There is no exact definition of exotic options. In popular terms, these are any financial options that contain clauses that do not allow them to be classified as vanilla options. The creation of new exotic options, as a rule, is associated with the need for banks or corporations to pack assets and risks for their sale or redistribution through derivative financial instruments.

NB! Exotic conditions can be very different:

• without a knowingly clear execution price at the time of the conclusion of the contract, but with a clause on the method of its determination ( Asianoptions - Asian options);
• activation or cancellation of an option depending on the price level within a specified period ( barrier options);
• the presence of indications of the price corridor and others.

There may also be clauses linking an option contract to certain values ​​of indices (which are not the underlying asset of the option), portfolio of assets, exchange rates, rates, and the like.

Many exotic options are of no practical interest to ordinary investors. Investors are primarily interested in products that are available for trading and have an acceptable level of liquidity.
Of the exotic options, these requirements can be met by barrier options And binary options. These options have peculiar pricing mechanisms that differ
extreme individuality in both barriers and binaries. This is manifested in the specificity and features of elasticity indicators (Greeks), which, in fact, provide the speculative attractiveness of financial options.

Separate barrier and binary options, even for an ordinary investor, may be more interesting than vanilla options.

Regardless of the issuer, it must be understood that exotic options are created in response to the specific needs of investors and have a distinct personality. Because of this, the classification of exotics is somewhat arbitrary and often ambiguous, which gives rise to a very dangerous hidden risk of misunderstanding. Moreover, similar instruments - due to small features - may have different names.

barrier options

Barrier option - a contract, the payments under which are made dependent on the achievement of a certain price level of the underlying asset at or within a specified period of time.

barrier options related European call or put vanilla options, supplemented barrier clauses. However, these contracts can be American, and other performance styles. In fact, there are quite a lot of them, because a new style may result from crossbreeding. For example, bermuda style execution, which is necessarily used for binars, is an intermediate option between American And European styles, giving the right to the holder of the contract to execute it within a specified period repeatedly at certain intervals.

Barrier (barrier or trigger) - a price level that activates a barrier option.

Distinguish coming alive(coming-to-existence) options when barrier activates contract, which in this case is inclusion option(knock-in) or included option. Accordingly, there are dying(extinguishing) options in which the barrier de-activates contract, which makes it an option shutdown(knock-out), or turning off option.

There are four main types of barrier options. Knock-in-up-and-in is activated if the price of the underlying asset rises to the barrier and exceeds it. Knock-in-Down-and-In is activated if the price of the underlying asset breaks the barrier from top to bottom. Knock-out-Down-and-Out is deactivated if the price of the underlying asset falls below the barrier. Knock-out-Up-and-Out deactivated if the price of the underlying asset rises and exceeds the barrier.

There are more detailed classifications of barrier options, which include determining whether the option is a put or a call and indicating where the barrier is in relation to the strike price.

Let's say triggering call option(Knock-in-Down-and-In-Call) comes alive when he is out-of-money. However, reverse triggered call option(Reverse Knock-in-Up-and-In-Call) revives when situation in-money. Detailed classifications are essentially descriptions ready-made strategies for investors.

If activation has not taken place by the expiration date, under the terms of some barrier options, the holder receives compensation (rebate).

More complex types of barrier options include options with double barriers (double-barrier options), with reservations regarding the effective date (option window- windows option), choice options(choice option), which do not specify the type of option (put or call), but determine the date when the buyer can make his choice, and the like.

NB! In some cases, the value of exotic options follows the put-call parity rules that apply to vanilla contracts. It is on the basis of this ratio, for example, that the price is calculated choice options (Chooser).

Hypothetically, put-call parity should be observed in suitable included And turning off European-style options if there is no loss compensation condition. For example, according to the meaning of the barriers presented above, the following condition must be met: long base + long Knock-out-Up-and-Out = long Knock-in-Up-and-In.

Binary Options

A binary option is a contract, the payment for which has a fixed amount and is made when the agreed condition is met for a certain period of time or on a specific date.

The condition, as in barrier options, is usually associated with the achievement of the underlying asset a certain price level. When an option is exercised, calculations of a known size take place. In these tools barrier often defined as strike price.

In binaries, the potential income and loss are fixed and known in advance.

The amount of loss on such contracts - regardless of the position in them (long or short) - is also limited. Some variations of binary options involve payouts compensation(rebate) to the losing side.

It should be remembered that European style inary options performed only at the specified time, even if the barrier is reached early. These options usually include derivatives such as all or nothing(all-or-nothing option), money-or-nothing(cash-or-nothing), asset-or-nothing(asset-or-nothing), digital option(digital option), option with fixed income (fixed return option - FROs).

European style binary options are only executed at the specified point in time.

Binary Options american style- which includes options with-touch (touch option), as well as options without-touch (no touch option),- upon arrival at the barrier immediately activated and executed.

Options with-touch ( touch option) is also called tangent options, and options no-touch(no touch option) are also known as touchy.

American style binary options are executed immediately upon arrival at the barrier.

There is an opinion that binary options are a bet “for luck” packed into a financial instrument - like in a casino. However, methodologically, these derivatives can be considered as barrier options that are activated when the barrier is reached at the in-the-money strike price. At the same time, the distance from the strike price to the barrier is such that it provides the necessary payment for the option.

Such an interpretation, despite its simplicity, is rather a purely theoretical refinement, although it opens up scope for checking the value of options using various models. In practice, for binary options, the barrier is defined as the strike price, so that a suitable option pricing approach is applied.

American and European style binary options vary widely in price and trading opportunities

European style binary options

Binary Options on the Chicago Board Options Exchange (CBOE), which have been traded on the S&P 500 and the VIX Volatility Index since 2008, are a classic example of options all or nothing European style.

NB! Externally, binary options on the S&P 500 and VIX indices are no different from European vanilla options, except that the value of the binaries at expiration does not exceed $100, no matter how deep they are in-money. The payout for a losing short position of an exercised option is also limited to the same amount, the net loss on which is the difference between $100 and the premium of the option contract when it is sold.

According to the theory, the value of these options is equal to the delta of the corresponding vanilla options- that is, having the same strikes. Possible discrepancies are due to the fact that option pricing is closely related to the assessment of the probability of reaching the strike price by the underlying asset. However, a purely probabilistic approach often leads to mispricing binary options.

The quotation is carried out in fractions of a unit. You can sell and buy an option at any time before the expiration date.

American style binary options

Options one-touch(one touch) and no-touch(no touch) on exchange rates - available to ordinary investors when working through Saxo Bank (6 major pairs at the beginning of 2013, plus gold and silver) - represent good example american style binary options.

Binary options on currency, gold and silver from Saxo Bank have an American style of execution

These options are exercised immediately after the spot rate reaches the strike price(barrier), so there is no need to determine the type of option (call or put). There is just the current spot market and option strike prices (barriers) that are higher and lower with a certain discreteness. For example, for EUR/USD, the usual step is 50 points (Fig. 4).

The closer the strike price tangent options to the current spot market, the greater the probability of winning, which means that the purchase is more expensive and the sale of these contracts is cheaper.

FROM touchy exactly the opposite.

binary options quotes in percents, which are multiplied by the amount of payments for the option specified by the investor when it is exercised.

