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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.

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 crosses the barrier, the barrier option knocks in and becomes a regular option. If the stock price never passes the barrier, the option is worthless no matter it is in the money or not. An out barrier option or knock-out option pays off only if the option is in the money and the barrier is never being crossed in the time horizon. As long as the barrier is not being reached, the option remains a vanilla version. However, once the barrier is touched, the option becomes worthless immediately. More details about the barrier options are introduced in [

The use of barrier options, binary options, and other path-dependent options has increased dramatically in recent years especially by large financial institutions for the purpose of hedging, investment and risk management. The pricing of European knock-in options in closed-form formulae has been addressed in a range of literature (see [

There are many different types of barrier binary options. It depends on: 1) in or out; 2) up or down; 3) call or put; 4) cash-or-nothing or asset-or-nothing. The European valuation was published by Rubinstein and Reiner [

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.

knock-out + knock-in = knock-less . (2.1)

Above all, barrier options create opportunities for investors with lower premiums than standard options with the same strike.

We start from the cash-or-nothing option. There are four types for the cash-or-nothing option: up-and-in call, down-and-in call, up-and-input and down-and-input. 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 down-and-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.

1) Consider the stock price X evolving as

d X t = r X t d t + σ X t d W t (3.1)

with

where K is the strike price, L is the barrier level and

under

We introduce a new process

It means that we do not need to monitor the maximum process

2) Standard Markovian arguments lead to the following free-boundary problem

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 [

for

for

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 [

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 [

1) Consider the stock price X evolving as

with

where K is the strike price, L is the barrier level and

under

We introduce a new process

It means that we do not need to monitor the maximum process

2) Let us determine the structure of the optimal stopping problem (4.2). Standard Markovian arguments lead to the following free-boundary problem (see [

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

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

Actually,

On the other hand,

Hence we conclude that

4) Based on the optimal stopping time (4.13), a direct solution for (4.2) can be expressed as

For the geometric Brownian motion the density

for

for

for

5) To get the solution to the optimal stopping problem (4.2), apply Ito’s formula to

where the function

and

The martingale term vanishes when taking E on both sides. From the optional sampling theorem we get

for all stopping times

for all

The first term on the RHS is the arbitrage-free price of the European knock-out cash-or-nothing put option

We write

Recall that the joint density function of geometric Brownian motion and its maximum

for

6) We will discuss the calculation about the local-time term

From the definition of local time

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

where

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

for all

Theorem 1. The arbitrage-free price of the American knock-out cash-or-nothing put option follows the early-exercise premium representation

for all

The proof is straightforward following the points 4, 5 and 6 stated above. Note that our problem is based on the stopped process

The cash-or-nothing call option can be handled in a similar way. The different part is the European value function in (4.27). The arbitrage-free price of the European down-out cash-or-nothing call option

The arbitrage-free price of the European knock-out asset-or-nothing option

where

Theorem 2. The arbitrage-free price of the American knock-out asset-or-nothing option follows the early-exercise premium representation

for all

for all

Proof. The proof is analogous to that of Theorem 1. Back to (4.22), it is easy to verify that the value of H vanishes since

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 barrier. Basic reasons to purchase barrier options rather than standard options include a better expectation of the future behaviour of the market, hedging needs and lower premiums. In the liquid market, traders value options by calculating the expected value of the pay-offs based on all stock scenarios. It means to some extent we pay for the volatility around the forward price. However, barrier options eliminate paying for the impossible scenarios from our point of view. On the other hand, we can improve our return by selling a barrier option that pays off based on scenarios we think of little probability. Let us imagine that the 1-year forward price of the stock is 110 and the spot price is 100. We believe that the market is very likely to rise and if it drops below 95, it will decline further. We can buy a down-and-out call option with strike price 110 and the barrier level 95. At any time, if the stock falls below 95, the option is knocked-out. In this way, we do not pay for the scenario that the stock price drops firstly and then goes up again. This reduces the premium. For the hedgers, barrier options meet their needs more closely. Suppose we own a stock with spot 100 and decide to sell it at 105. We also want to get protected if the stock price falls below 95. We can buy a put option struck at 95 to hedge it but it is more inexpensive to buy an up-an-out put with a strike price 95 and barrier 105. Once the stock price rises to 105 when we can sell it and this put disappears simultaneously.

The relationship between knock-in option, knock-out option and knock-less option (standard option) of the same type (call or put) with the same expiration date, strike and barrier level can be expressed as

This relationship only holds for the European barrier options. It has not been obtained for the American version when we get the American values from the sections above.

We plot the value of the American barrier binary options using the free-boundary structure in the above sections. Note that the value of

The American value curves in

Figures 5-8 show the values for the knock-out binary options.

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.

The authors are grateful to Goran Peskir, Yerkin Kitapbayev and Shi Qiu for the informative discussions.

The authors declare no conflicts of interest regarding the publication of this paper.

Gao, M. and Wei, Z.F. (2020) The Barrier Binary Options. Journal of Mathematical Finance, 10, 140-156. https://doi.org/10.4236/jmf.2020.101010