The Black-Scholes Model: The Formula That Changed Finance
Wilmott calls Chapter 5 “without doubt, the most important chapter in the book.” He is not exaggerating. Everything before this was setup. Everything after this builds on what happens here. The Black-Scholes equation was first written down in 1969, the derivation was published in 1973, and finance has never been the same since.
The two key ideas: delta hedging and no arbitrage. Put them together and you get a partial differential equation that tells you the fair price of an option. Let us walk through it step by step.
The Setup: A Very Special Portfolio
We have a stock following our lognormal random walk:
$dS = \mu S , dt + \sigma S , dX$
And an option $V(S, t)$ whose value depends on the stock price and time. Think of V as a function we want to find.
Now build a portfolio $\Pi$ consisting of one option and a short position in $\Delta$ shares of the underlying:
$\Pi = V - \Delta \cdot S$
The change in portfolio value over a small time step:
$d\Pi = dV - \Delta \cdot dS$
Using Ito’s lemma on V, we get:
$d\Pi = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right) dt + \left(\frac{\partial V}{\partial S} - \Delta\right) dS$
Look at this carefully. There are two types of terms: deterministic ($dt$) and random ($dS$). The random terms are the risk. They contain the unpredictable stock price movement.
Delta Hedging: Killing the Risk
Here is the insight that changed everything. The random terms in the portfolio have a coefficient of:
$\frac{\partial V}{\partial S} - \Delta$
If we choose:
$\Delta = \frac{\partial V}{\partial S}$
then the random terms vanish. Zero. Gone. The portfolio becomes completely risk-free.
This is delta hedging. You hold one option and short exactly $\frac{\partial V}{\partial S}$ shares of stock. The correlation between the option and the stock cancels out all the randomness.
The catch: this $\Delta$ changes over time because it depends on S and t, which are always moving. You need to continuously rebalance. In practice, you rebalance frequently but not continuously. This is dynamic hedging.
Fun historical note: delta hedging was first described by Thorp and Kassouf in 1967. Ed Thorp was also the guy who figured out how to beat the house at blackjack. Same person was using the Black-Scholes formula to make money before it was even published.
No Arbitrage: The Punchline
After delta hedging, our portfolio changes by a purely deterministic amount:
$d\Pi = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right) dt$
A risk-free portfolio. Now apply the no-arbitrage principle: a risk-free portfolio must earn the risk-free interest rate r. No more, no less.
If it earned more, everyone would borrow money at rate r, invest in this portfolio, and make free money. If it earned less, you would short the portfolio and put the cash in the bank.
So:
$d\Pi = r \Pi , dt$
Substituting everything in and dividing by $dt$:
$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$
This is the Black-Scholes equation. Read it. Memorize it. You will see it again and again for the rest of this series.
What the Equation Tells Us
A few observations that Wilmott highlights:
The drift $\mu$ disappeared
Notice that the expected growth rate of the stock is nowhere in the equation. It dropped out when we eliminated the random terms. This is profound: two people who completely disagree about where the stock is heading will agree on the option price, as long as they agree on the volatility.
Why? Because we eliminated all risk through hedging. Since there is no risk, there is no reward for bearing risk. Only the risk-free rate matters.
It is a partial differential equation
The Black-Scholes equation is a linear parabolic PDE, closely related to the heat equation from physics. This is good news because heat equations are well understood and easy to solve numerically.
Linear means: if $V_1$ and $V_2$ are solutions, then $V_1 + V_2$ is also a solution. This lets you break complex payoffs into simpler pieces.
Final conditions determine the option type
The equation itself knows nothing about whether we are pricing a call, a put, or something exotic. That information comes from the final condition (what happens at expiry):
- Call: $V(S, T) = \max(S - E, 0)$
- Put: $V(S, T) = \max(E - S, 0)$
- Binary call: $V(S, T) = H(S - E)$ (Heaviside function: 1 if $S > E$, 0 otherwise)
Different payoffs at expiry, same equation.
Both the stock and cash satisfy it
As a sanity check: $V = S$ (the stock itself) and $V = e^{rt}$ (money in the bank) both satisfy the Black-Scholes equation. This makes sense. These are the simplest “derivatives.”
The Assumptions (and Their Limits)
Wilmott is honest about what goes into Black-Scholes:
The stock follows a lognormal random walk. Not exactly true in practice. But the equation still works if volatility depends on time or on the stock price. You just might not get neat closed-form solutions.
The risk-free interest rate is known. In practice, interest rates are themselves random. The fixed-income world deals with this complexity.
No dividends. Easy to fix. With a continuous dividend yield D, the equation becomes:
$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-D)S\frac{\partial V}{\partial S} - rV = 0$
Delta hedging is done continuously. Impossible in practice. You hedge in discrete time. The lower the transaction costs, the more frequently you can rehedge. Chapter 47 covers this.
No transaction costs. In reality, there is a bid-offer spread on the underlying. Sometimes this matters a lot, sometimes not.
No arbitrage. Of course there are arbitrage opportunities in the real world. People make a lot of money finding them. But the assumption is about model-dependent arbitrage. If the model is right, there should not be arbitrage. If there is, the model is wrong.
Currency and Commodity Options
The equation extends naturally to other asset types:
Currency options: Replace dividends with the foreign interest rate $r_f$:
$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-r_f)S\frac{\partial V}{\partial S} - rV = 0$
Commodity options: Replace dividends with the negative cost of carry q:
$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r+q)S\frac{\partial V}{\partial S} - rV = 0$
Options on futures: Actually simpler than the original equation because the cost of carry is already baked into the futures price.
Other Derivations
Wilmott mentions that Black-Scholes can be derived in other ways:
- Martingale approach: The option value is an expectation under a special “risk-neutral” probability measure. Useful for Monte Carlo simulations.
- Binomial model: A discrete-time version using up/down trees. Take the limit and you get Black-Scholes.
- CAPM/Utility: From portfolio theory. Options are functions of their underlyings, so they do not add new risk-reward possibilities.
All roads lead to the same equation. That is reassuring.
Replication and Complete Markets
Another way to think about delta hedging: instead of hedging an option with stock, you can replicate an option by trading stock and cash. Hold $\Delta$ shares and some cash, rebalance continuously, and at expiry you will have exactly the option payoff. This is the idea of a complete market, where every derivative can be replicated with the underlying.
If options can be perfectly replicated, they are technically redundant. Why buy an option when you can manufacture its payoff yourself? In practice, transaction costs, stochastic volatility, and other real-world frictions make perfect replication impossible, and that is why options markets exist.
The Takeaway
Chapter 5 gives us the most important equation in quantitative finance. The logic is clean:
- Build a portfolio of one option and some stock
- Choose the stock quantity ($\Delta = \partial V / \partial S$) to kill all randomness
- A risk-free portfolio must earn the risk-free rate
- This gives us the Black-Scholes PDE
The drift disappears, volatility is the only market parameter that matters, and the equation is the same for all payoffs. The specific option type is determined by the boundary condition at expiry.
Every chapter from here on is either solving this equation, extending it, or questioning its assumptions.
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