The range of available options in terms of payouts starts from 100 units of the base currency and more. Options of any kind that are not exercised within their measured life are closed on the expiration date with settlement of non-activated options.

There is a conditional equivalence between these binaries:

Long with-touch ~ short without-touch

Short with-touch ~ long without-touch

All this means the following. If, for example, with a EUR/USD exchange rate of 1.30 tangent an option with a barrier of 1.32, expiring in 7 days, is quoted at 16.27 / 26.27 (%), then you can buy it for 26.27%. With the amount of payments established by the investor in 100 units. base currency (which in this case - the euro), the purchase will cost 26.27 euros. When the level of 1.32 is reached during the life of the option, it is immediately exercised, after which a payment of 100 euros is made in favor of the buyer, providing him with a profit of 73.73 euros (= 100-26.27). If the market fails to hit the barrier, the option will burn out, depreciating to zero, resulting in a net loss of €26.27.

For comparison, touchy the same lifetime with a barrier of 1.32 will have a quote of 73.73 / 83.73 (%). Short position in an option no-touch similar to long one-touch with a barrier of 1.32, providing an income to the account of 73.73 euros when selling the option. If the market touches the barrier, it will be removed and the option will be liquidated without any settlement between the parties to the transaction, bringing the trader a net income of 73.73 euros. If the market does not hit the barrier, then the option will be activated at expiration, as a result of which this short position will have to pay 100 euros. The final result will be negative, bringing a net loss of 26.27 euros - 100 euros debit with a loan of 73.73 euros.

Purchase performance no-touch and sales one-touch binary options with the same strike price is the same. In both cases, the transaction is made based on the assumption unreachable barrier.

The above information can be gleaned from the option board with binary options in trading platform Saxo Bank. The quote can be seen in two ways - as a percentage or in monetary terms. From a preliminary order to buy or sell, you can find out Greeks and get an idea of exposure option - its current equivalence of a position in a spot asset (Fig. 5).

Fig.5. Orders to buy and sell binaries in SaxoTrader (Saxo Bank)

When determining a different amount of payments - say, 500, 1000 units, and so on - the cost of investments increases accordingly. As well as the size of the winnings in absolute terms. But, of course, not in relative terms, the evaluation of which is given on the option board and in the preliminary order.

To be continued.

Postscript: although there are banners and links throughout the site, I’ll write again that you can buy it using this link (if you’re too lazy, let us know about your desire in the comments). Take it, it's weight. The book is capacious, understandable and only 150 pages long - an excellent guide for beginners, and experienced option traders too. It turned out!

Exotic options appeared as derivatives of classic financial instruments. The main incentive for the development of new types of options is the desire of traders to control operational risks and possible payouts.

Barrier option, as one of the exotic types, is the most popular in its group. The reason for such popularity is ease of use and relatively low starting requirements.

What is a barrier option

As already noted, the barrier option is one of the exotic types. The essence of the contract in this case is to receive payment or not receive it, provided that the value of the asset reaches the specified level (barrier).

This level performs a kind of role of a switch - a trigger, so you can still find the second name of options of this type in the network - trigger options. The switch, depending on the initial conditions, can enable the option or deactivate it.

In addition to the trigger level, the barrier option does not differ from the classic European option with a fixed expiration time. Like other BOs, these contracts can be classified according to the type of underlying asset, for example, a currency option with a barrier condition, stocks, indices. However, it is more informative to classify these contracts by trigger type:

1. trigger to enable the option (knock-in);
2. switch to disable the option (knock-out).

Each of these types is subdivided into two more subspecies depending on the direction of the trend - up and on, down and on, up and off, down and off.

Simple barrier option

Knock-in

If the value of the asset reaches the specified barrier, then the key to activate the contract will work. This ability is also called the "revival" of the option. For example, you bought a call option worth one hundred units, and the option will only be activated if the asset reached level 95 (barrier) during its lifetime.

Knock-out

This type of contract protects the trader from a bad deal. If the price during the contract has reached the specified barrier, then the option is automatically canceled (“dies off”).

Using Barrier Options

All subtypes of barrier contracts can be used for Call and Put deals. Barrier put and BO call options significantly expand the player's opportunities in the market. Their structure allows not only to expand the trader's behavioral range, but also to reduce the cost of the option, since the premium for purchased barrier options is much lower.

So, for example, in order to secure their capital, a trader can buy multidirectional options. Depending on the behavior of the market, the disadvantageous option is disposed of. But the use of barrier contracts allows you to insure yourself even more, since well-placed triggers activate or deactivate contracts in time.

Barrier Options Variations

Barrier options at a price that will satisfy any client arose as an interaction between market needs and the capabilities of brokers. Naturally, one option option was not enough. There are such barrier options, where the barrier is valid only for a while, and then disappears, as well as many other options.

The most common variations of barrier options are:

1. money back - crediting the seller;
2. Explosive options - the assignment of the contract is equal to the internal price;
3. double barriers - the level of the switch is calculated by a formula from several assets.

Of course, the list of variations does not end there, companies offer more and more interesting combinations.

Currency option with a barrier condition

Currency options with a barrier condition were used in the United States as early as the mid-seventies of the last century, but became wildly popular in Japan in the early eighties. The theoretical calculations of well-known financiers contributed to the breakthrough of this type of trade.

Today, barrier options are among the top most popular transactions among the non-exchange market. More than 10% of the total volume of currencies is occupied in this type of trade.

A barrier option is an example of the rational use of one's capital and risk control. It is believed that any subspecies of this type of transaction is opened at the “out of money” moment. For any put option, regardless of subtype, the switch must be in the money. That is, the options themselves are aimed at making money for the trader.

What is such a financial instrument of enrichment as a deposit-backed currency option? What features exist here? What should you focus on when making deals?

general information

Let's deal with the terminology. What is an option? This is a kind of contract that allows you to exchange one currency for another if the deal is winning, and abandon it if it is not profitable. In this case, you can act according to two schemes: "call" (call) and "put" (put). In the first case, it is possible to purchase a predetermined amount of currency on a specific day and at a predetermined price. "Put" provides for the sale on the same terms. Deposit-backed currency options are offered by banks to their clients. They are carefully organized. It includes such instruments as a currency option and a deposit. Why is such a product offered at all? Thus, banks give their customers the opportunity to earn a higher percentage than is offered by conventional deposits. But at the same time, it is necessary to understand that savings are also subject to currency risk.

About the benefits

A standard currency option has the following advantages:

1. Opportunity to receive more significant income than under the terms of a simple deposit.
2. Flexible choice of term, currency pairs, rate and profitability.
3. A short period of placement, usually up to three months.
4. Possibility of remote registration.

Who is it suitable for?

Holders of cash savings, assets and liabilities, who have their own vision of how rates will develop, can try options on currency pairs. Of course, if they are not afraid of risks. When choosing this type of investment, it is necessary to give preference to the base and alternative currency pairs, a carefully chosen period of placement, as well as the exchange rate. What should be invested in? The base currency is used for this purpose. But it can be converted into alternative funds according to the previously set rate. The income received from this transaction is an option premium. It is formed from the difference in exchange rates. The currency options market is quite developed, so if you want to find a niche that is convenient for you, it is not difficult.

