LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Implied volatility is the volatility that makes the Black–Scholes formula match an option's market price. Line those numbers up across strikes and expiries and you get a surface. On a real equity book the surface is skewed and bent through time. A model that forces one volatility onto every contract cannot quote a barrier, or any other exotic, without disagreeing with the vanillas already trading.
Why it matters
Black–Scholes prices a European call or put from spot, strike, the riskless rate, volatility, and time to expiry. The market or the contract fixes four of those inputs, and supply and demand fixes the premium. Volatility is the one nobody observes. A backward look at returns is not a traded level. Backing it out of the premium is what "implied" means.
Change the strike or the expiry and that number changes. Black–Scholes assumes it does not. Price a book from one constant and a counterparty can buy the cheap contract, sell the rich one, and hedge the direction. In the lecture's simulation that trade has a positive average gain, the desk's wealth falls, and the counterparty's rises. Matching the vanilla surface closes that hole. Credit, liquidity, and a maker's margin still sit on the quote. They do not repair a book that disagrees with itself.
The idea in plain words
With the other inputs fixed, Black–Scholes depends on volatility alone, and that map can be inverted. In the lecture a guess of 0.20 produced a premium of 10.45 and missed the market. A guess of 0.50 overshot. Minimize the squared gap between the two premiums. The volatility at the minimum is the implied volatility. Raise that example's premium to 15 and the result is 32.4 percent. Drop the premium to 10 and the implied volatility falls. The two labels name one point when the map and its inverse are both well defined.
A surface is those points on one underlying. The lecture uses five strikes and five maturities. At-the-money calls and puts share an implied volatility in theory, by put-call parity. Equity puts out of the money sit above calls out of the money, which the lecture reads as downside fear. A commodity book can lean the other way. Strikes are sometimes written as moneyness, such as 80 to 110 percent of spot. A strike slice is the skew. An expiry slice is the term structure. Neither is flat.
How to use it
One volatility cannot match two premiums that imply two different volatilities. The lecture's sketch uses calls struck at 90 and at 110. A trader who buys one from the desk, sells the other, and hedges direction keeps the volatility gap. Simulated many times, that trade shows a positive average gain for the trader. The at-the-money number, the put number, and the call number still disagree.
The usual repair is a model that can bend. Local volatility makes volatility a function of spot and time. The lecture names a discrete form associated with Emanuel Derman and a continuous form associated with Bruno Dupire. Stochastic-volatility examples are Heston, SABR, and jump diffusion. In the lecture's Heston setup, variance is its own mean-reverting equation, with a speed, a long-run level, and a volatility of volatility. The two Brownian drivers are correlated, which is the leverage effect in that lecture: volatility and price tend to move oppositely, and not symmetrically.
Before fitting, a 90 put and a 110 call with spot at 100 imply 22.71 percent and 17.54 percent. Skew is already there. Calibration picks parameters so each vanilla, passed back through Black–Scholes, matches the market grid. Those parameters can then price the barrier.
A usable model reproduces the vanilla surface, stays consistent on extrapolated prices, and quotes quickly. Ten million paths over an hour can be tidy and still useless. It should assume little about the true distribution. Constant volatility fails that test as a trade against the desk.
What usually goes wrong
One miss treats a single implied volatility as a property of the underlying. Another slides that number up or down and expects a skew. A constant lifts the whole surface. It does not bend it. A rich model that cannot be fit before the client leaves fails the other way. Fitting removes one arbitrage. It does not set credit, liquidity, or the maker's margin, and it does not choose the hedge.
Questions people ask
How do you imply volatility without a closed inverse?
Hold the other four inputs fixed and minimize the squared gap between the market premium and the Black–Scholes premium. The minimizing volatility is the implied volatility.
Why can one volatility be arbitraged?
Strikes that the market prices with different implied volatilities will not both match the market under one volatility. The lecture's 90 and 110 calls are the picture, and the simulated wealth paths favor the counterparty.
What did the unfitted Heston pair show?
With spot at 100, the 90 put and the 110 call implied different volatilities, 22.71 and 17.54 percent. Skew appeared before any fit to the market grid.