Non-Linear Derivatives in Foreign Exchange Hedging

Transcription

1 Non-Linear Derivatives in Foreign Exchange Hedging Stefan D. Helgason Copenhagen Business School, 2013 CM Finance and Accounting Master s Thesis September 10, ,041 characters 72 pages Thesis Supervisor: Bent Jesper Christensen

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5 Contents 1 Introduction Introduction Hedging Pegged Foreign Exchange Rates Cost of Hedging Structure Method Theoretical Foundation Stochastic Calculus Theory Itô Calculus and Itô s Lemma Geometric Brownian Motion Variance Gamma Cholesky Decomposition Multifactor Correlation Structure Interest Rate Models Vasicek Hull and White Cox, Ingersoll and Ross The Affine Term Structure of Interest Rates Option Theory Risk-neutral Valuation and the Measure Q Black-Scholes Garman-Kohlhagen Data Description Case Investor The Danish Pension Market Case Portfolio v

6 vi CONTENTS Data Sample Market Imperfections Empirical imperfections - implied vs. observed deposit rates Cost of imperfections Hedging Floating Currencies Variance Gamma Options Computing Option Prices Estimating Option Delta The Model General Discussion The Algorithm Generating State Prices Determining Derivatives Prices and Rates Computing Portfolio Returns Results Hedging Pegged Currencies Market Implied Volatilities Simulating Pegged Currencies Mean-Reversion and Irregular Return Patterns The Model Dimension Reduction Foreign Exchange Forward Hedging Foreign Exchange Option Hedging Results Discussion 61 8 Conclusion 65 A Generating Gamma-Distributed Random Variates 73 B Projecting Summary Statistics 75 C Source Code 77 C.1 Identifying Optimal Hedge Ratio C.2 Computing V.G. Option Prices using Monte Carlo Integration

7 List of Figures 1.1 EURDKK foreign exchange rate evolution Danish Krone implied interest rate differential Geometric Brownian motion vs. Variance Gamma Cox, Ingersoll and Ross calibration curves Covered Interest Parity EURDKK forward point differential EURDKK forward implied rate differential Option pricing, V.G. and GBM Options vs. forwards in floating-rate foreign exchange hedging Option price vs. option delta EURDKK 3-month volatility smile Modified EURDKK 3-month volatility smile V.G. with non-constant skewness Options vs. forwards in fixed-rate foreign exchange hedging vii

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9 List of Tables 3.1 Summary statistics and calibration results Cox, Ingersoll and Ross calibration results Covariance Matrix Skewness Calibration Summary statistics and calibration results Reduced Covariance Matrix ix

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11 Chapter 1 Introduction 1.1 Introduction As an integral component in many institutional investors investment planning, capital budgeting and (as is our focus here) investment portfolio returns, foreign exchange hedging is a field of study that has been covered quite extensively in the financial literature (see for instance Stulz (1984) and Bailey, Ng, and Stulz (1992)). In such literature, it is often suggested that an optimal foreign exchange hedging policy exists that minimizes the ex-ante variance of the realized portfolio returns (however in general the principles from modern portfolio theory (Markowitz, 1959) applies to foreign exchange hedging just as they do for any investable security, so that an investor may chose his exposure based on his desired ex-ante positioning along the efficient frontier). The hedging policy often deemed optimal is termed the minimum-variance hedge ratio, and the solution to this hedging problem is often solved using a regression-based model, incorporating the variances and covariances of the hedging portfolio and the hedging currency. However, such regression-based models, although convenient as easily implemented, does suffer from being overly simplistic for a range of applications (for instance, they apply only for linear derivatives and assume that investors operate in a foreign exchange environment that is completely free-floating). As a result, some investors may need to turn to analytical frameworks in order to solve their hedging problems, however the generalized principles derived from the simpler regression-based models remain valid Hedging Pegged Foreign Exchange Rates Addressing the issue of hedging under non-free-floating (i.e. pegged) exchange rate regimes, the idea for this thesis was originally born from contemplating foreign 1

12 2 CHAPTER 1. INTRODUCTION exchange hedging for an investor whose liabilities or performance evaluation were DKK (Danish Krone) based, and thus wanted to hedge his foreign currency exposure back to DKK in an optimal manner. The DKK is pegged against the Euro under the ERMII (European Exchange Rate Mechanism II) agreement at a central rate of 7.46 with a nominal rate band of ± 2.25%, however as can be seen from Figure 1.1, the Danish central bank has implemented a de-facto much narrower band of less than 1%. When dealing with pegged exchange rates such as the EURDKK, because we re not dealing in markets that are allowed to float freely, the regression-based methods in large part fails to work. The primary reason for this is that the variance and covariance structure becomes relatively predictable, especially when the exchange rate is near the bounds of the peg. As a result, mechanical hedging using a regression-based model fails to capture the intricacies of the exchange rate behavior. 7.7 EURDKK Exchange Rate EURDKK Foreign exchange rate EURDKK ERMII central rate EURDKK ERMII bound Date Figure 1.1: EURDKK foreign exchange rate evolution since the Euro was introduced to financial markets on January 1, Note that the bounds of the peg is far from reached in the period, indicating that a much narrower policy band is implemented from policymakers Cost of Hedging When an investor contemplates the method by which he wishes to hedge his foreign exchange risk, the cost of doing so is critical. The regression-based models of determining optimal hedge ratios fail to account for the various costs associated

