Application of Proxy Hedging in an Illiquid Market

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University of Amsterdam

MSc Stochastics and Financial Mathematics

Master Thesis

An Application of Proxy Hedging in

Emerging Markets

Author: Heather Robertson Supervisors: Dr. A.J. Van Es Mr. A. Sorochan Mr. N. Vermeijden 25 September 2019

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Abstract

Hedging is a regular process in the financial sector, particularly for companies that hold large portfolios that are made up of fluctuating assets. This study revolves around a portfolio consisting of Frontier currencies with a fixed matu-rity time. The concept where it is not possible to hedge traditionally in some markets is made clear in this thesis and an alternative concept, Proxy Hedg-ing, is studied. These market restrictions are removed when using these proxy assets and the setup of a necessarily complete market is considered. The risk involved with these portfolios is discussed and a reduction method considered when using a suitable portfolio of proxies. The use of Stochastic Differential Equations in studying the movement of Foreign Exchanges is prioritized, then applied using a suitable Geometric Brownian Motion. The concept of scenario-based studies promote the idea of simulation, thus the Monte Carlo Method is discussed and used for estimating the future value of a chosen portfolio, its corresponding proxy portfolio and the risk associated through the future time interval.

Keywords: Proxy Hedging, Portfolio Theory, Illiquid Markets, Frontier

Currencies, Stochastic Differential Equations, Monte Carlo Method, Value at Risk, Conditional Value at Risk.

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Preface

This thesis concludes my master’s degree in “Stochastic and Financial Mathe-matics” from the University of Amsterdam. I am grateful to TCX Investment Management for giving me the opportunity to intern with them for the past six months. I gained a lot of knowledge that I will be able to build on, and the experience has been invaluable. I would like to express my gratitude to Dr. Bert van Es for his supervision, time and support. I would also like to thank Niels Vermijden for taking time, not only at work, but whilst on honeymoon, to read, comment and assist me. I would like to thank my parents, James and Diana, for their continuous support throughout my academic career, especially in the proofreading of these pages. I would not be here without them. Finally, I also would like to thank Antonio Mendes Anasagasti for standing by me in the final stages of this paper - his unwavering support was much appreciated.

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9.4.1 Histograms representing the Estimated Portfolio Gains Process at T for the original portfolio, the proxy portfolio and the combined portfolios . . . 89 10 Conclusion 91 10.1 Further Work . . . 92 11 Bibliography 94 A Financial Derivatives 97 B Stochastic Calculus 101 C Additional 102 C.1 Convex Optimization. . . 102

C.2 Currency Exchange Rate Assumptions . . . 102

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1 Introduction & Preliminaries

The exploration of proxy hedging is derived from the fact that not all financial as-sets can be directly hedged due to the unavailability of hedging instruments in the same market. This unavailability is not generally seen in a liquid or developed mar-ket, so an asset in an illiquid or underdeveloped market would need a proxy hedge in a more liquid market where hedging instruments are available. Often “Illiquid Mar-kets” are associated with Frontier markets. These are described by the FTSE Russell as “markets constituting a segment of the equity market that is typically missing within institutional portfolios”. These developing countries are associated with high rates of economic growth, but comparably small and relatively illiquid financial markets. Their attracted attention emerges due to their diversification opportunities and large growth potential associated with being in an early stage of development. Thus equities listed in Frontier markets have become increasingly investable. The one benefit comes from early-entry into some asset classes, described by the belief that potential opportunity at entry is deemed greater than that at a later stage. All markets were once considered Frontier before undergoing vast economic growth and continuous development of both infrastructure and the construction of trading platforms and regulations.

Citing the FTSE Russell [1], “the New York Stock Exchange was created under a But-tonwood tree in order to facilitate trading amongst brokers in just a handful of listed companies”. This was around 220 years ago. It was not possible at that time to foresee the growth associated with the United States, and that it would in fact, be one of today’s largest economies, (if not the largest). China is a great example of unexpected growth, from what was considered a Frontier market in 1980, to the second largest economy in the world. There are many Frontier markets which did not even exist as little as 10 years ago. This extreme growth could be supported by higher levels of technology and exchange platforms, which are already in use for developed countries. Not all emerging countries have this rapid growth. Two well noted examples of slow or even regressional growth are Argentina and Venezuela. If the trend of time relates directly to increasingly more liquid markets and global growth, the Frontier markets of today could become the Developed Markets of tomorrow.

The term illiquidity has already been used above, this is where “Frontier market secu-rities tend to be more thinly traded than their Developed and even Emerging Market

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counterparts”. Many investors with a large amount of readily available funds to invest are seeking this high growth potential associated with the Frontier market, but since illiquidity leads to capacity constraints, there are many limitations in actually investing in said markets. A further constraint is that once the investment has been made, it is often difficult to change or liquidate the investment, especially on short notice. Thus this could lead to higher trading costs associated with wider bid-ask spreads than when com-pared to those in more developed, liquid markets. Another consideration when investing in these markets is the observed volatility. This observed volatility may be understated due to such illiquidity. This is linked directly to situations where there is no new or updated price to observe. An example would be when the buyers and sellers do not meet an agreement on the current traded price, “either because the buyers are offering too little or the sellers want too much”. Liquidity is more of a concern for shorter term investors than others. When considering the constraint linked to changing or liquidating a position in a Frontier market, it can be seen that long term investors such as plan sponsors or foundations, may not be as concerned with this limitation All the above reasons have, in part, given rise to the need for Proxy Hedging.

One of the Oxford Dictionary’s definitions of a Proxy is “A figure that can be used to represent the value of something in a calculation.” Thus the use of Proxy Hedging can therefore be seen as a mechanism undertaken by traders and fund managers who have underlying positions in illiquid markets, where the change in economic conditions may result in a negative or potentially negative valuation of those positions. Hence the position to undertake a hedge in another asset that is closely correlated to the illiquid asset, but has the depth of a market or “liquidity” to permit the necessary hedges to be concluded is of great importance. It can be shown that the loss incurred on holding an asset in a specific market can be offset by “shorting” a different, but strongly correlated asset which could be in either the same or a different market. Shorting is the process of selling an asset which you do not yet own, but you hope to buy back at a lower price in the future.

The correlation between two different assets can only be back-tested and an assumption as to whether it will continue in the future has to me made. This leads us to consider how we would estimate future values, for not only the correlation, but also the value of the assets. Another reason that this is of great importance in the financial world, is the estimation of risk associated with a continuously changing valuation of assets, such that the risk itself can be measured and a contingency plan put in place.

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In this thesis, we venture into the prediction of future values of an asset considering both the randomness of future change, which can be described using Brownian Motion (BM), as well as a time dependent change over a specified interval. In our studies, these equations are called stochastic differential equations (SDEs) and they are often used to model assets such as stock prices, interest rates and foreign exchange rates.

