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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 support and resistance levels), 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 area of ​​definition 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 fluctuations in the price process. 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 trend change, but also as natural barriers for the corresponding exotic options.

diplomaRworkcomposedfromthreemajorparts: 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 qualification 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. Much useful information has been gleaned from articles by former Goldman Sachs Quantitative Strategies 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 of 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. The payment on the barrier option of inclusion (in barrier option, knock-in option) occurs 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 normal 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 asset's spot price 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 the same increase 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 greater 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). 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 large 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. 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 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 must 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 by both corporate organizations and financial institutions, is the 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 instrument 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 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 forfeited. 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 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 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 the 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 cost 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 functions of various types. 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 other 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 commercial regulations in 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 commercial regulations in 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.

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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17:17 — REGNUM

Since then, there have been many articles in the press discussing this topic. The random sample shows that the majority of authors believe that the decision is unfair to the bank, undermines the foundations of the emerging derivatives market in Russia, creates legal uncertainty for participants in the derivatives market, etc. A closer look, however, reveals that many of these statements are the result of a hasty and very superficial reading of the judgment, and the threats to the Russian derivatives market are greatly exaggerated. Although the format of this article does not allow for a detailed - with reference to accumulated international experience - justification as to why such a hasty reaction may turn out to be erroneous, it is nevertheless quite appropriate to outline some elements of the legal analysis of this decision, albeit outlined, in the following few paragraphs. .

The arguments of the authors, who are critical of the Arbitration Court's decision, boil down mainly to the following: (i) a currency option is a fairly simple transaction, and it is inappropriate for a company like Transneft to claim that the company did not understand the risks associated with the transaction when entering into it, (II) the court showed excessive paternalism in relation to the company, protected it from the consequences of its own actions and unjustifiably imposed fiduciary obligations on the bank, which merely acted as an ordinary market counterparty to the transaction, (III) when analyzing the integrity of the bank, the court assessed not so much the contractual documentation, how much the bank's pre-contractual presentations about the product, and (iv) the decision hinders the development of the Russian market for hedging instruments. Let's look briefly at each of these arguments.

“An option is a simple transaction understandable by Transneft”?

This argument is either the result of deliberate cunning, or a demonstration of deep ignorance of the subject. There is nothing simple in an option, especially if you sell this option. An option, including an option on exchange rates, is considered one of the most complex financial instruments, especially if one of the currencies is an emerging market commodity currency, and the transaction is concluded under a barrier condition. The complexity of a financial instrument is manifested in its pricing, which reflects a whole tangle of difficult-to-calculate risks, variables and assumptions. Generations of financial mathematicians have worked on financial models for use in option pricing. The world-famous Fisher Black and Myron Scholes, who developed the most widely used price model since then for calculating the value of a “simple” (without additional elements) option, received the Nobel Prize in Economics for their development.

It is significant that in the structure of large banks there is a special subdivision for options trading, which has its own staff of mathematicians specializing in this product, building and applying their own mathematical models. Based on the basic models for calculating the value of an option, bank dealers develop their “add-ons” for various types of the underlying asset (including exchange rates), as well as for various types of elements that complicate the forecasting of option payouts. For example, the barrier condition exacerbates the difficulty of the already complex mathematical calculation of the value of an option to such an extent that it makes the Black-Scholes model not directly applicable, since such an instrument is an option on an option according to the "risk profile". Instead, dealers (banks), using complex mathematical packages, create their proprietary "distributed" financial models (ie, combining elements of various price mathematical models). Companies, even large ones, whose core business is not trading in complex financial market instruments, as a rule, are not able to repeat such an examination.

The value of an option is derived not only from the price parameters for the purchase and sale of the underlying asset, but also from the degree of probability that the option will be exercised. Five variable elements affect the price of an option: (I) the strike price, (II) the time to expiration of the option, (III) the level of interest rates, (IV) the price action of the underlying asset, and (V) the spot price of the underlying asset. asset. At the same time, the value of an option reacts to fluctuations in the price of the underlying asset in a completely different way than other types of derivative financial instruments: it is sensitive to changes in any of these five variables (the strike price can also change in non-vanilla option contracts). To characterize the sensitivity of the value and other properties of the option to a change in certain values, various coefficients are used, named by the letters of the Greek alphabet (“Greeks”) - delta, gamma, theta, vega, rho and lambada. Unlike an option, the value of an interest rate swap, for example, depends on only one variable, the interest rate described by the yield curve; The cost of a bond futures depends only on the spot price of the underlying bond and the current repo rate, while the price of a currency futures depends on the spot rate and interest rates for each of the currencies of the currency pair. But the main thing is that unlike other financial instruments, the relationship between the value of an option and the change in any of the five elements is not linear. This complex multi-variable sensitivity of an option makes it the most difficult financial instrument to manage its risks.

To illustrate the “simplicity” of an option as a financial instrument, let me give you a typical popularized (!) basic explanation of the behavior of the above properties of an option: “when the option is deep in the money (in-the-money) or deep out of the money” (out-of -the-money), its delta does not change dramatically and its gamma is therefore rather negligible. But when an option approaches at-the-money, its delta can change suddenly and its gamma is large. A long option position has a positive gamma, while a short position has a negative gamma. Large gamma options are more difficult for market makers because hedging them requires constant adjustment to maintain a neutral delta, and this leads to significant transaction costs. The larger the gamma, the greater the risk that the option book will be affected by sudden market movements. A position with a negative gamma represents the highest risk and can only be hedged by a long position in other options.” Or another example: “managing an option book requires choosing between gamma and vega, as well as choosing between gamma and theta. A long position in an option is a long position in vega and gamma. If volatility declines, the market maker may opt to maintain a positive gamma if he believes that the decrease in volatility can be offset by adjusting the gamma in the direction of the market. On the other hand, he may prefer a position with a negative gamma by selling options, i.e., by selling volatility. In both cases, the cost of rebalancing the delta should offset the decline in volatility.” I am convinced that any author who claims that an option is a simple financial instrument will not even be able to repeat the truths given above for option dealers, let alone explain their meaning. Specialists, for whom the above passages are clear, will not call an option a simple financial product. But understanding the interaction of these coefficients (Greeks) is the minimum necessary skill for understanding and managing risks in options. It is no coincidence that the classification of financial instruments adopted in accordance with European directives classifies currency options, especially those with a barrier condition, as complex financial instruments that require the dealer to take special care of the client's interests.

The ability or inability of a company to understand the risks involved in entering into a transaction is a matter of fact, which is established and assessed by the seller of such a product (bank) or, if there is a dispute, by the court. At the same time, the more complex and risky the product, the more information the dealer must obtain about the client, his knowledge and experience in handling similar instruments and making deals in the past, about his business and the need for such a product, as well as provide the client with complete and clearly stated information. about the product to ensure that the customer understands the risks involved.