13 1.1. INTRODUCTION 3 with carrying out the hedge. Beyond the fees that market participants may pay their brokers for trading OTC or exchange-traded derivatives, the rates markets themselves may impose costs onto the investor that in an arbitrage-free world ought not happen. Taking again the regression-based models into consideration, they suggest hedging using linear derivatives, of which the most widely used is the foreign exchange forward. From international finance theory we know that such a forward can be fairly valued using the covered interest parity, stating that in an arbitragefree environment, the forward exchange rate at time t maturing at time T > t for the exchange rate between the base currency b and pricing currency p is a function of the current exchange (S b,p t ) rate and the risk-free rates in the base (rt,t b ) and the pricing (r p t,t ) currency respectively, i.e. (Sercu, 2009) t,t = S b,p 1 + r p t,t t. (1.1) 1 + rt,t b F b,p Rearranging, we obtain an expression for the pricing currency implied risk free rate, given that we can observe the value of F b,p t,t in the market, i.e. r p t,t = F b,p t,t (1 + rt,t b ) S b,p 1. (1.2) t Should the covered interest parity hold, we would expect that the implied riskfree rate r p t,t and the observed risk free rate r p t,t should not deviate much from one another (i.e. the implied rate differential should be close to zero). In reality, though, we ve observed significant rate differentials in among others the EURDKK forward markets since the advent of the financial crisis of the late 2000s. Figure 1.2 shows the 3-month implied interest rate differential for the pricing currency (DKK) for the EURDKK exchange rate using the respective currencies overnight index swaps (OIS) as risk free rates. The immediate consequence of this rate differential is that foreign exchange forward rates are skewed compared to their theoretical values. Specifically, a higher observed risk free rate (i.e. a lower implied rate) indicates that the numerator in Equation 1.1 is lower than would be in an arbitrage-free environment, this decreasing the value of the entire equation. This means that when investors hedge their foreign exchange exposure by shorting forwards, the lower rate means that the distance to the forward being in-the-money is greater than it would have been, thus imposing a direct cost to the hedger.

14 4 CHAPTER 1. INTRODUCTION EURDKK observed vs. implied risk free rate Percent Date Figure 1.2: Implied interest rate differential in the pricing currency for the EUR- DKK exchange rate. Positive (negative) number indicates higher (lower) observed risk free rate compared to implied rate. Data obtained from Bloomberg and Nordea Analytics software packages. At this point it should be noted that market standards are such that all hedging activities involving the DKK is channeled through the Euro. This means that no matter the currency to which a portfolio is exposed, in order to hedge to DKK the investor will need to hedge into Euros before hedging into DKK. In practice this double hedging may be carried out behind the scenes by the bank with which the forwards are traded, but the results (i.e. a more expensive forward rate) remain unchanged, regardless. In order to estimate the cost per each basis point of implied rate mispricing, we note that, in terms of the covered interest parity, the higher the number in Figure 1.2, the lower the yield on a domestic risk free placement (while the yield on the foreign risk free placement remains unchanged). The implied rate differential can therefore also be interpreted as a direct percentage measure of the extra cost to investors from forward mispricing on an annualized basis. To illustrate the magnitude of this mispricing, at around end-of-year 2012, the 3-month DKK OIS implied interest rate differential was around 20 bps. In the same period, the yield on a 3-month Italian government t-note traded at similar levels. Thus, in order to counteract the extra cost from forward mispricing, investors would have to add to their balance sheets securities with a similar credit quality as Italian government debt (Baa2/BBB).

15 1.2. STRUCTURE Structure The aim of this thesis is to address the problems outlined in the previous section. At its core, we aim to examine how investors may reduce the quite sizable cost that they must incur when hedging in the foreign exchange forward markets. To this end, we aim to build a framework that incorporates non-linear derivatives (foreign exchange options), enabling investors to analyze and optimize their foreign exchange hedging policies. The solving of this problem is structured around hypothetical case investor whose liabilities are DKK nominated and whose assets are a diversified portfolio of both domestic and international financial assets. Even so, the concepts derived and the framework built is generalizable to any investor facing similar market conditions as one whose liabilities are in DKK. Chapter 3 provides an in-depth description of our hypothetical investor and his portfolio construction. However before doing to, in Chapter 2 we introduce some theoretical concepts that the reader should familiarize himself with. In this chapter, we introduce the basic numerical elements of the framework derived in this thesis. Safe for the sections that treat the variance gamma (V.G.) stochastic process, most sections in this chapter should be a preexisting part of most readers frame of knowledge, and as as result we won t introduce the concepts beyond the extent to which they are used in this thesis. However to the end of full disclosure, a brief summary of each concept is supplied. In Chapter 4, we take a closer look at the dynamics that drive primarily the foreign exchange forward mispricing, in particular in the EURDKK market. We do so to motivate to the reader that the analysis of the DKK forward market is correct, and that the issue we examine is in fact a relevant one to investors. In addition hereto, we examine the extent to which this problem exists for other currencies so as to estimate the relative uniqueness of the situation. Although this chapter in large part builds upon what we learned in the previous section to this chapter, the particular placement of the chapter done so in the interest of continuity in relation to the following chapters. As the reader may recall, market practice in the Danish foreign exchange forward market dictates that investors hedge their foreign exchange risk to Euros before hedging finally into DKK. This sequential procedure we mirror in Chapters 5 and 6, of which the former deals with the hedging of foreign exchange exposure into Euros while the latter deals with hedging from Euros into DKK. The two situations

