<Dl> <Dd> P (0, S) K N (− d 2) − P (0, T) N (− d 1) (\ displaystyle P (0, S) KN (- d_ (2)) - P (0, T) N (- d_ (1)) \,) </Dd> </Dl> <Dd> P (0, S) K N (− d 2) − P (0, T) N (− d 1) (\ displaystyle P (0, S) KN (- d_ (2)) - P (0, T) N (- d_ (1)) \,) </Dd> <P> Here σ is the standard deviation of the log - normal distribution for P (S, T). A fairly substantial amount of algebra shows that it is related to the original parameters via </P> <Dl> <Dd> S σ P = σ α (1 − exp ⁡ (− α (T − S))) 1 − exp ⁡ (− 2 α S) 2 α (\ displaystyle (\ sqrt (S)) \ sigma _ (P) = (\ frac (\ sigma) (\ alpha)) (1 - \ exp (- \ alpha (T-S))) (\ sqrt (\ frac (1 - \ exp (- 2 \ alpha S)) (2 \ alpha))) \,) </Dd> </Dl>

Efficient and exact simulation of the gaussian affine interest rate models