International law enforcement practice in the field of derivative financial instruments proceeds from the fact that such experience and knowledge of the client should not be established in the abstract, but in relation to a specific type of transaction. The experience of dealing with derivative financial instruments in the past is not transferable to other instruments with an excellent risk profile. For example, the fact that a client concludes even similar transactions is far from being an “indulgence” for a dealer, because, firstly, it does not in itself indicate an understanding of the risks by the client, and secondly, it may not be applicable when concluding a more complexly structured transaction, for example , transactions with a barrier condition. Complication of the transaction by a barrier condition can completely distort the expected payments on it.

It can be seen from the court decision that the parties had previously entered into transactions with derivative financial instruments, but with different conditions and purposes (they were aimed at hedging the risk of a fall in the dollar exchange rate), and also without a barrier condition; at the same time, the bank evaluates such transactions as “similar”. If the company had significant foreign currency balances, then you can use options to hedge the risk of a fall in the foreign exchange rate by purchasing a put option. At the same time, the company, as follows from the decision, stated that the disputed transaction was not a hedging transaction for it, and the structure of the transaction described in the decision confirms this. If, as in the case of the disputed transaction, the bank had previously sold “double” call and put options to the company, where the bank acted as the seller of the put option and the buyer of the call option (and on this basis, the court referred to the earlier concluded “similar” transactions), then setting the barrier at a level well above the current spot rate makes the put option practically meaningless for the client as a hedging instrument (since the probability of its execution tends to zero), and the transaction essentially boils down to the sale by the company a call option jar with unlimited risk to the client. In this regard, it is noteworthy that the argument that the company had experience in concluding similar transactions was brought by the bank and carefully examined by the court. Based on the results of the study, the court concludes that “[t]he evidence presented in the case file and the explanations of the parties, the court found that the Claimant had not previously entered into transactions with derivative financial instruments, the terms of which would be similar to those of the disputed transaction.”

The text of the court decision does not provide an opportunity to form an informed idea of ​​the level of understanding by the company of the risks that it assumed when concluding the transaction. An important factor in this analysis is the correspondence between Sberbank and Transneft when discussing the terms of the controversial transaction: whether it demonstrates the level of discussion that shows that both parties "spoke the same (preferably with elements of Greek) language." If, instead of a professional discussion of the elements that are essential for the option, the discussion boiled down to the fact that “over the past three or four years, the volatility of the ruble has been insignificant, therefore, by selling a call option with a high barrier, you risk little, but earn a lot”, then this will expose the same asymmetry in the expertise in option trading between Sberbank and Transneft as the asymmetry in the distribution of risks between the seller and buyer of the option. In this regard, it is interesting to note the court's remark that “[d]ues of the Respondent about the Claimant's understanding of the disputed transaction, knowledge of the formulas and risks associated with the transaction are not supported by the case materials. The correspondence provided by the parties on the transaction prior to the date of its conclusion does not indicate a proper explanation to the Plaintiff of complete and objective information about the transaction, the mechanism for calculating the amounts of payments under it and the associated risks. It is also indicative that after the conclusion of the transaction, the bank had to additionally explain to the client how the options will be settled.

“Russian courts in similar disputes have previously not taken into account the arguments about the lack of understanding by the affected party of currency risks”

To substantiate this argument, the authors of the publications refer to court decisions on the claims of borrowers on foreign currency mortgage loans. Drawing an analogy with such lawsuits, the authors argue that the Russian financial market has been living in conditions of volatility in the exchange rate of the national currency for 25 years, and when the parties take on the currency risk, the Russian courts do not heed the complaints of persons who have suffered from an unfavorable change in the ruble exchange rate. This analogy is simplistic.

From the point of view of the classification of financial instruments adopted by the international community, deposits, bonds or loans denominated in a currency other than the currency of the state of the place of the transaction are classified as simple financial instruments. Indeed, the structure of such a financial transaction is quite simple - the face value and interest (coupon) rate by their nature do not differ from the same indicators inherent in similar debt instruments denominated in the national currency. The main thing is that when placing a deposit or attracting a loan in foreign currency, the depositor or borrower already knows at the time of the transaction how the payment obligations under the transaction will be formed. A completely different thing is a currency call option with a barrier condition. First, at the time of its conclusion, the parties do not know whether the option will be exercised (ie, whether the spot rate will exceed the value of the strike rate at the date of option exercise). In this regard, it is noteworthy that in the disputed transaction, the strike rate was equal to the spot rate on the date of the transaction, and not the forward rate (which was higher taking into account the higher interest rate on rubles), which means that the client has sold an option that was already in-the-money for the bank at the date of sale. In the case of a barrier option, the parties do not even know whether the option will be exercisable, that is, whether it will come into effect. Secondly, if the option is exercised, at the time of its conclusion, the parties do not know the amount of the option payout, which depends on the future value of the spot rate of the base currency. It is important to note, however, that with regard to a call option, the size of such payments is potentially unlimited. That is why international experience in the field of regulation of financial markets requires the client to have special knowledge and experience in option trading as a precondition for the dealer to complete a transaction to purchase an option from the client. Especially when it comes to buying an “uncovered” (not secured by the underlying asset) option — the so-called “naked call option”, which creates one-sided — unbalanced in the event of an increase in the price of the underlying asset — risks for its seller.

The sale of just such a “bare” call option was structured by Sberbank in a transaction with Transneft, and so far Russian judicial practice has not expressed its attitude to such transactions. The analogy between taking out a mortgage loan and selling an unsecured settlement currency call option by a non-professional in this respect does not stand up to criticism and runs counter to the global practice of regulating financial transactions.

"Judicial paternalism towards a bank client"

In international regulatory practice, transactions are divided into inter-dealer and client transactions. In inter-dealer transactions, the parties do not have an obligation to take care of the interests of the counterparty. In client transactions, the dealer (professional market participant) is obliged to take care of the interests of the client. Elements of such diligence include: (I) classifying the client according to his level of competence in dealing with relevant financial instruments, (II) evaluating the proposed instrument for its suitability to achieve the client's goals in the event that the bank advises the client on the transaction (suitability ), (III) assessing the risks posed by this type of financial instrument in terms of the client’s financial capacity and risk appetite (appropriateness), (IV disclosure to the client of the risks posed by the financial instrument and explaining such risks in a “full, clear and in a misleading way”, (V) an assessment of the financial competence of the client in terms of his understanding of the risks posed to him by the financial instrument, and (VI) the execution of the transaction on the best terms for the client.

There is also an intermediate category of transaction participants - a "professional" client (other clients are referred to as "retail clients"). The criteria for assigning a client the classification "professional" is very strict. The level of care required when entering into a transaction with a professional client is significantly lower than with an ordinary client, and in many ways approaches inter-dealer transactions. In this case, if the bank advises the client on the transaction (i.e. sends him personally recommendations and (or) other information about the proposed transaction), then the bank is obliged to assess the suitability of the transaction for the client (suitability), regardless of the classification of the client either retail or professional. A personal recommendation is a recommendation structured according to the specific circumstances of the client (eg, as in a disputed transaction, setting the option par value in the amount of a bonded loan). It is noteworthy that European regulators did not previously extend such a requirement to transactions with professional clients, but subsequently, in the face of increasing complexity of financial instruments, they extended this standard to all dealer-client transactions, to transactions with large and experienced consumers of financial services.