16 6 CHAPTER 1. INTRODUCTION are distinct since the case investors asset portfolio is constructed such that the problems outlined in the previous section to this chapter relates only to the hedging from Euros to DKK. It will be in these two chapters that the final framework will be derived. The policy recommendations themselves will be presented in the conclusion to this thesis. 1.3 Method The majority of the methodological concepts used in this thesis are described in Chapter 2, however we do wish to comment briefly on the techniques we use to derive our results. In deriving the results in this thesis, a range of possibilities exist for the sake of computing. One obvious direction that many academics and students pursue is the Microsoft Office Excel solution which provides a range of useful tools for the purpose. However, Excel is limited in the sense that the software overhead is quite large, resulting in computation times being rather long. For application in subjects like the one examined in this thesis, it is critical that computation times are not too long, and as a result we went for using the programming language C++ for all programming needs. C++ has the benefit of not having the huge overhead and resulting loss of speed that Excel suffers from, while retaining some of the outof-the-box range of possibilities of Excel through open-source extension frameworks. All source code is supplied in the appendices. All time-series data used in model calibration and return estimation is primarily fetched from the Bloomberg Professional software package. However, for some markets or time periods, Bloomberg data may be insufficient. In these cases we look to a range of alternative or specialized data providers for our data needs. A reference to the data provider will be made in the text whenever such alternative data provider is used.

17 Chapter 2 Theoretical Foundation As mentioned in the introduction to this thesis, it is generally assumed that the reader is well-acquainted with the basic elements of finance theory, and as a result safe for the sections that treat the variance gamma (V.G.) framework most of the material covered in this chapter should to a large extent be prior knowledge to the reader. Nevertheless, in the interest of thoroughness we cover here a range of standard as well as non-standard methods in numerical finance theory that we employ in the solving of the problem posed in the introduction. The reader should familiarize himself with especially the concepts and parameters surrounding the variance gamma (V.G.) stochastic process, since this knowledge is critical in understanding how we characterize and simulate financial asset returns as well as foreign exchange rate paths. Beyond this, the coverage of the standard methods will be relatively brief. 2.1 Stochastic Calculus Theory In stochastic calculus theory (Lawler, 2006), a stochastic process is a process by which, given the the probability space (Ω, F, P ), a variable X : Ω R is a random variable indexed by time. The probability space defines the structure of the random movements of X. The state space or sample space Ω is a non-empty set of all possible outcomes of X. The event space F is a σ-algebra, i.e. is a set of events with zero or more outcomes, each of which is a subset of Ω. The probability P assigns nonnegative probabilities to each of the events in F. As we consider a stochastic process (X t ) t T, where T R is a set, the event space F contains a collection of sub-σ-algebras (F t ) t T such that for all s < t, F s F t, called a filtration. That is, the event space F t contains information of all events leading up to t; alternatively, (F t ) t T is said to be an increasing sequence of information. Two central concepts in stochastic calculus theory that relates to filtration that 7

18 8 CHAPTER 2. THEORETICAL FOUNDATION readers should familiarize themselves with are the Markov property and the Martingale property. The Markov property states that a stochastic process (X t ) t T is a Markov chain if for s < t, P (X t F t ) = E(X t X s ) is satisfied; that is, the Markov property states that in order to make predictions as to the future state of the stochastic process, it is necessary to consider only the present state of the process, not those states preceding it (a stochastic process fulfilling the Markov property is said to be memoryless). A Martingale is a stochastic process with E( X t ) < that for all s < t satisfies E(X t F s ) = X s ; that is, a stochastic process is a Martingale if the expected state of the process is the current state of the process Itô Calculus and Itô s Lemma One of the central results in stochastic calculus is Itô s Lemma or Itô s Formula (Lawler, 2006). Consider a very basic form of stochastic process X t whose parameters a and b are functions only of the underlying variable X t and time t. Such a process is usually called an Itô process and can be written in differential form as dx t = a(x t, t)dt + b(x t, t)dw t, (2.1) where W t is a standard Wiener process. The term a(x t, t)dt can be interpreted as the drift of the process (purely deterministic) while the term b(x t, t)dw t can be interpreted as the stochastic (non-deterministic) component of the process. In relation hereto, a can be interpreted as being the drift parameter and b 2 the variance of the process. Itô s Lemma or Itô s Formula states that given a variable X t that follows the Itô process in (2.1), a function f dependent only on X t and time t follows the process (Hull, 2009) df(x t, t) = ( f a + f X t t ) f b 2 X 2 t dt + f X t b dw t. (2.2) Note that the process in (2.2) is in itself an Itô process with drift f X t a+ f and variance ( f X t b ) Geometric Brownian Motion f b 2 t 2 2 Xt 2 The geometric Brownian motion (GBM) is a Markovian stochastic process whose logarithm follows the more general Brownian motion with drift. The geometric