What do you need to do to make money?

Suppose the reader is interested in a currency option transaction. What do you need to do to participate in it? On the day of registration of the option, you should:

1. Deposit a certain amount in a pre-selected base currency. As such, the Russian ruble, euro, US dollar, British pound sterling, Chinese yuan or Swiss franc are most often used. Options may vary slightly between financial institutions.
2. The second currency is selected.
3. The placement period is set. As a rule, a range from seven to ninety days is available.
4. The exchange rate is negotiated, as well as the premium for the currency option. Remember that the larger the amount you claim, the higher the risks.

When the day of determining the option comes, then there are several options for further action:

1. If the exchange rate is less profitable or equal to the one determined at the time of the conclusion of the contract, then on the day of return the person claims to receive the amount of his investments, interest on the deposit, as well as an option premium in the base currency. Alas, this is a failure.
2. If the rate is more profitable than the one determined by the person, then he receives all the amounts of money due to him in an alternative currency.

Let's say a word about the award

Probably, now many readers are interested in one question. How is the option premium determined? The task of paying out cash is entrusted to the trader who purchases the option. Such an option of investing money together with the interest accrued on the deposit will allow you to get a high income. Of course, if there were corresponding movements in the market. Let's look at a small example to explain how this system works. Suppose a person has an assumption that the value of the euro will increase from 90 to 95 rubles in a week. Since it assumes the growth of the currency, the choice is made in favor of call options. Then they are bought from a broker, often accompanied by certain surcharges. But let's not talk about them for now. If the rate really rose, the value of the option also increases with it. At the same time, no additional investments are required from a person. The option itself can be stored until the end of its term, or it can be exercised at any convenient moment before the end of the agreements.

About individual moments and development

The tool considered in the framework of the article is constantly changing and improving. There is even something that was not there before. One of the latest developments is binary currency options. It's such an amazing transformation valuable papers that it is difficult to see their ancestors in them. What is their feature? Initially, any asset is completely thrown out of circulation. The main object here is the price. Traders are interested in its behavior over a certain period of time, and that is what they bet on. If they think that the price will rise, then they take a “call”, if they think it will fall, they take a “put”. You see, even these words have lost their original meaning. At the same time, the price of an asset does not depend on its behavior. In this case, the most interesting is the percentage of the value at the time of the agreed day to the amount of the rate. This is what the premium is based on. The more bet, the greater the win, as well as the risk of losing money. If the investor's bet has not played, then the premium remains with the broker. As you can see, binary currency options provide that only one side will win.

How to make a profit?

The considered financial instrument attracts a large number of speculators of all stripes. Well, this is not surprising, because it allows you to get huge incomes. And at the same time, the probable losses are clearly recorded. But if this is such a profitable business, then why are there those who are ready to sell currency options? And the answer is simple: the amount of premium that the seller of the option receives is commensurate with the risk of unlimited losses. Does it make sense to get involved in this whole story? Theoretically, it is quite possible to become a successful bidder. To do this, you only need to be the first to know about all fluctuations in the exchange rate, be able to predict its changes and have information that others do not know about. Otherwise, it will be very difficult to compete. It should be understood that financial activity has significant risks. Here are some examples:

1. The price of transactions is overstated in comparison with other types of investments.
2. You need to have a deep knowledge of finance, as well as relevant experience. Currency options are difficult to implement, and it is impossible for everyone to succeed in this area.
3. It should be borne in mind that such an investment is extremely time-bound. Therefore, many contracts remain unfulfilled.

As you can see, making money in this case is not so easy as it might seem at first glance. Is it worth risking your money, time and nerves - everyone decides for himself. The main issues have already been considered, let's pay attention to additional points.

Administrative difference

Initially, currency options began to be used in Europe. But over time, they gained great popularity in America. Now they are also widely used in Asia. At the same time, depending on the geopolitical attachment, their own characteristic features appeared. The most popular now is the American model of interaction. Its feature is the possibility of early execution of rights. The European model is aimed at obtaining medium incomes with minimal risks. Asians perceive an option as a specific commodity, which should have its own price. For this, exchanges are widely used.

What to play?

People who have decided, but have not yet begun to act, are for the most part interested in the question: what are the best currency pairs for binary options? Unfortunately, there is no 100% correct answer here. To do this, you need to be aware of the many different trends, specific moments and other elements of successful investing.

Let's look at a small example. Here we have the euro, US dollars and yuan. What can be said in this case? The most predictable pair is the dollar and the yuan. The Chinese government supports the export economy. And for this you need a cheap own currency. Therefore, they constantly depreciate the yuan against the US dollar. But this is not all possible field of activity. Other interesting currency pairs for binary options are euros and dollars, as well as the money of the European Union and the yuan. True, it is impossible to accurately predict here. So, now the euro exchange rate against the dollar is growing, although just a year ago it was steadily falling. And it was seriously believed that the euro would become cheaper than the dollar.

Conclusion

You must be aware that financial activities involve enormous risks. There are always losers and winners in it. And if you do not have the necessary knowledge, experience, acumen, then there is a very high probability of falling into the first category, the losers. This topic is quite broad and requires considerable preparation. One could also talk about barrier and range currency options and other financial instruments. But to fully cover this topic, even speaking in passing, the size of the article is not enough. Even one book will not be enough. For people who want to conduct successful financial activities, it is possible to obtain a full-fledged higher education which now takes four years. Yes, not all the information that is taught is useful, but consider whether it is worth competing with those who have studied well and have a good understanding of financial processes.

Cap Call Options and Barrier Options

In order to increase profits, in the strategy below we use barrier options, the obligation on the 1.5300 USD call option for 2 million dollars will arise only if the USD/HF rate reaches the level of 1.6100 during the life of the strategy. For example, if the spot is at 1.6000 at expiration (and has never traded at 1.6100), you will exercise the 1.4660 USD call option and the 1.5300 call option will not be exercised.

Options of this family are exercised or not exercised not only depending on the ratio of the strike price to the spot price, but also depending on whether the spot price has crossed some barrier during the contract or not. The barrier is indicated at the conclusion of the contract. Barrier options are possible with both one barrier and two. The main classes of barrier options are in and out options.

This is a barrier option that ceases to exist if EUR/USD is at the level of 0.8000 at least once during the life of the option.

The cost of a barrier option is always less cost corresponding European option , and it is generally more difficult to hedge it.

In 1994, Derman and Kani developed a binomial tree model, previously used for long-term US stock options, to value barrier options under a volatility curve (each period has a different level of volatility). Gradually, binomial models evolved into trinomial ones, making it possible to more accurately estimate option premiums.

Let's take a closer look at barrier options. IN English language there are many different names for options of this type3. We

Revive - the barrier option comes to life when the spot touches the barrier. If, during the life of the option, the spot has not touched the barrier, the option expires and cannot be exercised, even if it is on its own. An option is a financial contract that stipulates the right of the buyer and the obligation of the seller to buy or sell a specified par value at a specified price or with a specified payment within a certain period or on a given date. The contract may stipulate not only the date, but also the time of execution, price levels at which the contract loses its force or enters into force, etc. An out-of-the-money option (Otm) is a call with an exercise price above the current market price of the forward on the underlying asset. A put with an exercise price below the current market price of the forward on the underlying asset. The premium of an out-of-the-money option is equal to the time value of the option.

