Spot is 100, volatility is 40%, one year to expiry, rates zero. At which strike is vega largest, and for the fixed 100 strike, at which spot level is gamma largest? Derive both.

Spot is 100, volatility is 40%, one year to expiry, rates zero. At which strike is vega largest, and for the fixed 100 strike, at which spot level is gamma largest? Derive both.

Approach: Both greeks carry the same normal density factor n(d_1). Differentiate the logarithm of each with respect to the variable you are moving and set it to zero.

108.33. Vega is S*sqrt(T)*n(d_1), and as a function of strike only d_1 varies, so the maximum sits where d_1 = 0, which means ln(S/K) + (r + sigma^2/2)*T = 0 and K = S*e^{(r + sigma^2/2)T} = 100*e^{0.08} = 108.33. Gamma as a function of strike peaks at the same place, since it is n(d_1)/(S*sigma*sqrt(T)) and shares the density factor. Gamma against spot is a different question: taking the log derivative of n(d_1)/S in S gives -d_1/(S*sigma*sqrt(T)) - 1/S = 0, so d_1 = -sigma*sqrt(T) and the peak sits at S = K*e^{-(r + 3*sigma^2/2)T}, which for the 100 strike is 100*e^{-0.24}, about 78.66. Both results say the same thing: the greeks peak slightly away from the money in the direction the lognormal density is thickest, and the displacement grows with sigma^2*T rather than with sigma.

Follow-up: At what spot does the dollar gamma, meaning gamma times the square of spot, peak for the same 100 strike?

Key concepts: vega, gamma, d_1, normal density, moneyness.