ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments -

Transcription

1 AARHUS SCHOOL OF BUSINESS AARHUS UNIVERSITY MASTER THESIS ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments - AUTHORS: DANIELA ZABRE GEORGE RARES RADU SIMIAN SUPERVISOR: JOCHEN DORN SEPTEMBER 2010

2 Table of Contents 1 Introduction Problem statement Methodology Data Delimitations Notations Digital/Binary options General description of digital cash-or-nothing options Payoff of digital options Pricing of European digital options Barrier options General description of barrier options Payoff of European barrier options Pricing barrier options Binary barrier options General description of binary barrier options Payoff of binary barrier options Pricing binary barrier options General description of corridor products General characteristics of corridor products Types of corridors Motivation for issuance of and investment in corridor products Miscellaneous corridor products Digital ranges General description Replication Valuation of digital ranges Barrier ranges General description I

3 6.2.2 Replication of barrier ranges Valuation of barrier ranges Hedging aspects Dynamic hedging approach Digital ranges Barrier ranges Static hedging Static hedging of digitals Static hedging of binary barrier options Static hedging of digital ranges Static hedging of barrier ranges Conclusion and recommendations BIBLIOGRAPHY II

4 List of tables Table 1: Payoff of binary barrier options Table 2: Example of a FX digital range Table 3: Valuation at time zero for the digital range presented in Table Table 4: Example of a FX barrier range Table 5: Replicating portfolio for the coupon part of a barrier range Table 6: Valuation at time zero for the barrier range presented in Table List of figures Figure 1: Payoff digital call... 8 Figure 2: Payoff digital put... 9 Figure 3: Digital range - payoff corresponding to a fixing date Figure 4: Digital range value evolution across the product s lifetime Figure 5: Relationship between the maximum and minimum interest rates for a digital range Figure 6: Barrier range value evolution across the product s lifetime Figure 7: Relationship between the maximum and minimum interest rates for a barrier range Figure 8: Delta for a digital range Figure 9: Gamma for a digital range: different perspectives Figure 10: Vega for a digital range Figure 11: Rho for a digital range Figure 12: Theta for a digital range Figure 13: Delta for a barrier range Figure 14: Gamma for a barrier range Figure 15: Vega for a barrier range Figure 16: Rho for a barrier range Figure 17: Theta for a barrier range Figure 18: Approximate replication of a digital option with a vertical spread Figure 19: Matching the adjusted payoff function for a down-and-out binary in two points under the barrier Figure 20: Matching the adjusted payoff function for an up-and-out binary in two points above the barrier Figure 21: Vega of the digital range derived with respect to... iv Figure 22: Vega of the digital range derived with respect to... iv Figure 23: Summation between Vega of digital range derived with respect to and Vega of digital range derived with respect to... v Figure 24: Vega of digital range derived with respect to and... v III

5 Figure 25: Vega of barrier range derived with respect to... vi Figure 26: Vega of barrier range derived with respect to... vi Figure 27: Summation between Vega of barrier range derived with respect to and Vega of barrier range derived with respect to... vii Figure 28: Vega of barrier range derived with respect to and... vii Figure 29: Graphical representation of the adjusted payoff function of a down-and-out cash-or-nothing binary option... x Figure 30: Graphical representation of the adjusted payoff function of an up-and-out cash-or-nothing binary option... xi IV

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7 1 Introduction The market for exotic options has been growing in popularity since its inception and so did the research in this area. Subsequent developments in financial engineering made it possible to include exotic options in more complex structures designed for investors with specific risk profiles. This thesis will focus on only one class from the wide variety of structured products available on the market, namely corridor products. These are maybe among the most attractive and most heavily traded instruments, next to structures with embedded barrier options. Corridor products offer investors the possibility of earning a coupon whose magnitude depends on a certain underlying price staying within a predefined range. These kinds of products are suitable for those investors who expect stability in the market through the lifetime of the corridor type instrument they invest in. If their expectations are met, the return on their investment is considerably higher compared to alternatives such as direct investment in bank deposits or bonds. As with all structured products, corridor type instruments are conceived starting from simpler parts, and therefore, a logical approach to be used for valuation and hedging purposes is to separate the individual components and analyze them one by one. Corridor products are most often designed as combinations of digital, barrier or binary barrier options, reason for which this paper will first analyze such basic components before dealing with our products of interest. 1.1 Problem statement The purpose of this paper is to analyze two types of corridor products, which are most popular in the literature on the subject matter as well as mostly used by practitioners. Our 1

8 approach will be focused on developing valuation formulae, analyzing the sensitivity parameters and proposing different hedging strategies. The central idea from which the authors of this paper began their research was to employ options such as digitals, binary barriers and barriers in the valuation and the hedging processes of the products of interest. Two different products will be studied, each of them bearing interesting features. The first corridor product a digital range is a deposit which brings its owner an enhanced return for those days when the underlying price remains within certain limits. While the interest rate earned in those days when the reference rate is within the barriers is accrued in the case of a digital range, for barrier ranges our second product the mentioned condition needs to be fulfilled for the entire lifetime of the product in order to bring a satisfactory payoff. The objective of the thesis is to provide closed form solutions for pricing digital and barrier ranges. Further on, the sensitivity of the products value to several parameters will be analyzed and alternative hedging approaches will be investigated. 1.2 Methodology The paper is divided into two main blocks: the first one will look at valuation issues, while the second will focus on hedging aspects. The first steps will consist in finding closed-form valuation formulae for digital options, barrier options and binary barrier options in the Black-Scholes framework (see Appendix A for the assumptions). Chapters 2, 3 and 4 introduce the main features of the aforementioned options and lay the foundation for the theoretical understanding of the structure and behavior of corridor type products. The general description of corridors as well as the reasoning behind their usage is presented in Chapter 5. In Chapter 6, we will use the previously obtained results as building blocks for elaborating the valuation formulae for digital and barrier ranges. Having found the closed form solutions, we will perform simulations based on real market data and a limited number of 2

9 assumptions in order to analyze how the prices of the two corridor products behave across the structures lifetime. While the obtained formulae are valid for a wide range of underlying assets, the given examples are built on the EUR/USD foreign exchange rate. The following chapter will look at the hedging particularities of the two instruments. The analysis in the first section of the chapter will cover the products sensitivity indicators with respect to the Black-Scholes parameters and variables (more commonly known as the Greeks ) and their behavior as maturity approaches. The next section explores alternative means for hedging the products, indicating the advantages and shortcomings of different strategies. Simulations were performed in Microsoft Excel, by coding under Visual Basic for Applications (VBA). The results can be found in the file Thesis.xlsx on the enclosed CD. The last chapter provides concluding remarks and recommendations for future research. 1.3 Data The data was obtained from Bloomberg and Thomson Datastream databases. We gathered information regarding the EUR/USD spot exchange rate, the 3-month Euribor rate and the 3-month T-Bill rate (which were used as risk-free rate and respectively foreign risk-free rate in the simulations). We extracted the forward curve (with daily compounding) for EUR deposits, which was used for pricing the digital range. We also obtained future implied volatilities as they were estimated at certain dates for call options on the EUR/USD FX rate with exercise prices set at the corridor limits. Further on, based on the market data, a set of realistic assumptions were proposed, such as input values for the upper and lower corridor limits, the maximum and minimum interest rates to be paid out within the structures. 3

10 1.4 Delimitations This paper develops an analytical approach, centered on pricing the instruments using closed-form formulae. The background for our analysis is set to be the Black-Scholes world. Therefore, our results rely on the classical assumptions of constant volatility and interest rates and are subject to the errors arising from their inconsistency with real world conditions. Further research can be carried on by relaxing some of the assumptions. Furthermore, this thesis is not meant to present corridor products in an exhaustive manner, due to space and time constraints. We chose to shed some light on the issues arising from the pricing and hedging of two main categories which are found in the literature regarding the subject matter: digital ranges and barrier ranges. We focused our attention on these two products, as we were interested in dealing with corridors through the perspective of options such as digitals or barriers. The corridor type instruments cover a wide range of exotic features and the analysis and discussions around the topic could fill the pages of many books. Further delimitations will be considered throughout the paper when appropriate as well as in the Conclusions and recommendations section at the end of the thesis. 1.5 Notations The literature on corridor notes as well as generally the literature of the financial world uses different notations for the same terms/notions. Before continuing with the lecture of this master thesis please take a look at the list of notations we will mainly be using further on (provided below). List of notations: generic notation for a barrier; 4

11 the Black-Scholes value of a plain vanilla European call option at time 0; cash or nothing; down-and-in (used for barrier options and binary barrier options); down-and-out (used for barrier options and binary barrier options); upper limit of a corridor; K strike price; lower limit of a corridor; number of fixings when the underlying is inside the range during the life of the contract; total number of fixings; the Black-Scholes value of a plain vanilla European put option at time 0; operator for the probability under the measure; the continuously compounded risk-free interest rate; the continuously compounded dividend yield; risk-neutral probability measure; the price of the underlying asset at time 0 (spot price); the price of the underlying asset at time ; the spot rate of the underlying at time running minimum of the underlying asset up to time ; running maximum of the underlying asset up to time ; the price of the underlying asset at maturity ; running time; time zero for the Digital Range structure; 5

