Critiquing Black-Scholes early exercise on futures options

An audit, not a prettier forecast

The archive rebuilds a futures-option premium twice: once with a Black-Scholes Model closed form, and once by working backward through a binomial tree that uses the same price, time, rate and volatility inputs.

Editorial interpretation: treat that pairing as option premium analysis used as an assumption audit. Implied volatility analysis enters only as the shared volatility input both calculations must hold fixed. The question is whether early-exercise rights and futures financing make the European closed form cheap on deep in-the-money options, not whether the tree is a better forecast.

A one-step lattice and its delta

A one-step lattice that sends the underlying from 100 to 105 or 95 with equal probability values an at-the-money call at 2.5 percent of par, because the two expiry payoffs are 5 and 0.

In that one-step example, delta is 0.50. The 10-point gap between 105 and 95 in the underlying maps to a 5-point gap in terminal call values, so a 5-point underlying move changes the premium by 2.5.

A five-point tree and time decay

Working backward through a multi-period five-point tree prices that same call at 3.75 percent of par before financing. Discounting at an 8 percent short rate for 0.50 years lowers the premium to 3.60.

Holding the underlying at 100, the tree premium falls from 3.75 to 2.5 over two periods and then to zero at expiry, a faster time-value rundown than the paths held at 105 or 95.

When the discrete tree meets the closed form

Once the number of binomial intervals is large, about 25 to 30 or more, the discrete tree is described as approaching the same results as the Black-Scholes formula for options on futures.

A closed-form futures-option reading

With futures at 61, strike 60, 91 days to expiry, 16 percent annualized volatility and an 8 percent short rate, the closed-form futures-option formulas produce a call of 2.42, a call delta of 0.5868, a same-strike put of 1.44 and a put delta of -0.3933.

Editorial interpretation: those four readings are the option premium analysis table the tree is asked to challenge, using the same volatility figure an implied volatility analysis would lock in.

Early exercise and futures financing

Option buyers pay cash while futures can be margined with interest-bearing collateral, so a deep in-the-money American futures option can be worth more exercised than held once time value is nearly gone and delta is near 1.0.

Black-Scholes cannot represent early exercise, while a binomial tree can stop at an exercise node. On American futures options the closed form can therefore underprice in-the-money contracts by a tick or two relative to the tree.

Where the gap shrinks, and what speed costs

Gaps between the two premiums shrink as expiry approaches or as options move out of the money. Leftover out-of-the-money differences are attributed mainly to rounding rather than to different economics.

The early-exercise-aware tree is slower on period desktop hardware, up to about one to two seconds with many intervals, while the closed form remains the faster way to fill large premium tables.