Bachelier formula

The Bachelier formula is a function that maps 4 positive numbers f,K,T,σ and a special integer φ = ±1 to a value BACHELIER(f,K,T,σ,φ) according to the formula:
(f,K,T,σ,φ) -> BACHELIER(f,K,T,σ,φ) := Dφ(f-K)N(φd) + DσT¹ᐟ²n(d)
where
D is the discount factor for a maturity equal to T.
N(x) is the cumulative distribution function (CDF) of the of the standard normal distribution
n(x) is the probability density function (PDF) of the of the standard normal distribution
d = (f-K)/(σT¹ᐟ²)

While the above formula is just a definition, one can prove that if V(0) represents the price (i.e. present value) as of time t = 0 of a European call (if φ = 1) or put (if φ = -1) with expiry T (in annual units) and strike K on any tradable asset of which the forward price with maturity T observed at t = 0 is denoted as f(0;T), then V(0) is given by the formula:
V(0) = BACHELIER(f(0;T),K,T,σ,φ)
provided f(t;T), i.e. the forward price observed at time t, is diffused over t as df(t;T) = σ(t;T)dw with respect to the forward martingale measure.