Static replication is always the most elegant way of pricing. The no-arbitrage argument will let us say that if, at some time in the future, two portfolios have the same value for sure, then their price should be equal any time before this. We will show how double-knock-out (DKO) options could be used to build a DNT. We will need to use a trick; the strike price could be the same as one of the barriers. For a DKO call, the strike price should be lower than the upper barrier because if the strike price is not lower than the upper barrier, the DKO call would be knocked out before it could become in-the-money, so in this case, the option would be worthless as nobody can ever exercise it in-the-money. However, we can choose the strike price to be equal to the lower barrier. For a put, the strike price should be higher than the lower barrier, so why not make it equal to the upper barrier. This way, the DKO call and DKO put option will have a very convenient feature; if they are still alive, they will both expiry in-the-money.

Now, we are almost done. We just have to add the DKO prices, and we will get a DNT that has a payout of (U-L) dollars. Since DNT prices are linear in the payout, we only have to multiply the result by K*(U-L):

dnt2 <- function(S, K, U, L, sigma, T, r, b) {

    a <- DoubleBarrierOption("co", S, L, L, U, T, r, b, sigma, 0,
        0,title = NULL, description = NULL)
    z <- a@price
 
    b <- DoubleBarrierOption("po", S, U, L, U, T, r, b, sigma, 0,
        0,title = NULL, description = NULL)
    y <- b@price
 
   (z + y) / (U - L) * K
}

Now, we have two functions for which we can compare the results:

dnt1(0.9266, 1000000, 0.9600, 0.9200, 0.06, 0.25, 0.0025, -0.025)
[1] 48564.59

dnt2(0.9266, 1000000, 0.9600, 0.9200, 0.06, 0.25, 0.0025, -0.025)
[1] 48564.45

For a DNT with a USD 1 million contingent payout and an initial market value of over 48,000 dollars, it is very nice to see that the difference in the prices is only 14 cents. Technically, however, having a second pricing function is not a big help since low volatility is also an issue for dnt2.

We will use dnt1 for the rest of the chapter.