19 2.1. STOCHASTIC CALCULUS THEORY 9 Brownian motion models the underlying process X t as an Itô process with a(x t, t) = µx t and b(x t, t) = σx t, i.e. dx t = µx t dt + σx t dw t (2.3) or equivalently, This is the geometric Brownian motion. dx t X t = µdt + σdw t. (2.4) If we define the function f(x t, t) = ln X t by convention of the Itô formula, each of the derivatives in (2.2) becomes f X t = 1, X t (2.5) f = 0, (2.6) t 2 f X 2 t Plugging (2.5)-(2.7) into (2.2) we get df(x t, t) = = Xt 1 = 1Xt 2 = 1. (2.7) X t Xt 2 ( ( 1 µx t ) ) (σx 2 X t Xt 2 t ) 2 dt + 1 σx t dw t. X t Reducing we get df(x t, t) = ( µ 1 2 σ2) dt + σdw t. (2.8) Recall that since we defined f(x t, t) = ln X t then we see that ln X t follows Brownian motion with drift. Then the solution to (2.3) is ln X t = ( µ 1 2 σ2) t + σw t, (2.9) and therefore X t = exp {( µ 1 2 σ2) t + σw t }, (2.10) The geometric Brownian motion as derived here is the foundation for a range of well-known applications in finance theory, most prominently the Black-Scholes formula for pricing European-style stock options. Later in this chapter, we will look at a related formula, the Garman-Kohlhagen formula for pricing foreign exchange options, which is based upon the Black-Scholes formula and utilizes geometric Brownian motion for describing the underlying price process.

20 10 CHAPTER 2. THEORETICAL FOUNDATION Variance Gamma The variance gamma (V.G.) stochastic process was introduced towards the end of the 1980s and beginning of the 1990s (Madan and Seneta, 1990) as an extension to the Brownian motion introduced above, as a way of overcoming some of the GBM s (and its derivative Black-Scholes formula) shortcomings in relation to empirical observations that conflicted with the assumption of normality. The V.G. framework introduces two additional parameters to the GBM, θ and ν, which control indirectly for respectively skewness and kurtosis in the process dynamics. Several interpretations of the variance gamma process exist. Perhaps the most common, and the one we will cover here, is to interpret the process as a geometric Brownian motion subject to a stochastic time change. Specifically, under this interpretation we evaluate the geometric Brownian motion at gamma distributed increments. Formally, as earlier we define the Brownian motion as b(t, µ, σ) = µt + σw t. (2.11) Further, consider a gamma process γ(t, µ, ν) where µ is the mean rate and ν is the variance of the process. We then define the variance gamma process as Y (t, θ, σ, ν) = b(γ(t, 1, ν), θ, σ). (2.12) ν The price dynamic for the underlying is then obtained by noting that the Wiener process W t can be interpreted as a standard Brownian motion with mean of 0 and a standard deviation of 1. By replacing the standard Brownian motion with the variance gamma process we obtain (Madan, Carr, and Chang (1998) and Madan and Seneta (1990)) X t = X 0 exp {µt + Y (t, θ, σ, ν) + ωt}. (2.13) The constant ω is chosen such that E[X t ] = µt, and can be shown to equal (Madan and Seneta, 1990) ω = ln ( 1 θν σ2 ν 2 ) /ν. (2.14) In order to calibrate of the process to financial time-series data, we need closed-form expressions for each of the process parameters. In reduced form, the moments of

21 2.1. STOCHASTIC CALCULUS THEORY 11 the process Y (t, θ, σ, ν) is given by (Seneta, 2004) σ 2 = ˆσ 2 + ˆθ 2ˆν, (2.15) β = 3ˆθˆν ˆσ, (2.16) κ = 3(1 + ˆν), (2.17) where accents indicate parameters to be estimated (otherwise they indicate an observed parameter, i.e. the target value). We now solve the three equations with respect to the three unknowns ˆσ, ˆθ and ˆν and obtain ˆσ = 1 + ( β κ 3 σ 2 ) 2 ( κ 3 1) (2.18) ˆθ = βˆσ κ 3 (2.19) ˆν = κ 3 1. (2.20) Figure 2.1 illustrates the differences between the Variance Gamma process and the geometric Brownian motion process. The figure shows four example simulated price paths for the two processes. In each example, both processes are subject to the same underlying Wiener process. Noting this, one can see that the price paths of the two processes can be quite divergent even though the variance and standard deviation of the two are the same. Notice especially the affine jumps present in the Variance Gamma price paths, not present in the geometric Brownian motion case Cholesky Decomposition Whenever we need to draw random numbers from a multivariate distribution, we need some numerical method of correlating the random numbers that we generate from a range of univariate distributions. A multitude of methods exist that will accomplish this task, however perhaps the most widely used is the Cholesky decomposition (Hull, 2009). Consider a vector y of random realizations from a multivariate normal distri-

22 12 CHAPTER 2. THEORETICAL FOUNDATION (a) (c) (b) (d) Figure 2.1: Four examples of simulated price paths under Variance Gamma (red) and geometric Brownian motion (black) illustrating the differences between the two processes. In each example, both processes are subject to the same underlying Wiener process. bution N (µ, Σ), where µ is a vector corresponding to the systematic means of the distribution and Σ is the covariance matrix, corresponding to the unsystematic or random components of the distribution. The vector y can then be decomposed such that it satisfies the equation y = µ + AZ, (2.21) where Z is the standard normal distribution N (0, I), I being the identity matrix. In the above equation, A refers to the square root of Σ, i.e. the multivariate equivalent to σ in the univariate case (Σ being the multivariate equivalent to σ 2 in the univariate case). The Cholesky decomposition in linear algebra defines the matrix A such that AA T = Σ. Specifically, since Σ is a positive-definite symmetric matrix, a lower triangular matrix A is facilitated as a solution. Finding the matrix A that satisfies this equality is an iterative process. The algorithm and source code used in conjunction with this thesis can be found in Appendix C.