In most financial cases, we believe the above mentioned assets will follow a Markov Process over a period of time such that the stochastic process movement is not based on the historical values, but rather on their present values. Using past data can help us with estimating parameters, calculating historical and estimated risk as well as back-testing resulting portfolio returns or estimating future expected returns. An extension of using these SDEs is applying them in a discrete sense to estimate the future path of such assets, using a method of simulation - such as the Monte Carlo Method - which will be explained and used further on in this paper.

For the duration of this paper we consider a portfolio, whose value is based in US Dollar (USD), where the composition is made up of different amounts of foreign currencies. The exchange rate simply put is the price of one country’s currency in terms of an-other country’s currency. These foreign exchange rates are represented in terms of the USD throughout this paper. The currencies and their derivatives make up markets as explained in Chapter2. The basic assumptions and theory adapted to our specific port-folio is found in Chapter3. Thus the value of the portfolio is dependent on the foreign exchange spot rate of these currencies compared to the US Dollar. This “spot” rate de-notes the exchange rate at that exact moment in time. We can see that these exchange rates can be modelled as stochastic processes as future values are unknown, they are investigated in Chapter4. Furthermore, we take special consideration that the portfolio is made up of currencies whose financial instrument markets are illiquid and therefore the need for proxy hedging should be considered. A discussion on hedging is detailed in Chapter5. This leads to both correlation analysis in Section5.5 and economic analysis when choosing the correct proxy. In Chapter 6, we explore the risk associated with the portfolio and the effect that a proxy hedge may have on this measure. Some risk measures we will consider are Standard Deviation, the Value-at-Risk and the Conditional-Value-at-Risk. In conclusion, an example of expected correlation between two foreign exchange rates is presented in Chapter 8 followed by a proxy hedging application based on the given portfolio of foreign currencies in Chapter9.

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2 Financial Markets

There are many markets in the financial sector which are classified overall under the description of a financial market. These are defined by a gathering of people with the in-tention to trade financial assets, securities or derivatives at low transaction costs. These can include Capital Markets, Money Markets, Commodity Markets, Foreign Exchange Markets and Derivatives Market. The latter two are of most importance to this paper.

2.1 Foreign Exchange Market

This market, is solely used for the trading of currencies which determines the worldwide exchange rate for all currencies. It is the largest and most liquid market in the world and determines the foreign exchange rate which is a relative value by setting its price in comparison to other currencies. Exchange rates are made up of a primary and a secondary currency, global foreign exchange convention barring a few exceptions, is to quote most currencies against the US Dollar so $1 = 𝑋. The most common exceptions are the Euro and GBP, where both of these are quoted as primary currencies. Although it is possible for each country’s Central Bank to intervene, without this possibility, it is considered as the market closest to perfect competition. A specific characteristic worth mentioning is that it is almost continuously operational since it can trade from 22:00 GMT on Sunday (Sydney) until 22:00 GMT Friday (New York).

2.2 Derivative Markets

Similar to the Foreign Exchange Market, but based on trading derivatives (the full description in Appendix A) which are financial instruments based on an underlying variable/s whose values are derived as a function of these variables’ value. There are many types of derivatives and some are mentioned in the AppendixA. We make special note of forwards and options.

2.3 Market Efficiency

Bachelier [2] and Fama [4] both mentioned that the claim for the efficient market hy-pothesis is that the price of an asset traded in an efficient market reflects all information Ω𝑡which is available publicly about itself. Timmerman & Granger [5] make a distinction

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based on the information set Ω𝑡:

• Weak market efficiency: Where values in Ω𝑡 contain information on only the past and present prices 𝑝𝑠 of the asset for 𝑠 ∈ (−∞, 𝑡].

• Semi-strong market efficiency: Extended such that values in Ω𝑡 include all public information that may contribute to the changes of the asset price, this includes prices as mentioned above.

We assume from here on that all information sets considered are of the semi-strong form of market efficiency. This implies that worldwide exchange rates and derivative prices in these markets reflect all public information in their values. It is said that without non-public or insider information - a market participant cannot achieve returns consistently exceeding the time-average growth rate of the market (“to beat the market”) by choosing the price at which he buys and sells an asset. Peters & Adamou [6] refer to this as “ordinary efficiency” to differentiate from stochastic market efficiency.

The difference between the two relates to the investment in an asset or portfolio of assets and relies on movements or fluctuations in the market price at each future time step. These changes are unknown to the market participants at any time prior to the actual time that it changes. Thus, we can define stochastic market efficiency as the impossibility of a market participant, without insider information, to beat the market by choosing the amount to invest in an asset or portfolio of assets. For the duration of this thesis, we will assume that stochastic efficiency is held in all markets considered. This especially includes the foreign exchange market and the markets for any asset used as a proxy hedge included in the analysis. The efficient market hypothesis (Bachelier [2]; Fama [4]) claims that the price of an asset traded in an efficient market reflects all the information publicly available about the asset.

2.4 Discounting Future Cash Flows

Ordinary efficiency makes special consideration to the fact that a return should increase or decrease parallel with the time-average growth of the market. In terms of the same currency it is implied that with a positive interest rate, one unit of currency today should be worth more tomorrow. If the growth is linear, represented by (1 + 𝑟) with r defining the positive interest rate, then the value of the one unit of currency today is worth 1 × (1 + 𝑟) tomorrow. Thus we need to consider a suitable discount factor such that the future expected values can be transformed back into the value in which it would

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be expressed at present. This discount factor should give information about the risk-free interest rate. Thus the interest rate can be seen as the proportion by which your money expands from present to a future time. By assuming that the markets are stochastically efficient, we are able to use specific bond yields as a discount rate.

“A bond is a securitized form of a loan. Bonds are the primary financial instruments in the market where the time value of money is traded” as defined by Damir Filipovic [7]. It is possible to choose an appropriate zero-coupon government bond with a value of 𝐷(𝑡, 𝑇 ) at present time t, where T represent its maturity date. Some assumptions include:

• The value of the bond at maturity is 𝐷(𝑇, 𝑇 ) = 1 in the unit of currency in which the bond was issued. Under a positive interest rate, it is clear that 𝐷(𝑡, 𝑇 ) <

𝐷(𝑇, 𝑇 ) and so for every 𝜓-units of the same currency at time 𝑇 , its value at time 𝑡would be 𝐷(𝑡, 𝑇 ) × 𝜓.

• The market of these bonds are frictionless for every 𝑇 > 0. That means there are no associated costs, risks or restrictions on the transactions.

