Constructing hourly index futures lattices from live volatility

A construction site for the book

This TradersWeek editorial treats an index-futures book as a construction site. The Black-Scholes Model supplies the continuous-dividend futures backbone. The binomial probability model lays each hourly node. The volatility forecast is the only input that should be redrawn as new prints arrive.

Recover a futures price first

Option-pricing constructions that treat an index as a continuously dividend-paying stock, including the Black-Scholes Model family, first recover a futures price. Only after that step do they value the option or the reachable price set. The recovered futures price is the starting point for the recomputed tree of reachable futures prices over a chosen time slice.

Place the hourly nodes

The binomial probability model for a futures step uses a equal to 1. The up factor is e raised to volatility times the square root of the time slice, with a matching down factor. Those exponential multipliers set by volatility and the square root of the step length define the next nodes. The up-move probability is (a-d)/(u-d).

What widens the estimated range

For a given time slice, annualized volatility has the largest effect on the width of the estimated range. The risk-free rate has little effect. Extending the slice widens the range only with the square root of time.

An hourly lattice from one session

A 16 September 1993 hourly lattice for December S&P 500 futures started from the prior 462.30 close. A 10 percent volatility forecast and a one-hour step of 0.000114 years produced first-hour nodes of 462.79 and 461.81, with multipliers near 1.001069 and 0.998932. Those same inputs gave an up-move probability of 0.50187 and a down-move probability of 0.49813, treated as nearly even for practical purposes.

Rebuilding the first two hours at volatility forecasts of 0.1, 0.2, 0.3 and 0.40 showed a one-hour band of 463.78 to 460.82 at 0.30. A fitted 0.284 matched the 10:30 print of 460.90.

After 11:30 printed 461.10, the back-fitted volatility forecast dropped to 0.04 and the next-hour band narrowed to 461.29 to 460.90. Later hours needed about 15 percent then 10 percent volatility to stay near the 460.35, 460.50 and 460.60 prints.

Redraw only the volatility forecast

A volatility forecast can be taken from history or back-fitted by iterating until two successive observed prices sit on the tree. That forecast is then used to rebuild a lattice when the latest interval implies a new volatility. The same construction can run on intervals from minutes to days. In this editorial reading, the latest print changes the volatility forecast, not the Black-Scholes Model backbone or the binomial probability model rules.