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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 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.

Thus, 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 high school graduation qualifying 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 trend change, but also as natural barriers for the corresponding exotic options.

diplomaRworkconsists offromthreemajorparts: 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. A lot of 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

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 option potentially becomes higher, but this same rise simultaneously 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 strike price, and the same barriers in the 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, participants market relations 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 these risks in a timely manner, understand them, 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's challenge, 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 its competitors if the foreign currency appreciates significantly.

Currency 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 Identification of risk - 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. Corporates will benefit from consulting with investment banks that 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 By 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 purchased), assuming that the market exchange rate on the expiration date of the option contract will be higher than the strike on the option being purchased (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 the corresponding 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. 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 the 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 By 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 present support and resistance levels - internal boundaries within the definition area of ​​a random price process - as moving boundaries that define a narrower area of ​​the function definition, which serves as a parameter of the distribution 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 quantification 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 parametrizing 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 currency pair quotes, 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 rules V comparison With 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 rules V comparison With 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 participants in financial markets 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 John Hull's book, as well as using the mathematical calculations obtained earlier, a trading rule was constructed 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.

The study of the issue of derivative financial instruments is now receiving more attention in Russian higher education institutions, however, there is still not much educational literature in Russian. In light of this, this thesis bears a certain novelty and is of interest in educational purposes, and in its practical application on real data.

Summing up the results of the study, it is worth noting its positive result - the complex analysis 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. Informational 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 with 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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exotic options appeared as derivatives of classical 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.

Many traders prefer to use barrier binary options (trigger) as effective financial market instruments. What are barrier options and are they interesting for an investor? They belong to the so-called exotic species. Their main difference is the presence of special conditions and trading parameters, we will consider the main ones in this material.

What are barrier options?

Trading conditions for exotic binaries may be different, but they all contain special clauses that distinguish them from standard instruments. Such transactions involve closing or, conversely, entering the market, relying on an additional parameter in the form of a certain asset value limit.

American, European and other styles are suitable for trading. General condition is the limit of reaching the indicator that activates the contract. The classification gives a general idea of ​​how to trade barrier options. There are several types:

• Knock-in-Up-and-In - activation occurs when the specified level is reached and exceeded.
• Knock-in-Down-and-In is the opposite activation system when the price drops to the agreed limit from top to bottom.
• Knock-out-Down-and-Out - completion of the transaction if the price falls below the planned limit.
• Knock-out-Up-and-Out - deactivation when exceeded.

Barrier binary options an object of trade, the activation of which takes place when a certain limit is reached. They can turn on (revive) and deactivate (dying). The inclusion payment is made when the spot price is in the money and reaches the set level before the expiration date.

When this indicator crosses, the transaction is carried out in the standard way, depending on the type - Call or Put. At the same time, the initial amount of execution and expiration are preserved.

Trading Principles

Let's compare the usual classic Call and Call "up out". An increase in the value of an asset will cause an increase in the yield of the first. In the case of the second transaction, two options are possible: the indicators grow - the payment increases. But when approaching the limit values, the situation may change. When the agreed level is reached, the transaction is canceled, and the trader receives a loss in the amount of the contract.

The essence of trading is to guess whether the value will reach a certain limit. It serves as a kind of trigger that turns on the contract or deactivates it. Otherwise, the principles of operation are similar to standard binaries.

Trigger contracts are often cheaper than standard ones. The premium paid will also be less. Trading is complicated by the fact that it is necessary to calculate the risks in the movement of the value of the underlying asset. The basic rules of trading have been developed for a long time and were used mainly in the stock market. When mastering the technique of working with a trigger, you can reach a very solid income on binary options.

One of the advantages of such trading is cost optimization due to the low premium.

Under such financial activity a wide variety of strategies have been developed. This allows the trader to choose tactics depending on their own trading style.

How are barrier options different from regular options? This is the possibility of obtaining an assignment of payment, for example, if the concluded contract is no longer valid. The amount is negotiated at the conclusion of the transaction and in most cases does not exceed the amount of the initial premium.

Benefits of Barriers

Payments on trigger trades allow you to evaluate the movement of the market. You can predict the correct indicator by taking the initial payment and the average price in the future as a basis. Usually the volatility payment is equal to the forward cost. Savings on the purchase lies in the absence of the need to pay for non-obvious market outcomes.

Barrier instruments most fully meet the requirements of hedging. For example, a trader wants to sell an asset if its value rises, but he needs to hedge against a fall. There are two options here:

• Buying an ordinary Put at a price below the market price. However, with growth, the need for hedging becomes irrelevant.
• Acquisition of a put “up out” with a strike lower and limit level above the spot price. As a result, if the specified value is exceeded, the transaction is cancelled.

Another positive moment: the amount of premium payments is less than with standard types of contracts. For example, shutdown is not paid if the level reaches the trigger value. Assuming a low turn-off probability, the benefits are the same, but the premium is smaller.

Trading example:

Suppose an investor buys a Call Knock-out. The amount of the award is 40 rubles. The underlying asset is a share of Rostelecom. Strike - 400 rubles, trigger - 200. Timeframe - month. Assume that the value of the asset on the 15th of the month broke through the set value, falling to 150 rubles.

As a result, the contract is canceled and the trader loses the right to it. The loss is 40 rubles. even under the condition that a month later the share will cost 500 rubles. When trading conventional Call contracts, the investor would receive shares at a strike price of 400 rubles, since he guessed the general trend movement.

However, despite the additional parameters of the barriers and the increasing complexity of the forecast, the premiums for them are usually much lower than usual. This allows you to get more high income with the same investment.

When trading, it is important to consider the proximity of the trigger to the current level. The closer, the easier it is to break through. Accordingly, the price of Knock-in will decrease. By changing the position of the value indicator, it is possible to optimize the premium/probability of breaking through the barrier ratio.

Where to trade?

Due to the specifics, trading using trigger contracts is carried out mainly OTC. Access to them by small private investors is significantly limited. On Russian market only some of the largest brokers offer platforms for trading exotic species.

The limited nature of such trading is due to low demand from investors, the unpreparedness of Russian investment banks to work with such products, and low market development. Is it possible to make money on barrier options?

One of the most important tasks is to assess the fairness of the premium. Since the conditions are different, it can be difficult to do this. Complex mathematical formulas confuse even experienced traders. However, many of them, having mastered strategies using a trigger, increase their income by investing less.

As can be seen, transactions with the establishment of barriers have a number of irreplaceable advantages. But they are not suitable for everyone. After gaining some experience in trading, you can try yourself in this market by choosing a suitable asset, determining the possible direction of the trend and setting the optimal limit.