Foreign exchange derivatives (foreign ex hange derivatives) include forward contracts, currency swaps, vanilla options, and numerous exotic options. Among exotic options, barrier and middle options are very popular. Foreign exchange derivatives are traded primarily on the OTC market. Their standard currency denomination is the US dollar.

Barrier forward - the standard term of a patent is zero, then there will be a point when it should expire with the option value also almost at zero. The critical decision is when the outage occurs or when the renewal end year is reached, after which the absence of any returns is unacceptable. This is something that can be defined, but very likely depends on the industry and the associated product. Considering a decrease in the value of the options involved in a patent may thus justify setting some form of barrier year for patents to generate revenue and recover costs from them.

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• Introduction
• Chapter 1. Theoretical part
• 1. Barrier Options
• 2.1 Types of currency risk
• 2.2 Risk Reversal
• Conclusion
• Applications

Introduction

Barrier options are an example of exotic options that are not as widely available on the market as the classic types of financial derivatives. Barrier options became more prominent in the derivatives market in the 1960s, when they began to be actively used as a currency hedging tool. As a rule, the cost, or premium, of a barrier option is lower than the cost of a similar one, i.e. to buy or sell a classic option, but the barrier option premium is based on expectations regarding the future change in the value of the underlying asset. Therefore, estimating the cost of barrier options allows you to build a forecast in the foreign exchange market. There are not so many materials in Russian on this topic, so the issue of assessing barrier options in the light of exchange rate forecasting is relevant.

object studies of this thesis are barrier options, namely call and put inclusion options.

Subject studies are premiums on barrier options, on the basis of which a forecast is made regarding future changes in the exchange rate and a trading rule is built.

Targetthesiswork can be formulated as follows: solve the problem of the calculus of variations to find floating boundaries (natural levels of support and resistance), which would be the upper and lower boundaries of the barrier levels; based on historical data, namely daily quotes of the euro-dollar currency pair, build a trading rule for a portfolio consisting of barrier options. This will require valuation of barrier inclusion options and analysis of their premiums. That's what it is a taskresearch.

barrier option currency risk

The foreign exchange market is a complex price process that involves risk and expectations of players. Consequently, to insure against extreme emissions in the market and to minimize losses, options were invented, the definition of which is a priori based on the market's expectations regarding the future dynamics of the exchange rate. According to the theory of classical finance, the price process follows a certain law and is subject to noise; From the theory of behavioral finance, the concept of regime change is known. Therefore, the price volatility observed in the market is closely related to the attempt to clean up the noise of a trend change in order to clearly understand the law by which the market moves.

In this way, methodologyresearch thesis implies the following: in the dynamics of the price process, it is necessary to find where market expectations embedded in option premiums change the generator, i.e. The market is moving into a different mode. To do this, a number of mathematical transformations are carried out within the framework of solving the problem of the calculus of variations and finding floating levels. The inner boundaries of the definition area of ​​the distribution function of the price process are taken as natural levels of support and resistance, the breaking of which indicates changes in the direction of price process fluctuations. In addition, these boundaries are taken as the levels of the upper and lower barriers for barrier options. Since only barrier put options were considered in the paper, strike prices for call and put contracts were set as a barrier and plus or minus 100 basis points for the respective option types. For up options, the strike price was set above the barrier, for down options, respectively, lower.

As mentioned above, the cost of options represents the probability of a trend change: as long as the price has not crossed the threshold values, this is noise, only when they are crossed can we speak of a new regime. This assumption underlies the trading rule presented in the paper.

workinghypothesis final qualification work can be formulated as follows: the inner boundaries of the definition area of ​​the probability density function of the price process are perceived not only as support and resistance levels, the breaking of which indicates a change in trend, but also as natural barriers for the corresponding exotic options.

diplomaRworkincludesfromthreemajorparts: introduction, main part and conclusion. The introduction talks about the relevance of this work, defines the object and subject of research, sets the goal of writing a final qualifying work on this issue, formulates the research task and describes its methodology. In addition, the introduction establishes the working hypothesis of the thesis and describes the structure of the work with a brief, within 2-5 sentences, indicating the content of each chapter. The main part consists of two chapters. The first chapter provides a theoretical basis for further research. The first chapter is divided into two paragraphs, each of which consists of several subparagraphs. The first paragraph provides a general theoretical framework for barrier options: the first subparagraph defines these exotic options, presents payment schedules for them; in the second - what is the use of barrier options; the third subparagraph talks about the features of barrier options in comparison with ordinary options. The second paragraph of the first (theoretical) part of the final qualification work is devoted to currency risk hedging instruments and is also divided into three subparagraphs. The first subparagraph contains the classification of currency risks, as well as decision-making techniques in relation to risk hedging; the second and third subparagraphs describe more complex hedging instruments than ordinary options (plain vanilla), however, which are actively used in the derivatives market by both corporations and financial institutions. The second part of the course work presents the practical part. It consists of two points: first, the solution of the problem of the calculus of variations with floating boundaries is described in detail, and the solution of the problem is checked on the basis of real numbers. The second paragraph is devoted to the analysis of barrier options, the calculation of premiums on them. The boundaries of the domain of definition found in the first paragraph are used as the corresponding levels of barriers. An example of the return of a portfolio consisting of barrier options is given in comparison with the market return. In conclusion, conclusions are drawn on the entire final qualifying work, detailed answers are given to the questions posed in the introduction.

The following sources were used in writing the course work. The theoretical basis is the book of John Hull (John C. Hull) "Options, Futures, and Other Derivatives" . The book covers both derivatives markets and risk management features, including credit risk and credit derivatives, forwards, futures and swaps, weather and energy derivatives, and more. The book is based on a smooth transition from theory to practice, which makes it useful for both students and professionals or investors. Data on the spot rates of the euro-dollar currency pair was taken from the Bloomberg Terminal information source. Many useful information was obtained from articles former employees Goldman Sachs quantitative strategies by Emanuel Derman and Iraj Kani , . In addition to the book by John Hull, the analytical basis for writing the coursework was the work of Hans-Peter Deutsch (Hans-Peter Deutsch) and Thomas Björk (Tomas Bjork). Much information has been found on complex hedging instruments in the works of Professor Uwe Wystup of the Frankfurt School of Finance and Management.

This thesis has a practicalapplicability. The construction of the model is based on real data taken from a large authoritative database, which makes the model tied to the real situation in the financial markets. In addition, the algorithm used to calculate the future exchange rate in this paper can be applied to other data series, including the stock market. In fact, this paper presents a ready-made model for analyzing and forecasting the market. Thus, this final qualifying work carries a certain practical meaning.

Chapter 1. Theoretical part

Conventional vanilla options (plain vanilla options) have well-defined properties, and they are traded quite actively on the exchange. Exchanges or brokers regularly update their price quotes or their implied volatility values. However, the over-the-counter derivatives market has a wide range of non-standardized products created by financial engineers - exotic options. Although these types of options make up a small percentage of an investor's portfolio, they are important because of their much higher returns than vanilla options.

