Mertons (1973) and Reiner and Rubinstein (1991) formulae for pricing Standard barrier options: 
where 
,
,
,

"In" Barriers 
Down and in call S >H Cdown&in(X>H) = C+E =1, =1 Cdown&in(X<H) = AB+D+E =1, =1 
Up and in Call S<H Cup&in(X>H) = A+E =1, =1 Cup&in(X<H) = BC+D+E =1, =1 
Down and in put S>H Pdown&in(X>H) = BC+D+E =1, =1 Pdown&in(X>H) = A+E =1, =1 
Up and in put S>H Pup&in(X>H) = AB+D+E =1, =1 Pup&in(X>H) = C+E =1, =1 
"Out" Barriers 
Down and out call S >H Cdown&out(X>H) = AC+F =1, =1 Cdown&out(X<H) = BD+F =1, =1 
Up and out Call S<H Cup&out(X>H) = F =1, =1 Cup&out(X<H) = AB+CD+F =1, =1 
Down and out put S>H Pdown&out(X>H) = AB+CD+F =1, =1 Pdown&out(X>H) = F =1, =1 
Up and out put S>H Pup&out(X>H) = BD+F =1, =1 Pup&out(X>H) = AC+F =1, =1 
where 
X = Strike price T = Time to maturity r = risk free rate N = The cumulative normal distribution function S = Sport price H = Barrier K = Predetermined cash payoff 

See Also 
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