The picture to hold in mind
Shares + cash = the derivative, always
Here's the picture to keep in your head for everything below: hold shares and in cash, keep adjusting the mix as prices move, and that little portfolio tracks the derivative's value at every instant. No guessing, no risk, no cash sneaking in from outside. Everything below is just making that precise.
Three ideas you need in your pocket
Before we touch the proof itself, get comfortable with three requirements a trading strategy has to satisfy. They sound almost too simple to matter — but once they click, the five-step proof is really just bookkeeping.
Rule 1
No peeking at the future
A strategy is previsible if you decided how much to hold at time t using only what you knew before t. It's like placing your bet the instant before the number is called, not after you've already seen it land.
Rule 2
No money sneaks in or out
Self-financing just means you're not cheating. Once the strategy starts, you don't add outside cash and you don't quietly pull any out — every bit of movement in the portfolio has to come from the market itself.
Rule 3
Same bill, every time
A strategy replicates a derivative if it hands over exactly the payoff that was promised at the end. Not roughly, not on average across a few tries — exactly, every single time.
Consequence
A complete market
Put those three together for every possible payoff, and you've got a complete market — one where every derivative has exactly one price that doesn't leave an arbitrage sitting around for someone to grab.
Equivalent measures: same possible, different likely
Before the two engines can do their work, we need to be precise about what "changing probabilities" is allowed to mean. This is the definition every result below leans on.
Two measures P and P′ are equivalent — written P′ ∼ P — if they agree on which outcomes are possible, even when they disagree on how likely those outcomes are:
In words: for every event A, P′ assigns it zero probability exactly when P does too. Anything P calls impossible, P′ also calls impossible, and vice versa — everything else can be reweighted however you like.
That's exactly why equivalence is awkward to prove directly — checking the definition means checking every event in the sample space lines up. The Cameron–Martin–Girsanov theorem below sidesteps that by handing you an explicit Q, instead of asking you to verify the definition from scratch.
A continuous-time warm-up
Let be a standard Brownian motion under P, so for all t. Let be a Brownian motion with drift. Is there a measure Q, equivalent to P, under which is itself a standard Brownian motion?
The answer: yes if σ = 1, no if σ ≠ 1 (proving that needs the theorem below, so we won't do it here). That one asymmetry is the whole ballgame: a change of measure can move the drift wherever you like, but it can never touch the volatility.
What actually does the heavy lifting
Two theorems carry the whole argument. Neither is proved here — both proofs are beyond what this method covers — but you need their exact statements, not just a one-line summary, because the fine print is where the five steps get their power.
Engine 1 — the Cameron–Martin–Girsanov theorem
This is the theorem that lets us swap the drift of a Brownian motion for a more convenient one, without changing what's possible.
Forward direction
Let be a standard Brownian motion under P, and let be any previsible process. Then there exists a measure Q, equivalent to P, under which the shifted process below is itself a standard Brownian motion.
Converse direction
Run it backwards: if is a standard Brownian motion under P and Q is any measure equivalent to P, some previsible must exist that produces exactly that same shift.
Put the two directions together and you get the fact that Step 1 runs on: every equivalent change of measure is a drift shift of exactly this form — so it can dial to anything you like, but the volatility is untouched by construction.
Why we actually need it
Here's the concrete problem Girsanov solves. Model the share price as geometric Brownian motion under P:
Discount it and take the expectation under P:
That's only a martingale if μ = r — and there's no reason a stock's real-world drift should equal the risk-free rate. Pricing needs the discounted price to be driftless, so we use Girsanov to move to a measure Q where it actually is driftless. That's exactly what Step 1 of the five-step method is doing.
Engine 1, restated for diffusions
Everything above generalises to any diffusion, which is the form Steps 3–4 actually use. Write the model as an SDE under P:
If is a martingale under P, its drift term must vanish identically:
So "no drift" and "martingale" mean the same thing here — a fact Step 1 leans on directly.
Engine 2 — the martingale representation theorem
This is the theorem that actually produces the trading strategy in Step 4. Suppose is a martingale w.r.t. P, meaning for every t < s:
Suppose is another martingale w.r.t. P. The theorem states there exists a previsible process such that:
if and only if there is no other measure equivalent to P under which is a martingale. That condition is exactly what "complete market" meant back in the Three Core Rules section — it's why the representation only works when the market has no redundant sources of risk. (The theorem also holds when is a vector of martingales, i.e. several risky assets at once; the proof itself is beyond what we cover here.)
Engine 2, made concrete
In the diffusion setting, φ isn't just guaranteed to exist — it has a formula. Suppose both X and Y are martingales under P, driven by the same Brownian motion:
That ratio of volatilities — provided almost surely — is φ. This is exactly the object Step 4 identifies as the number of shares to hold.
Five moves that build the strategy
Go through these one at a time — don't rush step 4, that's where the actual payoff is. Click to expand each step. Each equation line drops in one at a time as you open a step, and the terms that just changed glow so you can see exactly where the argument moves.
φ has a name you already know: Delta
Here's a nice payoff for all that work: run the same five steps on an ordinary call option, and just turns out to be — the Delta traders have been tracking this whole time anyway. Try dragging the share price below and watch the portfolio rebalance itself.
Strike is fixed at K = 100 and the risk-free rate at r = 5%, so you can isolate one variable at a time.
Δ = φ_t–
ψ_t · B–
V_t–
Dividends swap one asset, not the method
One wrinkle worth knowing about: if the share pays a continuous dividend, its raw price isn't quite what you're trading anymore — value keeps leaking out as dividends. The fix is smaller than you'd expect, and it only touches step 1: trade a version of the share with the dividends reinvested, then convert back to real shares right at the end.
show the dividend-paying version
Ordinary case — trade the share itself.
Dividend case — you trade the reinvested version instead, then convert back with to get real shares of .
The alternative is guessing
Fair question at this point: why go through five steps when there's a shorter route? You could write down a differential equation and guess a solution that fits the boundary conditions instead. Here's honestly how the two compare.
You don't have to guess
With the differential equation, you propose a solution and then check that it fits the boundary conditions — that's a real guess. Here, the price just falls out as an expectation you can compute directly.
The hedge comes along for free
You also don't have to go derive Delta separately afterward. The same steps that prove the price is right hand you the exact hedge as a side effect.
Handles the awkward payoffs
Path-dependent payoffs — Asian options and their relatives — are where this really pays off. They fall out of the same five steps just fine, while the differential-equation route can genuinely struggle with them.
But to be fair to the other side
For a quick, basic hedging argument, guessing a differential equation is often just faster to state and easier to follow at a glance. This route takes more setup to get going.