23 2.2. INTEREST RATE MODELS Multifactor Correlation Structure In relation to the above section on the Cholesky decomposition, we note that, in part because the Cholesky decomposition in equivalence only corresponds to a second-order Taylor-series approximation (i.e. it does not account for higherorder information), the Cholesky decomposition only produces a unique multivariate distribution so long as it is jointly normally distributed. In fact, taking into account higher-order information, there exist an infinite number of multivariate distributions that satisfy the Cholesky decomposition. In relation to the two stochastic processes we described earlier (GBM and V.G.), only in conjunction with the geometric Brownian does the Cholesky decomposition produce a multivariate distribution that is unique to the matrix A. For the variance gamma stochastic process (i.e. a multifactor process that is not normally distributed), the same does not apply. As mentioned, if we produce a multivariate non-normal distribution by way of the Cholesky decomposition, we disregard the possible correlation implicit in higher-order information. Even so and having noted this, the Cholesky decomposition is used throughout whenever we wish to draw random numbers from normal as well as non-normal multivariate distributions. In relation to the variance gamma stochastic process and on the subject of correlating using the Cholesky decomposition, we note that this process is made up of two separate stochastic terms (one standard normally distributed and one gamma distributed). Throughout this thesis, whenever we correlate variance gamma processes, we apply the Cholesky decomposotion to the normally distributed term only. This means that we incur small errors on realized correlation, however small enough that we accept these. 2.2 Interest Rate Models The below sections on interest rate models are adapted from a previous work by the author of this paper, see Helgason (2012). In this section we will be going over the basics of the numerical framework under which we aim to derive the target model. As mentioned in the introduction, to this end we employ a short term interest rate model, that is, a stochastic process model that assumes that the entire term structure of interest rates can be determined by the instantaneous short rate 1. An interesting feature of the short-term models 1 The short rate is defined as the interest rate at which money can be borrowed for an infinites-

24 14 CHAPTER 2. THEORETICAL FOUNDATION that we will be covering is that they possess an affine term structure, meaning the continuously compounded rate of return R(t, T ) at time t t < T can be determined from the instantaneous short rate, where T is the time to maturity. As the aim of this section is to simply provide an introductory overview for a working knowledge of short-term models, we refer to Brigo and Mercurio (2006) for a thorough exposition. Consequently, we cite the same work as the prime source of information contained within this chapter. Below, we go over three of the most common one-factor short rate models. The models two-factor equivalents will not be covered Vasicek In general, short rate models aims to model the future evolution of interest rates by modeling the short rate (as defined earlier). In general such models assume that the short rate r(t) at time t can be described by a stochastic process. Vasicek (1977), for instance, assumed that the short rate could be defined as having the dynamics dr(t) = k[θ r(t)]dt + σdw (t), (2.22) where k and θ, σ and r(0) are positive constants with W(t) denoting a Wiener process. One defining feature of the Vasicek model is that it is mean-reverting to a longterm equilibrium level. The constant θ defines this equilibrium level (notice that if the current level of r(t) is greater than θ the drift term θ r(t) is negative, and this drift would as a result contribute to a pull towards the equilibrium level θ). The constant k defines the rate of mean reversion in the dynamics. σ denotes the standard deviation of the stochastic term and r(0) defines the short rate at t = 0. By integrating (2.22), Brigo and Mercurio (2006) showed that, for each s t, r(t) = r(s)e k(t s) + θ ( 1 e k(t s)) t + σ e k(t u) dw (u) (2.23) s meaning that E{r(t)} = r(s)e k(t s) + θ ( 1 e k(t s)) (2.24) V ar{r(t)} = σ2 [ ] 1 e 2k(t s) (2.25) 2k imally small period of time

25 2.2. INTEREST RATE MODELS 15 which, by implication, means that the models allows for negative rates. While negative interest rates are theoretically possible (and has been and continues to be observed), this feature is less than optimal, since negative rates continue to be an extremely rare event Hull and White In a series of papers published in 1994 (see Hull and White (1994a) and Hull and White (1994b)), Hull and White explored an extension to the Vasicek model. Specifically they modified (2.22) so that r(t) could now be described by the dynamics dr(t) = k[θ(t) r(t)]dt + σdw (t), (2.26) where k, σ and r(0) remains positive constants as per the Vasicek model. The only modification is that θ now is time dependent. They considered this extension due to the deficiency of the Vasicek model to exactly fit the initial term structure. Making θ time-dependent, they added another degree of freedom in fitting the model to the initial term structure. Otherwise, the model s dynamics are identical to that of Vasicek and is associated with the same advantages and drawbacks, most notably, of course, that its still allows for negative rates in the short rate Cox, Ingersoll and Ross We now consider a model proposed by Cox, Ingersoll and Ross (henceforth CIR). The model builds upon Vasicek model, but its aim was specifically to correct the possibility of negative interest rates. The model they proposed defined the short rate dynamics as dr(t) = k[θ r(t)]dt + σ r(t)dw (t), (2.27) where k, θ and σ are positive constants following the definitions as per the Vasicek model, constrained by 2kθ > σ 2. CIR thus only modifies the original Vasicek model by adding r(t) to the second term. This, however, ensures that rates can never become negative with positive probability. For obvious reasons, this is a very convenient property ensuring realism in the development of the short rate, and thus the entire term structure of interest rates.