Exotic derivatives were required for various reasons. Sometimes it's really a hedging need in the market, sometimes it's for tax, accounting, legal or regulatory reasons that cause treasuries, fund managers or other financial institutions to resort to exotic options. In addition, exotic derivatives often reflect future movements in certain markets. As part of the course work, we will be interested in barrier options.

1. Barrier Options

1.1 What are barrier options

Payments for ordinary options depend on one indicator in the market - the strike. Barrier options - a type of options, the payment for which depends not only on the strike, but also on whether the price of the underlying asset reaches a certain level over a certain period of time or not. Investors use them to get information about the future market situation, since barrier options carry more information than just information about market expectations contained in standard options. In addition, their premiums are usually lower than regular options with the same strikes and expiration dates.

A standard European option is characterized by the time to expiration and the strike price. On the exercise date, the holder of a standard call option receives the difference between the spot price and the strike price if the spot price is higher than the strike price, and zero otherwise. Similarly, the holder of a standard put option gets the difference between the strike price and the spot price if the spot price is below the strike price, and zero otherwise. The owner of a call option benefits from an increase in the spot price, the owner of a put option from a decrease in the spot price.

Barrier options are a modified form of standard options that include both puts and calls. Barrier options are characterized by their strike price and barrier level, as well as discount (cash rebate), associated with reaching the barrier level. As with standard options, the strike price level determines the payment at expiration. However, the barrier option contract specifies that the payoff depends on whether the spot price reaches the barrier before the option expires. In addition, if the barrier is reached, some contracts imply that the holder of the option will receive a discount Derman E., Kani I. The Ins and Out of Barrier Options: Part 1, p. 56 .

Barriers are of two types:

· Upper barrier (up barrier) - above the current price, it can be reached by price movement from below;

· Down barrier - below the current price, can be achieved by lowering the price.

Barrier options can be of two types: on options and off options. An in barrier option (knock-in option) pays only when the spot price is "at the money" and when the barrier is reached before expiration. When the spot price crosses the barrier level, the barrier option is triggered and becomes a regular call or put option of the same type with the same strike price and expiration. If the spot price does not reach the barrier, the option will expire.

An out barrier option (knockout option) pays off if the spot price is in-the-money and the barrier level is never reached before expiration. Since the spot price of the asset does not reach the barrier, the cutoff barrier is a normal option (call or put) with the appropriate strike and expiration. Thus, barrier options can be up-out (up-and-out), up-in (up-and-in), down-out (down-and-out), down-in (down-and-in) . Types of barrier options and payments on them, provided that the barrier is reached, are presented in Table 1.

Table 1

		Below spot
		Below spot
		Above spot
		Above spot
		Below spot
		Below spot
		Above spot
		Above spot

1.2 Why Use Barrier Options

There are three main advantages of barrier options over standard options:

· Barrier option payments can more accurately reflect future market behavior.

Traders value options based on option theory. In liquid markets, you can estimate the value of an option by calculating the expected payoff on it and averaging all possible market outcomes, where the average price is the forward price in the future. The theory is that the volatility payment is approximately equal to the forward price.

Buying a barrier option, you can not pay for those market outcomes that seem not obvious. Conversely, it is possible to increase the income received by selling a barrier option, the payments on which depend on the least probable market outcomes.

· Barrier options are more eligible for hedging than regular options.

For example, an investor has decided to sell the underlying asset if its price rises in the next period, but he wants to hedge against a fall in price. To do this, an investor can buy a put option with an exercise price lower than the current one, which will hedge the fall, but if the price of the asset rises, the need to hedge the fall ceases to exist. Instead, the investor can purchase an up-out put option with a strike below the spot price and with a barrier above the spot price - thus, if the price rises to the level of the barrier, the put option will cease to exist, since there will be no need for it anymore.

· Barrier option premiums tend to be lower than regular options.

Investors often choose barrier options because the premium on them is lower than on regular options. For example, cut-out options will not pay if the spot price hits the cut-out barrier—thus, they are cheaper than a similar option with no "turn-off" capability. If the probability of a shutdown occurring is low, the investor pays a lower premium and receives the same benefits. In addition, the investor has the right to pay a large premium and receive a return (cash rebate) if the option is turned off.

Likewise, the premiums for inclusion options are lower than for regular options with the same strike and expiration.

1.3 Features of barrier options

Managing the risks of an option portfolio is much more difficult than managing the risks of, for example, a stock portfolio. An investor can hedge options by selling the delta of the underlying asset and buying the option position. In this case, delta is the theoretical hedging factor. Option value and delta depend on both market conditions and volatility. Ordinary call options have delta values ​​between 0 and 1 and a strike that rises when volatility increases Derman E., Kani I. The Ins and Out of Barrier Options: Part 1, p. 58 .

Barrier options, although similar to regular options, are a more complex product because their payments depend on many factors in the future. As with conventional options, an investor can hedge their delta by using a theoretical model to calculate the value of the option and its delta.

The price sensitivity of barrier options can be very different from conventional options. For example, you can compare an up-out call option with a normal call option. As the price of the underlying asset rises, a regular option will always rise in value. In the case of a barrier option, two opposite options are possible. If the price of the underlying asset rises, the payment on the barrier call potentially becomes higher, but that same rise also causes the value of the entire contract to be canceled as it approaches the cut-off barrier. Due to these different directions of movement, the price near the barrier becomes very sensitive, and the delta can quickly change from positive to negative.

There are two main ways in which barrier options differ from standard options when the price of the underlying asset is near the barrier. First, the delta of a barrier option can be significantly different from the delta of the corresponding regular option. For example, a barrier call option can have delta values less than zero or more than one. The up-out call option, whose value is reset to zero when the barrier is reached, has a negative delta near the barrier due to the rapid price decline in this area.

Second, the value of a barrier option decreases as volatility rises. The probability of turning off the up-in call option discussed earlier becomes higher near the barrier as volatility rises.

In some cases, the strike level of the include option is such that any non-zero payment at expiration guarantees that the barrier will be reached. These European barrier options are similar in payment and value to standard European options with the appropriate strike price and expiration date. Any up-in call option with a strike price above the turn-on barrier has the same value as a standard call option because when turned on, the barrier call becomes a standard call. For the same reason, any down-in put with a strike below the barrier has the same value as a standard put.

The same happens with turn-off barriers, if their level of execution is such that any non-zero payment guarantees the turn-off of the option - then the option is reset to zero. Thus, an up-out call with a strike price above the barrier has no price. A down-out put with a strike level below the barrier also has no value.

There is a simple pattern between European on and off options, as well as between standard options. If an investor holds both an on option and an off option of the same type - call or put - with the same expiration date, the same exercise price, and the same barriers in a portfolio, he is guaranteed to receive the payment of the standard option, whether the barrier is reached or not. Thus, the value of a down-in call (or put) with a down-out call (or put) is equal to the value of the corresponding standard call (or put) option. The value of an up-in call (or put) option together with an up-out call (or put) option is equal to the value of the corresponding standard call (or put) option.