12 running time in the Digital Range structure; expiry date; expiry date for the digital options from the Digital Range structure corresponding to the -th fixing date; up-and-in (used for barrier options and binary barrier options); up-and-out (used for barrier options and binary barrier options); Visual Basic Application (component of Microsoft Excel software); V(S, t) the value of a contract as a function of the value of the underlying asset S at time t; standard Brownian motion under the probability measure ; fixed amount of cash; a stochastic process describing a Brownian motion with drift and volatility ; the value of a riskless bond at time ; forward rate applicable for the period between and the Digital Range structure s expiry date; zero coupon bond; the continuously compounded true rate of return of the underlying asset; the volatility in the log-returns of the underlying; implied volatility for an option struck at L; implied volatility for an option struck at H; put/call operator: takes the value +1 for a call and -1 for a put; the stopping time; Cumulative Distribution Function for the standard Normal Distribution; 6

13 2 Digital/Binary options 2.1 General description of digital cash-or-nothing options Cash-or-nothing digitals or binaries 1 are options with a discontinuous payoff profile which pay out a fixed amount of cash only if the underlying asset satisfies a predetermined trigger condition but nothing otherwise. 2 Such a condition usually assumes that the underlying trades below/above a certain price level, which is referred to as the strike. Weithers (2006, pg. 209) classifies digital options taking into consideration the moment when they can be exercised. Therefore, there are two main types of digital options: European (at maturity or at expiry binaries) and American (one-touch) options. The European digital call/put pays off if the underlying is traded above/below the strike price at the maturity date. The American digital call/put (also known as one-touch option) pays off once the spot price first rises/falls above/below the strike price during the life of the option. The payoff of a one-touch binary may be due as soon as the trigger condition is satisfied or at expiry (one-touch immediate or one-touch deferred binaries). 3 Another common classification of digital options available in the literature categorizes them as: barrier. One touch: the investor gets the payoff if at any time the price hits the No touch: the payoff of a no-touch is paid if the trigger has not been reached by the time the option expires. 4 Double one touch: this option pays off if at any time the price touches any of the barriers of a predetermined range. Double no touch: the buyer gets the payoff if the price does not touch either of the barriers of the predetermined range. 1 Another type of digitals are the asset-or-nothing options, which pay out the price of an underlying, as long as the underlying triggers a certain event (such as reaching a certain price level) during the life of the option. However, we shall only use the cash-or-nothing digitals in the present paper, and, thus, our focus will rely solely on them. 2 SOCIÉTÉ GÉNÉRALE CORPORATE & INVESTMENT BANKING The ABC of equity derivatives and structured products. 3 Ibid. 4 WYSTUP, U FX Options and Structured Products, Chichester, John Wiley & Sons. 7

14 Motivation for usage of digital options There are many ways in which digital options can be used by market participants, depending on the scope followed and their expectations regarding the behavior of the market. Binary options can be bought/sold by investors who are rather interested in speculating over the rising or falling of the underlying. Other investors use them as a hedging strategy enabling the capturing of compensation in case their view on the market is not confirmed. Furthermore, digital options can be found within the composition of different structured products Payoff of digital options The payoff for European digital options is given by: { } where is a fixed amount of cash, {+1, -1} takes the value +1 for a call and -1 for a put option. The payoff of a one-touch (American) option is given by: {} where = inf { 0 }, and {+1, -1} takes the value +1 for a call and -1 for a put. Figure 1: Payoff digital call Source: own contribution. 5 Ibid. 8

15 Figure 2: Payoff digital put Source: own contribution. 2.3 Pricing of European digital options The pricing will be done within the Black-Scholes framework. Therefore, we start with the assumption that the underlying asset return follows a log-normal random walk. The standard model for the evolution of stock prices is given by the geometric Brownian motion = + ( 1 ) while a riskless bond follows the deterministic process Which, due to Ito s Lemma, is equivalent to = dt ( 2 ) = e where is a standard Brownian motion under the probability measure 6, is the price at time of the underlying asset, is the continuously compounded risk-free interest rate, is the volatility in the log-returns of the underlying. We assume that = 1. If we consider the riskless bond as the numeraire, then the relative price process is the discounted asset process. Differentiating this process yields: 6 The probability measure represents the true market distribution of the asset. 9

16 = + d ( 3 ) = r dt + In order to switch to the risk-neutral world and therefore to be able to price in the Black- Scholes framework, one needs to make the transformation from a real-world measure to a risk-neutral measure. Using the Girsanov theorem (as shown in Appendix B), the previous equation can be written as = r dt + dt where is a standard Brownian motion under the probability measure (h = + dt). In order to make the process driftless, we choose =. Our discounted price process becomes a martingale under the new risk-neutral probability measure, as follows: = ( 4 ) Using equations ( 2 ), ( 3 ) and ( 4 ), we obtain that in a risk-neutral world, a non-dividend paying underlying is described by the following process: = r dt + For a dividend paying stock 7, the underlying process under the risk-neutral measure is defined by: = r q dt + ( 5 ) where is the dividend yield/foreign currency risk-free interest rate. 7 For the remainder of this paper, we will assume that the underlying has a dividend yield different from zero. 10

17 Defining the process = and using Ito s transformation we have: = = r q S = r q 1 2 dt + = r q 1 2 T t + Thus, the price at maturity of the underlying asset is expressed as: = e ( 6 ) We will derive the following results: = d ( 7 ) = d ( 8 ) where is a fixed amount of cash, / denote the price at time t of a cashor-nothing call or put, and d = ln K is the maturity date and is the strike price. σ + r q 2 T t σ T t Proof: The value of a digital call at time (0 < ) can be expressed as follows: 11

18 = e { } = e { } = e > = e ln > ln = e ln + r q σ 2 T t + σ > ln = e ln T t > K = e T t > + r q σ 2 T t T t Where is the operator for expectation under the probability measure, and is the operator for the probability under the measure. But we know that 0,1 hence: = e d Similarly the value at time for a digital put can be expressed as: = e 1 { } = e 1 { } = e < = e ln < ln = e ln + r q σ 2 T t + σ < ln = e ln T t < K = e T t < + r q σ 2 T t T t Knowing that 0,1 we obtain that: = d 12

19 3 Barrier options 3.1 General description of barrier options Barrier options are weakly path-dependent options (i.e. their value depends on the path that the underlying followed until expiry; however, the only information we need is to know whether the triggering event has occurred or not). They are either terminated or activated if the underlying asset s spot price reaches and crosses a predefined trigger level (barrier). They almost bear the same specifications as a standard option (call or put) such as the strike (exercise) price, the expiration date, settlement features, to which the barrier feature is added. One of their advantages over plain vanilla options is the fact that they command a lower premium. Since the terminology used in relation to these products is still in the process of standardization, we will list below some terms and the meanings with which they will be associated throughout this paper. 8 An in option comes into existence when the barrier level has been crossed, and it is said that the option has knocked in An out option ceases to exist when the barrier level has been crossed, and it is said that the option has knocked out If the barrier level is above the initial value of the underlying asset, we will have up options If the barrier level is below the initial value of the underlying asset, we will have down options The term regular is used for options which have the barrier set in the outof-the-money direction The term reverse is used for options which have the barrier set in the inthe-money direction 8 WILMOTT, P Paul Wilmott on quantitative finance, Chichester: John Wiley & Sons. 13

20 Motivation for usage of barrier options Among the main reasons why barrier options are traded there is the fact that they are cheaper than the corresponding vanilla options. They are also quite simple to understand and valuate. For example, fund managers use these options to decrease their hedging costs. Furthermore, barrier options are attractive to those market participants who have strong views on how the market will behave on a certain time horizon. Therefore, investors who expect small fluctuations of the market will choose to buy knock-out options while those who believe that the market will behave erratically will opt for knock-ins. Last but not least, barrier options are very often integrated in structured products. 3.2 Payoff of European barrier options Making the following notations = min and = max, we will list below the individual payoffs for the four main types of barrier options 9 (up-and-in, up-andout, down-and-in, down-and-out). If < = max, 0 { } = max, 0 { } If > = max, 0 { } = max, 0 { } where {+1, 1} is the usual call/put indicator, is the barrier level and =. 9 ZHANG, P. G Exotic Options: A Guide to Second Generation Options, Singapore, World Scientific Publishing 14

21 Being given the payoffs from above we can now write the following observation about knock-in options: as long as the barrier is touched during the life of the options, the investor will receive an European option at the maturity of the barrier option, while, if the barrier is not touched during the life of the option, the investor will receive zero (sometimes, the investor receives a compensation i.e. a rebate 10 ). For knock-out options, the European option is received as long as the barrier is not breached up to maturity. 3.3 Pricing barrier options Starting with the same assumptions as for digitals (Black-Scholes framework, underlying asset returns following a log-normal random walk, risk-neutral world), we have that the underlying process is described by the following equation under the risk-neutral measure : = r q dt + The solution for this partial differential equation, as shown before, is given by: = e The notations for the parameters in the last two equations are the same as in 2.3. Further on, we make the additional notations: = min = max = = = + σ = + σ = + σ 10 Generally throughout the paper we will assume that the rebate equals zero. 15