26 16 CHAPTER 2. THEORETICAL FOUNDATION The Affine Term Structure of Interest Rates All of the above mentioned one-factor short rate models features a so-called affine term structure. This means that one may infer the shape of the entire term structure of interest rates from the instantaneous short rate. The models thus provide closedform analytical solutions to valuing the most common securities and derivatives (typically options). For the purpose of this paper, we will focus on inferring the term structure of interest rates as well as on zero coupon bonds. In this section, we show the analytical framework for valuing zero-coupon bonds and their corresponding yields from the at any time prevalent short rate. In general, models with the dynamics as defined by Vasicek in (2.22) are said to possess an affine term structure. Since we have already established that both Hull and White and the CIR models are extensions to the Vasicek model, we can conclude that all three are affine in r(t). Specifically this means that the continuously compounded rate R(t, T ) at time t for at zero-coupon bond maturing at time T can be analytically determined by an expression of the form R(t, T ) = α(t, T ) + β(t, T )r(t), (2.28) where α and β are deterministic functions of time. Under the CIR framework, the price of a zero-coupon bond at time t maturing at time T can be expressed as P (t, T ) = A(t, T )e B(t,T )r(t) (2.29) where [ ] (k+h)(t t)/2 2kθ/σ 2 2he A(t, T ) =, (2.30) 2h + (k + h)(e (T t)h 1) B(t, T ) = 2(e (T t)h 1) 2h + (k + h)(e (T t)h 1), (2.31) h = k 2 + 2σ 2. (2.32) Given this framework for pricing a zero-coupon bond, the functions α and β used

27 2.3. OPTION THEORY 17 in determining the yield R(t, T ) of such a bond are defined as α = lna(t, T ), T t (2.33) β = lnb(t, T ). T t (2.34) Given these formulas for determining the entire term structure of interest rates, we can by the same token determine the forward rate F (t, T, S) at time t expiring at time T and maturing at time T + S, i.e. S is the forward tenor. Given the zero coupon bond prices P (t, T ) we define the forward rate as F (t, T, S) = 1 [ ] P (t, T ) S P (t, T + S) 1 (2.35) The analytical framework presented here for determining the term structure of interest/forward rates as well as zero-coupon bond prices will prove very valuable later on when valuing the portfolio of assets that make up the strategy that we are concerned with in this paper. This will be the primary application of the formulas derived in this section. 2.3 Option Theory Risk-neutral Valuation and the Measure Q When valuing financial derivatives (i.e. financial contracts that derive their value from the price of an underlying asset) we do so under the assumption of a probability measure. One such measure is the measure P which describes the real-world (called physical) probabilities of, for instance, the price on an underlying asset moving up or down (or, for that matter, staying the same). When we value derivatives under the P measure, we simply assume that the price at time t of a derivative f(x t, t) maturing at time T, is simply the expected value of that derivative, i.e. f(x t, t) = E[f(X T, t) F t ], (2.36) where E denotes the expectation under the probability measure P. However, since market participants have different levels of risk tolerance (they typically require different levels of compensation for risk), a measure like P, which makes no assumptions as to the risk tolerance of market participants, falls short.

28 18 CHAPTER 2. THEORETICAL FOUNDATION In relation hereto, the fundamental theorem of asset pricing states that in a market free of arbitrage there exists some probability measure Q, equivalent to P, such that the underlying process becomes a martingale (Delbaen and Schachermayer, 1994). We call this measure an equivalent martingale measure. The word equivalent here means that the new measure Q does not modify the original measure P such that any event in P with zero probability has positive probability in Q. Formally, given the probability space (Ω, F, P) on a process X where P is the physical probability measure, we would like to replace P by some equivalent probability measure Q such that the discounted value of X is a martingale. A concept related to the above is that of numéraire change. A numéraire is the unit by which we measure value. Suppose that a risk-free money-market account yields according to the process B t = e rt, (2.37) where r is a constant reflecting the risk free rate. Then, in terms of Itô s formula, we can define f(x t, t) = Xt B t, meaning that the numéraire is now the risk-free moneymarket account; in other words, we discount the process X t by the risk-free rate of interest. It follows from this that for a process to be a martingale under the Q measure, the drift os such a process must equal r. Thus, we have that the price at time t of a derivative f(x t, t) maturing at time T is f(x t, t) = E Q t f(x t, t) = E Q t f(x t, t) = E Q t [ f(x T, T ) B ] T B F t, (2.38) [ t ] f(x T, T ) e rt e rt F t, (2.39) [ f(xt, T )e r(t t) ] Ft, (2.40) where E Q t denotes the expectation under the Q measure Black-Scholes While the risk-neutral valuation framework derived above is true in general for any kind of financial derivative and can be approximated using for instance the Monte Carlo method, for some derivatives we may derive convenient closed-form expressions for their prices. One such expression, which is an important result in numerical finance, is the Black-Scholes formula for pricing European-style options. We define the functions describing the payoff of respectively call and put options

