The Barrier Binary Options

We extend the binary options into barrier binary options and discuss the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we show that the price function of a knock-in American binary option can be expressed in terms of the price functions of simple barrier options and American options. For the knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. By the properties of Brownian motion and convergence theorems, we show how to calculate the expectation of the local time. In the financial analysis, we briefly compare the values of the American and European barrier binary options.


Introduction
Barrier options on stocks have been traded in the OTC (Over-The-Counter) market for more than four decades. The inexpensive price of barrier options compared with other exotic options has contributed to their extensive use by investors in managing risks related to commodities, FX (Foreign Exchange) and interest rate exposures.
Barrier options have the ordinary call or put pay-offs but the pay-offs are contingent on a second event. Standard calls and puts have pay-offs that depend on one market level: the strike price. Barrier options depend on two market levels: the strike and the barrier. Barrier options come in two types: in options and out options. An in option or knock-in option only pays off when the option is in the money with the barrier crossed before the maturity. When the stock price

Preliminaries
American feature entitles the option buyer the right to exercise early. Regardless of the pay-off structure (cash-or-nothing and asset-or-nothing), for a binary call option there are four basic types combined with barrier feature: up-in, up-out, down-in and down-out. Consider an American (also known as "One-touch") up-in binary call. The value is worth the same as a standard binary call if the barrier is below the strike since it naturally knocks-in to get the pay-off. On the other hand, if the barrier is above the strike, the valuation turns into the same form of the standard with the strike price replaced by the barrier since we cannot exercise if we just pass the strike and we will immediately stop if the option is knocked-in. Now let us consider an up-out call. Evidently, it is worthless for an up-out call if the barrier is below the strike. Meanwhile, if the barrier is higher than the strike the stock will not hit it since it stops once it reaches the strike. For these reasons, it is more mathematically interesting to discuss the down-in or down-out call and up-in or up-output.
Before introducing the American barrier binary options, we give a brief introduction of European barrier binary options and some settings for this new kind of option. Figure 1 and Figure 2 show the value of eight kinds of European barrier binary options and the comparisons with corresponding binary option values. All of the European barrier binary option valuations are detailed in [6]. Note that the payment is binary, therefore it is not an ideal hedging instrument so we do not analyse the Greeks in this paper and more applications of such options in financial market will be addressed in Section 5. Since we will study the American-style options, we only consider the cases that barrier below the strike for the call and barrier above the strike for the put as reasons stated above. As we can see in Figure 1 and Figure 2, the barrier-version options in the blue or red curves are always worth less than the corresponding vanilla option prices. For the binary call option in Figure 1 when the asset price is below the in-barrier, the knock-in value is same as the standard price and the knock-out value is worthless. When the stock price goes very high, the effect of the barrier is intangible. The knock-intends to worth zero and the knock-out value converges to the knock-less value. On the other hand in Panel (a) of Figure 2, the value of the binary put decreases with an increasing stock price. As Panel (b) in Figure 2 shows, the asset-or-nothing put option value first increases and then decreases as stock price going large. At a lower stock price, the effect of the barrier for the knock-out value is trifle and the knock-in value tends to be zero. When the stock price is above the barrier, the knock-out is worthless and the up-in value gets the peak at the barrier. The figures also indicate the relationship knock-out knock-in knock-less.
Above all, barrier options create opportunities for investors with lower premiums than standard options with the same strike.

The American Knock-In Binary Option
We start from the cash-or-nothing option. There are four types for the cash-ornothing option: up-and-in call, down-and-in call, up-and-input and down-andinput. For the up-and-in call, if the barrier is below the strike the option is worth the same as the American cash-or-nothing call since it will cross the barrier simultaneously to get the pay-off. On the other hand, if the barrier is above the strike the value of the option turns into the American cash-or-nothing call with the strike replaced by the barrier level. Mathematically, the most interesting part of the cash-or-nothing call option is down-and-in call (also known as a downand-up option). For the reason stated above, we only discuss up-and-input and down-and-in call in this section.
We assume that the up-in trigger clause entitles the option holder to receive a digital put option when the stock price crosses the barrier level.
where K is the strike price, L is the barrier level and P t x . The process X is strong Markov with the infinitesimal generator given by We introduce a new process which represents the process X stopped once it hits the barrier level L. Define It means that we do not need to monitor the maximum process where the continuation set is expressed as and the stopping set is given by and the optimal stopping time is given by The proof is easy to attend by applying the definition of optimal stopping time.
3) Summarising the preceding facts, we can now apply the approach used in [10] and [15] to obtain a representation for the price of the American knock-in binary option as follows: is the probability density function of the first hitting time of the process (3.1) to the level L. The density function is given by (see e.g. [16]) , where φ is the standard normal density function given by ( ) Therefore, the expression for the arbitrage-free price is given by (3.14) and can be solved by inserting the price of the American cash-or-nothing put option.
The value of the American cash-or-nothing put option is given by [6] ( ) The other three types of binary options: cash-or-nothing call, asset-or-nothing call and put follow the same pricing procedure and their American values can be referred in [6].

