Approximating Option Delta and Digital Prices from Call Prices

Summary

The document asks whether a European call price can be used to approximate Black–Scholes quantities without first solving for implied volatility. The targets are the call delta, represented by the cumulative normal probability at the first standardized moneyness measure, and the corresponding cash-or-nothing digital value, associated with the second. The desired approximation may be rough, but should preserve limiting behavior, such as tending to zero as the call price tends to zero.

The motivation is to use such an estimate as an intermediate step toward recovering implied volatility from the observed option price. No approximation, derivation, data, or comparison is supplied, so the document frames a research question rather than establishing a method. Any proposed mapping would need to account for inputs such as spot, strike, rates, and maturity, and its accuracy may vary across moneyness and market conditions.

Key ideas

• The question seeks estimates of Black–Scholes delta and digital option value directly from a call price.
• The proposed estimates are intended to avoid calculating implied volatility first.
• A useful approximation should respect limiting behavior as the call price approaches zero.
• The document gives a motivation but no formula, evidence, or validated method.

Tags

# Rough approximation of option delta (or binary option) given option price wihtout implied volatility?


# Rough approximation of option delta (or binary option) given option price wihtout implied volatility?












Under the Black-Scholes framework, is there any approximation of option delta (i.e., $N(d_1)$) or binary options price(i.e., $N(d_2)$) given the option price $C$? Of course, you can calculate those once you know the implied volatility (IV). I am looking for an approximation without using IV. A very rough approximation is fine as long as an approximation holds the correct limit behavior (e.g., $N(d_1)$ or $N(d_2)$ goes to 0 as $C$ goes to 0).

I am looking for these approximations as an intermediate step to estimate IV because you can obtain IV once you know $N(d_1)$ or $N(d_2)$.

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