Constructing option premiums against a Black-Scholes baseline

A cash premium is not a valuation

Editorial framing: treat option construction as two layers. The first layer is an explicit Black-Scholes-family fair-value forecast from the underlying price, volatility, risk-free carry, remaining life and dividends. The second layer is an option-premium and implied-volatility check of how far the listed market sits from that baseline.

A listed option cannot be judged cheap from a small cash premium alone. The underlying’s volatility, the risk-free carry rate, remaining life and dividends given up versus holding the stock are required inputs.

The Black-Scholes baseline

The Black-Scholes Model, presented in 1971, is a lognormal pricing formula used as the quantitative baseline for a theoretical option value versus the listed premium. It replaced an earlier normal-distribution treatment with a lognormal security-price assumption and originally omitted dividends.

A later modification added dividend effects because options on dividend-paying securities were described as worth less than otherwise identical options on non-dividend names.

An empirical-distribution variant, motivated by more mass in the center, longer tails and a slight upward secular skew, was presented as tracking listed premiums more closely than the lognormal baseline.

Rich, fair or cheap, then the implied-volatility gap

Option-premium analysis compares a listed call or put with a model fair value after volatility, carry, dividends and remaining life are included, then labels the premium rich, fair or cheap. Model-based work in the archive used those labels as overvalued, fairly valued or undervalued, and noted that an already cheap or rich premium can become more so.

Implied volatility is the volatility and regime information embedded in the gap between the traded premium and the model value, including cases where a small cash price still overstates remaining time value.

After the October 1987 break, out-of-the-money put premiums were observed as high as 15 percent, and the model comparison was used to rank which of those premiums looked most stretched.

A May 1988 comparison of model values with listed IBM quotes described the call as 25 percent cheap to the model and the put as 41 percent cheap, read as an implied-volatility signal of a low-volume, trendless tape.

Time value is not guaranteed by a printed price

Moneyness is whether the underlying sits below, at or above the strike, which determines how much of the premium is intrinsic versus time value. Time value is the decaying portion of an option premium that can shrink even when the underlying later moves in the expected direction.

A short-dated, far out-of-the-money index put that still printed 15/16 was treated by the model as having no remaining time value. Over the next 13 days the premium fell to 1/4 even though the index declined 6.46 points.

From fair values to constructions

After static security data were stored, updating the date, T-bill rate and latest prices produced printed fair values. A worksheet then reported expiration and annualized return, break-evens, commissions and margin for combination, covered-call, ratio-write and hedge-wrapper constructions.

A combination buys or writes both calls and puts, typically at different strikes. A covered-call holds the underlying and writes one call per block of stock. A ratio-write holds the underlying and writes more than one call per block of stock. A hedge-wrapper holds the underlying and writes both a call and a put.