22 Moreover, the value of a call/put option at time 0 will be denoted by /, while the value of an up-and-in call/put option at time 0 will be denoted by / (and similar for the other three types of barrier options), and the barrier by. The pricing of barriers in this chapter will be based on barriers with zero rebate and will be done at time zero. Given that + = e max, 0 { } + e max, 0 { } = e max, 0 = we can prove in a similar way that: + = + = + = I) If <, the following results will be demonstrated: a) If < then: = e x K e x σ T e y y + K e y + σ T y + σ T ( 9 ) = e y + K e λ y + σ T ( 10 ) b) If > = 0 ( 11 ) = ( 12 ) = e x + K e x + σ T + e λ y K e λ y + σ T ( 13 ) 16

23 Proof a) For < = e max, 0 { } = e {, } = E e S e K {, } We make the following notation: =. Thus, = S c = E e S e K, = e E S e K, We also denote ln = x and ln = b. Since K <, we obtain x <. The price of the up-and-in call option at time 0 becomes: = 1, 1, = Further on, we will use the following lemma: Lemma 1: Let there be > 0 and <, with the stochastic process = + and its maximum defined by =, where is a standard Brownian motion under the probability measure. Then, we have that:, = =

24 We can now write and using the previous lemma as: I = e E K 1, = K e Where: b + ν T σ T = x σ T + e e B σ ln + r q = S 2 T σ T = ln S B + λ 1 σ T σ T e b ν T σ T = e = B S y + σ T ln B S λ 1 σ T σ T e x 2 b ν T = e σ t ln K S 2 ln B S ν T σ T = B S ln B S S K λ 1 σ T = B σ T S y + σ T Thus, I = K e x σ T + B S y + σ T y + σ T We now solve for I 18

25 I = e E S e 1, = S E e e 1, = S e E e 1, Our objective is to eliminate the exponential term from the expectation by changing the probability measure. We will employ the Girsanov Theorem (Appendix B). Let = σ t. = = = I = S e E e 1, = S e E e 1, = S e E 1, Once again, we will use Lemma 1 and we make the following notation: ν = +. I = S e b + ν T σ T + e where x 2 b ν T σ T b ν T σ T e 19

26 B σ b + ν T ln σ T = S + r q + 2 T = ln S B + λ σ T σ T σ T = x e = e = y e x 2 b ν T = e σ t ln K S 2 ln B S λ σ T σ T = B ln B S S S K λ σ T = B σ T S y Thus I = S e x + B S y y And = = e x K e x σ T e B S y y + K e B S y + σ T y + σ T The demonstration for the pricing formula of the up-and-in put option at time 0 follows. = E e max, 0 { } = E e {, } = E e K S e {, } 20

27 We make the following notation: =. Thus, = S. = E e K S e, = e E K S e, We also denote ln = x and ln = b. Since K <, we obtain x <. = e E K, e E S e, = I I The following lemma will be used: Lemma 2: Let there be > 0 and <, with the stochastic process = + and its maximum defined by =, where is a standard Brownian motion under the probability measure. Then, we have that:, = 2 We can now write and using the previous lemma as: Where: I = e E K 1, = K e e x 2 b ν T σ T 21

28 e x 2 b ν T = e σ t ln K S 2 ln B S ν T σ T = B S ln B S S K λ 1 σ T = B σ T S y + σ T And, thus: I = K e B S y + σ T We now solve for I I = e E S e 1, = S E e e 1, = S e E e 1, Our objective is to eliminate the exponential term from the expectation by changing the probability measure. We will employ the Girsanov Theorem (Appendix B). Let = σ t. = = = 22

29 I = S e E e 1, = S e E e 1, = S e E 1, Once again, we will use Lemma 2 and we make the following notation: ν = +. I = S e e x 2 b ν T σ T Where: e x 2 b ν T = e σ t ln K S 2 ln B S λ σ T σ T = B ln B S S S K λ σ T = B σ T S y Thus: I = S e B S y And: = e B S y + K e B λ S y + σ T 23

30 Proof b) For > For a up-and-out call one of the conditions to be met is that >, otherwise it expires worthless. But if > > > {, } = 0 = E e {, } = E 0 = 0 = The price for an up-and-out put at time 0 is given by: = E e max, 0 { } = E e {, } Since we know that <, and our condition is that <, the other condition ( < ) is redundant. Hence, we can restate the payoff function simply as { }. We make the following notation: =. Thus, = S. = E e K S e = e E K S e We also denote ln = b. = e E K 1 e The following lemma will be used: Lemma 3: E S e 1 = I I Let there be > 0, with the stochastic process = + and its maximum defined by =, where is a standard Brownian motion under the probability measure. Then, we have that: 24

31 = We can now write and using the previous lemma as: Where: I = e E K 1 = K e b ν T σ T e b ν T σ T b ν T B σ ln r q σ T = S σ T 2 T = ln S B λ 1 σ T σ T = x + σ T e And, thus: b ν T σ T = e = B S y + σ T ln B S λ 1 σ T σ T I = K e x + σ T B S y + σ T We now solve for I I = e E S e 1 = S E e e 1 = S e E e 1 25

32 Our objective is to eliminate the exponential term from the expectation by changing the probability measure. We will employ the Girsanov Theorem (Appendix B). Let = σ t. = = = I = S e E e 1 = S e E e 1 = S e E 1 Once again, we will use Lemma 3 and we make the following notation: ν = +. Where: I = S e b ν T σ T e b ν T σ T b ν T B σ ln σ T = S r q + σ T = x 2 T = ln S B λ σ T σ T e b ν T σ T = e ln B S λ σ T = B σ T S y Thus: I = S e x B S y 26

33 And: = e x + K e x + σ T + e B S λ y K e B λ S y + σ T II) If >, the following results can be demonstrated similar to when <. a) If < then: = e x K e x σ T e y + K e y σ T ( 14 ) = 0 ( 15 ) b) If > then: = e y K e y σ T ( 16 ) = e x + K e x + σ T + e y y K e y σ T y σ T ( 17 ) 27

34 4 Binary barrier options 4.1 General description of binary barrier options Binary barriers combine features of the two types of options presented in the previous chapters, in the sense that they pay a fixed amount of some asset as long as some preestablished conditions are triggered (such as that a barrier must (or must not) be touched during the life of the product and/or that the price at expiry should be above/below a certain level (which can be different or identical to the barrier)). Rubinstein and Reiner (1991) describe 28 types of binary barrier options, differentiated by criteria such as the position of the price at time zero towards the barrier (down/up), the exercise date (at hit or at expiry) or the type of payoff (cash-or-nothing or asset-or-nothing). In order to have positive payoffs, some options require not only that the barrier be breached, but also that the underlying asset finishes above/below a given level. Given that the scope of the present thesis is to shed some light on corridor products, we are only interested in the so called binary barrier cash-or-nothing at expiry options, which we will be using in the valuation process. The binary barrier cash-or-nothing at expiry option pays out a certain amount of cash at the maturity date if the barrier has been hit (or not) during the life of the product. For ease of notations, the expression binary barrier option will refer to this type of option for the rest of the paper. 4.2 Payoff of binary barrier options Making the following notations = min and = max, we will list in the table below the individual payoffs for the main types of binary barrier options. 28

35 Table 1: Payoff of binary barrier options Type of option Payoff Condition Binary UI at expiry CON X (t) > B Binary UO at expiry CON X (t) < B Binary DI at expiry CON X (t) > B Binary DO at expiry CON X (t) < B Note: if the condition is not met than the payoff is zero. Source: own contribution. 4.3 Pricing binary barrier options I) If <, the following results will be demonstrated: = e x σ T + X e y + σ T ( 18 ) II) If >, then = e x σ T e y σ T ( 19 ) Where = = + σ = + σ Proof: Using the same notations as for barrier options, the value of a European binary upand-in cash-or-nothing can be expressed as follows: = E e X { } We make the following notation: =. Thus, = S. 29

36 = E e X = e X E We also denote ln = b. The following lemma will be used: Lemma 4: Let there be > 0, with the stochastic process = + and its maximum defined by =, where is a standard Brownian motion under the probability measure. Then, we have that: = + + We can now write the binary up-and-in at expiry cash-or-nothing option using the previous lemma as: Where: = e X E 1 = e X b + ν T σ T + e b ν T σ T b + ν T B σ ln σ T = S + r q 2 T = ln S B + λ 1 σ T σ T σ T = x σ T 30

37 e Thus: b ν T σ T = e = B S y + σ T ln B S λ 1 σ T σ T = X e x σ T + B S y + σ T The result for the binary down-and-out cash-or-nothing can be demonstrated in the same manner. = e X { } We make the following notation: =. Thus, = S. = E e X = e X E We also denote ln = b. The following lemma will be used: Lemma 5: Let there be < 0, with the stochastic process = + and its maximum defined by =, where is a standard Brownian Motion under the probability measure. Then, we have that: =

38 We can now write the binary down-and-out at expiry cash-or-nothing option using the previous lemma as: Where: = e X E 1 = e X b + ν T σ T e b + ν T σ T b + ν T B σ ln σ T = S + r q 2 T = ln S B + λ 1 σ T σ T σ T = x σ T e = e = y σ T Thus: = X e x σ T B S y σ T 32