In Russian conditions, there is no similar classification of participants in transactions. The current breakdown of participants into three categories - professional market participants, qualified investors and others - is based on formal criteria and is a sign of weak development of the derivatives market and its regulation. It seems that the classification of products according to the criterion of complexity should take into account the degree of market maturity. It would be quite natural if the same financial instrument required a different degree of “paternalism” on the part of the bank (or court) in jurisdictions with different levels of derivatives market development. Recall that not so long ago in the Western markets such seemingly simple instruments as interest rate swap and currency interest rate swap were treated by the regulator as complex financial instruments. As we gain experience in the use of derivative financial instruments and manage the risks associated with them, as well as a more meaningful categorization of market participants depending on their real experience, rather than formal criteria, it will be possible for us to weaken the degree of required care for the interests of the client on the part of the bank or dealer. In the meantime, the dispute between Sberbank and Transneft should calm down the sales departments of the leading financial institutions on the Russian market and give them a greater share of responsibility for providing recommendations to clients. In the absence of meaningful (rather than formal) regulation of the status of market participants with varying skill levels in a young market such as Russia, all clients who are not professional market participants for the purposes of dealing in complex derivatives should be treated a priori as “retail clients”. » according to the international classification, i.e. enjoy the degree of protection provided in international practice for unqualified investors. Over time, with the development of the market and the accumulation of experience by its participants, some of these tools will move from the category of complex to the category of simple ones, and in relation to them it will be legitimate to weaken the required degree of protection for those who do not need it.

It seems that the leading banks and dealers in the Russian market should be guided by the highest standards of international market practice and provide customer care in line with international best practice, rather than capitalizing on gaps or shortcomings in the Russian regulatory framework. Disregard for such standards, as practice shows, is fraught with significant legal and reputational risks.

Presentation or contract documentation?

In international practice, the dealer's obligation to ensure the suitability of a financial instrument for the client (suitability) has two aspects: (i) ensuring the suitability of the instrument itself for the client's purposes, and (ii) ensuring the proper way of offering and selling it to the client. This implies that the dealer must ensure that the customer properly understands the risks of the tool and makes an informed decision to accept them (provided that the tool itself meets the suitability criteria). In particular, the dealer's description of the proposed tool must be "comprehensive, understandable and not misleading". Violation of these requirements may result in a defect of will on the part of the client when deciding to make a transaction, since it does not allow him to make an "informed" decision about the risks and benefits of the proposed instrument.

Assessing whether the bank's actions prior to the transaction were appropriate is a matter of fact. Without access to all the materials of the case, it is very difficult to comment on the fairness of the assessment of such actions by the court, however, contradictions, inaccuracies and errors in the presentation materials, underestimation of the risk of reaching the barrier rate and other flaws in the discussion of the transaction with the client, given in the court decision, as well as the formal a few days before the date of the transaction - the disclosure of the risks of the transaction, of course, is not in favor of the bank. Contractual documentation can by no means eradicate all of the indicated flaws in dealer sales practice, since it concerns completely different issues. Of course, there may be stipulations that the client understands the risks, acts at his own risk, does not rely on the advice of the bank when concluding a transaction, etc. The extent to which such provisions of the contract can serve as a basis for releasing the dealer from liability for creating false expectations in the client depends on many factors (including actual circumstances) and is decided differently in different legal systems. But in any case, these factors are at the discretion of the court, whose decision becomes a guideline for future standards for the sale of financial instruments in the relevant market.

"The decision of the Arbitration Court adversely affects the development of the hedging institution in the Russian market"?

In a recent speech to reporters, Bela Zlatkis, deputy chairman of the board of Sberbank, said, commenting on Sberbank’s intention to appeal the decision of the Moscow Arbitration Court, that “everything will be decided to everyone’s understanding, to the correct position, to the development of the market ... all the more so now, of course, it is very important hedging in the Russian market, because many are under sanctions, for many hedging is not available, and this, of course, is very unpleasant.”

The development of risk hedging instruments is a necessary element of the modern financial market. Such a development, however, assumes that the regulation of the market and, if necessary, the judicial system will keep its participants from excesses and ensure a balance of interests of all its participants. Therefore, one has to disagree with the bank's position that the court decision creates an obstacle to the development of hedging instruments in the Russian market. The fact is that the controversial transaction can hardly be called a hedging transaction. On the contrary, this transaction created risks for the company. Moreover, the risks are not commensurate in size with the financial benefit for the company from the purchase of the instrument offered by the bank. Packed in a beautiful box with a bow, the bank sold the client a time bomb. It is noteworthy that earlier similar transactions with options were characterized by both parties as hedging: payments on the call option sold by the company with the growth of the dollar were offset by an increase in the value of foreign currency balances against the ruble (after all, the company bears expenses mainly in rubles), and potential losses from depreciation of the dollar would be compensated through the exercise of the put option purchased from the bank. Unlike earlier transactions, however, the disputed transaction was essentially a sale by the company to the bank of a put option, since once the exchange rate reached the barrier value, the probability of exercising the put option was negligible.

Such an instrument—which was essentially the sale of a call option by the client—has nothing to do with the hedging itself. The only thing that connects this instrument with bonds is the par value match. At the same time, such a coincidence from the point of view of economic correlation is absolutely arbitrary. In addition to marketing attractiveness, linking the face value of the option and the bond does not carry any semantic load. The bond is denominated in rubles and does not create currency risks for the company, whose income and expenses lie mainly in the ruble zone. The conclusion of a controversial transaction gave the client an unlimited risk of a fall in the ruble exchange rate, which has no correlation with the indicated bonds. The client's losses on the option were not offset by a mirror increase in income (real or even accounting) in any other positions from the devaluation of the ruble - such losses were initially included in the transaction as a pure, unbalanced and unlimited currency risk for the client. As noted above, its hedging function was largely leveled by the inclusion of a barrier condition in the deal.

Noteworthy in this respect is Sberbank's argument that "Claimant ... did not suffer significant losses under the transaction, since Claimant had sufficient foreign currency to complete the transaction, and therefore Claimant had no currency risk." In this wording, this argument, of course, seems absurd in the light of the fact that, as a result of the movement of the exchange rate, the bank issued a demand to the company for the payment of 67 billion rubles!!! It would probably be more correct to argue that the presence of amounts in US dollars on the company's accounts, comparable in size to the par value of the option, at the time of the transaction, compensated for possible losses on the transaction as a result of the depreciation of the ruble. However, the mere fact that the company has currency at the date of the transaction does not make this option "secured", and the transaction itself - hedging: the company could have its own plans to use the currency in its core activities, and the terms of the transaction or the presentation materials of the bank were not specified ( as far as it can be judged from the text of the court decision) reserving some amount for a controversial transaction. As foreign currency is spent on capital investments or operating expenses, the specified option would lose "coverage" and an ever-increasing delta between the company's foreign currency funds and the nominal value of the option would increasingly turn this option into uncovered.

