Black–Scholes Delta of a Call as Implied Volatility Becomes Very Large

Summary

The note answers a narrow options question: what happens to the Black–Scholes delta of an out-of-the-money call as implied volatility tends to infinity. It presents the call delta formula and observes that, for a remaining time to expiry, the formula’s normal-distribution argument tends to positive infinity, so the delta tends to one.

The explanation is an asymptotic result within the stated Black–Scholes setup. It assumes the option has not yet expired; it does not discuss the expiry boundary, alternative pricing models, or market behavior when volatility is merely high rather than unbounded. The document offers no empirical evidence or broader trading rule.

Key ideas

• The Black–Scholes call delta depends on spot, strike, rates, dividends, volatility, and time to expiry.
• For an unexpired call, the formula’s argument tends to positive infinity as volatility tends to infinity.
• Under that limit, the call delta tends to one.

Tags

# Implied volatility interview question


# Implied volatility interview question












If an implied volatility of an out of the money call option goes to infinity,what happens to the delta of the said call option?

## Answer by emcor (score 3, accepted)

https://quant.stackexchange.com/a/18362

The Black-Scholes delta: $$\partial_SC=N\left(\dfrac{\ln\left(\frac{S_0}{K}\right) +(r - q + \frac{1}{2}\sigma^2)(T - t)}{\sigma\sqrt{T - t}}\right)$$ As you can see this delta would go to$1$ if $\sigma\to\infty$ (and $t<T$).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.