• The value of the bond at time t is 𝐷(𝑡, 𝑇 ) = 𝑒−∫︀𝑇

𝑡 𝑟𝑠𝑑𝑠 where 𝑟_𝑠 is defined as the

implied interest rate at time 𝑠 ∈ [𝑡, 𝑇 ]. The idea behind the exponent is due to continuously compounded interest rates, that is where

(︁1 + 𝑟

𝑛

)︁𝑛

→ 𝑒𝑟 _{as 𝑛 → ∞.}

The observation that 𝑟 changes with respect to time shows that it is also a stochas-tic process in its own right and can be modelled as such. This is shown in Section 4.2.1.

• the bond D(t,T) is differentiable in T

The properties mentioned above coinciding with a zero-coupon bond, appropriately chosen to fit the correct market of currency and time frame required, can be used simultaneously as a discount curve. Suppose we need to make a payment 𝜓𝑇 at time 𝑇 , then the value today of 𝜓𝑡 is what we need to hold from now, thus we have

𝜓𝑡= 𝐷(𝑡, 𝑇 ) × 𝜓𝑇 = 𝜓𝑇𝑒−

∫︀𝑇 𝑡 𝑟𝑠𝑑𝑠

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Figure 1:Two zero-coupon Government bonds, each in a different country with its own

currency, which can be seen as a discount curve. The orange one denoting a South African Bond whose currency is the ZAR whilst the grey denotes a Myanmar Bond whose currency is the MMK. The x-axis denotes the time of maturity, explaining that the time today of

𝑡 remains consistent. This can be explained by the line graph represented by the

term-structure 𝑇 ↦→ 𝐷(𝑡, 𝑇 ). By the slope of the graphs, it is possible to note that Myanmar has a higher implied interest rate so the value of money in MMK here is expected to grow quicker than the value of the corresponding amount in ZAR over the same time period.

2.5 Portfolio Construction

A portfolio can be explained in simple terms as holding a range of multiple investments for a specified time period with the intention of making a return. This return is seen as the net gain made or received for holding the portfolio for a specified amount of time. These investments can be numerous items including money accounts (cash), assets or financial derivatives. As mentioned in the introduction, the portfolio we are using is measured in USD, but is constructed by holding a range of multiple foreign currencies. Batten and Thornton [8] showed the possibility that “the exchange rate can be deter-mined in the same manner as are the prices of other assets, such as stocks, bonds or real estate” which allows us to interchange an asset such as stock for an exchange rate in a portfolio. Portfolios can also be seen as a basket, if these foreign currency cash amounts were in weighted proportions rather than absolute figures. It is clear to see

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that the changes in the portfolio are linked directly to changes in future exchange rates, even if the absolute cash amounts denoted in the respectable foreign currencies does not change. Due to the nature of currencies in Frontier countries, depreciation of the cur-rency is often expected. When this happens, the portfolio value in USD would decrease and the future value would be less than its present one. Often, the exchange rate and discount rate are linked within a country. Hence a change in one realizes a change in the other and vice versa. It is explained in more detail by Batten and Thornton [8]. Thus there is the need to estimate the value of such a portfolio by discount as this should already be accounted for within the exchange rate.

The aim to optimize such a portfolio would be to maintain or increase the USD value from today, up until maturity, of said portfolio, such that the holder of the portfolio does not receive less than what would be expected if the portfolio were to be realized immediately. This introduces us to the risk associated with the possible decrease in value of such a portfolio. The following Section2.6details the management of the above risk.

2.6 Downside Risk Strategies

Risk associated with a portfolio can be seen as a deviation of its actual value in the future, when compared to either its value today or its expected one in the future. A more detailed definition of the risk is the probability that the value of a portfolio fails to meet the required financial expectations of the holder. Downside risk, as its name suggests, is related only to the portion of risk where the portfolio has a higher value today or a higher expected value in the future when compared to its true value. Another point which is considered, is that the different outcomes of the portfolio can be seen as different events, where each event can be assigned a probability to which it occurs. Many extreme events, such as a financial crisis or natural disaster, for example the financial crisis of 2007, have a large negative effect. Generally these are events of low probability. However, we take special care to account for these events, as a low prob-ability event may still occur, and most companies need to be financially capable of sustaining its consequences. There are multiple ways to measure this risk, including its absolute expected size. These are explained in detail in the Risk Measure Chapter6. We can model the returns of a portfolio at a future time with a distribution, say a

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normal one for simplicity of explanation. Then any left-tailed event (a negative return) represents a loss to the holder. A protection strategy, such as a hedge, may be considered to lessen the negative effect. Any such strategy should take into account the following considerations:

• Protection policies can be seen as insurance, however many of these will entail a cost.

• The plan would need to be developed to coincide with the company’s capacity to absorb a loss.

• A good understanding of the use and parameters involved to optimize the strategy. • The accessibility to these strategies and the amount of time or supervision needed

to maintain them.

• Utilization of appropriate and cost effective strategies will need to be considered. • Whether any other risk is associated with a hedge strategy.

Figure 2:A graph showing the expected return of the portfolio at a future time, modelled

with a distribution to represent the probabilistic outcomes. A tail choice parameter is the probability that we believe that the portfolio can end within the space of outcomes less than a specific value. To find the amount at risk, we can use a parameter of:

If this left-tail parameter is 𝛼 = 0.05 or 5%, we are able to represent this by the blue shading. If it is 𝛼 = 0.25, the returns may fall within the grey shading or the blue one. If

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3 Modelling

The aim of this chapter is to become familiar with the financial environment that we will be working with, together with a brief motivation of the notation that we use. Although we only introduce Stochastic Differential Equations (SDEs) in the following Chapter4, we bear in mind that these describe how the underlying asset value, price or rate moves with respect to time and a random factor. This randomness can be accounted for by the well known Brownian Motion (BM) whose properties are explained in Definition 3.2. Furthermore, the correlation - or connection - between the foreign exchange rate and its corresponding proxy asset value is demonstrated in the modelling by including the same portion of randomness (the BM) for each of these connected processes at the same time in the future. This is of most use when modelling and simulating the future values of these specific processes.

This Chapter also formally defines the portfolio, adapting it to our requirements and consolidating some of its most useful characteristics. Lastly, we consider the effect of the complete markets that we work with and how they effect our measures. Once we have introduced this notation, we continue to show the existence and need for an equivalent martingale measure P*_.

3.1 Motivation & Preliminaries

We start by introducing the mathematical framework of the environment that we are working with. This is well defined by Karatzas [9] and Protter [10].

Definition 3.1 (The Probability Space). Let (Ω, ℱ, P) be the probability space where

Ω denotes the set of all possible outcomes, ℱ denotes the filtration which is understood as a set containing at least {𝜑, Ω} and where P is the real-world measure which assigns probabilities to the occurrence of each of the events in Ω. Furthermore, the filtration (ℱ𝑡)𝑡∈𝑇 is a family of 𝜎-fields indexed by 𝑡 satisfying

ℱ_𝑠⊂ ℱ_𝑡⊂ ℱ for 𝑠 ≤ 𝑡 & 𝑠, 𝑡 ∈ [0, 𝑇 ].

