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HomeAbout UsPublications Differences in Theoretical and Actual Prices of Double Knockout and Binary Range FX Options.
By: G.Ioffe On a number of occasions participants have observed a significant difference between the theoretical values of Double KnockOut ("2KO") options and their market quotes as well as the theoretical values of Binary Range Options ("Range") and their market quotes. In general, it appears that the market quotes are significantly higher than the theoretical values and the difference becomes even more magnified when the barriers are close. The magnitude of these differences are at times as much as 510 times and as such are difficult to explain by factors such as market spreads and arbitrage. Differences between theoretical values and market quotes have also been observed in inthemoney knock out options as the spot price approaches the barrier. Frequently, the explanation for this is that marketmakers add a cushion of safety into the quote due to the greater risk and difficulties of delta hedging this option. The wider discrepancies between the theoretical values and actual quotes of 2KO and Range options, in our opinion, can be partially explained by the lack of widely available pricing models for 2KO and Range options. Due to this, many professionals have adopted approximation methods to calculate the theoretical prices for 2KO and Range options. We describe one approximation method that utilizes pricing models for knockout options. This approximation formula uses the following notation:
H low  low barrier,
The approximation formula calculates the price of 2KO call and put options as: Since KOC(K, H low) = C(K)  KI(K, H low) and the approximation replicates 2KO call option as: The first three terms in the right part of equation (2) comprise portfolio(P) of long European call C(K), short knockin call KIC(K, H up) and short knockin call KIC(K, H up). The fair value of a 2KO call represents all the positive payoffs, at expiration, when the underlying asset stays within both barriers until expiration. Such payoffs result from all the combination of paths that the underlying asset can take while crossing neither barrier by expiration and end up inthemoney.
To generate such paths:
Note that the paths that cross both barriers were subtracted twice and thus to generate all the paths that lead to a positive payoff for a holder of 2KO call option: All the paths that end up inthemoney, result in a positive payoff for the holder of a European call option. All the paths that end up inthemoney and cross barrier Hup, result in a positive payoff for the holder of a knockin call option KIC(K, H up). All the paths that end up inthemoney and cross barrier H low, result in a positive payoff for the holder of the of a knockin call option KIC(K, H low). These observations lead to equation (3): From equations (2) and (3) it is logical to conclude that the accuracy of approximation (1) depends on how close the fair value of all the paths that cross both barriers is approximated by the term KI(K, H up) * KI(K, H low )/ C(K). Intuitively it is clear that we should compare the underlying asset price probability of crossing both barriers Prob( S(t) > H up, S(t) < H low ) with the product of two probabilities: probability of underlying asset price crossing the upper barrier Prob( S(t) > H up) and probability of underlying asset price crossing the lower barrier Prob( S(t) < H low). To cross the upper barrier, the underlying asset price moves away from the lower barrier and then S(t) is less likely to cross lower barrier. When the distance between the lower and upper barrier is small it leads to Formulas (2), (3) and (4) explain why when H up  H low is small, the approximation formula (1) calculates a larger value than the theoretical model. We use the theoretical model and approximation formulas to calculate 2KOC and 2KOP values. To calculate the value of a Binary Range Option we use the identity that derives the price of a Range option as a function of 2KOC and 2KOP option. To prove this identity, consider the payoff from a portfolio consisting of: This portfolio will always have a zero payoff amount at expiration. Identity (5) can be used to realize possible arbitrage opportunities as well as to calculate the price of Binary Range Options. For Example:
For call options:
When the approximation method is used in formula (1)
For put options:
When the approximation method is used in formula (1)
Finally we use identity (5) to calculate the price of a Range option: Conclusion: In this paper, Ioffe:
Gena Ioffe 
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