29 2.3. OPTION THEORY 19 as C(X T, T ) = max(x T K, 0), (2.41) P (X T, T ) = max(k X T, 0), (2.42) where K denotes the contractually defined strike price of the option. Taking the call option as an example, using the risk-neutral valuation framework derived in the previous section, we have that the value at time t of an option expiring at time T is [ C(X t, t) = E Q t e r(t t) max(x T K, 0) ] Ft, (2.43) equivalent to C(X t, t) = e r(t t) E Q t [max(x T K, 0) F t ]. (2.44) In order to solve the above equation under the risk neutral pricing framework, we consider once again the geometric Brownian motion from (2.10): ln X t X 0 = (µ 1 2 σ2 )t + σz t t, (2.45) where Z t t = Wt, i.e. Z t is a standard Brownian motion evaluated at unit intervals. Dividing E(ln Xt X 0 ) = (µ 1 2 σ2 )t by σ(ln Xt X 0 ) = σ t, we obtain the z-score for the expected value of the geometric Brownian motion, i.e. z GBM = (µ 1 2 σ2 )t σ. (2.46) t In fact, we can find the z-score for any level of return r above the expectation (µ) by subtracting it from the numerator of the above, i.e. z GBM+r = r + (µ 1 2 σ2 )t σ. (2.47) t Setting r = ln K X t, we obtain the z-score for the return associated with the strike price of the option. We can then use the cumulative normal distribution function N to find the probability of the option finishing in the money at expiry, time T : P t (X T > K F t ) = N (z GBM+r ) = N ln K X t + (µ 1 2 σ2 )T σ T. (2.48) The economic interpretation of this expression is that it is the probability of the

30 20 CHAPTER 2. THEORETICAL FOUNDATION option being exercised, that is, the probability of the price of the underlying exceeding the strike price at maturity; is this case, the holder of the option exercises his right and pays the strike price. Thus, the at time t expected future value of the strike price K paid at time T is then found by multiplying K by N (z GBM+r ), i.e. E t (K F t ) = KN (z GBM+r ). Similarly, we can find the at time t expected future value of the delivery of the underlying. However, since delivery is not paid in cash but rather in kind, we cannot simply multiply X t by P(X T > K F t ) as was the case with the strike price K. We need to consider the fact that the exercise event is not independent of X T, such that we obtain a risk-adjusted probability of X t > K (Nielsen, 1993). As it turns out, the risk adjusted probability of X t > K equals ˆP t (X T > K F t ) = N (d 2 + σ T ), (2.49) where d 2 = z GBM+r and ˆP denotes the probability using risk-adjusted probabilities. Putting all of the above together, we obtain a closed-form expression for the price of a European-style call option. Recall that the price of such as option is C(X t, t) = e r(t t) E Q t [max(x T K, 0) F t ]. Using the expressions we obtained above, we have C(X t, t) = e r(t t) ( X t e r(t t) N (d 1 ) KN (d 2 ) ), (2.50) C(X t, t) = X t N (d 1 ) KN (d 2 )e r(t t), (2.51) where d 1 = d 2 + σ T and µ = r under the risk-neutral measure Q. Equation (2.51) is the Black-Scholes formula for pricing a European-style call option Garman-Kohlhagen The Garman-Kohlhagen formula for pricing European-style foreign exchange options is an extension to the Black-Scholes formula described above. In simple terms, the Garman-Kohlhagen formula modifies (2.51) such that we have two interest rates, one for the domestic currency and one for the foreign currency. Using the same notation as above, the Garman-Kohlhagen formula for pricing EUropean-style foreign

31 2.3. OPTION THEORY 21 exchange options is given by C(X t, t) = X t N (d 1 )e r f (T t) KN (d 2 )e r d(t t), (2.52) where r d and r f is the risk-free rates of interest in the home and foreign currency respectively. Further, Garman-Kohlhagen modifies d 1 and d 2 such that d 1 = ln Xt K + (r d r f 1 2 σ2 )t σ t, (2.53) d 2 = d 1 σ T. (2.54)

32

33 Chapter 3 Data Description In solving the problem that we posed in the introduction to this thesis, we base the majority of our analysis on a theoretical case investor, such that both the extent to which the market imperfections affect investors and the effects of the policy recommendation made in this thesis can be properly illustrated. In this chapter, we motivate first the choice of case investor; specifically, we aim to apply the analysis to an investor fitting certain characteristics. Following this, we motivate why such an investor is an interesting case by briefly describing the industry in which such an investor might operate. Then, we characterize numerically the case investors portfolio holdings and general portfolio policies (such as rebalancing). 3.1 Case Investor As mentioned in the introduction, the market imperfections relating to foreign currency forward mispricing is a known phenomenon in the Danish Krone denominated market. Further, the Danish Krone is pegged to the Euro under the ERM II agreement. The combination of these facts renders it possibly quite rare in international comparison and quite an interesting case to analyze 1. As a result, the investor we contemplate in this thesis is a professional investor whose liabilities are Danish krone denominated. A typical example of such an investor would be a large domestic institutional pension fund. The pension fund case is particularly interesting since they amount of funds under management is of such a size that a change in hedging policy by one of the major players can have significant effects on the entire market, and especially the returns for perhaps millions of beneficiaries pension savings portfolios. 1 The extent to which this phenomenon is observed in other (pegged or floating) currencies is out of scope for this thesis. 23