Some of the assumptions and descriptions used throughout this paper include:

• The foreign exchange market includes the exchange rate of currencies when com-pared to each other, where these currencies can be traded at any time 𝑡 ∈ [0, ∞)

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and are all measurable with respect to the filtration ℱ. Suppose we are only in-terested in 𝑑 + 1 currencies, all of them described by the relative exchange rate when compared to the USD. We denote each currency’s exchange rate at time 𝑡 by 𝑋𝑖

𝑡 where 𝑖 ∈ [0, 𝑑]. Thus 𝑋𝑡𝑖 is adapted to ℱ𝑡 and for the foreseeable future, it is possible to buy or sell these currencies with little effort. Our assumption here tends from the implication that these markets are fully complete and liquid, this is where the values contain all public information with little or no transaction costs and there exists a value for every currency compared to any other currency in the foreign exchange market.

• We start with the simplistic case of considering the effect of time movements, where each time step is discrete and can be represented as a sequence of future time flows 0 ≤ 𝑡1 < 𝑡2 < · · · < 𝑡𝑛 = 𝑇 , where 𝑇 ∈ (0, ∞) can be any future finite date which gives the time interval [0, 𝑇 ] in which we are concerned with market movements.

• During this thesis, many of these stochastic processes can be seen as dynamic models. This is due to the possibility that at any time step in the future, the value may fluctuate and these changes are unknown. Specifically, we use dynamic models in our portfolio theory as the building blocks to compensate for changes in the expected future value of both the foreign exchange rates, as well as the portfolio itself.

• These dynamic models, or stochastic processes, are assumed to follow the “memo-ryless” property, which entails that their future change or movement only depends on the value today and not on any of its past movements. This is also called the Markov property and can be seen under the real world measure P as

P(︀𝑋𝑡+1= 𝑥𝑡+1|𝑋𝑡= 𝑥𝑡, . . . , 𝑋1= 𝑥1 )︀

= P(︀𝑋𝑡+1= 𝑥𝑡+1|𝑋𝑡= 𝑥𝑡 )︀

This is of great use when using simulations to estimate the path of future values, as in Chapter 7.

Brownian Motion

As mentioned in the introduction of this Chapter, the future value of stochastic processes is unknown and these small random motions can be estimated by using a specific concept

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of “randomness” - the Brownian Motion (BM). It is a continuous-time stochastic process and a key term for describing more complicated stochastic processes. This is seen in detail for the foreign exchange rate stochastic processes in Chapter 4.

Definition 3.2 (One-Dimensional BM). A one-dimensional Brownian Motion seen as

𝑊 = (𝑊𝑡)𝑡∈[0,∞), can be represented when all values 𝑊 are R-valued and measurable with respect to the filtration ℱ defined in the probability space. It also needs to satisfy the following conditions:

1. 𝑊0 = 0, which means that there is no unexpected randomness at present time. 2. The path 𝑡 ↦→ 𝑊𝑡 is almost surely continuous, which allows this factor to be

observed at any time 𝑡 > 0.

3. The time step (𝑊𝑡− 𝑊𝑠), for 𝑡 > 𝑠, has a normal distribution with mean zero and variance (𝑡 − 𝑠). Thus 𝑊𝑡− 𝑊0 = 𝑊𝑡∼ 𝒩(0, 𝑡).

4. The Brownian Motion increments between time steps are stationary and indepen-dent. Hence they are equal in distribution, so we have 𝑊𝑡+𝑠− 𝑊𝑡= 𝑊𝑑 𝑠∼ 𝒩(0, 𝑠).

Definition 3.3 (Multi-Dimensional BM). A d-dimensional BM process is a R𝑑-valued

process 𝑊𝑡 = (𝑊𝑡1, 𝑊𝑡2, . . . , 𝑊𝑡𝑑) where 𝑊𝑡1, 𝑊𝑡2, . . . , 𝑊𝑡𝑑 are independent Brownian Motions.

Interest Rate Parity

The Covered Interest Rate Parity “verges on a physical law in international finance” as described by Borio, McCauley, McGuire, Sushko in [12]. It describes the interaction between the interest rate changes in each currency in the cash money markets to describe the interaction of the forward exchange rate and spot exchange rate in terms of the same currency. The spot rate is described as the rate at which immediate settlement on a commodity, a security or a currency can occur. In our case, the latter is the relevant description used in this paper. Mathematically the relationship is

𝐹 𝑆𝑡0

= (1 + 𝑟(𝑡0, 𝑇))

(1 + ˜𝑟(𝑡0, 𝑇)) (3.1.1) where 𝐹 is the forward exchange rate in units of US Dollar per foreign currency at time T, S is the corresponding spot exchange rate, 𝑟(𝑡0, 𝑇) is the US Dollar interest rate, and ˜𝑟(𝑡0, 𝑇) is the foreign currency interest rate, both over the time interval between the

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forward and spot rates of (𝑡0, 𝑇). The basis for using the US Dollar here is an assumption maintained throughout the paper, where all currencies and assets are denoted in a value compared that of the USD. The Covered Interest Rate Parity involves future or forward contracts, as defined in the AppendixA, where the future value is given within the contract. However, the Uncovered Interest Rate Parity predicts the future foreign exchange rate instead of using a set, given value. Thus

E[𝑆𝑇]

𝑆𝑡0

= (1 + 𝑟(𝑡0, 𝑇))

(1 + ˜𝑟(𝑡0, 𝑇)) (3.1.2) where 𝑆𝑇 is the spot rate at time T, 𝑆𝑡0 is the spot rate today and both interest rates correspond to their rate through the time interval [𝑡0, 𝑇]. The Interest Rate Parity is based on the no arbitrage theory where it should not be possible to make money without taking on some risk. Thus the risk taken on by the changing interest rate should be proportional to the gain in exchange rate over the same period.

There is no theoretical difference between covered and uncovered interest rate parity when the forward and expected spot rates are the same. This is seen when comparing Equation (3.1.1) and Equations (3.1.2), for 𝐹 = E[𝑆𝑇]. This is important when choosing the characteristics of the foreign exchange rate models since changes in the interest rate of either or both countries will proportionally effect the changes in the exchange rate. One of the largest problems in estimating this forward rate is that we cannot always observe or predict the future interest rates.

