LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Black-Scholes and the fundamental theorem of asset pricing look like two ways to price a European option. One solves a hedging equation. The other takes a discounted risk-neutral average of the payoff. Under the model's own assumptions they are the same function. The link is the Feynman-Kac theorem, borrowed from physics. Matching on a chart is not that proof, and neither argument claims the real market obeys the assumptions.
Why it matters
A European contract depends on the underlying only at a future date T, the maturity. The payoff is a max function of that future price and the strike. If the spot finishes at 102.5 in the lecture's sketch, the per-share payoff is 102.5, and listed contracts are typically in lots of 100. You do not know the future spot, so you cannot plug it in today. You need a function V that prices the contract at any earlier time, not only at expiry.
Violated assumptions are expected. The lecture's point is different: assumptions do not by themselves make an argument internally consistent. The question is whether the hedging price and the no-arbitrage expectation are the same object inside one framework.
The idea in plain words
The Black-Scholes route builds a replicating portfolio with no arbitrage and no transaction costs, arrives at a partial differential equation, and solves it. The solution uses the spot you can see now, plus the other inputs, and returns a price at the current time t. At maturity you do not need it. The payoff is the price.
The fundamental theorem says that if there is no arbitrage, the fair present value of the claim is the discounted risk-neutral expectation of that payoff. Simulate many paths, evaluate the payoff at T, discount, and average. As the path count grows, the error in the lecture's picture shrinks toward zero by the law of large numbers. Each fresh batch of paths settled on the same theoretical value.
Plot both recipes at the same spot and the same time, including t = 0.5 and a check at t = 0.2, and the prices lay on top of each other for the parameters tried. That is evidence for those inputs, not a proof for every input. The proof identifies both recipes with one V.
How to use it
The lecture's derivation starts from risk-neutral geometric Brownian motion: in expectation the spot earns the riskless rate. That is all "risk neutral" means here. Let the discounted value be the option price multiplied by a continuous discount factor back to today. Cash at one time is not cash at another, so both prices are compared in time-0 dollars.
Because the option price depends on the random spot and on time, the differential of that product uses Itô's formula, in product-rule form. Split the discount factor and the option value into two processes. The cross term from the purely deterministic discount factor is zero. The discount factor's own differential is minus the rate times the factor times dt.
Itô's formula on the option value drops the usual higher-order pieces: dt squared and dt times dW vanish, while dW squared behaves like dt. Substitute the risk-neutral spot dynamics and the same rules kill every term except the volatility contribution to the second derivative. Collecting terms, the drift piece is the Black-Scholes operator. If V solves that equation, the drift is zero. What remains is a Brownian shock. Its expectation is zero, so the discounted value is a martingale: given what you know at t, the risk-neutral expectation of the discounted value at a later time is the discounted value now.
What usually goes wrong
Reading a matching plot as a general proof is the first miss. The curves agreed on the selected spots and times, including a second maturity fraction, and the lecture still demanded an identity that covers arbitrary parameters.
A second miss is to hear "correct" as "true of listed prices." The lecture means correct relative to this framework and to the fundamental theorem. Real contracts face frictions and risks the replication story removes. Building a model is what you do because the world is not the assumption list.
Questions people ask
What is unknown at the start?
The spot at maturity. The payoff formula is known. Today's price is the hard object, a function of the spot you can observe now.
Do the two formulas merely look similar?
On the lecture's plots they matched, including at t = 0.5 and t = 0.2. The martingale argument is what identifies them for a general parameter set.
Does this say the market is arbitrage-free?
It says that, inside the model, the hedging PDE and the discounted risk-neutral payoff are one price. Market frictions sit outside that claim.