39 5 General description of corridor products 5.1 General characteristics of corridor products Corridor products 11 are structured products which offer enhanced yields to investors who believe that the underlying will stay within a certain range for a predefined time period. These customized products are particularly structured for those investors who view a different path for a certain rate than that given by the forward rate curve. Forward curves have not historically proven to be good predictors of rates, and this offers opportunities to exploit the arbitrage. (Navatte and Quittard-Pinon, 1999) This kind of products offers the investor the possibility of capturing returns higher than the money market interest rate. The interest rate is paid when the structure matures, but it is accrued at certain dates according to the fixing schedule, whenever the reference rate (i.e. the underlying asset) is situated within the bandwidth the issuer and the investor have agreed upon before the start of the operation. For those fixings when the reference rate is outside the range, no interest is paid or nominal amounts are paid (or sometimes a lower than the market interest rate is offered as compensation), which are traditionally called rebates. For these structures, the capital is generally guaranteed. (Knop, 2002) Types of corridors 12 In the same manner as with options and other structured products, corridor products can be characterized in relation to their European or American feature. European style corridors have the resurrecting feature, that is to say, even if the underlying crosses the range limits until maturity, all the fixings 13 inside the range are considered for the payoff. For a fixing schedule 14 {,,, }, we can specify the payoff as follows: 11 Sometimes known as range accruals 12 WYSTUP, U FX Options and Structured Products, Chichester, John Wiley & Sons. 13 Fixings (also known as fixing rates) are official quotations for prices of assets such as gold, FX rates, interest rates, etc. published by sources such as the European Central Bank, Reuters, Bloomberg, etc. 33

40 1 { } where N represents the total number of fixings, L the lower barrier, H the higher barrier. American style corridors are non-resurrecting, i.e. fixings inside the range are accumulated only for the period up to the first breach of the range limits. In this case, the payoff can be defined as follows: 1 { } { } where is the stopping time, defined as: = inf{, } American style corridors with complete knock-out specify that the entire accumulated amount is lost if the exchange rate leaves the range. The payoff structure is defined as follows: 1 { } { } 5.2 Motivation for issuance of and investment in corridor products Corridor type products are known under different terminologies such as digital ranges, accrual notes and corridor notes. All these are structures used by those investors who have a strong perception of the way in which the market will evolve. The investor bets against the general expectations of the market evolution and, hence, he can position himself in such a way that he will benefit from the situation, by buying or selling against the market. (Knop, 2002) 14 The fixing schedule is established with a daily, weekly or monthly frequency. 34

41 The general risks involved with investing in such structures come from the evolution of interest rates and the volatility of the underlying. With FX corridor products, the major risk is given by the significant fluctuations in foreign exchange rates. Another aspect is given by the relative lack of liquidity, as a result of the customized nature, but this is generally an issue with all structured products. Investor perspective When buying a corridor type structure, the investor bets on a stable market, and the maximization of his profits will take place when the reference rate is mostly within the predefined range. Therefore, we could say that he is selling volatility and that the risk emerges in case of volatile situations. Issuer perspective By contrast with the investor, the issuer risk comes from stable situations, since his position is that of a speculative buyer in volatility (if un-hedged). He might end up paying above market interest rate if the reference rate will remain mostly within the range. 35

42 6 Miscellaneous corridor products Since the financial markets offer the possibility of trading for a wide variety of products that imitate the structure of corridors, it is impossible to include in a master thesis an exhaustive classification, description and valuation of them. Therefore, we chose to present in this chapter two main categories of the mentioned products, following to price them further on. Our choice was based on the possibility of decomposing their structure using options of the digital and barrier type presented in the previous chapters. 6.1 Digital ranges General description Digital ranges are products similar to straight bonds in the sense that they offer a preestablished fixed interest rate. However, they are different from straight bonds with respect to the fact that the total amount of coupons depends on the number of days (according to the fixing schedule) when a certain predefined reference rate (e.g. FX rate, according to the fixing source 16 ) stays within a pre-specified corridor, i.e. the corridor is resurrecting (European style corridor). At maturity, the fixings respecting the condition are counted and the coupon received will depend on the investor s level of success. Such a product guarantees the capital and also a worst case (minimum) coupon of at least 0%. Corridor notes are appealing to those investors who expect the reference rate to be mainly stable and to stay within a certain range for most of the days for which the contract is valid also known as corridor notes (CHRISTL, J Financial Instruments: Structured Products Handbook. Oesterreichische Nationalbank.) or corridor deposits (WYSTUP, U FX Options and Structured Products, Chichester, John Wiley & Sons.). We will be using the digital range term, since, as it will be seen further on, in order to evaluate these products, digital options will be used. 16 Usually the ECB or FED. 17 CHRISTL, J Financial Instruments: Structured Products Handbook. Oesterreichische Nationalbank. pg 40 36

43 Table 2: Example of a FX digital range Maturity Nominal amount 18 EUR 1 Spot rate Upper limit Lower limit Corridor style Money market interest rate Minimum interest rate Maximum interest rate Reference rate Total number of fixings N 91 Fixing source Coupon date March 1, 2010 to May 31, 2010 (91 days) EUR/USD EUR/USD EUR/USD Resurrecting 0.655% (actual number of days)/ % (actual number of days)/ % (actual number of days)/360 EUR/USD FX rate ECB At maturity Note: The minimum interest rate is paid in any case. The extra coupon paid at maturity in EUR is 1,4% 0,3% 91360, where is the number of fixings within the range. Source: own contribution. Advantages Possible higher coupon than the market Guaranteed capital and known up-front worst case coupon Fixings outside the corridor do not lead to a worst case coupon Disadvantages Possible lower coupon than the market They are expensive due to their resurrecting feature 18 For ease in further calculations, we assume that the notional is worth EUR 1 in our examples, even though in the real world these products have significantly higher notional. 37

44 6.1.2 Replication 19 Digital ranges can be decomposed into the redemption amount and the coupons. While replicating the redemption amount is straightforward (using a zero coupon bond), the replication of coupons is more complicated (especially if there is a non-zero rebate). At any time within the fixing schedule, we need to check if the reference rate is within the boundaries of the corridor. Furthermore, we need to take into consideration the coupon payment schedule, as the fixings within the range accrue for payment at the next coupon date (we will consider here that there is only one coupon date, at maturity). Therefore, the algorithm of the process can be described as: h,, h =, = where is the period for which the digital range is issued 20. The payoff structure of a digital range for one of the dates corresponding to the fixing schedule can be depicted as in Figure 3 below. Figure 3: Digital range - payoff corresponding to a fixing date Source: own contribution. 19 This section follows the ideas of CHRISTL, J Financial Instruments: Structured Products Handbook. Oesterreichische Nationalbank. and WYSTUP, U FX Options and Structured Products, Chichester, John Wiley & Sons. 20 For the maturity dates of each of the digital options, we will use. 38

45 The payoff profile for a digital range corresponding to all the dates in the fixing schedule can be built in different ways: a long position in a series of digital calls ( =, = ) and a short position in a series of digital calls =, = ) plus a long position in a series of digital puts ( =, = ) and a long position in a series of digital calls ( =, = ) for every day when the interest accrues (according to the fixing schedule) a long position on a bond ( = ), a long position in a series of digital calls ( =, = ) plus a short position in a series of digital calls ( =, = ) a long position on a bond ( = ), a short position in a series of digital puts ( =, = ) and a short position in a series of digital calls ( =, = ) It is important to mention that the digital options used in the replication of digital ranges pay off at the coupon dates, not at the exercise dates. This creates the need for an adjustment in the valuation process, as one needs to take into account the time value of the individual payoffs. We will return to this aspect in the part dedicated to the valuation of digital ranges. The pricing of the structure carried on in the next sub-chapter will use the decomposition made under the first bullet, since the authors of this thesis found it to be easier to work with, from an analytical point of view. All these being said we can now decompose the product from Table 2 as follows: 39

46 + Table 2 = = , = 1,4% = , = 1,4% = , = 0,3% = , = 0,3% Valuation of digital ranges For pricing the structure of digital ranges, we will use the results obtained in chapter 2, i.e. the valuation formulae for cash-or-nothing digital options at time zero: = d = d where ln K d = + r q σ 2 T σ T However, we will adjust these values, as suggested by Knop (2002, pg. 63) due to the fact that the payoff of the digital options does not convert into cash at the options expiration date, the settlement taking place at the structure s maturity date. Therefore, we will multiply the value of each option with, where is a suitable forward interest rate applicable to the period elapsed between the digital options maturity dates ( and the structure s expiry date (. Due to the fact that we employ digital calls and puts with two different strike levels, we will use the implied volatilities corresponding to vanilla options struck at the lower limit, 40

47 respectively at the higher limit of the corridor. Furthermore, for simplicity reasons, we assume a constant volatility term structure. The value of digital ranges at time = 0 can be specified as follows: = + =, = =, = + =, = + =, = = + ln S L + r q σ 2 σ ln S H + r q σ 2 σ + ln S L + r q σ 2 σ + ln S H + r q σ 2 σ 41

48 = + + With d = d = The more general formula for the value of a digital range at an intermediate time can be written by accounting differently for the digital options before and after time, in the following manner: =

49 = + +,, +,, With d, = d, = d, = d, = Where is the index for an intermediate day between the inception and the maturity of the structure, is the spot rate of the underlying at time, represents the forward rate applicable for the period between and, represents the forward rate applicable for the period between and. is the structure s inception date. For our example will always be set to zero. An observation needs to be done here. As one can see, we use the risk-free rate (corresponding to the structure s maturity) in order to discount the value of our product to time zero. The situation changes when we perform the valuation of our structure at an intermediate time, as we will need to use another discount rate, namely, in order to adjust for the time differential between the structure maturity and the intermediate time. For the hypothetical example illustrated in Table 2, using market data for the forward rates, risk-free rate, foreign risk-free rate and implied future volatilities, the value of our digital range as at time should have been as follows: 43