Conclusion

Of course, the recognition of a new type of financial transaction as invalid causes reasonable concern among market participants in such instruments. However, some decisions - even such radical ones as invalidating the deal - can be useful for the development of the market. Of course, the share of responsibility in concluding a transaction lies with both parties, but practice knows many examples when, due to differences in the position or level of competence of the parties, such responsibility is distributed unevenly. Banks and other financial market professionals have traditionally experienced increased responsibility for the client. “The banker is a member of a profession practiced since the Middle Ages. Since then, a code of professional ethics and customs has matured, on the observance of which his reputation, success and usefulness depend on the community in which he works ... and if in his activity the banker neglects this code - which can never be expressed in legislation, but has the power more powerful than any law, it sacrifices its credibility. Such trust is his most valuable asset; it is the result of many years of honest and honorable dealings… A banker is obliged at all times to conduct himself in such a way as to justify the confidence in him of his clients and preserve it for his successors… The thought that we should do only first-class business, and do it only in a first-class way, never leaves our minds… “It is these criteria that should determine the behavior of professional market participants, especially when it comes to its flagships. Whether these criteria were met in the controversial transaction between Sberbank and Transneft is largely a matter of fact to be established by the court. But many of the rules formulated in the decision of the court of first instance, in the opinion of the author, do not at all run counter to the goal of improving the market for derivative financial instruments. On the contrary, in the absence of detailed regulation at the level of legislation or professional standards, such rules set guidelines that market participants must adhere to in order to avoid excesses. This is a normal symptom of growing pains.

The price model is built on the very generous assumption that the change in the price of the underlying asset follows a so-called "standard underlying" or "normal" distribution. Financial mathematicians here would give an explanation that the main characteristic of such a distribution is the standard deviation, hence the importance of the concept of underlying asset price volatility.

The only exception to a number of Greek letters, sometimes replaced by the Greek kappa.

From a speech by Dsojon Pierpont Morgan (Jr.) to the US Senate Banking Committee on May 23, 1933.

(https://fraser.stlouisfed.org/scribd/?item_id=33957&filepath=/files/docs/publications/sensep/sensep2_pt01.pdf#scribd-open)

Barrier Options- are options, the payouts of which depend on the price reaching the underlying asset, which occurred during the existence of the option. The corresponding price level can be seen as a trigger, turning the option on or off. This is why barrier options are often referred to as trigger options.

With the exception of such an additional parameter as the trigger, barrier options are ordinary European options (plain vanilla). Barrier options are almost always cheaper than European options of the corresponding series: they have the same maximum return, but the probability of receiving it is lower as a result of some peculiarities in the price movement of the underlying assets.

Barrier options, along with Asian ones, are the most popular among numerous exotic derivatives and account for 10% of the total volume of foreign exchange options on the OTC market. Unlike other exotic options, they are often delivered rather than settled.

It should be noted that barrier options were traded on the American markets even before 1975. However, they gained the greatest popularity in the late 80s in Japan (the times of the so-called "inflated assets").

Types of barrier options

1. Trigger that includes an option - knock-in
2. The trigger that disables the option is Knock-out.

Each of them is subdivided into two more subtypes depending on the movement of the price direction:

• "Up & In" - the option comes into effect when the price rises to the agreed level;
• "Up & Out" - the option expires when the price rises to the agreed level;
• "Down & In" - the option comes into effect when the price falls to the agreed level;
• "Down & Out" - the option expires when the price falls to the agreed level;

All four options can be applied to both classes of barrier options - Call and Put. Thus, there are eight combinations, which are conventionally divided into reverse (reverse) and normal (normal) trigger options.

Barrier options are denoted as follows:

0.9020 EUR Call/USD Put RKI @ 0.9125 mkt 0.9077

Where: 0.9020 - strike price; EUR Call - an option to buy EUR against USD; RKI - reverse option (Reverse Knock-In); 0.9125 - trigger value; mkt 0.9078 – spot price level that determines the validity of the quote (it is not always present, because depending on the decision of the currency player, the quote can be set regardless of the current price level).

Using Barrier Options

Simple American and European options can be used to form various option strategies, which not only significantly expands the range of modeling possibilities, but also significantly reduces costs - after all, barrier option premiums are lower.

An example of the Bull Spread strategy (see Fig. 1). It is built by buying a more expensive call option with strike S1 and then selling a cheaper option with strike S2 ​​(where S2 > S1). The maturity date and the volume are the same. It turns out the curve of the financial result, which has the following form:

(Fig. 1 - Bull Spread Option Strategy)

Barrier Options Variations:

• Money-Back Options (options "Money back") - with a concession, the amount of which is equal to the initial premium. Most often, these are Knock-out options, and they are quite expensive. Can be used as a loan instrument that gives the buyer of an option to the seller.
• Exploding Options - Reverse Knock-out options with a rebate equal to the amount of intrinsic value.

Double Barrier Options are options with second market barriers. In this case, the trigger value is set as a formula that uses the prices of several assets in one or more markets. Another name for them is Dual-Factor Barrier Options.

It is characterized by the fact that at least one of the terms of the contract takes into account the historical values ​​of various parameters or imposes restrictions on them.

Options with an average price (Average Options)

This type of option takes into account the average prices of the underlying asset throughout the entire period until the option is exercised. The term "average" in this context means either a simple arithmetic average or a weighted average.

There are two main varieties: middle price option (Average rate Option, ARO) and middle strike option (Average strike Option, ASO). In the first case, the price of the underlying asset for the period (settlement price) is averaged, in the second case, the strike price.

Asian options payouts are based on the average characteristic (asset price or strike), so the payouts are less volatile than regular (vanilla) options. Thus, Asian options are an inexpensive way to hedge periodic cash flows.

Average Rate Options (ARO, Asian Options)

For this type of spot option, the price of the underlying asset on the exercise date is replaced by the arithmetic average of the asset prices reached in the period until the option expires. A geometric mean option is an average option whose arithmetic mean of the underlying asset prices has been replaced by the geometric mean.

Exchange trading in Asian options began in the late seventies, in the form of bonds with an embedded option. Asian options as an independent derivative financial instrument appeared later.

Since the late 1980s, the Asian option has gained a reputation as one of their most popular exotic products, despite the fact that this instrument is traded mainly on the over-the-counter market. The popularity of Asian options is explained by the fact that hedging with Asian options is cheaper than hedging with conventional options and at the same time more effective. They also have less volatility and therefore less risk. The latter is widely used to reduce the translational risk in accounting when conducting international transactions (currency transactions) to avoid losses when exchange rates change.

Options with an average strike price (Average Strike Option, ASO)

This is an option whose exercise price is replaced by the arithmetic mean of the prices of the underlying asset that were observed in the period until the option expires.

Most often, ASOs are used when the required target value is determined based on future average prices, and the position needs to be hedged right now.

The averaging period in ASO options does not have to coincide with the expiration time of the option, it can be chosen arbitrarily depending on the needs of the buyer (seller). If the period is close to the maturity date, then the ASO option premium tends to zero. Conversely, if the averaging period is chosen within the time limits of the option contract, then the premium will be approximately equal to the premium of a European option in the money with the same maturity date.

