Constructing option value from historical and implied volatility

A construction bench rather than a quote machine

Editorial: treat a pricing calculator as a construction bench rather than a quote machine. Build a theoretical premium from a transparent historical-volatility recipe, then invert the same model on the live mid-premium so a single option sits in a two-regime frame before any grant or listed mark is read as a standalone value.

A listed call is the right to buy, and a listed put the right to sell, at a stated strike price on or before a stated expiration.

A standard equity option covers 100 shares, so the cash amount for one contract is the quoted premium multiplied by 100.

Strikes and the expiration clock

In the source period, listed strike increments widened with the underlying price, including 2.50-point steps below 25 and 5-point steps above 25.

Listed stock options last-trade on the third Friday of the expiration month, while index options expire on the Thursday before that stock expiration.

Building the theoretical premium

A Black-Scholes calculator turns spot, strike, interest rate, dividend, time to expiration, and a volatility assumption into a theoretical premium.

That closed-form pricing engine maps those inputs into a theoretical option premium. The theoretical premium is the model output shown as a fair contract price per share before it is compared with the live quote.

Historical volatility as a sample recipe

Annualized historical volatility can be built as the standard deviation of daily log returns multiplied by 15.8, the square root of a 250-day trading year.

That 15.8 figure is the annualization factor, the square-root-of-trading-days scalar that converts a daily return dispersion into a yearly volatility input.

The lookback chosen for that calculation can change the annualized figure, so historical volatility is a sample-dependent input rather than a unique market fact.

Implied volatility from the same engine

Clearing the volatility field and solving the same calculator on a bid-ask midpoint recovers implied volatility, the market-implied counterpart to the historical input.

Implied volatility is the volatility that makes the same pricing engine reproduce an observed bid-ask midpoint.

Gaps, delta, and later marks

A large gap between historical and implied volatility leaves open whether a private forecast is more informative than the market-implied figure.

Model Greeks such as delta describe how the theoretical premium moves with the underlying, and the same inputs can be restressed to sketch later marks. Delta measures how the theoretical premium changes when the underlying price changes.