34 24 CHAPTER 3. DATA DESCRIPTION The Danish Pension Market As of 2011, the retirement savings in Danish pension funds amounted to roughly 3.4 trillion DKK or approximately 460 billion Euros 2. According to Mercer (2012), the Danish retirement income system is among the highest-rated and best-performing systems in the world, and is the only system to receive a A-rating under the criteria stipulated in the report. According to the same report, however, in terms of growth in assets, Denmark ranks quite poorly at around 11% 20% in asset growth. Even so, the total amount of retirement savings/assets held with privately owned pension providers is the highest in Denmark among all the countries surveyed, at 10% of GDP. 3.2 Case Portfolio The case portfolio is chosen such that it represents a wide array of asset classes and currencies. Further, it is chosen such that is represents approximately the asset allocation of a typical danish pension fund. We have modeled the case portfolio in part after data in Mercer (2012). In here, researchers have compiled data as to the makeup of international retirement savings assets, which shows that in the case of Denmark, quote a large percentage of retirement savings assets are invested in fixed income securities. Additionally, we assume that the pension funds in general has a skew towards domestic fixed income securities. As such, the case portfolio presented in 3.1 is heavily allocated towards domestic mortgage bonds, also due to the relative size of the Danish mortgage bond market, compared to other developed economies. Beyond this, the portfolio is exposed to various asset classes denominated in Euros, US Dollars, British Pounds and Swedish Kronor. The detailed specifications on currencies and allocation is broken down in Data Sample In constructing table 3.1 we sampled for each asset class weekly data spanning from January 29, 1999 through December 28, 2012 (for a total of 727 observations per asset class). The data was sampled based on total return indices that fit the regional, asset type and currency characteristics. The sample period was chosen such that the length of the data sample was maximized without compromise data 2 Obtained from Finansrådet s report on the Danish pension fund sector.

35 3.2. CASE PORTFOLIO 25 quality. The means were computed based on a regression on log returns. Higher moments were calculated using standard methods, with the regression-based mean as central moment. We note that the choice of sample period spans approximately two credit cycles (approximated by the 2001 asset bubble and the 2008 credit crisis). The lack of a longer sample period means that, while being the best estimate, the parameters we estimate may possible incorporate more volatility and skew than what we on average would expect to observe. In computing the summary statistics for securities and foreign exchange rates, we estimate first the mean of each distribution by regressing log returns, such that we account properly for compounding. Beyond this, standard formulas are used for calculating higher moments. The Variance Gamma parameters are computed using (2.18) (2.20) on summary statistics of the relevant length. The data in table 3.1 is annualized (projected to a horizon of one year), however all calculations are based upon a time-series containing weekly observations, as specified above. Notice that Variance Gamma parameters listed in the table are annualized as well. The actual parameters used in calculations going forward may not be annualized. For the interest rate curves for each of the five currencies used in this thesis, we have calibrated the Cox-Ingersoll-Ross framework to the Overnight Indexed Swap (OIS) term structures as at December 31, The result from the calibration to the interest rate curves are shown in 3.1. The precise methodology we used in calibrating the CIR process will be covered in a following chapter. Figure 3.3 shows the covariance matrix of the entire set of assets, rates and instruments. The covariance factors are computed based upon log returns using standard methods.

36 26 CHAPTER 3. DATA DESCRIPTION Summary statistics V.G. parameters Asset class Currency Alloc. µ σ β κ ˆµ ˆσ ˆθ ˆν (a) Danish equity DKK 5.00% 7.80% 19.45% % 18.75% (b) US equity USD 5.25% 1.40% 19.43% % 19.13% (c) European equity EUR 6.75% -0.71% 23.53% % 23.08% (d) Global equity USD 3.00% 2.80% 18.69% % 18.25% (e) Danish gov t DKK 5.00% 5.31% 4.39% % 4.39% (f) Danish mortgage DKK 29.00% 5.57% 3.49% % 3.44% (g) US gov t USD 6.00% 6.40% 6.53% % 6.42% (h) European gov t SEK SEK 3.00% 2.38% 3.72% % 3.68% (i) European gov t EUR EUR 6.00% 4.68% 3.94% % 3.94% (j) European gov t GBP GBP 6.00% 5.57% 5.76% % 5.76% (k) IG USD USD 9.38% 5.55% 8.60% % 8.16% (l) IG EUR EUR 3.13% 2.29% 12.11% % 11.95% (m) HY USD USD 4.69% 7.35% 7.99% % 7.68% (n) HY EUR EUR 1.56% 6.59% 12.11% % 12.05% (o) EM USD USD 6.25% 10.22% 8.92% % 8.63% (p) EURUSD % 0.37% % 0.37% (q) EURGBP % 10.44% % 10.35% (r) EURSEK % 8.44% % 8.43% Table 3.1: Summary of panel data used in estimating the returns of the combined investment portfolio, including portfolio allocation, summary statistics and fitted Variance Gamma (V.G.) parameters. Data is estimated from weekly observations from January 29, 1999 through December 28, 2012 (N = 727) and projected to a horizon of 1 year (annualized; see Appendix B).