2. Hedging toolkit

Barrier options, as well as any other derivative financial instruments, are subject to risk caused by the uncertainty of exchange rate fluctuations, which in turn are already dependent on macroeconomics, geopolitics and speculative interventions. Any economic agent associated with foreign exchange transactions at the macro level - companies in the real sector or financial institutions - are faced with the task of hedging their foreign exchange positions. The second paragraph of the first chapter will consider some more complex than plain vanilla hedging tools that are used to some extent by market participants today.

2.1 Types of currency risk

According to economic theory, market participants face three main types of risks - currency, credit and interest. Corporations and financial institutions are exposed to both the above and many other risks associated with their activities, but it is important to identify and understand these risks in a timely manner, and minimize possible losses. Competent policy of the Treasury allows companies to insure against exchange rate fluctuations.

Obviously, there would be no currency risk if all transactions were carried out in a single currency. For example, there is no such risk between European countries that are part of a monetary union. However, any big company, and even more so a financial institution, due to its size, go beyond the boundaries of one country, a monetary union and are exposed to currency risk.

Currency risk management is not as straightforward as it might seem at first glance. Hedging 100% of FX positions may seem like the most logical solution to the Treasury problem, but it is important to note that even with a full hedging, there is a risk that a company will not be in the best market position relative to competitors if the foreign currency appreciates significantly.

Foreign exchange risk falls into two broad categories:

1. transactional risk- the risk that the national currency will become cheaper or more expensive during the validity of the contract from the moment of its signing until the final payment. For example, at the time of the conclusion of the contract, the exporter agreed on a sale price of 100,000 pounds, and the euro-pound exchange rate was 0.6600. When the final payment date came, the rate rose to 0.7000. A 6% change in the exchange rate resulted in a loss of 8,658 euros for the exporter under this contract.

2. Translational risk - the risk that the value of assets and liabilities denominated in foreign currencies will change due to exchange rate fluctuations, which will be reflected in the balance sheet of the organization. If an exporter has assets in the UK that are worth £330,000, he will show them on his balance sheet at 0.6600 as 500,000 euros. However, if the exchange rate strengthens to 0.7000, the asset will be worth 471,429 euros.

Corporations and financial institutions need to build their own currency risk management policies, namely to find a balance between hedging, flexibility and costs. The currency risk hedging policy should include:

o Risk identification - when certain foreign exchange transactions are made, it is important to correctly assess the fluctuations in exchange rates throughout the duration of the contract

o Risk assessment - the risk should be measured with the greatest accuracy, so that the company could realistically assess the scale of foreign exchange positions in order to budget certain foreign exchange fluctuations into the corporate budget

o Choosing a hedging technique - after companies have assessed the possible losses by risk, it is necessary to choose the most appropriate hedging techniques for their currency positions. It will be useful for corporations to consult with investment banks, which can offer a wide range of hedging products. Also, as mentioned earlier, it may make sense to keep a portion of the currency position unhedged.

o Implementation of hedging techniques - exporting companies must ensure that they understand the correct hedging techniques.

2.2 Risk Reversal

Very often, corporations need so-called zero-cost financial instruments to hedge their transnational cash flows. Since there is a premium to be paid when buying a call option, the buyer can sell another option to finance the purchase of the call option. A frequently used and quite liquid product in the foreign exchange markets is Risk Reversal.

Schedule 1 . Graphs payments on long (left) And short (on right) risk Reversal

The Risk Reversal strategy combines buying a call option and selling a put option, or selling a call option and buying a put option with different exercise prices. This combination can be used as a cheaper hedging strategy than conventional European call and put options.

According to the terms of the Risk Reversal strategy, the owner or investor has the right to buy a certain amount of currency on a certain date at a predetermined rate (strike on the option being acquired), assuming that the market exchange rate at the end date of the option contract will be higher than the strike on the option being acquired (long call / put ). However, if the exchange rate is below the strike price for the option being sold (short call/put) on the expiration date of the option contract, the investor is obliged to buy the amount of currency that corresponds to the strike on the option being sold. Thus, the purchase of the Risk Reversal strategy provides a complete hedge against the growth of the base currency. The investor will exercise the option only if the exchange rate is higher than the strike on the purchased option (long call / put) on the expiration date of the contract.

The strategy of the investor acquiring Risk Reversal is that she or she wants to limit her possible losses. Risk Reversal is used when the currency pair is highly volatile and the market is dominated by bearish expectations regarding exchange rate fluctuations.

It is also interesting to note that Risk Reversal is often used by traders as a measure of market sentiment. Positive Risk Reversal, i.e. when calls are more expensive than their respective puts due to the greater implied volatility of the calls, shows the bullishness of market participants for this currency pair. With a negative Risk Reversal, puts are more expensive than calls, indicating bearish expectations.

Tool Benefits

Full hedge against base currency appreciation

Tool with zero cost (zero-cost)

Tool Disadvantages

When the base currency weakens, the investor's income is limited by the strike of the sold put option

2.3 Target Accrual Redemption Forward (TARF)

In addition to plain vanilla options, investors often use exotic instruments to hedge their positions. An example of such a tool, which is often used as corporate organizations, and financial institutions, can serve as Target Accrual Redemption Forward (TARF).

Under the terms of TARF, an investor sells EUR and buys USD at a much higher exchange rate than the spot or forward exchange rates. The key feature of this product is that the investor has a general target profit level, upon reaching which all subsequent calculations are turned off.

The essence of the tool is to set a strike above the spot in order to allow the client to quickly accumulate profits on each fixing date and complete the transaction within 6 weeks (see Appendix No. 2). The investor will start to lose money if fixings at the euro-dollar rate are higher than the strike price.

Schedule 2 . Graphs payments on bullish (left) And bearish (on right) target Accual Redemption forward

Let the current EUR/USD spot rate be 1.4760, the investor enters into a one-year TARF whereby he or she sells 1 million euros weekly at 1.5335 with the following turn-off condition: if the sum of all the investor's weekly profits reaches the profit target, all subsequent payments are cancelled. Let the target value of accumulated profit be 0.30, which is accumulated weekly according to the following formula: profit = max (0, 1.5335-euro-dollar fixing).

From the table of weekly calculations in Appendix No. 2 to this work, it can be seen that the target profit value of 0.30 was reached in the sixth week. On the fifth week, the accumulated profit was 0.2625, the fixing of the exchange rate on the sixth week was 1.4850. Accordingly, the investor in the sixth week will receive not 0.0485 profit (1.5335-1.4850), but 0.0375, which is not enough for him to achieve the target value. After that, the transaction ceases to exist.

Chapter 2. Practical part. Building Models

In the practical part, a transition will be made to mathematical tools on the issues raised, namely, to solving the problem of the calculus of variations and estimating premiums for barrier options. In addition, a trading rule is built based on the calculated premiums.

2.1 Solution of the problem of the calculus of variations with floating boundaries

As mentioned in the introduction, the task of the final qualifying work is to plot support and resistance levels as a function of the current value of the price process. Usually, the price process is understood as a quote - or, more precisely, its logarithm. However, in this particular case, it is the values ​​of the quotes of the currency pair that will be used, since the values ​​of the upper and lower levels obtained as a result of solving the problem will be the corresponding boundaries for barrier options.

