Z > [ln(K/S) – (r – .5σ²)(T-t)]/σ*sqrt(T-t) = z₀

meaning that the area of integration is z₀ to ∞, or -∞ to  –z_0, the part of the normal distribution where exercise takes place.  The option price formula can be viewed as a conditional expectation multiplied by a probability,

                e^-r(T-t)[E[S_T|S_T > K]*N(d2) – K*N(d2)]

The conditional expectation E[S_T|S_T > K] can also be written as

                Se^{(r-.5σ^2)(T-t) + σ*sqrt(T-t)*Z}

where the Z in the last term ensures that only values of the underlying S where exercise takes place are considered.  The greater the volatility σ, the greater the possible up move, the greater the area of integration, and hence the greater the conditional expectation of S.

The option price formula can now be written as

                e^-r(T-t)[(Se^{(r-.5σ^2)(T-t) + σ*sqrt(T-t)*Z})*N(d2) – K*N(d2)]

The e^{(r-.5σ^2)(T-t) + σ*sqrt(T-t)*Z} term can be combined with N(d2) since it is also a standard normal probability.  N(d1) is then just N(d2) +  σ*sqrt(T-t), which gives N(d1) a greater expectation based on σ.  The option price formula can now be written as

                e^-r(T-t)[(Se^r*N(d1) – K*N(d2)] or

                S*N(d1)- e^-r(T-t)K*N(d2)

N(d1) simply takes into account the higher conditional expectation of the random underlying S versus the static K.