ARO and ASO payouts can be denoted as follows:

ARO = max( w[ AR(t) - X], 0 )

ASO = max( S(t) - AS(t)], 0 )

where AR(t), AS(t)– average values ​​of price and strike;

X– execution price (strike);

S(t)– spot price at the moment t;

w is a binary variable equal to 1 for a call and -1 for a put.

Barrier Options, Trigger Options

barrier option is an option, the payoff for which depends on whether the price of the underlying asset has reached a certain level over a certain period of time or not.

The corresponding price level can be seen as a barrier that either “turns on” the option or “turns it off.” The first case corresponds to class of barrier options knock-in, the second - knock-out.

The difference between a knock-out option and a simple option is that when the price of the underlying asset reaches a certain barrier, the option ceases to exist. In the case of a knock-out call option, the barrier lies below the strike price.

If the option ceases to exist, then the owner, depending on the terms of the contract, either does not receive anything or receives a fixed amount of money, called compensation.

For knock-in options the opposite is true.

Knock-in and knock-out options are each subdivided into two more subtypes depending on the direction of price movement.

All four options apply to both classes of options - call and put. Thus, there are eight possible combinations, which are divided into ordinary and reverse barrier options.

Conventional barrier options are considered to be "out of the money" at the time they are written. if exercised at that point, the option holder receives no premium. On the contrary, in order to reach the barrier, the price of the underlying asset must move in the money direction, which is a sign of inverse options.

Above all, it is not necessary that the barrier be determined by the price of the underlying asset. If the barrier, determined by the price of another asset, is called external.

barrier options widely used in hedging. Their use provides not only greater freedom of action compared to standard options, but lower hedging costs due to the low premium for barrier options.

Barrier options may contain an additional option called a yield. It represents a cash payment if the price of the asset has not broken through the agreed barrier during the period of the option. The concession cannot exceed the premium for the option and it increases its value, because. further reduces the risk of losing money.

Barrier options are always cheaper than ordinary European options of the corresponding series, since the maximum return on them is the same, but the probability of receiving it is lower. Due to the lower premium and many of the features similar to conventional options, b carrier options as well as Asian became the most popular among exotic derivatives.

Ladder and step options ( Ladder Options & step Options)

Combining the characteristics of a discrete look forward (i.e. the right to buy/sell an asset at the lowest price in a period) with a barrier option gives ladder option (Ladder Option) and step option (Step Option). The ladder option also allows the buyer to fix the already “earned” option profit, at moments not fixed in advance, but when the price breaks through a certain level. When the price of the barrier is reached, the option owner fixes the profit (if any) and names a new barrier. Thus, ladder option has even lower risk compared to the barrier and, accordingly, is traded at a lower premium.

Step Options, on the contrary, allow, as it were, to “average out” a loss-making option position, i.e. if the price of the underlying asset falls to a certain level, the “Down” step option fixes a new, lower strike price.

Thus, ladder and step options are widely used in hedging in anticipation of unfavorable price dynamics of the required asset.

Binary Options (Binary Options, Bet Options, Digital Options)

By binary options either a certain amount is paid or nothing is paid. A binary option gives its holder the right to receive a fixed amount if the current price of the asset on the exercise date is higher (binary call option) or lower (binary put option) than the strike price.

Depending on the type of payout, there are several types of binary options:

One Touch option (Onetouch)

This option is exercised at the moment when the spot price of the underlying asset reaches the agreed level.

Option "None of the two touches" (Doublenotouch)

In this case, the option is exercised if during the period of its validity the spot price of the asset did not go beyond the specified limits.

All or Nothing option (Allornothing)

If the spot price is at the set level on the option maturity date, then the holder receives the agreed fixed amount, if not, then he receives nothing.

There are two types of this type of option: cash-or-nothing options(if payment is made in cash) and asset-or-nothing options(if the payment is made by the underlying asset and if the price of the underlying asset on the execution day will be more/less than the strike price). In the latter case, the size of the premium will be significant, and hence the subscriber. and the option holder assume significant risk.

The payoffs for cash-or-nothing and asset-or-nothing options can be represented as follows:

PCON = (C if w S>=w X)or (0 if w SwX)

PAON = (S(t), if w S>=w X) or(0 if w SwX)

where X– execution price (strike);

S(t)– spot price at the moment t;

w

With- a fixed amount of money.

Binary Options and all their varieties are pure speculative instruments. In this regard, they are rarely used to insure positions.

Contingent Premium Option

contingent premium options first appeared on the commodity exchanges, after which they became popular in the foreign exchange markets, and also attracted a large number of investors playing on the Nikkei 225 index.

A feature of this type of options is that the buyer pays nothing when purchasing them. The premium on the option is paid when it is exercised, and not initially, as is the case when purchasing a simple option. Furthermore, contingent premium option contains a condition according to which the option is automatically exercised if the price of the underlying asset equals or exceeds the strike price in the period before the option is exercised. A contingent premium option is worth more than a simple option due to the above conditions, which are most beneficial to the option holder.

The payoff for a variable premium option is formally expressed as follows:

P = ((w[ S(t) - X] - Q) if w S(t) wX) or (0, otherwise)

where X– execution price (strike);

S(t)– spot price at the moment t;

w– a binary variable equal to 1 for a call and -1 for a put;

Q is the price of the option.

The risk of this option is that when it expires with a small in-the-money value, its intrinsic value will not be sufficient to pay the premium. In other words, the holder is more likely to prefer a regular option when the underlying asset is in the money. variable premium option usually in demand by investors with a clear idea of ​​the future price dynamics of the underlying asset.

It can be seen that conditional premium option is an mix of regular and binary options. Those. this type of option can be duplicated by taking a long position on a regular option and a short position on a binary one. If a binary option is not available, then dynamic hedging can be achieved by using the underlying asset and cash. A portfolio hedged in this way will have a delta and gamma equal to zero. Difficulties can only arise when the strike is close to maturity due to the volatility of the delta and unstable gamma in a binary option.

Options on extremes (Options on Extremes, Extremes)

Extreme Options are options with which you can fulfill the dream of any trader: buy at a minimum, sell at a maximum.

Options Lookbacks

A lookback option gives its owner the right to purchase or sell the underlying asset at the price that suits him most, which was reached in the period before the option is exercised (the strike price or the maximum / minimum for the option's validity period).

Lookback options are also divided into fixed and floating strike options. The payouts for these options are as follows:

Pfixed = (max[M(t)- X, 0], call),(max[X - m(t), 0 ], put)

where X– execution price (strike);

M(t)– maximum spot price;

m(t)- minimum spot price.

Pfloating = (max[S(t)- m(t), 0 ], call),(max[M(t)-S(t), 0 ], put)

where S(t)– spot price of the underlying asset;

M(t)– maximum spot price;

m(t)- minimum spot price.

