Double Barrier Options 
Use
A double barrier option ceases to exist ( Knock-Out ) or comes into existence ( Knock-In ) if the price of the underlying reaches a lower barrier L or an upper barrier U before the option becomes due. The formulae below refer to the double-knock-out option only The price of a double-knock-in call is the same as the price of a portfolio from a bought standard call and a sold double-knock-out call, both having the same strike price and the same amount of the term remaining. The double-knock-in put can be constructed in the same way. Double-knock-out options are valued with the formula from Ikeda and Kunitomo (1992).
Scope of Functions / Valuation
The following applies in all subsequent cases:
S = Current underlying price
X = Strike price
N(x) = Probability that a value
( )
occurs for a standard normal distribution
q = Dividend yield as continuous interest
r = Risk-free interest rate as continuous interest
T = Remaining term in fractions of a year
L = Lower barrier
U = Upper barrier
( )
= Volatility of the underlying
n = Summed index
Call-up-and-out-down-and-out
The following is counted at option expiration T: c(S, U, L, T) = max(S-X, 0) if L < S < U for all points in time before T, otherwise 0.
( )
where:
( )
( )
( )
( )
( )
,
( )
( )
,
( )
( )
and
( )
specify the exponential growth constants for the lower (L) and upper (U) barriers, respectively. The SAP System uses only
( )
and
( )
.
Put-up-and-out-down-and-out
The following is counted at option expiration T: c(S, U, L, T) = max(S-X, 0) if L < S < U for all points in time before T, otherwise 0.
( )
where:
( )
( )
( )
( )
( )
,
( )
( )
,
( )
( )
and
( )
specify the exponential growth constants for the lower (L) and upper (U) barriers, respectively. The SAP System uses only
( )
and
( )
.
According to Ikeda and Kunitomo, it is sufficient to let the sum run from -5 to 5 for the call and put. The values are then sufficiently accurate.