LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A barrier option is a European payoff with an extra rule: the path can cancel the payout. An up-and-out call pays only if the underlying finishes between the strike and the barrier and never touches the barrier. The desk's recipe is to pick a model that can match listed option prices, price the vanillas, fit the parameters, simulate the barrier, and only then adjust the quote.
Why it matters
A European call or put depends on the underlying at expiry alone. A barrier, a cliquet, and other exotics depend on the route. Black–Scholes cannot carry that book. Its implied-volatility surface is flat, because one volatility prices every strike and expiry. The market surface, on the lecture's five-by-five grid, is not flat. Raising or lowering the single volatility lifts or drops the whole sheet. It does not create skew. Quote the flat sheet against a skewed market and a listed put can be far too cheap. The counterparty keeps the difference. Fitting a flexible model to the listed grid is how the desk avoids handing over that difference before it even prices the exotic.
The idea in plain words
One up-and-out path finishes above the strike and pays the usual call amount. Another finishes above the strike but touched the barrier, so it pays zero. The contract is cheaper than the vanilla because some paying paths now pay nothing. A hypothetical in the lecture puts a stock near 170, a hope of 190 within ten days, and a barrier at 200. The buyer accepts knockout risk for a lower premium.
Black–Scholes means one thing here: the underlying follows geometric Brownian motion, so volatility is a constant. Local volatility replaces that constant with a function of spot and time, which can bend the surface. The Heston setup gives variance its own mean-reverting randomness and extra parameters, including a long-run variance. In the lecture, moving that long-run variance from 0.06 to 0.08 lifts the model skew. Before the fit, a Heston slice can already look closer to the market than the flat Black–Scholes line. It is not the market until the parameters are chosen.
Three routes price the vanillas used in the fit. Hedge variance risk, then spot risk, and solve the partial differential equation by finite differences. Use the Carr–Madan fast Fourier transform when the characteristic function is known. Or average discounted simulated payoffs. The lecture uses simulation for the exotic because you can see barrier hits. Calibration searches for the parameter set that matches the market surface. A fresh 100,000-path run on every trial does not scale, so a faster vanilla method is preferred for the fit.
How to use it
Write the up-and-out payoff as a risk-neutral expectation inside the fitted Heston model, then estimate it by simulation. Keep paths that finish between the strike and the barrier without a hit. Throw out paths that touch the barrier, and paths that finish below the strike. Discount the payouts and average them.
Ten paths are not a price. The lecture's first ten-path run printed 0.4273. The next printed 1.0078. More paths tighten the average. In a later picture the barrier came out near 2.21, while a vanilla call in the same Heston setup came out near 10.3, and a 10,000-step vanilla run near 10.6. The gap is the discount for knockout risk. Those figures are one parameterized example, not a market quote.
The average is still not the number sent to the client. The lecture adds a read of current conditions, a charge for counterparty risk, a liquidity charge, and a margin. The aim of that stack is a quote which, once hedged, has a positive expected value for the desk across many trades. The aim fails if the vanillas underneath were inconsistent, because the theoretical margin is then an arbitrage the other way.
What usually goes wrong
Simulating the barrier inside untuned Black–Scholes brings back the flat surface. Stopping after one visual tweak leaves the model skew only roughly on the market. Monte Carlo on every calibration trial is too slow. Sending the raw average omits the charges. Long-memory models such as rough volatility slow the loop further, because each trial rebuilds a covariance. The lecture notes SABR-type asymptotic shortcuts and neural nets as other routes.
Questions people ask
Why price the vanillas before the exotic?
The listed prices are the only market you can fit. The barrier is not on that grid. Without the fit, the barrier number is not tied to the options a counterparty can already trade.
Why did ten paths give 0.43 and then 1.01?
Each average is one small sample. The lecture treats both as too noisy to quote and adds paths until the average settles.
Is the model average the client quote?
No. Counterparty risk, liquidity, and a margin are added after the average. Those add-ons do not fix a model that misses the listed surface.