Options Lookforward (Lookforwards)

Lookforward gives the holder the right to buy/sell on the maturity date a fixed amount of the underlying asset at the strike price and sell/buy the asset at the highest/lowest spot price for a specified time interval. Unlike a lookback, a lookforward can be out of the money if, after purchasing the options, the market for the underlying asset begins to decline/rise and the trend continues until the maturity date.

Extreme option holders exercise them under the same market conditions. For ordinary options, this situation is impossible. Thus, the amount of premium for which sellers agree to write options will be higher than usual. This leads to a low volume of trade in lookbacks and lookforwards.

Range Extremes

Range extremes combine the characteristics of lookback and lookforward options. They make it possible to get either the difference between the minimum and maximum spot prices for the period, or some percentage of this difference.

Cliquet Options, Ratchat Options

Initially, a Cliquet option behaves like a simple option with a fixed strike price. But over time, on predetermined dates, the strike price takes on the value of the price of the underlying asset. Every time the strike price changes, the intrinsic value of the option is automatically fixed. If on a certain date the price of the underlying asset is below the previous level, the intrinsic value of the option is not fixed. The execution price in this case takes the value of the current price of the underlying asset. The intrinsic value will be re-locked when the price of the underlying asset surpasses the level of the previous lock date.

2. Options depending on the choice of the buyer

This group includes such options for which at least one parameter is chosen at the discretion of the option holder. American options can also be included in this group, since the exercise date remains with the buyer of the option.

Compound Options

Complex Options are options on options. Option Cpound gives its owner the right to purchase another (underlying) option in the future. When considering an option C ompound, it is necessary to take into account the existence of two expiration dates: the expiration date of the option C ompound and the exercise date of the option that is the underlying asset.

A compound option has two strike prices. The first strike price is a forecast of the future premium of a simple option, the second strike price is the expected price of the underlying asset.

The payoff for a compound option when it is redeemed is expressed as follows:

Pcompound = max( yO - y X, 0 )

where O – the price of the underlying option with a strike X and time to expiration T;

y is a binary variable equal to 1 for a call on a call and a call on a put, and -1 for a put on a call and a put on a put.

Thus, the higher the strike price of the underlying option X, the lower the value of the compound call option and the higher the value of the compound put option. If X=0, the compound option is equivalent to the underlying option.

Complex options can be hedged by using the option underlying the contract in the same way as a standard option. Another hedging approach would be to use an underlying asset.

The sensitivity of compound options is inherently more complex than that of standard options, which are inherently difficult to use in dynamic hedging. In this case, theta must take into account the elapsed time, because it affects the time remaining until expiration of the compound option and the time remaining until expiration of the basic option.

Chooser Options

Option chooser allows the buyer in the future to choose between the right to exercise either a simple call option or a put option with the same prices and expiration dates. The payoff function for this option looks like this:

Vchooser = max( C, P}

where C – the price of a standard call option with a strike X(tc) and time to maturity tc,

P – the price of a standard put option with a strike X(tp) and time to maturity tp,.

t- selection time t tc andt tp.

For a regular option X(tc) =X(tp) and tc=tp=t.

The choice of the option holder is based on his expectations regarding the price dynamics of the underlying asset. An option option has much in common with the straddle strategy (call and put portfolios with the same strike price and expiration date), but it is cheaper because the holder needs to decide between a call and a put on the selection date.

Choice Options most often used when some significant event is expected, which can determine a new market trend, but it is not yet known in what direction. The selection date is set immediately after the event, when the result is already known, but the market has not yet had time to decide where to move.

Shout Options

shout option gives its holder the right to equalize the exercise price with the current price of the underlying asset at any time before the option's exercise date, by "shouting out" a new exercise price. The holder of such an option can "shout out" only once during the life of the option in order to set a minimum payout equal to the current price of the underlying asset minus the strike. Thus, the holder fixes his minimum payout.

Shout option payouts are formalized as follows:

Pshout = max ( w[S(t)-x], w[S(ts)-x], 0}

where S(ts)- the price of the underlying asset at the time of the "shout".

If the price of the underlying asset at the time of the “shout” is the maximum over the life of the option, then the value of the Shout option is equal to the value of the Lookback option. Due to the uncertainty in determining the optimal time for fixing the price " shout-options» cheaper than lookback options.

Quasi-American Options (Bermuda Options)

While all options are time dependent to some degree, some options are more time dependent than others. The dependence of European options on time is very rigid - the option is exercised only on a certain date. On the other hand, there are American style options that can be exercised at any time before the option is exercised.

Quasi-American Options in terms of their features, they are somewhere between American and European types of options, which is why they are also called “Bermuda” or “Mid-Atlantic”. The owner of a quasi-American option has the right to exercise it only on certain dates specified in advance in the option contract in the period before the option is exercised. The “window” for execution can be either a specific day or several days. Moreover, there may be several windows for option exercise. In this case, if the owner chose not to exercise the option during the first time window, he has the right to exercise it in the next period specified in the contract. Obviously, as the time of all periods for exercise approaches the life of the option, the quasi-American option becomes more and more like the American one. The payoff function of this option is equivalent to the payoff function of a simple option.

3. Options dependent on asset correlation

This group includes options, the price of which depends on the parameters of several assets. Thus, these options also depend on the correlation of these assets. Such options are called multiactive. The higher the negative correlation of assets, the higher the premium of multi-asset options.

Rainbow Options

rainbow option called an option n assets ("flowers"), payments for which are made according to the best / worst financial result of several assets:

The simplest of the rainbow options is the two-color option, i.e. for two assets, which is executed according to the best result:

Prainbow = max(S1,S2, X }

There are also calls and puts on the maximum of the two assets.

Prainbow = max(0, w*max(S1,S2) - wX }

and calls and puts on the minimum of the two assets

Prainbow = max(0, w*min(S1,S2) - wX }

Options "Best or worst of assets" (Best or worst of assets)

It's not really an option in the normal sense of the word (like a contract) because it doesn't have a strike price. This option is similar to a clause in a sale and purchase contract, where the beneficiary has the opportunity to sell the most expensive (buy the cheapest) from the proposed list of assets.

pbest_worst = max_ or _min(S1(t),S2(t),..., Sn(t) }

Exchange Options

Exchangeable options give the holder the right to exchange one asset for another upon expiration of the contract (exchange the first asset for the second). A standard exchange option can be interpreted as a call option on the first asset with a strike equal to the future price of the second asset at the exercise date, or a put option on the second asset with a strike equal to the future price of the first asset at the exercise date. In the case of an exchangeable option, there is no distinction between a call and a put.

Payouts on an exchangeable European-style option when exchanging the first asset for the second:

pexchange=max(S1(t) -S2(t), 0 }

Can also apply different weights to assets

pexchange=max(n1S1(t) -n2S2(t), 0 }

and exercise price

pexchange=max(n1S1(t) -n2S2(t) - X, 0 }

After a simple transformation of the payoff functions, we can see that exchangeable options can be easily duplicated using options better-of-two-assets or worse-of-two-assets together with the underlying asset:

max(S1(t) -S2(t), 0 } = max(S1(t),S2(t) } - S2(t) = S1(t) -min (S1(t),S2(t) }

Spread Options

Payments for spread options are based on the difference between the actual price (yield) spread of the two assets and the imputed one (embedded in the option as the strike price). The option payoff can be calculated as:

Pspread = max(w*[n1S1(t) -n2S2(t) - X], 0 }

These options can dynamically hedged underlying asset and cash.