50 Table 3: Valuation at time zero for the digital range presented in Table 2 Notional 1.00 Asset price ( S ) Time to maturity in days (T) r min 0.30% r max 1.40% Running time 0.00 Risk-free rate ( r ) 0.655% Cost of carry ( cc ) 0.525% Volatility (K=L) % Volatility (K=H) % L H N Value Note: Computations done in MS Excel (Worksheet Digital range, File Thesis.xlsx, Enclosed CD). Source: own contribution. Furthermore, the dependency of the digital range value on the time to maturity and the underlying price can be observed in Figure 4. Figure 4: Digital range value evolution across the product s lifetime Note: Graph realized in MS Excel (Worksheet Digital range, File Thesis.xlsx, Enclosed CD). Source: own contribution. As it can be seen, through its lifetime, the highest value for our product is reached when the underlying trades around the initial spot price. As the spot leans towards any of the barriers, the structure starts declining in value, as the chances for leaving the range increase. 44

51 Relationship between the maximum and the minimum interest rates for a fixed price of the digital range: The relationship is shown with consideration to = 0, taking into account that the following condition needs to be respected: 0 < < <. = d + d / d d Figure 5: Relationship between the maximum and minimum interest rates for a digital range Note: The values are calculated with respect to the fixed price taken from Table 3; please see computations done in MS Excel (Worksheet Digital range, File Thesis.xlsx, Enclosed CD). Source: own contribution. The above figure shows how the absolute level of the maximum interest rate varies in relationship to a chosen guaranteed minimum interest rate, given that the price of the digital range is fixed. This provides the possibility of adjusting the payoff of the product according to the investor s risk profile. 45

52 6.2 Barrier ranges General description Also known as range protected deposits (or tunnel deposits), barrier ranges are products similar to straight bonds in the sense that they offer a pre-established fixed interest rate. However, they are different from straight bonds with respect to the fact that the total amount to be paid as coupon depends on the underlying staying within a pre-specified corridor. If the limits of the corridor are breached, a lower interest is paid (i.e. lower than the one which would apply if the underlying price would never leave the corridor during the coupon period). In comparison to digital ranges, where the interest accrues for those fixings inside the corridor irrespective of the corridor being breached, barrier ranges only pay off the above market interest if no trigger event occurs up to maturity, and the below market interest rate (applied for the whole life of the product) otherwise. Barrier ranges are appealing to those investors who expect a specific reference rate to be mainly stable and to stay within a certain range throughout the period for which the contract is valid. There is a wide array of possibilities to design products in the barrier range category. One could come up with a product which knocks out/in only if both barriers are breached or the payoff structure could be tailored such as to offer a best case interest rate if no trigger event occurs, a lower interest rate if either of the barriers is touched and sometimes even a worst case interest rate if both barriers are crossed during the lifetime of the instrument. The classical example to be found in literature is that of a product paying two different interest rates: the first is an enhanced one (the one greater than the money market interest rate) which can be captured if the barriers are never crossed, while the other one is a lower than the market interest rate to be paid if the corridor is breached. This classical barrier range structure can be decomposed with the use of double binary barrier options. Hui (1996) prices one such option (up-and-down out binary option) which pays a fixed cash 21 As presented in CHRISTL, J Financial Instruments: Structured Products Handbook. Oesterreichische Nationalbank. 46

53 amount if the barriers are not crossed and zero otherwise. According to Hui (1996), double barrier binary options are not combinations of single-barrier, binary barrier or digital options. Consequently, the pricing for the classic barrier range cannot be done by employing combinations of simple options and therefore we chose not to deal with it, since, as specified in the introduction of this paper, the main focus is on using the mentioned basic types of options in analyzing corridor products. Instead, we will analyze a similar product with a slightly different payoff structure, as explained below. Our example (please see Table 4) is a case of a barrier range with three possible payoffs. The investor receives a higher than the market interest rate provided that the reference rate remains within the range limits up to maturity. In case one of the barriers is breached at some point <, a lower than the market return is received at maturity. The worst case scenario is when both barriers are crossed, which will result in an even lower return. Table 4: Example of a FX barrier range Maturity March 1, 2010 to May 31, 2010 (91 days) Nominal amount EUR 1 Spot rate EUR/USD Upper limit EUR/USD Lower limit EUR/USD Coupons 1% if L <= <= H throughout the lifetime of the coupon 0.55% if one of the barriers is touched 0.1% if both barriers are touched 22 Payment at coupon date (at maturity) Market interest rate 0.655% Reference rate EUR/USD FX rate Note: L lower limit; H upper limit. Source: Own contribution. 22 For this product, the worst case interest rate must satisfy the restriction: =. This restriction is of high importance for pricing the product, as it will be seen in the next subchapter. 47

54 Advantages Possible higher coupon than the market Guaranteed capital and known up-front worst case coupon Disadvantages Possible lower coupon than the market High volatility of the underlying increases the risk of ending up with the worst case coupon Replication of barrier ranges In order to valuate this product, we will decompose it into the redemption amount and the coupon. The redemption amount will be priced as a zero-coupon bond, while the coupon will be replicated by a portfolio consisting from a zero-coupon bond with face value of, a long position on a down-and-out cash-or-nothing binary barrier option (with barrier L) and a short position on an up-and-in cash-or-nothing binary barrier option (with barrier H). The logic behind the replication of the coupon can be better understood with the help of the following table. Table 5: Replicating portfolio for the coupon part of a barrier range Payoff if no trigger event occurs Payoff if only the upper barrier is touched Payoff if only the lower barrier is touched Payoff if both barriers are crossed Bond DO binary 0 0 UI binary 0 0 Total payoff = 2 Source: own contribution. 48

55 Therefore, the replicating portfolio will be created as follows: a long position on a bond ( = ), a long position on a bond ( = ), a long position on a down-and-out cashor-nothing at-expiry binary barrier option ( =, = ) and a short position on an up-and-in cash-or-nothing at expiry binary barrier option ( =, = ) The replicating portfolio corresponding to Table 4 is: + Table 4: Example of a FX barrier rangetable 4 = = 0.55% = , = 1% 0.55% = , = 1% 0.55% Valuation of barrier ranges In order to price the structure of barrier ranges, we will use the results obtained in subchapter 4.3, i.e. the valuation formulae for standard binary barrier options, with the mention that the pricing will be done at time and that we will use one implied volatility for the option struck at and another implied volatility for the option struck at. Therefore, our formulae for the binary barriers become: 49

56 = X e x σ T t X e B S y σ T t = X e x σ T t + X e B S y + σ T t Where x = + σ x = + σ y = + σ y = + σ = = The value of barrier ranges at time can be expressed as follows: = + + =, = =, = = = e x σ T t L y σ T t x σ T t H y + σ T t 50

57 For the hypothetical example illustrated in Table 4, using market data for risk-free rate, foreign risk-free rate and implied future volatilities, the value of our barrier range as at time should have been as follows. Table 6: Valuation at time zero for the barrier range presented in Table 4 Notional 1.00 Asset price ( S ) r min 0.55% r max 1.00% Time to maturity in days ( T ) Risk-free rate ( r ) 0.655% Cost of carry ( cc ) 0.525% Volatility (K=L) % Volatility (K=H) % L H Value Note: Computations done in MS Excel (Worksheet Barrier range, File Thesis.xlsx, Enclosed CD). Source: own contribution. Furthermore, the dependency of the barrier range value on the time to maturity and the underlying price can be observed in Figure 6. Figure 6: Barrier range value evolution across the product s lifetime Note: Graph realized in MS Excel (Worksheet Barrier range, File Thesis.xlsx, Enclosed CD). Source: own contribution. As one would expect, the barrier range exhibits the highest prices when the underlying takes values within the range. 51

58 Relationship between the highest, the lowest and the worst interest rates for a fixed price of a barrier range The relationship is shown with consideration to = 0, taking into account that the following conditions must be respected: 0 < < < < and = 2 = { 1 + }/ e x σ T L S y σ T x σ T H S y + σ T + Figure 7: Relationship between the maximum and minimum interest rates for a barrier range Note: The values are calculated with respect to the fixed price from Table 6; please see computations done in MS Excel (Worksheet Barrier range, File Thesis.xlsx, Enclosed CD). Source: own contribution. The table gives hypothetical alternatives for the interest rates that the structure can offer at a fixed price. In this particular situation we notice that the enhanced yield potential is limited as the two restrictions from above must be fulfilled. 52