Short Rate

The short rate (𝑟𝑡)𝑡∈[0,𝑇 ]is generally described as the interest rate over an infinitely small period of time. This leads to short-rate models, which are used to predict the future interest rate path based on the progression of the short rate. As mentioned previously, the short rate cannot be observed directly either. Since it plays a large role in arbitrage-free pricing, the consideration of an alternative method to incorporate this is largely beneficial. This leads us to money-market accounts, which are described in more detail by Filipovic [7], as a “strategy of rolling-over just-maturing bonds”. Note that

𝐵𝑡= 𝑒

∫︀𝑡

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where 𝑟𝑠is the risk-free rate of return. The stochastic differential equation on the right-hand side above will be described in more detail in Chapter 4. It also follows that

𝐵𝑇 𝐵𝑡 = 𝑒 ∫︀𝑇 𝑡 𝑟𝑠𝑑𝑠 and equivalently, 𝐵𝑡 𝐵𝑇 = 𝑒 −∫︀𝑇 𝑡 𝑟𝑠𝑑𝑠

Furthermore, if the short rate is determined at the start of the period and remains constant, we have

𝐵𝑇

𝐵𝑡 = 𝑒 𝑟𝑡(𝑇 −𝑡)_.

Numeraire

A numeraire can be most beneficial in the financial world. It is described as any strictly positive (ℱ𝑡)𝑡∈[0,∞)-adapted stochastic process which can be denoted by (𝑁𝑡)𝑡 ∈ [0, ∞), that is taken as a unit of reference when pricing an exchange rate or an asset. The most basic and commonly used of numeraires is using one currency exchange as a base (the pri-mary currency), such as the US Dollar. Other typical numeraires include money-market accounts and zero-coupon government bonds with suitable maturities. It is possible to change the numeraire to facilitate the valuing of derivatives such as forwards or options. An example would be using a bond with the same maturity as the derivative. For exam-ple, a 𝑇 −bond, whose maturity is at time 𝑇 , where the implied interest rate, through the duration of the bond, can be observed using reverse calculations.

The Dot Product

The method in which we take two equal length sequences of numbers and multiplies such that the output is just one number. Suppose that we have two (𝑑 + 1)-vectors A and B, thus each containing (𝑑 + 1) values, the multiplication of the two described by the dot product · is

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐴0_𝑡 𝐴1 𝑡 ... 𝐴𝑑_𝑡 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ · ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐵_𝑡0 𝐵1 𝑡 ... 𝐵_𝑡𝑑 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ = 𝑑 ∑︁ 𝑖=0 (𝐴𝑖 𝑡× 𝐵𝑡𝑖). (3.1.3)

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3.2 Portfolio Theory

We introduced our portfolio at the beginning of this paper. However now we define it again more formally as the composition of different amounts of foreign currencies whose value is based in US Dollar (USD). We label this portfolio 𝐸 = (𝐸𝑡)𝑡∈[𝑡0,𝑇 ], consisting of (𝑑 + 1)−foreign currencies, which satisfy

𝐸𝑡= 𝜉𝑡· 𝑋𝑡 (3.2.1)

where 𝜉𝑡 denotes the amount held in each foreign currency at time 𝑡, 𝑋𝑡 denotes the foreign exchange rate based against the USD and · denotes the dot product described in Equation (3.1.3). Both are seen as (𝑑 + 1)−vectors. So

𝜉𝑡= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 𝜉_𝑡0 𝜉_𝑡1 ... 𝜉_𝑡𝑑 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , 𝑋𝑡= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 𝑋_𝑡0 𝑋_𝑡1 ... 𝑋_𝑡𝑑. ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

The process 𝜉𝑡is assumed to be adapted with respect to the filtration ℱ𝑡−1whilst 𝑋𝑡is assumed to be adapted with respect to the filtration ℱ𝑡. Thus the 𝜉𝑡are known one time step before the 𝑋𝑡. Since the filtration includes information from the past and present only, these processes cannot “see into the future” and reveal what is to come. However for the duration of this thesis, the amount of each foreign currency held, represented in 𝜉𝑡, will remain constant throughout time. Thus 𝜉𝑡 is predictable with respect to the filtration ℱ𝑡 since it is possible to know what its value will be at the next time step.

Note that for each 𝑖 ∈ [0, 𝑑], the foreign currency exchange rate 𝑋𝑖 _{is dynamic. This} is due to the continuous movements of the rate in the future and the inability of knowing where it will be. We consider each 𝑋𝑖 _{as its own stochastic process, the way we model} these are explained in Chapter 4. It is common to take 𝑋0

𝑡 to be non-random with

𝑋0

0 = 1. Since every foreign currency rate is described against the USD (as previously defined under the Numeraire Section 3.1), it makes sense to link this to the USD. Now any movement or value change can be related to the implied interest rate which can be interpreted by using a zero-coupon government bond or money-market rate to calculate the time value of money between each 𝑋0

𝑡+1 and 𝑋𝑡0 for 𝑡 ∈ [0, 𝑇 − 1]. We explicitly state that for each 𝑖 ∈ [0, 𝑑], the process (𝑋𝑖

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which its value at the next time step is based purely on today’s value and not any of the historical/past values at times (𝑡 − 1, 𝑡 − 2, . . . ).

Definition 3.4 (Self-Financing Portfolios). A portfolio process is self-financing if there

are no external cash flows influencing the portfolio, such that:

𝜉𝑡· 𝑋𝑡= 𝜉𝑡+1· 𝑋𝑡 𝑡 ∈0, 1, . . . , 𝑇 − 1.

Our portfolio 𝐸 of currencies are in line with Definition 3.4 as we have assumed that for the time period [0, 𝑇 ], there are no changes in the 𝜉𝑡. For any 𝑡 ∈ [0, 𝑇 − 1], it holds that

¯𝜉𝑡= ¯𝜉𝑡+1

A holder of a portfolio places high importance on whether their investment in foreign currencies are making a return. These returns are referred to as the gain or loss realized by the portfolio. These gains can be represented by their own process, say (𝐺𝑡)𝑡∈[𝑡0,𝑇 ], which is dependent on the movements from entry 𝑡0 of the portfolio to its hypothetical maturity time 𝑡. It is possible to calculate these at any time 𝑡 within the portfolio denoted time interval [𝑡0, 𝑇], as

𝐺𝑡= 𝑡 ∑︁ 𝑗=𝑡0

𝜉𝑗·Δ𝑋𝑗 (3.2.2)

where Δ𝑋𝑗 = 𝑋𝑗 − 𝑋𝑗−1 for 𝑗 ≥ 𝑡0+ 1 and Δ𝑋𝑡0 = 0. It is worth mentioning that a positive Δ𝑋 characterizes the appreciation (or depreciation) of the foreign currency/cur-rencies within the portfolio and therefore increases (or decreases) our USD denominated value of the portfolio. Since a gain and a loss are complete opposites, a negative gain can be measured as a loss. In some applications, a loss is modelled as a positive value even though its association of losing money is negative.