Cross-Currency Options

Three different currencies are required to create this option, the currency to be delivered on exercise, the currency of the strike price, and finally the currency used to express the option price. The correlation coefficient between the currency to be delivered and the currency of the strike price is very important.

Quanto Options

Quanto option payouts depend on both the price of the underlying asset and external risks to which the currencies involved are exposed. Quanto options are based on the purchase of an asset in a currency other than the currency of the option buyer's country. And since the owner will need to convert the amount of the payout into another currency, the amount of the payout should be adjusted accordingly.

There are two varieties of Quanto options:

Basket Options

Basket option is one of the most popular of all multi-factor options. Such an option is formed by a whole range of underlying assets (for example, stocks and currencies). Basket options offer an effective way to hedge your currency position. The premium on it is much lower than the composite premium on individual options due to the correlation between assets.

The payouts for basket options are as follows:

pbasket = max(w*[W1 S1(t)+W2 S2(t)+...+WnSn(t)-x], 0 }

where Wi– share of investments in i-th asset.

Dynamic hedging of the individual assets in the basket is achieved by using a partial delta.

A basket option is exercised as a whole, so its value is always less than or equal to the sum of the premiums of ordinary options on the underlying assets.

Compo Options

compo options are options on foreign assets denominated either in the currency of the buyer of the option or in the currency of the country of origin of the underlying asset. They are also executed in one of the two mentioned currencies, depending on the desire of the buyer.

According to materials:

1. Burenin A.N. Futures, forwards and options markets. - M., Scientific and Technical Society named after Academician S.I. Vavilov, 2003

2. Kanev M.A., Podoinikov S.A. Exotic options: basic concepts and features. Novosibirsk State University. - 1998.

3. Kozhin K. All about exotic options // Securities market. - 2002. - Nos. 15-17.

4.Sharp W., Alexander G., Bailey J. Investments: Per. from English. - M.: INFRA-M, 2001

5. Bris E., Bellallah M., Mai H. Options, Futures and Exotic Derivatives. – N.Y., 1998;

6. Hull J. Options, Futures and Other Derivatives. 4th ed., L: Prentice-Hall, 2000.

7.www.geocities.com/optionpage/ Yang Yuxin's Option Page

8.www.DerivativesStratigies.com. Andrew Webb. What's Exotic?

Exotic options got their name at a time when the models for determining their prices were the property of a few. Exotic option pricing is now a standard feature in financial software. To separate the exotic types from the European style options, the latter was given a new name - vanilla. If you hear the term "vanilla option", recognize it as old friends!

1. Classification

Exotic options are divided into barrier, binary, optimal and average options. We will consider the first two types.

The simplest barrier options are knock-in and knockout options, double knockins and double knockouts. These are the usual stakes and puts of the European style known to us, but with the ability to “come to life” and “die off”. Their "life" depends on a certain price level(s), called a barrier. When this level is reached, the life of the option begins or ends (depending on the type of option). For example:

If you bought a call option with an exercise price of 100 and a 95 barrier-in, you would only be entitled to buy shares at 100 if the stock traded at 95 during the life of the option. one deal;

If you sold a 100 IBM call option at 120 barrier-out, the option may be exercised and you must sell the shares at 100 only if the stock did not trade at 120 during the life of the option. (will cease to exist).

2. Binary options

Binary options are similar to roulette because they work on the principle of "guessed/not guessed". Many types of binary options do not have strike prices and denominations that we are used to (but there are barriers described above). As in roulette, their conditions are described by the amount of payouts. A typical example of such an option is the condition: if you pay $10,000 today, you are entitled to a payout of $25,000 if the USD/CHF rate does not touch 1.5000 within the next 5 months. This type of option is called "tangent". Tangent options are divided into tangent (one touch) and touchy (no touch).

A typical condition for a tangent option would be: if you pay $10,000 today, you are entitled to a payout of $25,000 if the USD/CHF rate touches 1.5000 within the next 5 months.

Another type of binary options are digital (digital) options. They combine ordinary options and tangents in their description. They have a face value, strike price and payout amount. They are exercised if they are in-the-money on the expiration date of the option. However, unlike common options, they have a fixed payout. So if you buy a 1.5200 US dollar call digital option (CHF put), you will receive $10,000 regardless of whether the spot rate on the expiration date is 1.5205 or 1.5700.

Note:

1) payments for tangent options can occur at any time during the life of the option, and for digital options only on the day the option is exercised;
2) No matter how strange the names of options may seem, the ideas behind them are very simple.

3. Barrier Options

Let's take a closer look at barrier options. There are many different names for this type of option in English. We prefer to refer to them as reverse knockin, reverse knockout, knockin and knockout, but we will use other (alternative) names as well.

Reverse knockins (Up-and-in stake or Down-and-in put)

Reverse knockin call - the option will come to life when it is in the money. For example, if you bought a 1.5000 USD call with a 1.6000 barrier-in, you will only be entitled to buy USD at 1.5000 if the price touches the 1.6000 barrier-in during the life of the option. If this happens, the option will be "deep in the money" at the moment of "revival".

Reverse knockin put - the option will come to life when it is in the money. For example, if you bought a 1.5000 USD put with a 1.4000 barrier-in, you would only be entitled to sell USD at 1.5000 if the price hits the 1.4000 barrier-in during the life of the option. If this happens, the option will be "deep in the money" at the moment of "revival".

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Reverse knockouts (Up-and-out count or Down-and-out put)

Reverse knockout call - the option will "die" (cease to exist) when it is "in the money". For example, if you bought a 1.5000 USD call at a 1.5500 barrier-out level, you will be entitled to buy USD at 1.5000 only if the price does not touch the 1.5500 barrier-out during the life of the option.

Reverse knockout put - the option will die out being in the money. For example, if you bought a 1.5000 USD put at the 1.4400 knockout level, you will be entitled to sell USD at 1.5000 only if the price does not touch the barrier-of-1.4400 during the life of the option.

So as soon as reverse is in the option name, you know the barrier is at the money.

Knockins (Down-and-in stake or Up-and-in put)

A knockin call option "comes to life" when it is "out of the money". For example, if you bought a 1.5000 USD call with a 1.4000 barrier-B, you would be entitled to buy USD at 1.5000 only if the price touches the 1.4000 barrier-B during the life of the option. If this does not happen, the option will be out of the money by the expiration date, even if the USD is above 1.5000. Note that the fact that an option has "lifed" does not mean that it will be in-the-money at expiration.

Knockin put - An option will "come to life" by being "out of money". For example, if you bought a 1.5000 USD put with a 1.6000 barrier-in level, you would only be allowed to sell USD at 1.5000 if the price reaches 1.6000 during the life of the option. If this happens, the option will be "out of the money" at the time of the "rise".

