The value of X after time t which relies on the winner process shows following a differential equation.

dXt = r Xt dt + σ Xt dWt

the payoff function of call option as g(x)

g(St) = max{ St-K, 0 }

C(St, t) =E[max{Xt-K,0} e-r(T-t)| Xt=St]

Xt = Xt e (r-(1/2)σ2)(T-t)+σ(WT-Wt)

where definition of the winner process,(WT-Wt) relies on the normal distribution which is average 0, variance (T-t), Z defined as stochastic variable relies on the normal distribution of the average 0, variance 1, replace with √(T-t)・Z

C(St・t)=E[max{St e (r-(1/2)σ2)(T-t)+σ√(T-t)Z ) -K, 0}] e -r(T-t)

C(St・ t)=∮-∞∞max{S, e(r-(1/2)σ2)(T-t)+σ√(T-t)Z )}f(z)dz e-r(T-t)

where

f(z) = (1/√2π)e -(1/2)z"2

to solve this integral

omit the calculation procedure.

where put the function N(x) of normal standard distribution

C(St,t) = St N(ht) - K e -r(T-t) N(ht - σ√(T-t)) -----------------(※)

put ht as

ht = 1/(σ√(T-t))log(St/K e -r(T-t))+ (1/2)σ√(T-t)

This equation (*) is well known pay off function of call option of the Black-Shoerls equation.