma \sqrt{T-t}\]

$\Phi(\cdot)$ = standard normal CDF.

Handwaving

Solving the PDE involves

Solving it isn’t super important though.

Interpretation

Option price structure

\[C(S,t) = \underbrace{S_t \,\Phi(d_1)}_{\text{benefit of stock if exercised}} - \underbrace{K e^{-r(T-t)} \,\Phi(d_2)}_{\text{cost of paying strike}}\]

What is $d_1$ and $d_2$?

Probabilistic interpretation

Put value follows from put–call parity:

\[P(S,t) = K e^{-r(T-t)} \Phi(-d_2) - S_t \Phi(-d_1)\]

Key insights

The big idea here is: stocks are dynamic, in $dS_t = \mu S_t dt + \sigma S_t d W_t$, the $dWt$ term is irreducible and can’t be eliminated. While options are also risky, under certain condititions, if you combine an option with the right fraction of stocks (Delta), then the random terms cancel out. That means that if you continuously rebalance via stock holdings (Delta hedge), you can replicate the option payoff with certainty. As a consequence, the option’s fair price is the cost of the replication strategy.

So why bother? Black-Scholes is a pricing model, not an investment strategy, but rather a pricing model for fair option prices. When an option is mispriced relative to the stock, then there is an arbitrage opportunity which forces option prices back into line. Black-Scholes tells us what happens in equilibrium.

Options allow traders to express their views on volatility, jumps, rail risk by trading options relative to that fair value.

The Greeks

The Greeks quantify how option value responds to risk factors.

Indicators and strategy

The Greeks tell us about the option market expectations, positioning, and risk sensitivity.

Some strategies may be

Practical Examples

NVDA Sept. 26 call option

The contract

The Greeks

Implications