The mathematical basis for the thesis work was the problem of the calculus of variations with two unknown functions Evstigneev V.R. Mathematical theory of support and resistance levels. Bulletin of NAUFOR. This choice is due to the following considerations. Assume that we are given a random price process defined on some area. Within this area, which can be represented as the area of ​​definition of the probability density function of a certain distribution, there are subdomain boundaries that are different for different values ​​of the price process. It is natural to identify this kind of internal boundaries with support and resistance levels.

It makes sense to represent the levels of support and resistance - the internal boundaries within the scope of the random price process - as moving boundaries that define a narrower domain of the function, which serves as a distribution parameter of the price process.

This type of problems is well known - these are problems of the calculus of variations with two required functions and problems of the calculus of variations for functions with moving boundaries. It is logical to accept that the support and resistance levels are the moving boundaries of the second function, which plays the role of a parameter for the first function, i.e. for the probability density function of the parametric distribution of a random price process.

Let's start solving a variational problem with two unknown functions.

First, let's define the following functionality:

The solution of the variational problem will be a pair of functions y (x) and m (x) such that the minimum value will be delivered to the definite integral of the functional F (…). The function y (x) here is the distribution function, and its derivative is, respectively, the probability density function, which is of interest to us. The function m(x) is our desired parametrizing function with floating boundaries, which must be found together with the density function. The values ​​b, c and l are arbitrary constants (scalar parameters of the density function).

The selected functional contains the main expression - the first term - and several restrictions. The main expression expresses the statistical entropy according to K. Shannon - taken with a minus and integrated, it gives a quantitative estimate of the uncertainty inherent in a given random process, provided that it is generated by a given distribution. This value is maximized according to the principle of maximum entropy. Therefore, the integrand for entropy enters the functional with a sign change, since the definite integral of this functional is minimized.

Solving the Euler-Lagrange equations for this problem for each desired function, we obtain a system of two ordinary differential equations - the second order with respect to the parametrizing function and the first order with respect to the density function.

Function solution for m(x):

From here we express m (x) - some parametrizing function - and we get:

The functions p(x) and m(x) are obtained as a pair solution of this system. Here p(x) is the first derivative of the function y(x), i.e. probability density function. The function m(x) must be obtained as a solution to the problem with floating boundaries. The changing boundaries of the definition area of ​​this function are considered as support and resistance levels.

The formal solution for the density function is given below.

It can be seen that it depends on the parametrizing function m(x). The function m(x) can be obtained by fulfilling the conditions on the boundaries of its non-constant domain of definition (at points "a" and "b"). These conditions are given below.

After the disclosure of all operators, they are reduced to the following restriction.

This restriction is obtained at the price of some simplification of the above conditions. The applied simplification requires that the first derivative of the expression containing the conditions be equal to zero at the boundary points, just as the expression itself is equal to zero at these points. To meet this requirement, it is necessary to assume a linear form of the boundary function w(x) at one boundary point and the corresponding function? (x) to another.

Such an assumption not only makes it possible to simplify the boundary conditions - it also allows further simplification of the constraint, since its right-hand side will obviously vanish. In this case, two boundary conditions are obtained, one of which (corresponding to one boundary point of the domain of definition of the function m (x)) refers to the parameterizing function m (x) itself, and the other (corresponding to the second boundary point) refers to its first derivative, as shown below.

This set of boundary conditions can correspond to the functions different kind. Let's choose one of them, which is obtained as a solution of a linear differential equation by the method of direct and inverse Laplace transformation.

Such a specification of the desired parametrizing function makes it possible to use the property of imposing partial solutions. As a result, after taking into account both boundary constraints, the function m (x) takes the following form.

Having obtained the parameterizing function explicitly, we can now explicitly express the probability density function p (x) itself.

However, this is a function for the probability density for the logarithms of exchange rate quotes, which, of course, is not suitable for the purposes of the thesis, since as a result of all transformations it is necessary to obtain boundaries expressed in quotes of a currency pair, and not in their logarithms. Therefore, it is necessary to obtain the inverse probability density function q (y), changing the domain of definition of the function accordingly.

Now you need to apply the above mathematical apparatus to real market data. For the simulation, daily quotes of the euro-dollar currency pair for 262 observed days were selected, i.e. from May 15, 2013 to May 15, 2014 with a sliding period of 12 trading days.

Schedule 3 . natural levels support (syn.) And resistance (beautiful.)

Chart 3 shows the boundaries of the area of ​​definition of the probability density function of the considered price process - the lines of support and resistance. The natural boundaries of the currency market are the boundary values ​​for a trend change, and the probability of a trend change is the cost of options. Thus, further calculations of premiums for barrier options should be made, which will be presented in the next paragraph.

2.2 Valuation of barrier options

By definition, it follows that barrier options are a type of options for which payment occurs only when the underlying asset reaches a certain level in a certain time. This particular level of value of the underlying asset is the turn-on or turn-off barrier. In the paper, only inclusion options will be considered, i.e. those options that become regular options when the barrier is reached. It makes no sense to consider barrier shutdown options, due to the fact that they cease to exist when the threshold value (barrier) is reached, and, therefore, it is not possible to predict the future rate. To begin with, let's recall the classic Black-Scholes-Merton formula for pricing ordinary options (plain vanilla) of Nobel laureates, especially since it is useful for calculating barrier option premiums in some cases.

where

A down-in call is an ordinary option that begins to exist only if the price of the underlying asset (in this case, the exchange rate) reaches a certain level - a barrier.

If the barrier is lower than or equal to the strike price, then the down-in call premium at the initial time is:

,

where

An up-in option is also a normal option when a barrier is reached. If the barrier is below or equal to the strike price, then the value of the call option at time is:

where

Below are the formulas for calculating barrier put options. As in the case of call options, we will only be interested in inclusion put options. The price of the option to sell up-in, if the barrier H is higher than or equal to the strike price:

If the barrier H is less than or equal to the strike price K, the option premium looks like this:

As with all barrier options, a down-in put option only comes into existence when the price reaches the barrier level. When the barrier is below or equal to the strike price, the down-in put premium is:

In all the above formulas, the following values ​​of the variables were used. The risk-free rate for the national currency (US dollar) was taken as the value of the 12-month dollar LIBOR rate as of May 15, 2014 - 0.53460. For foreign currency (EUR) the risk-free rate was 12-month EURIBOR on the same date - 0.587.

The time to exercise of the option was calculated taking into account trading days, not calendar days. The number of trading days in a year is considered to be 252 days. Due to the fact that barrier options were considered in the chapter, 2 days were taken before the expiration of the option - the first day when the option broke through one or another barrier, and on the second day the contract was executed.

When evaluating barrier options, the volatility on the sliding window was used. For each nth value, from May 16, 2013 to July 11, 2013, the volatility was calculated on a sliding window of 10.

According to the working hypothesis for solving the thesis problem, the natural levels of support and resistance obtained as a result of solving the problem of the calculus of variations are, respectively, the upper and lower barriers for options. The strike prices for up options were set by adding 100 basis points to the upper barrier (resistance level); strike prices for down options were given as the lower barrier (support level) minus 100 basis points. It should also be noted that in order to improve the visual perception of the support and resistance lines, they were slightly transformed: the average value of this vector was subtracted from the initial vector of currency quotes and multiplied by the leverage equal to 100. Having calculated the premiums for all four considered types of barrier options - an in call, down-in call, up-in put and down-in put - let's try to build a trading strategy. To do this, we empirically determine the boundaries, noticing the breaking through of which the investor decides to buy an option or sell it. Chart 4 shows call down-in and put-up premiums for 2013-2014. The threshold values ​​c and c1 are found empirically, so that the price process is cleared of noise and the passage of the quote through the upper or lower border signals a change in trend.

