A delta-hedged portfolio is riskless. A riskless portfolio earns $r$. Two lines of Itô and one no-arbitrage step give the most famous PDE in finance.
We want the fair price $V(S,t)$ of an option whose payoff at maturity $T$ depends on the underlying $S$. The underlying is assumed to follow geometric Brownian motion under the real-world measure:
$$dS = \mu S\,dt + \sigma S\,dW.$$
$\mu$ is the expected return, $\sigma$ the volatility, $W$ a standard Brownian motion. $V(S,t)$ is a deterministic function of two variables — the randomness it inherits comes entirely from $S$.
"The big leap is to notice that an option and its underlying share are two instruments driven by the same source of randomness. A suitable combination kills the noise."
— Wilmott, Ch. 6 (paraphrase)The miracle of Black–Scholes is that the answer doesn't depend on $\mu$ at all.
Form the portfolio
$$\Pi \;=\; V \;-\; \Delta\, S,$$
where $V$ is one long option and $\Delta$ is a (to be chosen) number of short shares. Both legs are exposed to $dW$. The plan is to choose $\Delta$ so the two exposures cancel.
Over an infinitesimal interval, with $\Delta$ held fixed (this is the self-financing condition):
$$d\Pi = dV - \Delta\,dS.$$
Means: any change in the value of $\Pi$ comes from the changes in the prices of its constituents, not from cash injected or withdrawn. We're not topping up the position from outside; we're letting the existing positions revalue. That's why we don't get a $\,S\,d\Delta\,$ term.
Now apply Itô to $V$, expand $dS$, and look at what coefficient sits in front of $dW$. If we can zero that coefficient with a clever choice of $\Delta$, the portfolio's value becomes a smooth function of time — no randomness left.
Itô's lemma applied to $V(S,t)$, with $dS = \mu S\,dt + \sigma S\,dW$:
$$dV = \Big(V_t + \mu S\, V_S + \tfrac{1}{2}\sigma^2 S^2\, V_{SS}\Big)\,dt \;+\; \sigma S\, V_S\,dW.$$
So
$$d\Pi = dV - \Delta\,dS = \underbrace{\Big(V_t + \mu S\, V_S + \tfrac{1}{2}\sigma^2 S^2\, V_{SS} - \Delta\,\mu S\Big)\,dt}_{\text{deterministic}} + \underbrace{\Big(\sigma S\, V_S - \Delta\,\sigma S\Big)\,dW}_{\text{noise}}.$$
The single coefficient of $dW$ is $\sigma S(V_S - \Delta)$. Choose
$$\boxed{\;\Delta = \frac{\partial V}{\partial S}\;}$$
and the $dW$ term vanishes. This is delta-hedging: hold $V_S$ shares short against one long option and your portfolio's tiny moves are deterministic.
With $\Delta = V_S$, the $\mu$ term in $d\Pi$ becomes $\mu S\,V_S - V_S \mu S = 0$. The drift of the underlying never appears in the equation for $V$. Two market participants with opposite views about $\mu$ will agree on the option price.
After the hedge, $\Pi$ is risk-free over the next instant. By the no-arbitrage principle (deck 01) a riskless portfolio must earn the risk-free rate:
$$d\Pi \;=\; r\,\Pi\,dt \;=\; r\,(V - \Delta\,S)\,dt.$$
Setting this equal to the deterministic part of the previous slide:
$$\Big(V_t + \tfrac{1}{2}\sigma^2 S^2 V_{SS}\Big)\,dt \;=\; r\,(V - V_S\,S)\,dt.$$
(The $\mu S V_S - V_S \mu S$ terms have already cancelled.) Rearranging into a tidy form:
Combined three things — Itô's lemma, a self-financing delta hedge, and a one-step no-arbitrage argument — to derive a deterministic equation for the option price. Not a single probability or utility function appeared.
If somebody quotes you an option price that violates this equation, you can hedge against them and bank a riskless profit, in the same spirit as the forward-pricing argument from deck 01 — but now continuously.
Collecting:
$$\boxed{\;\frac{\partial V}{\partial t} \;+\; \tfrac{1}{2}\sigma^2 S^2\,\frac{\partial^2 V}{\partial S^2} \;+\; r S\,\frac{\partial V}{\partial S} \;-\; r V \;=\; 0\;}$$
This is a linear, second-order, parabolic PDE for the function $V(S,t)$ on the region $S > 0$, $0 \le t \le T$. It is solved backwards in time: a payoff at $t = T$ is specified, and we ask what value at earlier $t$ is consistent.
| Term | Interpretation |
|---|---|
| $V_t$ | Time decay — how the value changes as the clock ticks even with $S$ frozen. |
| $\tfrac{1}{2}\sigma^2 S^2 V_{SS}$ | Gamma term — the value gained from convexity in $S$ under random shocks. The Itô correction. |
| $r S\, V_S$ | Cost-of-carry — how the value drifts under the risk-free growth of the underlying. |
| $-rV$ | Discounting — what the option must lose to keep pace with a riskless cash account. |
The PDE knows about $\sigma$ and $r$, and about nothing else from the real-world model. Same equation for calls, puts, digitals, barriers; only the terminal condition changes.
