<Dd> C (F, τ) = D (N (d +) F − N (d −) K) d ± = 1 σ τ (ln ⁡ (F K) ± 1 2 σ 2 τ) d ± = d ∓ ± σ τ (\ displaystyle (\ begin (aligned) C (F, \ tau) & = D \ left (N (d_ (+)) F-N (d_ (-)) K \ right) \ \ d_ (\ pm) & = (\ frac (1) (\ sigma (\ sqrt (\ tau)))) \ left (\ ln \ left ((\ frac (F) (K)) \ right) \ pm (\ frac (1) (2)) \ sigma ^ (2) \ tau \ right) \ \ d_ (\ pm) & = d_ (\ mp) \ pm \ sigma (\ sqrt (\ tau)) \ end (aligned))) </Dd> <P> The auxiliary variables are: </P> <Ul> <Li> τ = T − t (\ displaystyle \ tau = T-t) is the time to expiry (remaining time, backwards time) </Li> <Li> D = e − r τ (\ displaystyle D = e ^ (- r \ tau)) is the discount factor </Li> <Li> F = e r τ S = S D (\ displaystyle F = e ^ (r \ tau) S = (\ frac (S) (D))) is the forward price of the underlying asset, and S = D F (\ displaystyle S = DF) </Li> </Ul> <Li> τ = T − t (\ displaystyle \ tau = T-t) is the time to expiry (remaining time, backwards time) </Li>

How we came up with the option formula