The term structure of volatility

Implied volatility varies by expiry as well as by strike, and the variation by expiry has a structure worth extracting. Take an illustrative set of at-the-money figures: one month 18.2 percent, three months 20.1 percent, six months 21.4 percent, twelve months 22.3 percent.

The first thing to notice is that these are not four forecasts of four different dates. Each one is an average over its own horizon. The three-month figure contains the first month inside it, so the two numbers overlap and cannot be read as independent statements.

Extracting a statement about the stretch in between uses the additivity from lesson 3. Variance multiplied by time adds, so the three-month total minus the one-month total leaves the total for months two and three. Compute it: 0.182 squared times one twelfth is 0.0027603, and 0.201 squared times a quarter is 0.0101003. The difference is 0.0073399, spread over two months, giving 0.0440395 in annual terms. Its square root is 20.99 percent.

Doing the same for the later segments gives 22.63 percent between three and six months and 23.17 percent between six and twelve. So the curve is not merely rising; it is pricing successively higher volatility for successively later stretches, which is what an upward slope means once the overlap is removed.

For collateral the relevant end of this curve is the front, because a lender's assumed exit is measured in sessions rather than months. An upward-sloping structure puts the smallest figure on the surface closest to the lender's horizon, which means a haircut built on the near-dated point is the smallest one the surface supports. It is also the point that moves the most, which is lesson 13.