The American Knock-Out Cash-Or-Nothing Options
where K is the strike price, L is the barrier level and is the maximum of the stock price process X. Recall that the unique strong solution for The process X is strong Markov with the infinitesimal generator given by We introduce a new process which represents the process X stopped once it hits the barrier level L. Define It means that we do not need to monitor the maximum process X behaves exactly the same as the process X for any time L t τ < and most of the properties of X follow naturally for L t X .
2) Let us determine the structure of the optimal stopping problem (4.2). Standard Markovian arguments lead to the following free-boundary problem (see [17]) where the continuation set is expressed as the stopping set is given by and the optimal stopping time is given by denoting the first time the stock price is equal to K before the stock price is equal to L. We will prove that K is the optimal boundary and K τ is optimal for (4.2) below.
3) We will show that (4.13) is optimal for (4.2). The fact that the value function (4.2) is a discounted price indicates that the larger τ is, the less value we will get. As to the payoff, it is either £1 or nothing. Therefore, the optimal stopping time is just the very first time that the stock price hits K, which is (4.13). To prove this, we define τ as any stopping time. We need to show that , , e I I e I I . Actually, On the other hand, Hence we conclude that K τ is optimal in (4.2). E e I , .
for K x L ≤ ≤ . The value function concerns with the convergence due to the sum of an infinite series. More precisely we will apply the optimal stopping theory to value (4.2) and get a better result. However, the result from (4.20) indicates some properties of the pricing (4.2). It is easy to verify that local time-space formula is applicable to our problem (4.2). 5) To get the solution to the optimal stopping problem (4.
where the function

( )
, The martingale term vanishes when taking E on both sides. From the optional sampling theorem we get The first term on the RHS is the arbitrage-free price of the European knock-out cash-or-nothing put option E V at the point ( ) , t x and can be written explicitly as (see [6]) Recall that the joint density function of geometric Brownian motion and its maximum ( ) is given by (see [16]) The second step is attained by Fubini's Theorem and Dominated Convergence Theorem. By the definition of derivative, the last step in (4.31) equals The density function is the density function for standard normal distribution. Therefore, (4.30) can be expressed as Substituting the result (4.34) into (4.26), we get the early exercise premium (EEP) representation for the American knock-out cash-or-nothing put option where the first and second terms are defined in (4.27) and (4.28).
The main result of the present subsection may now be stated as follows. Below, we will make use of the following function ( ) Theorem 1. The arbitrage-free price of the American knock-out cash-ornothing put option follows the early-exercise premium representation where the first term is the arbitrage-free price of the European knock-out cash-or-nothing put option and the second and third terms are the early-exercise premium.

Financial Analysis of the American Barrier Binary Options
The payment of the American barrier binary options is binary, so they are not ideal hedging instruments. Instead, they are ideal investment products. It is popular to use structured accrual range notes in the financial markets. Such notes are related to foreign exchanges, equities or commodities. For instance, in a daily accrual USD-BRP exchange rate range note, it pays a fixed daily accrual interest if the exchange rate remains within a certain range.
Generally, an investor buying a barrier option is seeking for more risk than that of a vanilla option since the barrier options can be stopped or "knocked-out" at any time prior to maturity or never start or "knock-in" due to not hitting the We plot the value of the American barrier binary options using the free-boundary structure in the above sections. Note that the value of ( ) , x V u K + in Equations (4.37), (4.41) and (4.42) separately is estimated by finite difference method (see [19]).
The American value curves in Figure 3 and Figure 4 are simulated from (15) by inserting different American binary option values. Figure 3 shows that the  Figure 4 shows the value of the American up-in cash-or-nothing put option (asset-or-nothing put is similar ). As we can see before the barrier, the option value is increasing and gets its peak at the barrier. Then the value goes down as the stock price continues to go up after the barrier level. Generally, the price of the American version options is larger than the European version. Figures 5-8 show the values for the knock-out binary options. Figure 5 illustrates that the value of the up-out cash-or-nothing put option is a decreasing function of the stock price below the barrier. However, in Figure 6 the up-out asset-or-nothing put first goes up and then down to the barrier. We can see the value of the down-out cash-or-nothing call option in Figure 7 is strictly increasing as the asset price above the barrier. The asset-or-nothing call value in Figure 8 is also in the similar situation but with different amount of payoff size. All of the out figures show that the smooth-fit condition is not satisfied at the stopping boundary K.     The results of this paper also hold for an underlying asset with dividend structure. With minor modifications, the formulas developed here can be applied to handle those problems.