Volatility

Pick a series of returns. If the months sit close together, the width is small. If the months include large gains and large losses, the width is large. Volatility is that width. It is not a direction, and it is not a promise that the next month will match the last one.

Historical volatility is the width of a past window you chose. The last year of monthly returns is one window. The last sixty trading days is another. Change the window and the number can change, because you are describing a different slice of the past.

The name says where the number came from: history. It does not say what the next window will do.

Realized volatility is that same backward measurement, stated as a standard deviation. On the return risk page, standard deviation is how far the returns sat from their average. Annualized, that width is the realized volatility of the sample.

A classroom series can have a 20% realized volatility. In the window you measured, the returns bounced enough to be described by a 20% width. The next window is a new sample. Twenty percent does not set it.

Implied volatility is solved from an option price. You start with the premium and ask: what width, inside a pricing model, would produce that premium? A rich premium solves to a wide implied volatility. A cheap premium solves to a narrow one.

Put the two classroom numbers side by side. Realized volatility of the recent sample is 20%. The option’s implied volatility is 28%. The option price is using a wider number than the recent sample. That gap describes two measurements. It is not a prediction that the future will land on either number.

The volatility surface lesson works the same solve with a premium: a model at 25% prints $5.20, and an option at $5.80 is consistent with about 29% under those assumptions.

Hold the underlying fixed and read implied volatility by expiration. A classroom term structure is 7 days at 35%, 30 days at 29%, 90 days at 26%, and 1 year at 28%. The stock is the same. The expiration changes the number. That relationship is the term structure.

Upward sloping means longer-dated options carry a higher implied volatility. Downward sloping means the short expirations are wider, as in the left side of this sketch. Humped means a middle expiration sits above its neighbors.

Hold the expiration fixed and change the strike. A classroom smile is $80 at 34%, $100 at 25%, and $120 at 35%. The wings sit above the middle. Plot implied volatility against strike and the line bends up on both sides. That bend is the smile.

One flat volatility for every strike is a teaching shortcut. A listed market can price a different width at each strike.

Skew is the same strike axis when the curve is not symmetric. A classroom equity skew is $80 at 38%, then down through $100, and $120 at 22%. Implied volatility falls as the strike rises. The low strike is wider than the high strike.

Smile names the bend. Skew names the slope. A listed curve can carry both. The picture puts the classroom smile and the classroom skew next to each other.

A surface puts the two axes together. Strike runs one way. Expiration runs the other. Each cell is an implied volatility. There is no single width for the stock.

On the surface lesson, the 30-day $90 cell is 32%. That is one option: one expiration, one strike. The cell next to it can be a different number. Read the map there instead of treating 28% as the volatility of the name.

The widths above can be treated as fixed inputs. Volatility of volatility drops that shortcut. The width itself wanders. A quiet month in the width, and a wild month in the width, are different worlds even when the average width matches.

That moving width is the stochastic-volatility step on the ladder. SABR is one classroom story for a forward rate whose volatility also moves. Nu, on that page, is how fast the volatility itself moves.

Vega is how much an option’s value changes when implied volatility changes, with the underlying held still. On the Greeks page, a call with vega 0.12 gains about $0.60 per share if implied volatility rises 5 points, and can lose that amount if implied volatility falls 5 points.

The asset-liability book later in this track uses a teaching vega of $150,000 per volatility point. Five points on that coefficient is $750,000. Ten points is $1.5 million. The coefficient is labeled so the arithmetic is visible. It is not a solved market vega.

A forecasting model can print a guess of future realized volatility. Give that guess a classroom number, 22%, and set it beside the other two. Realized volatility of the past window is 20%. Implied volatility in the option is 28%. The model’s guess of what realized volatility will be is 22%.

Three numbers, three questions. What already happened? What width is inside the premium? What did a model print for later? This page does not run that model. The 22% is an illustration of the question. It is not what will happen.

Equity volatility on this page is a percent of the price. Rate volatility is quoted two ways, and the two ways are different widths.

Normal volatility is a width in rate points. Twenty-five basis points is 0.25 percentage points of rate. Lognormal volatility is a percent of the rate. Twenty percent of a 4% swap rate is 0.80 percentage points, about 80 basis points. Eighty is wider than twenty-five. The quote type has to travel with the number.

The swaption volatility page is where that pair is used on a grid of expiries and tenors.

A volatility shock changes the width and leaves the underlying where it is. The stock did not move. The swap rate did not move. The option, the guarantee, or the hedge still changes value, because vega is not zero.

On the market shocks page, a volatility shock sits beside rate, credit, and equity shocks. On the shock lab, +10 points times $150,000 is the $1.5 million hedge line. Move the slider and that line changes. The underlying shock can stay at zero.

Volatility changes the value of anything with optionality. A listed option, a payer swaption, a guarantee, and a policyholder choice all depend on a width. If that width rises, the optionality is usually worth more. If it falls, the optionality is usually worth less. The underlying can be unchanged while that value moves.

Hedge risk therefore includes the width, not only the stock and not only the rate. A hedge that offsets a parallel rate move can still gain or lose when implied volatility moves. Read vega, the surface, and the shock line before treating the hedge as finished.