A final note is the implication of the US Dollar as a numeraire. Since we already took 𝑋0 _{= $1 per US dollar and all currency exchange rates are in terms of USD, the} use of a USD as a numeraire is practically inherent. The discounted process (𝐷𝑖

𝑡)𝑡∈[𝑡0,𝑇 ] and for 𝑖 = 0, . . . , 𝑑 can be defined by

𝐷𝑖_𝑡= 𝑋

𝑖 𝑡

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It is clear to see that this does not change the original process 𝑋𝑖 _{= (𝑋}𝐼

𝑡)𝑡∈[𝑡0,𝑇 ] of each foreign exchange rate. We further note that the 1 unit of currency in US Dollars grows over time depending of the short rate of the market. The implication is

𝑋_𝑡0 = 𝑒 ∫︀𝑡 0𝑟 ℎ 𝑠𝑑𝑠_𝑋0 0 = 1 𝐷(0, 𝑡)𝑋 0 0

when we have a stochastic short rate model (𝑟𝑡)𝑡∈[𝑡0,𝑇 ]and the zero-coupon bond 𝐷(0, 𝑡) described in Section 2.4, is with respect to the USD.

Lemma 3.1 (Arbitrage Opportunity). A self-financing trading strategy is an arbitrage

opportunity if the USD denominated value of the portfolio E satisfies 𝐸0 ≤0, P(𝐸𝑇 ≥0) = 1 & P(𝐸𝑇 >0) > 0

A market is called arbitrage free if these arbitrage opportunities do not exist.

This assumption is necessary to determine a fair and sensible pricing system in the financial world.

Remark. A probability measure P* is called a martingale measure (or risk-neutral measure) if the process (𝐷𝑖

𝑡)𝑡∈[𝑡0,𝑇 ] is a martingale under P

*_{. Then it holds that}

𝐷𝑖₀= 𝐷(0, 𝑡)E*[𝐷𝑖_𝑡]

where 𝐷(0, 𝑡) denotes the discount rate proposed by the zero-coupon government bond and E* _{describes the expectation measure under P}*_{. These martingales are explained in} Section4.1.

Forwards & Contingent Claims

These financial derivatives are described in more detail in the Appendix A with their uses. We draw attention to the capability of hedging the risk involved with exchange rate fluctuations. It is possible to “lock-in” a price today for delivery at a future value date using these instruments.

A forward contract on a foreign exchange rate (𝑋𝑡)𝑡∈[𝑡0,𝑇 ] has a linear payoff and is directly related to the spot price at the time of entering the contract and the interest rate differential, between the two currencies concerned, to the time of maturity. The

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strike price 𝐾 is the agreed rate in which the contract is fulfilled and the exchange occurs. The final payoff at maturity is

(𝑋𝑇 − 𝐾).

It is an ℱ𝑇-measurable random variable, but can be positive (defining a gain) or negative (defining a loss).

A contingent claim C is a nonnegative ℱ𝑇-measurable random variable. It is a financial derivative based on the underlying asset. C is measurable with respect to the 𝜎-algebra

𝜎(𝑋0, 𝑋1, . . . , 𝑋𝑇). Thus there exists a Borel function 𝑓 : (R𝑑+1)𝑇 + 1 → R such that

𝐶= 𝑓(𝑋0, 𝑋1, . . . , 𝑋𝑇).

The benefit of forward contracts and contingent claims is that they are generally liquid instruments and easily attainable in liquid or developed markets. However, this is not the case in many Frontier markets, which gives rise to the need for proxy hedging.

3.3 Completeness & Risk-Neutral Measures

Before considering how to model these dynamic exchange rate processes in Chapter 4, the market conditions need to be considered. The market should be arbitrage-free and complete, two notions that we will define here. The foreign exchange market and derivative markets in developed countries, described in Section 2.1 and Section 2.2 respectively, fulfill both of these conditions. The Fundamental Theorems of Asset Pricing describe the helpful implications of these where the term asset can be interchanged with a foreign exchange rate.

Radon-Nikodym Derivative

The basic property of the Radon-Nikodym Derivative shown below is needed to finish this chapter. We can briefly explain this using two measures P and P* _{where it holds} that P*(𝐴) = ∫︁ 𝐴 1𝑑P* ₌ ∫︁ 𝐴 𝑑P* 𝑑P𝑑P = ∫︁ 𝐴 𝑍𝑑P where 𝑍 = 𝑑P*

𝑑P is seen as the Radon-Nikodym derivative of P

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Equivalent Martingale Measures

We denote 𝒫 as the set of all equivalent risk-neutral measures on ℱ. By definition of the term equivalent, we consider any other probability measure that has the same null sets as the original P. To show this, let P and P* _{be two probability measures on the} same (Ω, ℱ), then they are equivalent (P ∼ P*_{) if and only if for any event 𝐴 ∈ ℱ we} have

P(𝐴) = 0 ⇔ P*(𝐴) = 0

The Radon-Nikodym Theorem above shows the existence of the derivative 𝑍 = 𝑑P* 𝑑P. Then using the definition of equivalency, we see that 𝑍 > 0 almost surely implies

1

𝑍 = 𝑑P𝑑P* >0 almost surely. Thus the following are equivalent: • P* _{is absolutely continuous with respect to P}

• P is absolutely continuous with respect to P*

Then for any risk neutral measure P*_{, similarly seen by Remark}_{3.2, for all values 𝑋}𝑖 _it follows, for 𝑡 ∈ [𝑡0, 𝑇], 𝑋₀𝑖 = 𝐷𝑖(0, 𝑡)E* [︁ 𝑋_𝑡𝑖 ]︁ .

3.3.1 Complete and Incomplete Markets

A market is considered to be complete if every contingent claim is attainable.

Definition 3.5(Attainable). A contingent claim (𝐶𝑡)𝑡∈[𝑡0,𝑇 ]with underlying (𝑋𝑡)𝑡∈[𝑡0,𝑇 ], as defined in Section3.2, is attainable if there exists a replicating self-financing trading strategy (Ψ𝑡)𝑡∈[𝑡0,𝑇 ], described by Definition 3.4, such that

𝐶𝑡= Ψ𝑡· 𝑋𝑡 where 𝑋0

𝑡 denotes the risk-less exchange rate process and the remaining represents the risky exchange rate processes.

From results in portfolio theory, it follows:

Theorem 3.1(Fundamental Theorem of Asset Pricing I). A market is free of arbitrage

if and only if the set 𝒫 is nonempty. Thus there exists at least one P*∼ P such that the

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Theorem 3.2 (Fundamental Theorem of Asset Pricing II). An arbitrage free market

is complete if and only if there exists exactly one risk neutral measure, then 𝒫 = {P*}

and dim 𝐿0_{(Ω, ℱ, P) ≤ (𝑑 + 1)}𝑇_.