59 7 Hedging aspects To structure and price exotic options is one thing, to hedge them is another, and often a more challenging and trickier one. (Zhang, 1998) The next part of this thesis will look at how hedging for the products presented above can be done. Basically, hedging supposes making an investment to reduce the risk of adverse price movements in an asset. The benchmarks for measuring risk in financial products are the Greeks, where every letter measures a different dimension of the risk involved by a position in a certain option. In order to transform the risks from investing in options into acceptable risks, one must manage the Greeks. There are different hedging strategies one can employ. Among these, delta hedging is with no doubt one of the building blocks of the derivatives theory. The procedure exploits the perfect correlation between the changes in the price of the underlying and the price of the option. The idea behind delta hedging is buying/selling the option and selling/buying a quantity Δ of the underlying, in order to permanently have a delta-neutral position. The rebalancing of the portfolio should be done on a continuous basis, which is why deltahedging is a dynamic hedging strategy (Wilmott, 2006, pg. 122). Delta hedging may work successfully in liquid markets, with considerably low transaction costs, in which case the Black-Scholes assumption of continuous hedging can be considered to hold. However, issues arise in those illiquid markets where frequent rebalancing of the portfolio may imply high transaction costs and therefore it could only be done at discrete times. Furthermore, some contracts have a delta that becomes extremely large around certain asset values or for certain time moments, phenomenon which requires buying or selling (prohibitively) large quantities of assets in order to delta hedge. In these cases, this hedging strategy could prove to do more harm than if the position would be left un-hedged. This is the situation with binary options as well as barriers, since the delta becomes explosive as the underlying asset price approaches the barrier/strike price (Wilmott, 2006, pg. 122). 53

60 There are also issues worth mentioning with respect to other Greeks such as gamma or vega. Gamma is related to the magnitude of the transaction costs that the dynamic hedger might incur. In other words, a large gamma (as in the case of corridor products, please see the graphs in the next section) suggests that the delta is very sensitive to small changes in the underlying price which causes a need to trade more often and, therefore, significantly higher transaction costs are involved. Furthermore, a position in the underlying has zero vega, while the options to be hedged do not. For an option whose vega is significantly high, small changes in volatility will have a great impact on its value but no impact on the replicating portfolio s value, which will increase the hedging errors. An alternative to dynamic hedging is given by static hedging procedures, which are employed at the beginning of the contract and left until expiry. Such static hedges usually suppose creating a portfolio of simple European options (such as plain vanilla calls and puts), which must imitate the payoff of the portfolio to be hedged (Andersen et al., 2002). In other words, we simply buy puts and calls at time 0 such that the pay-off function is matched at expiry (Poulsen, 2006). Summarizing, contracts such as digitals, barriers, corridors and other path-dependent products are difficult to dynamically hedge, either because of transaction costs in the underlying or because of very high values for the Greeks. Therefore, a static hedging strategy will also be considered further on in this thesis. However, our analysis will first take a look at dynamic hedging issues with regard to our two examples of corridor products in order to show the limitations of employing such a strategy and the issues in which static hedging provides valuable insight. 54

61 7.1 Dynamic hedging approach Digital ranges Remember the following notations: d, = d, = d, = d, = Using the formula for the price of a digital range, we derive the sensitivity of the structure with regard to different parameters, i.e. we compute the formulae for delta, gamma, vega, rho and theta. Further on, using VBA coding under Excel (please see Appendix F), graphs have been obtained which plot the Greeks and their dependency on different spot prices and time to maturity. Delta Δ = = 1 1 1,, 2 Figure 8 indicates that delta takes extreme values near the barrier levels early in the structure s life and requires taking large positions in the underlying. As the structure approaches its expiration, less digital spreads need to be hedged and the absolute value of delta decreases. 55

62 Figure 8: Delta for a digital range Note: Graph realized in MS Excel (Worksheet Digital range, File Thesis.xlsx, Enclosed CD). Source: own contribution. Gamma Γ = Δ = ,, 1,,,, 2 The structure s gamma (please see Figure 9) exhibits sharp changes in magnitude and sign close to the barriers, which indicates that the structure needs to be frequently re-hedged in order to maintain delta neutrality. 56

63 Figure 9: Gamma for a digital range: different perspectives Note: Graphs realized in MS Excel (Worksheet Digital range, File Thesis.xlsx, Enclosed CD). Source: own contribution. Vega As shown in Appendix C, Vega with respect to both σ and σ is the same as the summation of Vega with respect to σ and Vega with respect to both σ. = + 57

64 = σ 1 2 =, + = σ 1 2, +, =, = 1, + 2, +,, We point out that the change in the implied volatilities affects the value of the structure at time only with respect to the digital options which did not expire up to then. Figure 10: Vega for a digital range Note: Graph realized in MS Excel (Worksheet Digital range, File Thesis.xlsx, Enclosed CD). Source: own contribution. 58

65 The graph indicates how implied volatility affects the structure s value. And, since the digital range reaches its peak values along, then the largest drop in value with respect to an increase in volatility occurs when the underlying trades near the initial spot value. A higher volatility increases the chances that the underlying crosses the corridor limits. However, the above representation is subject to criticism for two major reasons. First of all Wilmott (2006, pg. 128) points out the fact that Vega is meaningful only for options which have single-signed gammas (e.g. calls and puts), and this is not our case. Thus, the actual volatility risk is not accurately measured by Vega. The other issue arises from the fact that we assumed the same σ and σ for options with different maturities. Taleb (1997) indicates that this approach underestimates risk due to the fact that the Vega of a portfolio of options with different maturities must be weighted, as short term volatilities do not have the same volatility of volatility as long term volatilities. Rho Since the discount rate for digital ranges at an intermediate time is the forward rate corresponding to the period and since we are interested in the evolution of the structure s sensitivity across its lifetime with respect to changes in the interest rate, Rho will be derived relatively to. We remind the reader that is the forward interest rate applicable to the period between and. Therefore, we have that: h = = The graph on the next page indicates that changes in the discount rate will induce linear changes in the product s value. Taleb (1997, pg. 113) criticizes the analysis of Rho due to the fact that it is based on the unrealistic assumption that there is a parallel shift in the term structure at all times. 59

66 Figure 11: Rho for a digital range Note: Graph realized in MS Excel (Worksheet Digital range, File Thesis.xlsx, Enclosed CD). Source: own contribution. Theta Theta measures the product s sensitivity to a small change in time to maturity. The difference between Theta and other Greeks is that the underlying variable (time to maturity) can move in only one direction, i.e. decrease. Thus, practitioners generally compute Theta as minus the partial derivative with respect to time to maturity. (Haug, 2007, pg. 64) h = = Digital range + 1 e, ln σ σ 2 e, σ 2 2π ln σ 2 60

67 Figure 12: Theta for a digital range Note: Graph realized in MS Excel (Worksheet Digital range, File Thesis.xlsx, Enclosed CD). Source: own contribution. The above graph indicates the time decay of the portfolio. When the underlying is inside the range, the positive Theta starts to decrease as maturity approaches. This happens because we are dealing with an accrual type product which locks in the interest accumulated in the past. Hence, all other variables hold constant, the potential gains in the structure s value become smaller with the passage of time. One can observe that the evolution of Theta is closely related to the evolution of Gamma (negative relationship), as Taleb (1997) points out Barrier ranges For this section, remember that: x = + σ x = + σ y = + σ y = + σ 61

68 = = Delta Δ = e σ + L = e 1 2π T t e σ e H σ e L y σ T t H σ y + σ T t Figure 13: Delta for a barrier range Note: Graph realized in MS Excel (Worksheet Barrier range, File Thesis.xlsx, Enclosed CD). Source: own contribution. 62

69 In the case of barrier ranges, Delta exhibits explosive behavior when the underlying approaches the barrier levels and the product is near its expiration date. Delta hedging this product would require trading an amount of underlying which would be significantly higher than the maximum payoff that the option could bring to its investor at expiry. In other words, delta hedging would cause greater damage than no hedging at all, because one would be hedging more than one could lose. Gamma Γ = Δ = e x 2π σ T t e x L y σ T t 1 2π σ T t π σ T t π σ T t e H y + σ T t y σ T t L e y σ T t H e Figure 14: Gamma for a barrier range Note: Graph realized in MS Excel (Worksheet Barrier range, File Thesis.xlsx, Enclosed CD). Source: own contribution. 63

70 The structure s Gamma displays violent swings in magnitude near the barriers close to expiry, suggesting that transaction costs will be significant if one would try to continuously maintain delta neutrality. Vega We remember that = = + = = + As shown in Appendix C, Vega with respect to both σ and σ is the same as the summation of Vega with respect to σ and Vega with respect to both σ. = σ 1 2π e ln σ = + = e σ + 4 L ln L σ y σ T t L 1 2π ln e σ 2 σ 2 = σ 1 2π e ln σ = e σ H ln H σ y + σ T t H 1 2π ln e σ + σ

71 Figure 15: Vega for a barrier range Note: Graph realized in MS Excel (Worksheet Barrier range, File Thesis.xlsx, Enclosed CD). Source: own contribution. The graph suggests that, close to the product s expiration, an increase in volatility most significantly affects the product s value when the spot trades near the barriers. Rho h = = + e x σ T + 2 σ ln L L y σ T t + T t e L 2π σ + x σ T t + 2 σ ln H H y + σ T t T t 2π σ e H e e 65

72 Figure 16: Rho for a barrier range Note: Graph realized in MS Excel (Worksheet Barrier range, File Thesis.xlsx, Enclosed CD). Source: own contribution. We observe that Rho exhibits similar behavior to the case of digital ranges. The change in interest rates would cause a linear, negative decrease in the structure s value. These results can be questioned due to the fact that the calculation of Rho assumes parallel shifts in the term structure of interest rates, which is not the case in practice. Theta Figure 17: Theta for a barrier range Note: Graph realized in MS Excel (Worksheet Barrier range, File Thesis.xlsx, Enclosed CD). Source: own contribution. 66