A PDE without boundary data has many solutions. The contract specification supplies what the equation needs.
Terminal payoff:
$$V(S, T) = \max(S - K, 0).$$
Boundaries:
Terminal payoff:
$$V(S, T) = \max(K - S, 0).$$
Boundaries:
A binary call pays $\mathbf{1}_{S>K}$ at $T$; a barrier option dies (or activates) when $S$ first touches a level; an Asian option depends on the path average. All have the same PDE, different terminal/boundary conditions. The PDE machinery is more powerful than any one closed form.
The PDE is linear, so $V_C - V_P$ satisfies it with terminal payoff $\max(S-K,0) - \max(K-S,0) = S - K$. The function $S - K e^{-r(T-t)}$ solves the PDE with that payoff — hence the identity $C - P = S - K e^{-r(T-t)}$.
The derivation is short. Every step has an assumption hiding in it.
| Assumption | What it bought us | Where it bites |
|---|---|---|
| Continuous trading, no transaction costs | Self-financing portfolio at zero cost; the $\Delta$ can be rebalanced freely. | Real markets have spreads and discrete rebalancing — gamma profit/loss. |
| Constant $\sigma$ | The PDE has only one extra parameter beyond $r$. | The market disagrees — implied vol has a smile (deck 07/08). |
| Constant $r$ | Discounting is trivial; futures = forwards. | Long-dated and rate-sensitive products need stochastic rates (deck 12). |
| Lognormal $S$, no jumps | Geometric BM has tractable closed forms. | Tails of real return distributions are heavier; crashes happen. |
| No dividends (or known yield $q$) | $rS V_S$ as the carry term. | Discrete dividend dates need a model patch. |
| Short selling and infinite divisibility | Both sides of every trade can always be executed. | In a crisis, no. |
"The Black–Scholes model is not a description of the market. It is a self-consistent language the market uses to quote options — a way of expressing the price in units of implied volatility." The PDE is right; the assumptions are wrong on purpose, and we manage that with the Greeks (deck 07).
The Black–Scholes PDE can be transformed into the standard heat equation by a sequence of substitutions.
The heat equation has a Green's function (the Gaussian kernel) and the solution is a convolution of the initial condition with it. Backing out the substitutions gives the closed-form European call:
$$C(S,t) = S\,N(d_1) - K e^{-r(T-t)}\,N(d_2),$$
$$d_{1,2} = \frac{\log(S/K) + (r \pm \tfrac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}.$$
Calls, puts, binaries, forward starts — if the payoff is simple enough. Deck 07 starts here.
Barriers and lookbacks reduce dimension by exploiting the scale invariance of the PDE.
Finite differences (decks 13–14), binomial trees (deck 03), Monte Carlo. Same PDE, different machine.
The PDE derivation is not the only road to the BS formula. Three independent routes agree exactly:
What we just did. Continuous delta-hedging plus no-arbitrage gives the BS PDE; solve it.
Change measure (Girsanov) so $S$ has drift $r$. The price is the discounted expected payoff:
$$V_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[\,\mathrm{payoff}(S_T)\,].$$
Take the Cox–Ross–Rubinstein discrete tree (deck 03) and let the step size go to zero with $u/d$ scaled by $\sigma\sqrt{\delta t}$. The limit is the BS formula.
Each derivation emphasises a different idea: PDE makes the role of hedging explicit; martingales make the role of measure change explicit; binomial limits make the role of replication explicit. They are different vantage points on the same theorem.
Black and Scholes' original paper used a CAPM-style equilibrium argument; Merton later supplied the cleaner hedging argument. Both are in the historical record. All three give $V_0 = SN(d_1) - K e^{-rT} N(d_2)$ for the European call — the formula is robust to how you derive it.
Slide $K$, $T$, $\sigma$, $r$ and toggle call/put. The left subplot shows $V(S,t)$ as a heatmap over the $S\!\times\!t$ rectangle. The right subplot shows the family of curves $V(S,t)$ for a handful of times $t$, watching the smooth curve harden into the kinked payoff as $t \to T$.
The heatmap is the price surface; lighter values are higher. Note how the contour lines bend toward the strike as $t\to T$ — that is the value collapsing onto its intrinsic kink. Push $\sigma$ up and the curves visibly puff out for $t < T$; push $T$ down and they crowd against the payoff. Push $r$ up and (for a call) the curves shift up — the cost-of-carry effect.