Although not explicitly covered, the principles will apply to regular barrier put options as well as reverse barriers. In the up and out call option examples the profiles shown in Figs.

This was done for purely aesthetic purposes and does not substantially alter the key learning points. This position would be representative of an institution that has structured a reverse convertible bond see Chap.

Bennett, C. Google Scholar. De Weert, F. Haug, E. Download references. You can also search for this author in PubMed Google Scholar. Schofield, N. Barrier and Binary Options. In: Equity Derivatives. Palgrave Macmillan, London. Published : 15 March Publisher Name : Palgrave Macmillan, London. Print ISBN : Online ISBN : eBook Packages : History History R0. Anyone you share the following link with will be able to read this content:. Sorry, a shareable link is not currently available for this article.

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.

Binary Option, Barrier Option, Arbitrage-Free Price, Optimal Stopping, Geometric Brownian Motion, Parabolic Free Boundary Problem. 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 [1] and [2]. 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 [3] [4] [5] and reference therein. There are two types of the knock-in option: up-and-in and down-and-in. Any up-and-in call with strike above the barrier is equal to a standard call option since all stock movements leading to pay-offs are knock-in naturally.

Similarly, any down-and-in put with strike below the barrier is worth the same as a standard put option. An investor would buy knock-in option if he believes the movements of the asset price are rather volatile.

Rubinstein and Reiner [6] provided closed form formulas for a wide variety of single barrier options. Kunitomo and Ikeda [7] derived explicit probability formula for European double barrier options with curved boundaries as the sum of infinite series. Geman and Yor [8] applied a probabilistic approach to derive the Laplace transform of the double barrier option price.

Haug [9] has presented analytic valuation formulas for American up-and-input and down-and-in call options in terms of standard American options. It was extended by Dai and Kwok [10] to more types of American knock-in options in terms of integral representations. Jun and Ku [11] derived a closed-form valuation formula for a digit barrier option with exponential random time and provided analytic valuation formulas of American partial barrier options in [12].

Hui [13] used the Black-Scholes environment and derived the analytical solution for knock-out binary option values. Gao, Huang and Subrahmanyam [14] proposed an early exercise premium presentation for the American knock-out calls and puts in terms of the optimal free boundary. 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 [6]. However, the American version is not the combination of these options. This paper considers a wide variety of American barrier binary options and is organised as follows. In Section 2 we introduce and set the notation of the barrier binary problem. In Section 3 we formulate the knock-in binary options and briefly review the existing work on knock-in options. In Section 4 we formulate the knock-out binary option problem and give the value in the form of the early exercise premium representation with a local time term.

We conduct a financial analysis in Section 5 and discuss the application of the barrier binary options in the current financial market. 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.

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

Figure 1. A computer comparison of the values of the European barrier cash-or-nothing call CNC and asset-or-nothing call ANC options for t given and fixed. Figure 2. A computer comparison of the values of the European barrier cash-or-nothing put CNP and asset-or-nothing put ANP options for t given and fixed.

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. with under P for any interest rate and volatility. Throughout denotes the standard Brownian motion on a probability space.

The arbitrage-free price of the American cash-or-nothing knock-in put option at time is given by. 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 3. 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 , where is the first hitting time of the barrier L as.

It means that we do not need to monitor the maximum process since the process behaves exactly the same as the process X for any time and most of the properties of X follow naturally for. for and , where is the probability density function of the first hitting time of the process 3. The density function is given by see e. for and , where is the standard normal density function given by for.

Therefore, the expression for the. arbitrage-free price is given by 3. 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 arbitrage-free price of the American up-out cash-or-nothing put option at time is given by. Recall that the unique strong solution for 4. Define , where is the first hitting time of the barrier L:. Standard Markovian arguments lead to the following free-boundary problem see [17].

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 is optimal for 4.

The fact that the value function 4. 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. To prove this, we define as any stopping time. We need to show that. Hence we conclude that is optimal in 4. For the geometric Brownian motion the density is known in closed form cf. for , where is given by cf. The result is straightforward. 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. However, the result from 4. It is easy to verify that local time-space formula is applicable to our problem 4. where the function is defined by. is given by. and refers to integration with respect to the continuous increasing function , and is a continuous local martingale for with.

The martingale term vanishes when taking E on both sides. From the optional sampling theorem we get. for all stopping times of X with values in with and given and fixed.

Replacing s by in 4. for all , where and for. We obtain the following early exercise premium representation of the value function. The first term on the RHS is the arbitrage-free price of the European knock-out cash-or-nothing put option at the point and can be written explicitly as see [6]. Recall that the joint density function of geometric Brownian motion and its maximum under P with is given by see [16]. for with. Note that.

Chapter 1 provided a brief explanation of two popular exotic options, barriers and binaries—but what is an exotic option? There is no accepted definition but broadly speaking, it can be thought of as an option whose payoff is different from a vanilla, non-exotic option, for example, a payoff that is different from those shown in Fig.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, access via your institution. Although not explicitly covered, the principles will apply to regular barrier put options as well as reverse barriers. In the up and out call option examples the profiles shown in Figs.

This was done for purely aesthetic purposes and does not substantially alter the key learning points. This position would be representative of an institution that has structured a reverse convertible bond see Chap. Bennett, C. Google Scholar. De Weert, F. Haug, E. Download references. You can also search for this author in PubMed Google Scholar.

Schofield, N. Barrier and Binary Options. In: Equity Derivatives. Palgrave Macmillan, London. Published : 15 March Publisher Name : Palgrave Macmillan, London. Print ISBN : Online ISBN : eBook Packages : History History R0. Anyone you share the following link with will be able to read this content:. Sorry, a shareable link is not currently available for this article.

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Abstract Chapter 1 provided a brief explanation of two popular exotic options, barriers and binaries—but what is an exotic option? Keywords Call Option Implied Volatility Strike Price Spot Price Barrier Option These keywords were added by machine and not by the authors.

Buying options Chapter USD eBook USD Hardcover Book USD Tax calculation will be finalised during checkout Buy Hardcover Book. Learn about institutional subscriptions. Notes 1. Sometimes referred to casually as an American barrier. Bibliography Bennett, C. Author information Authors and Affiliations Verwood, Dorset, UK Neil C Schofield Authors Neil C Schofield View author publications.

Copyright information © The Author s. About this chapter Cite this chapter Schofield, N. Copy to clipboard.

The arbitrage-free price of the European knock-out asset-or-nothing option at the point can be written explicitly as see [6]. 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. for all stopping times of X with values in with and given and fixed. 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. 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.

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