The payoff of a simple European or American style call or put option depends only on the value of the asset, not on the path taken to get there. This is a property known as path independence. In a Barrier call or put option, the payoff is path dependent.
Not only does it depend on the final asset value but it also depends on whether a certain barrier level was touched or not touched at some time during the life of the option. Single barrier options of many types exist and it is best to try to understand these options by considering several key features. The first feature is the underlying option which can be: Other possibilities exist, for example, an Asian option, but these will not be considered in this document nor are the functions relevant for any other cases.
If the Barrier is touched, the holder now owns a standard option. If over the life of the option the barrier is never touched, the option dies worthless though the holder may be entitled to a rebate — see discussion later on. As long as the barrier is never touched, the holder owns a standard option. If the barrier is ever touched, the option dies worthless though the holder may be entitled to a rebate - see discussion later on.
The final feature is the type of monitoring that is done at the barrier. During these windows, the barrier is monitored continuously. During these windows, the barrier is monitored at discrete dates. For the case of continuously monitored barriers and European style calls and puts, closed-form solutions exist see for example Rubinstein and Reiner . Similarly for binary options involving one continuously monitored barrier, see Rubinstein and Reiner  , or Hudson .
These cases are standard and are covered in many basic texts like Hull . For partial barrier options, where the barrier is continuously monitored, or for continuously monitored barriers involving American style options, a binomial tree is used. As is well known, the key to using trees for pricing barrier options is to adjust the tree methodology near the barrier.
If no adjustment is made, the methodology will work, but the convergence will be painfully slow i. We use a tree scheme where the value at the barrier nodes is adjusted smoothed. Our method leads to very good convergence results. In Hull  pgs , he describes one possible adjustment scheme. For discrete barrier options, where the barrier is monitored at a discrete instant in time, a different type of approach is used.
For all cases, even European style options, no efficient closed form solution is available it is true that in some cases, one may be able to write the solution down as multiple integrals over each discrete sampling point, but these integrals cannot be efficiently calculated.
We use a binomial tree approach and again make an adjustment at the barrier points. The adjustment is that derived by Steiner et al. We also suggest the reader may want to look at the papers Horfelt  and Broadie et al. Finally, we note the following convention: In the case of a standard down-and-out option for which the underlying price is less than the barrier value, all statistics are thus equal to zero, except the probability of breaching the barrier, which is equal to one.
Similarly in the case of an up-and-out option for which the underlying price is greater than the barrier value. To search for the correct function, refer to the following table: Use the rebate feature in the functions: Continuous monitoring, whole life of option. We note that the level of the barrier and amount of any rebate may vary with time. The barrier level and the rebate payments may vary with time. The barrier level may vary with time. All of the barrier option functions allow for the payment of a rebate.
All functions also output the fair value of the rebate portion of the option as a stat often this is stat 8 or 9. Thus, it is possible to use the above barrier functions to price various forms of digital barrier options. However, for the cases of partial or discrete barriers, one can value the binary options using the relevant barrier option function and the rebate feature.
Set the rebate s to the relevant amount s. To ensure the option part of the barrier option has no value, a simple trick is to set the option to a put with zero strike, guaranteeing a put option value of zero.
Set the rebate to the relevant amount. Thus, using the relevant knock-out type option e. The value of the underlying asset on the value date. The strike value related to the call or put.
The barrier level for continuously monitored barrier options. In the case of discretely monitored barrier options, this input will correspond to a monitoring table that will contain the monitored dates, the barrier levels, and the rebates . The type of option. The type of barrier option see the description above.
The amount of rebate see the description above. The annualized volatility of the underlying asset. If the underlying is an equity, rate2 is the annualized dividend yield. If the underlying is a forward price, rate2 should be set equal to the risk free rate1. If the underlying is an FX rate, and quoted on a domestic per foreign basis, rate1 should be the risk. For the case of an American Style Option, there is an extra input: Number of time steps: The American Style option is solved using an adjusted binomial tree.
The tree is adjusted at barrier nodes in order to minimize the numerical error see references below. This parameter specifies the number of time steps used by the adapted binomial method. In several of the barrier functions, when the input is called optimize , if the value is set to 1, 2, 3 or 4, the function internally attempts to calculate optimal time step values 1 will have fewer time steps than 2 and so on. If optimize is set to larger than 4, it runs with this many time steps. For the case of Discretely Monitored Options, there are two extra inputs: The discretely monitored case is solved through an adjusted binomial tree.
The higher the number of time steps, the more accurate the solution is. Unlike the optimize method mentioned above this method has no level of optimization. Its value is the actual number of time steps used in the building of the adjusted binomial tree. This switch is used to indicate whether the knock-in option has already been knocked-in in which case the holder has a standard call or put option. This switch is used to indicate whether an option has been knocked out. The total fair value of the option includes the fair value of the rebate; to obtain the fair less the rebate subtract the value of the rebate see below.
The rate of change in the fair value of the compound option per one unit change in the spot value of the underlying asset. This is the derivative of the option price with respect to the underlying spot value. The rate of change in the value of delta per one unit change in the spot value of the underlying asset. This is the second derivative of the option price with respect to the underlying spot value.
This is the negative of the derivative of the option price with respect to time, divided by This is the derivative of the option price with respect to volatility, divided by This is the derivative of the option price with respect to rate1, divided by This is the derivative of the option price with respect to rate2, divided by The fair value of the rebate see the discussion above.
This is the probability of touching the barrier. It is important to realize that the probability is with respect to the adjusted risk-neutral probability distribution, not the lognormal probability distribution of the underlying itself.
The risk-neutral probability distribution is associated with the basic hedging strategy in Black-Scholes type option models and the probability distribution depends on the risk-free rates rate1 and rate2. What is the probability of touching the barrier? Risk-Neutral probability of hitting barrier early exercise considered. For American style options, early exercise is taken into consideration in calculating the risk neutral probability of hitting barrier.
During the life of the knock-out option, the option might already be deep in the money and have been exercised before the underlying price reaches the barrier. Hence, this number is adjusted for possible early exercise. For European style options, early exercise is not allowed and this value is the same as probability of hitting upper barrier above. Consider a one year down and out barrier put option on one share of stock.
The rate of change in the fair value of the compound option value per day. The fair value of the rebate. In no event shall FINCAD be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of this document or the information contained in it.
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The single rebate required will be input as a separate input. See rebate discussion below. Risk-Neutral probability of hitting barrier.More...