73 h = = Barrier rang e 1 2π 2 T t e ln L σ T t + σ 1 L e ln L σ T t + σ 1 e ln H σ T t + σ 1 H e ln H σ T t σ 1 Figure 17 illustrates the time decay in the case of Barrier Ranges. Theta experiences peaks near the barriers close to expiration. This means that if the underlying and volatility remain unchanged and the barrier has not yet been breached, the passage of time works in the investors favor and the portfolio s value increases. The Theta drops sharply at the barrier level due to the fact that if either of the barriers is knocked out close to expiration, the accumulated time value of the potential payoff would disappear. Once again, we observe the close (inverse) relationship between Theta and Gamma. 67

74 Observations with respect to the usage of Greeks As we have seen in this section, the Greeks behave in a strange way when it comes to structured products which have embedded binaries or binary barriers. Blindly applying textbook procedures when one is hedging could lead to a worse outcome than not doing anything at all. Furthermore, Taleb (1997) criticizes these measures due to the fact that they are built in caeteris paribus assumptions, ignoring the joint effect of different variables and parameters. There are also the liquidity issues which need to be considered, since the usage of Greeks requires frequent rebalancing of one s portfolio. Moreover, the transaction costs which are incurred by re-hedging on a frequent basis are not to be ignored. We conclude that in the case of digital and barrier ranges a second hedging approach is worth being investigated. The next part of the paper will therefore look at a method known as static hedging. 7.2 Static hedging Static hedging of digitals Hedging binary options raises issues due to the fact that the payoff is discontinuous around the barrier: small changes in the underlying cause sharp changes in the option s value. For example, the simplest cash-or-nothing digital option exhibits zero delta almost for any possible underlying price. Nevertheless, the same delta becomes significantly high as the underlying approaches the barrier. Delta for digital options may suffer extreme changes when the underlying trades close to the barrier or when the option is close to maturity. Due to the discontinuities in the payoff of these options, dynamic hedging is difficult to realize. Therefore, in practice, static hedging is employed, by using spread strategies. In this manner, the payoff of a binary option can be approximately replicated with European options. 68

75 Nelken (2000) explains the replication procedure. Assuming a digital call paying $1 if the underlying is above $100 at maturity, we compare it with a portfolio consisting of a long position in a vanilla call ( = 100) and a short position in a call ( = 101). As Figure 18 shows, the replicating portfolio pays $0 below $100 and $1 above $101, but between $100 and $101, the spread is only an approximate hedge for the binary option. It is to be observed that the narrower the spread gets, the more exact the hedge becomes: e.g., a spread built from ten long calls ( =$100) and ten short calls ( =$100.10) is a better hedge for the above digital call option except for the region between $100 and $ Generalizing the procedure, the digital option struck at and paying $1 can be hedged by going long on European options struck at and short on European options struck at +. Consequently, we can draw the conclusion that binary options can be synthesized using an infinite number of vertical spreads of vanillas. = lim + 1 If a more symmetrical hedge is wanted and also if using put options, the formula can be written as in (Bowie and Carr, 1994): = lim Figure 18: Approximate replication of a digital option with a vertical spread Source: Nelken, 2000, pg

76 In the case of digital options, the static replication method as compared to dynamic hedging has the advantage of not having to rebalance the weights of the portfolio and therefore no additional rebalancing costs are incurred. However, static hedging of digitals has its limitation given by the applicability of the previous two formulae. In order to be valid from a financial point of view, two assumptions have to be made which would require a perfectly liquid market for vanilla options as a setting (which is generally not the case in practice). The first assumption requires the availability of an infinite number of plain vanillas with all strikes and maturities. Obviously, if an approximation for a static hedge is made with a limited number of vanillas, this could lead to errors similar to those generated from discrete rebalancing in dynamic hedging. Secondly, vanilla options with strikes around the strike price of the corresponding cash-or-nothing call/put (slightly higher and lower) should be available at the same time; this is generally not the case in organized exchanges (Zhang, 1998). As a result, formulas ( 23 ) and ( 24 ) are important from an intuitive mathematical perspective, rather than from a practical one. However, in order to remedy the impracticability of these mathematical equations, a Richardson extrapolation technique can be applied as in (Carr et al., 1998). The technique can be used to guess the value for an infinitesimal parameter, given a set of approximations indexed by a step size. However, the mentioned article presented an example which does not refer to FX options, and therefore, the magnitude of the numbers used is different. We considered that in order to obtain reliable results for our example, a re-scaling operation needed to be done. The formulae below include these adjustments. In order to see the original formulae and the logic behind the rescaling procedure please see Appendix D. Let be the value of vertical call spreads with strikes and +, with = 1, 2, 3. = For a five-decimal-place accuracy, the following three-point Richardson extrapolation may be used:

77 Replacing the values of for every = 1,2,3, an approximate 23 value for the binary call is obtained. Given the fact that a static hedging strategy still leaves place for errors, one might wonder if this approach outperforms the dynamic one. We find support in favor of the former in Tompkins (2002), who compares the hedge performance of static versus dynamic hedging for exotic options using Monte Carlo simulations. According to his findings, for digital cash-or-nothing options, the static hedging approach is preferred, due to the fact that it saves transaction costs and the hedging performance is better under stochastic volatility conditions Static hedging of binary barrier options We will now focus on static hedging the two types of binary barrier options we have used in the decomposition of the barrier range. We will derive the replication portfolios for the above mentioned options in the context of a general non-zero drift assumption. The ideas we will be using further on lie in the articles of Carr and Chou (1997 a), Carr and Chou (1997 b), Carr and Picron (1999) and Patel (2005). However, in the mentioned paper, the authors apply their method to replicating barrier options, while we will focus on replicating binary barrier 24 options. Although the two types of products are somehow similar, there are certain aspects one must take into consideration as it will be seen further on. The method proposed by Carr and Chou (1997 a, 1997 b) is set in the Black-Scholes framework and consists in replicating barrier options by taking positions in plain vanilla puts and calls with different strikes, but with the same maturity as the barrier option. The portfolio of vanillas to be built should match the adjusted payoff function of the barrier. 23 The approximation deteriorates near expiration when prices are near the strike. 24 More precisely, we will look at the O and. 71

78 Carr and Chou (1997 b) define the adjusted payoff function as the payoff function of a European security 25 that can be used to replicate a barrier option. We extend this definition to include binary barriers as well. Using the payoff function of a European security to replicate the payoff for the binary barrier options is useful given that, in theory, any European security can be replicated with a combination of puts, calls, forwards and bonds. Nonetheless an exact static hedge for a European security with a non-linear payoff can be realized only by employing an infinite number of positions in vanilla instruments. Consequently, in practice, one can only approximate the replication portfolio for such a European security and, therefore, the replication portfolio for a binary barrier. Thus, we will only be able to find approximate replication portfolios for the barrier range, since exact static hedges through this method cannot be obtained. Assuming only that there is no arbitrage, the markets are frictionless and that the payoff function is twice differentiable, Carr and Picron (1999) show that the payoff of a European security can be interpreted as the payoff arising from a static position in bonds, forward contracts and vanilla options as below: = + +, 0 +, 0 Also, if by we denote the time value of the same payoff, then: =, +, +,, +,, 25 That is, a security which entitles the holder to a payoff at the maturity, T. 72

79 Where, is the time value of a zero coupon bond maturing at,,,,,, are the time values of puts and calls struck at and maturing at, is the delivery price of the forward contract. The lemma below is used by Carr and Chou (1997 b) in determining the equivalent European payoff for barrier options. We will adjust it so that it becomes a tool to be used in replicating binary barriers with vanilla instruments. The original lemma follows: Lemma 6: by the payoffs: Considering > 0 and the two European securities with maturity defined =,, 0, h =,, 0, h Where = 1 and,, are the usual notations for interest rate, dividend yield and volatility, respectively. Then for any 0,, the values of the two securities from above are equal when =. We will use the previous lemma to show that a a can each be replicated with a European security with a specifically chosen payoff function 26. or Binary down-and-out cash-or-nothing Consider the following payoff functions for two European securities maturing at : =,, 0, h 26 In the case of binary barrier options, we choose the function =, where is a constant. 73

80 =, 0, 0, h We will prove that the combined value of these two securities equals the value of a downand-out cash-or-nothing binary with payoff X and barrier level set at <. In order to do that, we will consider the two possible scenarios for the price of the underlying during the life of the security. I) The underlying price never crosses the barrier level over the lifetime of the II) down-and-out binary. The underlying price does cross the barrier level at some point during the lifetime of the down-and-out binary. For scenario I (thus for, ), the combined portfolio payoff at maturity is given by = + = + 0 =, which equals the payoff of the down-and-out binary in the given scenario. For scenario II, we make use of the above presented lemma (for = = ), which states that at the time when the underlying price hits the barrier, the value of the European security with payoff is exactly the same as the value of a European security with payoff function: =,, 0, h This makes possible the usage of the following strategy at the moment when the barrier is touched: selling the payoff and buying the payoff at zero cost. This strategy is equivalent to saying that when the underlying price hits the barrier, the portfolio should be liquidated. In this second scenario, the combined portfolio payoff at maturity becomes zero, which is the same with the payoff of a down-and-out binary when the barrier had been reached previous to (or at) maturity: = + =,, 0, h = 0 74

