Exercise 4.8 Let (x) denote the density function (3.13) of the standard normal distribution. Prove that where d is given by (4.22). Using this, prove

Exercise 4.8 Let ϕ(x) denote the density function (3.13) of the standard normal distribution. Prove that

where d is given by (4.22). Using this, prove the following for the Black–Scholes formula (4.21):

Here, τ = T − t and cx denotes the partial derivative of c with respect to x.

K (d) = e(T-t) (d- oT t), S -