With zero rates, no dividends, spot 100, strike 100, one year to expiry and 40% volatility, compute the Black-Scholes delta of the call. Explain why it exceeds 0.5 and what happens as volatility rises further.

With zero rates, no dividends, spot 100, strike 100, one year to expiry and 40% volatility, compute the Black-Scholes delta of the call. Explain why it exceeds 0.5 and what happens as volatility rises further.

Approach: Write d_1 for the spot-strike case at zero rate, evaluate the normal cumulative there, then explain the result through the lognormal median against the mean.

0.579. At zero rate with S = K, d_1 = (sigma^2*T/2)/(sigma*sqrt(T)) = sigma*sqrt(T)/2 = 0.2, and delta = N(d_1) = N(0.2) = 0.5793. Delta exceeds 0.5 because the terminal price is lognormal: its median is S*e^{-sigma^2*T/2} = 92.3, below the forward of 100, so the distribution is skewed with a long right tail and the probability-weighted upside dominates. The N(d_2) = N(-0.2) = 0.4207 is the risk-neutral probability of finishing in the money, and delta sits above it by exactly the skew of the lognormal distribution. Pushing volatility to 200% with T = 1 gives d_1 = 1 and delta 0.841, and in the limit the spot call delta goes to 1 while its exercise probability goes to 0.

Follow-up: At what volatility is the one-year spot-strike call delta equal to 0.75, and what is the exercise probability there?

Key concepts: delta, lognormal distribution, d_1, median versus mean.