Appendix A: Integration of Black–Scholes Formula
In this appendix, we derive the price of a European call option given in Eq. (6.19). Let x = ln(PT). By changing variable and using g(PT) dPT = f(x) dx, where f(x) is the probability density function of x, we have
Because x = ln(PT) ∼ N[ln(Pt) + (r − σ2/2)(T − t), σ2(T − t)], the integration of the second term of Eq. (6.36) reduces to
where CDF[ln(K)] is the cumulative distribution function (CDF) of x = ln(PT) evaluated at ln(K), Φ( · ) is the CDF of the standard normal random variable, and
The integration of the first term of Eq. (6.36) can be written as
where the exponent can be simplified to
Consequently, the first integration becomes
which involves the CDF of a normal distribution with mean ln(Pt) + (r + σ2/2)(T − t) and variance σ2(T − t). By using the same techniques as those of the second integration shown before, we have
where h+ is given by
Putting the two integration results together, we have
