Barrier options are well-established products in over-the-counter derivatives market. However, open positions in these kind of products are often difficult to hedge, since the gamma is large and fluctuates if the underlying asset trades close to the barrier. This is particularly true for products where knocking-in or knocking-out is not based on continuously-observed prices of the underlying, but on some discrete fixings.
Furthermore, buyers of these products are often afraid that the options suddenly will be knocked out if the price of the underlying rebounces. A company considers buying foreign currency calls, but judges the premium too steep. It might consider buying a down-and-out call that comes with a substantial reduction of the premium. Whenever the option is knocked out, the company simply buys the foreign currency immediately at the low price or goes into a long forward contract. This will result in lower costs than if the option had to be exercised at the strike price in the end. However, if the option is knocked out and the foreign currency suddenly increases in value again, the company might be too slow to react.
Although this is not a likely scenario for currencies, with other underlying assets, where there might be some room for short-term price manipulation, this might be a real danger. Hence, both for banks and clients it might be better to have products where knocking out or in goes more smoothly. An example of such a product is a so called Parisian variant of the down-and-out-call option. This is knocked out when the price of the underlying has been below the barrier for a certain number of days. These products come in different forms. In some cases knocking out is based on the total number of days the price has been below the barrier since the inception of the contract, for others the number of consecutive days below the barrier is the relevant variable. There are also forms where the option gradually knocks out with the number of days below the barrier.
Pricing of these products is more complicated than for standard barrier options, where Black-Scholes formulae exist for continuously-observed underlying prices and very accurate trinomial trees or other numerical methods for the discretely-observed species. Most existing valuation methods are based upon solving a three-dimensional partial differential equation either by finite difference methods or through Fourier transforms. Both require heavy computational tools and are hard to explain to non-specialists. We have developed a methodology that is easy to explain and implement and is based on the familiar binomial tree. The basic idea is given in figure 1.
A call is knocked out if it has been below the barrier for three days since the inception of the contract. Four identical flat binomial trees describing the underlying asset price are built. It is assumed that each period in the tree represents exactly one day. The method can be easily extended if more refined trees are built.
Next, they are put in layers. The top tree is associated with prices of the product when the underlying price has never been below the barrier. The second represents prices where the underlying has been below the barrier for one period, the third layer is associated with two periods below the barrier and the lowest tree consists of points where the option has been knocked out. Hence, at this last level option prices are known to be zero or equal to the (discounted) rebate. At the final nodes of the top three layers the values are equal to the exercise values for the call.
Prices of the option are determined by the familiar backward procedure. However, nodes are not necessarily linked to nodes in the same tree, but in some cases to nodes in other trees. For example, for a point in the second tree--i.e. a point where the underlying has been below the barrier for one day--the price of the underlying can have two values in the next period. For each of these values see whether it is below or above the barrier. If it is below, make a link to that value in the third tree, otherwise remain in the second tree. Hence the original point can be linked to points in different trees. For all non-terminal points in the trees we make links to points in the same or a lower tree. The backward procedure is based on these links. The procedure can easily be adjusted for options that are knocked out when they have been below the barrier for three consecutive days. In this case they do not link to the same level if the new price is above the barrier but link to the top level.
Table: values of Parisian down-and-out call with underlying asset values S = 100, strike price X = 100, interest rate r = 0.06 and time to maturity T = 130 days (1/2 year). The tree has 1300 steps.
Table 1 gives values for the down-and-out call option which knocks out if the underlying asset has a certain number of consecutive end of the day fixings below the barrier. The number of days is given in the second column. Different levels of the barrier H and the volatility of the underlying asset sigma are used. In row 3 and 4 the analytical and numerical values based on a binomial tree for the standard call are given. As expected the number of days below the barrier before the option is knocked out is relevant. The bottom row gives the values of the discretely-observed standard barrier option. As is well known binomial tree procedures are very efficient. Since for this methodology the number of trees is one more than the number of days the asset is below the barrier, also this methodology is very efficient.
This week's Learning Curve was written byTon Vorst, professor of finance and economics at Erasmus University, in Rotterdam, Holland.