Schedule 4 . Building trading strategies for options count down-in And put up-in

The strategy in the market can be formulated as follows: buying a down-in call option when the upper limit is reached and selling an up-in put option when the lower level is crossed. In other words, if at the previous step the exchange rate was above the upper limit, then this is a signal for the investor to buy a down-in call option. The rule works similarly for the up-in put option: if the value of the euro-dollar currency pair quote at the previous step was higher than the lower threshold value, this is a signal for the investor to sell the option, because. the price is likely to go up. Let's compare how this trading rule works on different samples, namely before the 2008 crisis and in 2012-2014. The vertical axis shows the levels of returns (normalized as of March 1, 2006), while the horizontal axis shows the daily closing prices of the euro-dollar currency pair from January 2, 2006 to December 31, 2007.

Schedule 5 . results applications trading regulations in comparison from market profitability 2006-2007 gg.

Chart 5 shows a strongly growing market dynamics - the blue dotted line. The proposed device - a red solid line - allows you to show a virtual retrospective strategy the result is not worse than the market.

Now let's apply the trading rule to the post-crisis period of 2012-2014. - the strategy loses to the market, but the portfolio profitability of this strategy is steadily growing. Approximately from the 500th point, the strategy stops working - we see a flat line. Zeroing of the values ​​occurs due to the fact that the quote value of the currency pair cannot break through the upper or lower threshold value - there is no signal for a trend change.

Schedule 6 . results applications trading regulations in comparison from market profitability 2012-2014 gg.

Let's try to take this segment and change the threshold values ​​to 0.001 and 0.0038 (recall that the original threshold values ​​were set as 0.012 and 0.02). After the threshold values ​​were lowered, and, consequently, the sensitivity of the portfolio was increased, one can see how its profitability increased sharply in relation to the market.

Schedule 7 . results changes threshold values to trade rule (2012-2014 gg.)

Thus, we can draw a definite conclusion about the algorithm for estimating barrier options premiums and building a trading rule based on them: the mathematical apparatus has shown the universality of solutions over different time horizons and samples, but it requires constant empirical refinement. If the strategy does not bring growth in portfolio returns for a long time, then this is a signal to make changes to the threshold values.

Conclusion

In the study, analytical work was carried out to evaluate barrier inclusion options. Adhering to the main research methodology, the problem of the calculus of variations was solved to find the natural boundaries of the domain of definition of the probability density function of the price process. In the future, to solve the main problem of the thesis, these boundaries were used as barriers to options. Based on the calculated premiums for barrier options, a trading rule was built and threshold values ​​were set, the intersection of which indicates a significant change in the trend of the price process.

To solve the problems of the final qualification work, it was required to write an algorithm for evaluating barrier options. In the course of the research, a software algorithm for evaluating barrier options was developed in the mathematical package Mathcad. The universality of the algorithm will allow further research on other time horizons or other currency pairs.

In the theoretical part, a description of barrier options was given, a brief excursion into the history of derivative financial instruments was given. The above classification of barrier options depending on the inclusion or deactivation and the direction of price movement outlined a clear picture for understanding the essence of this type of options. In addition, in the theoretical part, a clear explanation is given for what purposes barrier options are used, what risks they allow to hedge and by whom they are used. In the theoretical part, the principles of choosing hedging instruments for corporations or financial institutions were also touched upon, and a typology of currency risks faced by financial market participants was presented. A transition was made from the general to the particular - one of the most popular tools used by counterparties today is described in detail - Risk Reversal and Target Accrual Redemption Forward; their indicative parameters are indicated.

The practical part presents a mathematical solution to the problem of finding the natural boundaries of the domain of definition of a function, i.e. price process. As part of this task, the problem of the calculus of variations with floating boundaries was solved - support and resistance levels, which in turn were used as barrier levels for options. Using the classical forms of valuation of barrier inclusion options given in the book by John Hull, as well as using the mathematical calculations obtained earlier, a trading rule was built that gave a certain result. It is worth noting the result obtained by comparing trading rules on a sample of exchange rates in the pre-crisis period of 2006-2007. and in the post-crisis period of 2012-2014. With the exception of strike prices, as well as levels of barriers obtained using mathematical transformations, the work is based on real figures obtained from an international information source - Bloomberg agency.

More attention is now paid to the study of the problems of derivative financial instruments in Russian higher educational institutions, but there is still not much educational literature in Russian. In light of this, this thesis carries a certain novelty and is of interest for educational purposes, as well as in its practical application on real data.

Summing up the results of the study, it is worth noting its positive result - a comprehensive analysis of barrier options led to certain conclusions. In conclusion, brief answers are given to the questions posed at the beginning of the thesis. Thus, it can be argued that the goal of the final qualifying work has been achieved.

List of used literature and sources

1 John Hull Options, Futures and Other Derivatives: Pearson/Prentice Hall, 2009. - 822 p.

2. Hans-Peter Deutsch. Derivatives and Internal Models: Palgrave, 2002. - 621 p.

3. Information portal Bloomberg

4. Tomas Bjork. Arbitrage Theory in Continuous Time: Oxford University Press, 2009. - 466 p.

5. Derman E., Kani I. The Ins and Out of Barrier Options: Part 1 // Derivatives Quarterly (Winter 1996) - pp.55-67

6. Emanuel Derman, Iraj Kani, Deniz Ergener, Indrajit Bardhan: Enhanced Numerical Methods for Options with Barriers: Quantitative Strategies Research Notes. -May 1995

7. Investopedia Website: a resource for investing education - www.investopedia.com

8. Uwe Wystup. FX Options and Structured Products: John Wiley & Sons, 2007 - 340 p.

9. BNP Paribas Corporate & Investment Banking - Interest Rate Derivatives Handbook 2009/2010

Applications

Annex 1. Risk Reversal Indicative Termsheet

The exporter wants to hedge against the weakening EUR at minimal cost. The exporter buys EUR put USD call and sells EUR call USD put.

Strike 1.3200 put and 1.4700 call

Contact time 3 months

Forward rate 1.3940

Volatility 22.75% for strike 1.4700

22.85% for strike 1.3200

Zero cost premium

Exporter hedges against weakening EUR below 1.3200

· However, if the exchange rate is above 1.4700, the exporter will sell at 1.4700

Application number 2. Target Accrual Redemption Forward (TARF)

1 week fix 1.4800 profit = 0.0535max (1.5335-1.4800, 0)

2 weeks fix 1.4750 profit = 0.0585 accumulative profit = 0.1120

3 weeks fix 1.4825 profit = 0.0510 accumulative profit = 0.1630

4 weeks fix 1.4900 profit = 0.0435 accumulative profit = 0.2065

5 weeks fix 1.4775 profit = 0.0560 accumulative profit = 0.2625

6 weeks fix 1.4850 profit = 0.0485 accumulative profit = 0.3110

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