In liquid markets, every contingent claim has its own unique arbitrage-free price and is freely available to buy, sell or speculate on. For the purpose of hedging, we need every underlying foreign exchange rate or asset and its respective derivatives to be traded or transacted in a complete market. This allows for every claim to be potentially attainable and a hedge to be undertaken. Since our portfolio 𝐸 consists of multiple foreign currencies, consideration of countries with Frontier markets need to be taken into account. In a country with a developing economy, the currency is not always easily trade-able due to restrictions such as capital restraints or if there is no availability in the market. This implies that the market is not fully liquid, and it creates a greater risk when holding such assets and simultaneously decreases the chance of protecting ourselves from the negative effect from a disruption in the market. Therefore, our idea is to use a proxy whose financial derivatives are contained in a complete, arbitrage-free market as an alternative possibility when attempting to hedge.

During this thesis we are working with dynamic models where we have more than one time frame to consider and the underlying foreign exchange rate can change at any time step. In the dynamic model sense, we define the theorems below using:

• The probability space (Ω, ℱ, P) and a discrete time set T = {0, 1, . . . , 𝑇 − 1, 𝑇 } where 0 < 𝑇 < ∞ and 𝑇 ∈ Z+_.

• Furthermore, the filtration ℱ = (︀ℱ𝑡 )︀

𝑡∈𝑇 includes every P-null set and thus their complements.

In both Fundamental Theorems of Asset Pricing, the “asset” can be interchanged with a foreign exchange rate. For the remainder of this paper, we assume to be working within complete markets and so there is only one risk neutral measure which can be used. These markets include the foreign exchange market and the highly liquid derivative markets based on more developed currencies such as US Dollar and Euro. Also, any asset that we consider as a proxy needs to meet the criteria that it is within a complete market as well as its derivative market. These assumptions remain throughout the following chapters, where we use P* _{as the unique risk neutral measure given. The proofs of Theorem} _3.1 and Theorem 3.2are omitted since they are easily found in most mathematical finance textbooks such as [14] and [15].

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4 Stochastic Processes

A stochastic process can be seen as a family of random variables whose values are chang-ing at any time point in the future. These changes can be described by its Stochastic Differential Equation (SDE) which includes the Brownian Motion process described in Section 3.1. Ito’s Formula, described in the Appendix B, is key in finding a suitable solution. Where it is possible, we use well known SDEs whose properties satisfy the conditions in which the solution exists, is unique and can be derived. In many risk mea-sures, which will be detailed in Chapter6, it is possible to use moments such as variance in the measurement of risk associated with a certain process or portfolio. The following SDEs are assumed under the risk neutral measure P* _{whose expectation is denoted E} for the duration of this paper.

4.1 Martingales, Variance and Covariance

In this Section, we describe martingales especially with the intention of applying them to foreign exchange rates and Brownian Motion. Some of their properties give us the ability to simplify the process to find a solution. These explicit solutions can help us in the derivation of risk, for example via the calculation of the variance 𝜎2 _{= E[𝑋}2_]−E[𝑋]2_.

Definition 4.1 (Martingale). A stochastic process X is a ℱ𝑡-martingale if 1. X is adapted to the filtration

2. E[︀|𝑋𝑡|]︀ < ∞for all 𝑡 ∈ [0, 𝑇 ] 3. E[︀𝑋𝑡|ℱ𝑠

]︀

= 𝑋𝑠 for 𝑠 ≤ 𝑡 and 𝑠, 𝑡 ∈ [0, 𝑇 ]

Definition 4.2 (Square Integrable Martingale). A martingale (𝑋𝑡)𝑡∈[0,𝑇 ] is square in-tegrable if for all 𝑡, we have

E[︀𝑋𝑡2]︀ < ∞. (4.1.1) Following these two definitions, it is clear that the Brownian Motion (BM) process (𝑊𝑡)𝑡∈[𝑡0,𝑇 ] (Definition 3.2) is not only a martingale, but square integrable too under the filtration ℱ if and only if ℱ includes the natural filtration ℱ𝑊_{. When looking for} the expected future value of foreign exchange rates, it is incredibly helpful to have a process which is a martingale. Due to the continuous nature of the foreign exchange rate, it is possible to view exchange rates at almost any time. Therefore, it is possible

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to break down the time interval [𝑡0, 𝑇] into n discrete times with equal spacing between them. Then each timestep delta equals (︀𝑇 −𝑡0

𝑛 )︀

. We formalize this definition below.

Definition 4.3 (The Mesh). If we have a discrete time frame of 0 ≤ 𝑡0 < 𝑡1 < · · · <

𝑡𝑛= 𝑇 of equidistant spacing, that is 𝑡𝑖+1− 𝑡𝑖= 𝛿, then the mesh can be defined as ||Π|| = max

0≤𝑖≤𝑛−1(︀𝑡𝑖+1− 𝑡𝑖)︀. (4.1.2) In the situation with equidistant points this equals 𝛿. It is possible to estimate a con-tinuous stochastic process with discrete time points. This is done by reducing the size of the time deltas so that we increase the number of discrete time points until the time steps are close to the continuous situation. Mathematically, this is done by choosing the time intervals such that the mesh in Equation (4.1.2) tends to zero, as 𝑛 → ∞. This also enables us to solve for the quadratic variation and covariation using the discrete process which will be as close to the true value as possible.

Definition 4.4 (Quadratic Variation). If we have a foreign exchange rate process

(𝑋𝑡)𝑡∈[0,𝑇 ]with a mesh defined as above, then the quadratic variation can be defined as the pairwise measurement of variance

⟨𝑋⟩_𝑡= lim ||Π||→0 𝑛−1 ∑︁ 𝑖=0 (︀𝑋𝑡𝑖+1− 𝑋𝑡𝑖 )︀2 .

Definition 4.5 (Quadratic Covariation). If we have the same (𝑋𝑡), described in Defi-nition4.4, and a proxy asset or proxy foreign exchange rate process (𝑃𝑡), defined on the same mesh, then the quadratic covariation can be defined as the pairwise measurement of covariance between 𝑋 and 𝑃 which is

⟨𝑋, 𝑃 ⟩_𝑡= lim ||Π||→0 𝑛−1 ∑︁ 𝑖=0 (︀𝑋𝑡𝑖+1− 𝑋𝑡𝑖)︀(︀𝑃𝑡𝑖+1− 𝑃𝑡𝑖)︀.

The properties of quadratic covariation for square integrable martingales are given in the AppendixB.

We return to another definition associated with BM. Although much time is spent throughout this paper on the characteristics of the BM, it is used thoroughly when:

• Modelling the SDE and finding the solution of said SDE in Section4.2 • Representing a correlation between two processes which is seen in Section 5.5

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• Deriving a discretization scheme seen in Section7.3.