81 Given the above arguments we can now conclude that the payoff of the down-and-out binary barrier can be replicated by using the adjusted payoff function of a European security in the following manner:,, = + =, 0, The next step is replicating this adjusted payoff function with bonds and vanilla European puts and calls, process which is done in two parts. First, we replicate the adjusted payoff above the barrier and then the one below the barrier. While the former is straightforward, the latter poses some problems due to the non-linearity of the payoff function in this region (please see Appendix E for the graphical representation of the adjusted payoff function). The adjusted payoff above the barrier for a down-and-out binary option can be simply replicated by a ZCB with face value. The replication procedure for the adjusted payoff below the barrier for a down-and-out binary option is as follows: we first assume one value below the barrier for the final underlying price, let that be at point. Our portfolio needs a position in a vanilla option such that the following observations are accounted for: the payoff of this vanilla option together with the payoff of the previously purchased ZCB must fit the payoff of the adjusted portfolio below the barrier, for a final underlying price; the new vanilla option should not alter the already matched payoff above the barrier: this can be solved by adding a put option to the portfolio with a strike below the barrier ( < ); The strike of the put should be greater than ( > ); Let be the required position in the mentioned put option and the maximum between and 0. Then, the following equation needs to be solved in order to determine : + =. 75

82 With these being said, we now have a portfolio of a ZCB and vanilla European options, which replicates the adjusted payoff everywhere above the barrier and at a single final underlying price below the barrier. A common sense observation is that the approximation from above can be improved by increasing the number of points at which our portfolio payoff is matched with the adjusted payoff 27. Continuing in the same manner for a value for the final underlying price (Figure 19), with < <, a second equation emerges: + + = Figure 19: Matching the adjusted payoff function for a down-and-out binary in two points under the barrier Note: The graph illustrates the approximation of the adjusted payoff function in two points below the barrier using put options (intuitively, we drew them as if being short positions; however, the -s found through solving the matrix from below will give the required positions). Source: own contribution. Increasing the number of points at which the adjusted payoff will be matched will give better results. For points matched, the obtained portfolio can be synthesized in the following matrix notation: = 27 As the number of points at which the adjusted payoff is matched goes to infinity, the approximation converges to the adjusted payoff function. 76

83 Where < < < < < < <. Solving the set of equations from above yields the solutions for,,. Hence, our replicating portfolio for the adjusted payoff function is set up as follows: A long position in a zero coupon bond ( = ) put options struck at and maturity put options struck at and maturity put options struck at and maturity Binary up-and-in cash-or-nothing The adjusted payoff function for an up-and-in binary option can be found in the same way as for the down-and-out case. = +,, 0, 0, For simplicity reasons, we can first derive the adjusted payoff function for an up-and-out binary and then, taking into account that an + = ( = ), we can replicate the up&in binary by purchasing a ZCB and shorting the replicating portfolio for the up&out binary at expiry. The adjusted payoff function for an up-and-out binary is given by 28 : =,,, 0, 28 The formula can be demonstrated following the same steps as for the down-and-out binary option. 77

84 Figure 20: Matching the adjusted payoff function for an up-and-out binary in two points above the barrier Note: The graph illustrates the approximation of the adjusted payoff function in two points below the barrier using call options (intuitively, we drew them as if being short positions; however, the -s found through solving the matrix from below will give the required positions). Source: own contribution. The replication of the adjusted payoff of the up-and-out binary with vanilla calls will result in a matrix having the following shape: = Where < < < < < < <. Solving for the set of equations gives the positions (, = 1, ) in call options needed in order to replicate the adjusted payoff of the up&out binary option as follows: A long position in a zero coupon bond ( = ) call options struck at and maturity call options struck at and maturity call options struck at and maturity 78

85 Conversely, using in-out parity, the up-and-in binary option can be (approximately) replicated by the following portfolio: call options struck at and maturity call options struck at and maturity call options struck at and maturity To the authors knowledge, up to the present, there is no article which compares the hedge performance of static versus dynamic hedging for binary barrier options. Perhaps a future line of research should address this direction. Clearly, the static hedging approach has the advantage of lower transaction costs compared to dynamic hedging strategies, as the weights of the hedge portfolio are fixed and there is no need for rebalancing. However, it is obvious that there are shortcomings when it comes to statically hedge binary barrier options. The first argument is given by the fact that the adjusted payoff functions can only be approximately replicated in their non-linear regions, as the availability of an infinite number of vanilla options is hardly a valid assumption. Nevertheless, it would be interesting to investigate how many options with different strikes (and how large the positions held in these options) would be necessary in order to achieve a satisfactory convergence. If the hedging portfolio requires holding significant positions in vanilla options, the cost of setting it up would also be high and thus the argument of low transaction costs in favor of static hedging would reduce its importance. Moreover, the Carr and Chou approach is based on the Black-Scholes assumptions, hence it is model dependent. One should study if this approach yields better results for binary barriers than dynamic hedging provided that some of the Black-Scholes assumptions are relaxed. 79

86 7.2.3 Static hedging of digital ranges Since we have shown in previous chapters how the value of a digital range can be computed using digital calls and puts, this section will be built on the results obtained in We have shown that options can be synthesized using an infinite number of vertical spreads of vanillas, but, by employing a Richardson extrapolation technique, a finite number of vanillas can suffice in order to obtain an approximate value for the binary option Following this, our formula for a digital range can be rewritten so that it is a function of vanillas, as shown below. Remember that a static hedge strategy is employed once, at the beginning of the structures life. Thus, we obtain: Where and = And similar if L is replaced by H 80

87 Since the components of a digital range can only be approximately hedged, it is natural to imply that the static hedging of the structure is also subject to errors. However, we find that the static approach can be taken into consideration as a potential candidate for hedging digital ranges Static hedging of barrier ranges In chapter 6.2.2, we have shown how our example of a FX barrier range can be replicated using zero coupon bonds and binary barrier options. Since the zero coupon bond brings a known fixed amount, we are interested in hedging the option components of our structure. Hence, we need to hedge a long position in a down-and-out binary and a short position in an up-and-in binary. We have demonstrated how to hedge the two mentioned types of binary barrier options in chapter 7.2.2, using the concept of adjusted payoff function. Thus, our hedge for the long position in a =, = can be carried on by selling a portfolio containing: A ZCB ( = ) A combination of put options (, ), where and satisfy the conditions from chapter , given that: = Where = 1.,,, 0, The short position in the =, = can be hedged by buying a portfolio composed of: 81

88 A combination of call options (, ), where and satisfy the conditions from chapter , given that: Where = 1. =,,, 0, Once again (as in the case of digital ranges) we remind that static hedging only provides an approximate hedge portfolio for our product and that the issues which are specific to the static replication of binary barriers are also valid for the barrier range structure. 82

89 8 Conclusion and recommendations Corridor products are highly popular among practitioners and academia, since the wide variety of such instruments presents interesting features. They are particularly attractive to those investors who have strong views about the way in which the market will behave, since these investments provide opportunities to obtain enhanced returns. On one hand, the characteristics of such products definitely offer certain advantages, but on the other hand, one must be aware of the problems that these instruments raise, both from a pricing point of view, as well as from a hedging perspective. Being aware of the richness of these types of instruments and of the intricacies that lie behind the different categories, we chose to restrain our focus on only two types of corridors throughout this paper. As explained in the introduction, this was due to the authors interest in products which can be decomposed in digitals and binary barriers as well as due to space and time constraints. In this paper, we have employed an analytical approach for pricing corridor products under the Black-Scholes assumptions. Initially, the focus was placed on understanding the simpler components, namely binary options, barrier options and binary barrier options. The general characteristics of corridor products were presented, and the details of structuring products such as digital ranges and barrier ranges were thoroughly analyzed. Replication portfolios were built and closed-form valuation formulae were developed. The pricing of corridors is straightforward in the Black-Scholes framework, but it relies on strong assumptions, hence it is model-dependent. An interesting topic for future research would be the pricing of these instruments when some of the Black-Scholes assumptions are relaxed. Next, we addressed the issue of hedging corridor products. Due to the exotic options which are embedded in the structures, the hedge process is an important issue, as the risks are complex. Based on market data, simulations were performed in order to analyze the 83

90 behavior of Greeks. The findings suggest that delta-hedging the products raises difficulties due to discontinuities at the barrier levels. In the case of digital ranges, the Vega measure has its limitations due to the fact that we assumed a constant term structure of volatility. As Taleb (1997) points out, the implied volatility behaves differently for options with shorter maturity compared to options which have longer maturity, and this needs to be accounted for by weighting the Vegas corresponding to each maturity bucket. The static hedging approach was also discussed. Possible implementations of static hedges for the two types of corridor products were proposed, but, since numerical methods are beyond the scope of the thesis, the testing of the implementations effectiveness remains for future research. Advantages and shortcomings of static hedging were also presented. According to the literature, the main argument in favor of static hedging is represented by the fact that transaction costs are lower, but this is debatable since the initial costs of setting up the static hedge portfolio must also be considered. In addition, the availability of a large number of traded vanilla options is required and there is also a liquidity issue which needs to be taken into account. 84

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93 OVERVIEW OF THE ENCLOSED CD Master thesis.pdf Thesis.xlsx - Digital range - Barrier range - Vega Digital range - Vega Barrier range - St hedging 1 - St hedging 2 - Data - Diverse 87