Knockouts (Down-and-out count or Up-and-out puts)

Knockout call option will "wither away" by being "out of money". For example, if you bought a 1.5000 USD call at a 1.4000 barrier-out level, you would lose the right to buy USD at 1.5000 if the price reaches the 1.4000 barrier-out during the life of the option. If this happens, the option will be "out of the money" at the time of "dying off".

Knockout put - the option will "wither away" being "out of money". For example, if you bought a 1.5000 USD put at the 1.5500 barrier-out level, you would lose the right to sell USD at 1.5000 if the price reaches 1.5500 during the life of the option. If this happens, the option will be out of the money by the time it is called.

Note that we talked about forward hedging earlier. This rule continues to this day, but the barriers are used by the spot, not by the forwards.

4. More complex types

Consider variations of these strategies.

Options with double barriers (double-barrier options): for example, double reverse knockin, double knockout, double touchy double no touch (range binary), etc. The idea is clear: instead of one barrier - two! Moreover, the spot is always inside two barriers, but the strike can be outside (see below). "Double" (two-barrier) options are very risky strategies, because it is extremely difficult to guess the range of market fluctuations. As a rule, investors are attracted to owning strategies with double barriers due to their cheapness. Hedgers are attracted to selling strategies with double hurdles. they have an additional chance of not being exercised by selling options as part of a hedging strategy.

An interesting sub-type of dual options are double barrier-outs in the money. For example, 100 USD call (JPY put) with 105-110 hurdles-out. With this strategy, the spot is between the barriers 105-110. Due to the relatively narrow distance between the barriers, the chances of being "obsolete" are high. Therefore, despite the fact that the option is "at the money", it will cost cheaply.

Rebate options are options with multiple rows of barriers. For example, if you invest $1,000 and for three months USD/JPY is in the range of 105.00-110.00, you will receive $2,000, but if the market "touches" 105.00 or 110.00, you are entitled to get back the premium you invested if the market does not " touches” 103.00 or 112.00. If the market trades at any of the last barriers, then you lose the premium. Those. options with discounts leave the opportunity not to lose investments or part of them by reducing the amount of payments in the first "internal" range.

Passport option - a contract that allows you to trade a certain amount of the underlying asset a set number of times against a deposit. In this case, the buyer is not responsible for losses from trading and loses the initial deposit as much as possible. For example, in exchange for 1 million dollars, you are granted the right to trade EUR/USD with a par value of no more than 100,000 euros, no more than twice a day for a month.

A window option is usually a barrier option that takes effect after a certain period of time. For example, a 100.00-110.00 double-barrier-out option, effective in one month and expiring in three months. In this example, it doesn't matter if the spot traded at 100.00 or 110.00 in the first month. The main thing is that these levels are not traded during the life of the option, stipulated by the contract.

Choice option - at the time of the conclusion of the contract, the type of option (call or put) is not indicated. The contract specifies the date on which the buyer determines what the type of option will be.

A compounded option is an option on an option. Another option is used as the underlying asset. At the time of exercising a compound option, its buyer has the right to buy or sell the original option (inside) at the strike price - the exercise price.

QUESTIONS

1) You are long 1.4500 USD put with a 1.4000 knockin barrier. During the life of the option, the spot rate fell to 1.4005 and is now at 1.4200. If the option expires today and the spot is at the current level at the time of expiration, how much will you earn?

2) You are short 1.4350 USD call with a 1.4100 knockout barrier. During the life of the option, the spot rate has always been above 1.4217 and is now at 1.4360. If the option expires today and the spot is at the current level at the time of expiration, how much will you earn?

3) Compare the cost of two positions: X - long 1.4800 USD stake; Y - long 1.4800 USD call with 1.5000 knockin barrier and long 1.4800 USD call with 1.5000 knockout barrier. The spot course is at:

a) 1.4700;
b) 1.4900;
c) 1.5500;
d) If you were to buy positions X and Y, which one should be more expensive?

4) What is your real position (in terms of ordinary options) if you have two positions on your account: X - long 1.4700 USD call with a barrier of 1.4600 knockin; Y - short 1.4700 USD call with 1.4600 knockout barrier and long 1.4700 USD call. The spot rate one week before expiration is at:

a) 1.4500;
b) 1.4900;
c) if you were to buy positions X and Y, which one should be more expensive?

5) The spot rate is at the level of 1.4400. You think the market is going to go lower and want to go long the 1.4000 USD put with the 1.4420 knockout barrier because the option is cheaper if you are closer to the knockout level. An option is worth less when:

a) is the expected volatility higher or lower?
b) time until option expiration 1 week or 1 month?
c) why?
6) The spot rate is at the level of 1.4230. You are long 1.4400 USD call with a 1.4200 knockout barrier. Since the option is likely to be called, you are more concerned with hedging the premium paid than making a profit. What are you going to do in this situation to hedge: buy or sell dollars?

7) The spot rate is at the level of 1.4230. You are long 1.4400 USD put with a 1.4200 knockout barrier. Since the option is likely to be called, you are more concerned with hedging the premium paid than making a profit. What are you going to do in this situation to hedge: buy or sell dollars?

ANSWERS

1) 0. The spot rate did not reach 1.4000, so the option did not become valid.

2) 10 pips. Since the spot did not reach 1.4100, the option is in effect and is 10 pips in the money.

3) a) both positions were not used;
b) Both positions are 100 pips in the money. At level 1.4900, reverse knockin will not be performed, while reverse knockout will be "in the money";
c) Both positions will be 700 pips in the money. At 1.5500, reverse knockin will be in the money, while reverse knockout will not be performed;
d) theoretically they should cost the same: they provide the same level of P/L at any spot level. However, market makers usually quote differently from the theoretical quotes and have a larger spread, which means that you will have to pay twice as much spread when buying two options than when buying one. Thus, an ordinary option will cost less.

4) a) both positions are long 1.4700 USD call: X: the USD call option became effective when the level of 1.4600 was reached; Y: USD call short position expired at 1.4600;
b) both positions were not executed; in position Y, both short and long options are in-the-money, offsetting each other, while position X has not taken effect;
c) theoretically they should cost the same: they provide the same level of P/L at any spot level. However, market makers usually quote differently from the theoretical quotes and have a larger spread, which means that you will have to pay twice as much spread when buying two options than when buying one. Thus, an ordinary option will cost less.

5) a) the option is cheaper when the expected volatility is higher;
b) the option is worth less when there is more time left before its expiration;
c) the higher the volatility, the less likely the option will remain valid, and the cheaper it should cost. The same logic applies to timing: the more likely the option is to be called, the cheaper the option. That is, as in other cases, the volatility and time factors affect the option premium.

6) You will sell dollars. You will be hedged in the same way as you would with a common option.

7) You will sell dollars. This seems counter-intuitive because you are hedging in the opposite way to a common option. Examples 6 and 7 highlight the difference between reverse knockout and regular knockout.