Definition 4.6 (Levy’s Characterization [7]). A stochastic process 𝑊 = (𝑊𝑡)𝑡∈[𝑡0,𝑇 ] defined on the probability space (Ω, ℱ, P) such that

1. P(𝑊0= 0) = 1

2. 𝑊𝑡 is a continuous martingale with respect to the filtration (ℱ𝑡)𝑡∈[0,𝑇 ] under the measure P*

3. The quadratic variation ⟨𝑊 ⟩𝑡= 𝑡 holds P*-a.s. then 𝑊 is a BM.

The following Lemma4.1is stated and proven due to the fact that it is possible to use a discrete time stochastic process, whose mesh becomes infinitely small, to show that its quadratic variation coincides with that of the same stochastic process in its continuous form.

Lemma 4.1 (BM Quadratic Variation). If we have the BM 𝑊 = (𝑊𝑡)𝑡∈[0,𝑇 ] on the

filtration ℱ𝑡 and the time intervals defined by the mesh on 𝑡 ∈[0, 𝑇 ], then

⟨𝑊 ⟩_𝑡= lim ||Π||→0 𝑛−1 ∑︁ 𝑖=0 (︀𝑊𝑡𝑖+1− 𝑊𝑡𝑖 )︀2 = 𝑡

Proof of Lemma 4.1. It suffices to show that 𝐵𝑛−1→ 𝑡almost surely for any 𝑡 ∈ [0, 𝑇 ].

Firstly, we define the time steps of the interval [𝑡0, 𝑡] as 0 = 𝑡0 < 𝑡1 < · · · < 𝑡𝑛 = 𝑡

satisfying 𝛿 = 𝑡 𝑛 and 𝐵𝑛−1 = 𝑛−1 ∑︁ 𝑖=0 (︀𝑊𝑡𝑖+1− 𝑊𝑡𝑖 )︀2 .

Thus 𝑡𝑖 = 𝛿𝑖 = 𝑖𝑡_𝑛 for 𝑖 = 1, . . . , 𝑛. We write

𝑊𝑡𝑖+1− 𝑊𝑡𝑖+1 = √︂

𝑡 𝑛𝑍𝑖,𝑛

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with 𝑍𝑖,𝑛 a 𝒩 (︀0, 1)︀ random variable and all 𝑍𝑖,𝑛 independent. Then we have 𝐵𝑛−1= 𝑛−1 ∑︁ 𝑖=0 (︀𝑊𝑡𝑖+1− 𝑊𝑡𝑖 )︀2 = 𝑛−1 ∑︁ 𝑖=0 (︀ √︂ 𝑡 𝑛𝑍𝑖,𝑛 )︀2 = 𝑡 𝑛 𝑛−1 ∑︁ 𝑖=0 𝑍_𝑖,𝑛2 → 𝑡 almost surely since E𝑍2 𝑖,𝑛= Var[︀𝑍𝑖,𝑛 ]︀

= 1 and so by the Strong Law of Large Numbers, we have 1 𝑛 𝑛−1 ∑︁ 𝑖=0 𝑍_𝑖,𝑛2 → E𝑍2 𝑖,𝑛 = 1.

4.2 Stochastic Differential Equations

A Stochastic Differential Equation (SDE) describes the way a process moves with respect to time and with respect to a random factor or “noise” term. Therefore a general SDE can be defined as,

𝑑𝑋𝑡= 𝜇(𝑋𝑡, 𝑡)𝑑𝑡 + 𝜎(𝑋𝑡, 𝑡)𝑑𝑊𝑡,

𝑋0 = 𝑥

(4.2.1) where 𝑊𝑡 is the R-valued Brownian Motion, both 𝜇(𝑋𝑡, 𝑡) and 𝜎(𝑋𝑡, 𝑡) can be either a R-valued constant or dependent on 𝑋𝑡, t or both. If both, then we have the functions 𝜇 and 𝜎 on R × [0, ∞) or for our thesis, R × [𝑡0, 𝑇]. This SDE explanation can be found in many stochastic processes for finance courses such as [32].

Definition 4.7 (A Solution). Using the SDE in Equation (4.2.1) with initial 𝑋0 = 𝑥, we say that (𝑋𝑡)𝑡∈[0,𝑇 ] is the solution if it satisfies

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2. 𝜇(𝑋𝑡, 𝑡) ∈ ℒ1 (︀ [0, 𝑇 ]; R𝑑)︀ and 𝜎(𝑋𝑡, 𝑡) ∈ ℒ2 (︀ [0, 𝑇 ]; R𝑑×𝑚)︀ 3. The SDE integrated becomes (P-almost surely):

𝑋𝑡= 𝑋0+ ∫︁ 𝑡 0 𝜇(𝑋𝑠, 𝑠)𝑑𝑠 + ∫︁ 𝑡 0 𝜎(𝑋𝑠, 𝑠)𝑑𝑊𝑠 and holds for every 𝑡 ∈ [0, 𝑇 ].

Definition 4.8(Strong Solution). A process 𝑋𝑡, described in Definition4.7, is a strong solution to the SDE if ∀𝑡 > 0, we have that both of the following two integrals exist:

∫︁ 𝑡 0 𝜇(𝑋𝑠, 𝑠)𝑑𝑠, ∫︁ 𝑡 0 𝜎(𝑋𝑠, 𝑠)𝑑𝑊𝑠 (*)

with (*) being an Itó Integral. An Itó Integral is an integral where for any 𝑓, ℎ ∈ ℒ2_{[𝑡, 𝑇 ],} finite constants 𝑎, 𝑏 and 𝑇 and 𝑡 ∈ [0, 𝑇 ] - we have

1. Linearity, where ∫︁ 𝑇 𝑡 (︁ 𝑎𝑓(𝑠) + 𝑏𝑔(𝑠) )︁ 𝑑𝑊𝑠= 𝑎 ∫︁ 𝑇 𝑡 𝑓(𝑠)𝑑𝑊𝑠+ 𝑏 ∫︁ 𝑇 𝑡 𝑔(𝑠)𝑑𝑊𝑠 2. Measurable with respect to the filtration ℱ𝑊

𝑇 , ∫︁ 𝑇 𝑡 𝑓(𝑠)𝑑𝑊𝑠 ∈ ℱ𝑇𝑊 3. An expectation of zero, E [︁∫︁ 𝑇 𝑡 𝑓(𝑠)𝑑𝑊𝑠]︁ = 0 4. Itó Isometry holds,

E [︁(︁∫︁ 𝑇 𝑡 𝑓(𝑠)𝑑𝑊𝑠 )︁2 ]︁ =∫︁ 𝑇 𝑡 𝑓2(𝑠)𝑑𝑡

Not all classes of SDEs have closed form explicit solutions, thus for this thesis we make an effort to entail that the solution to our SDEs, both for the short rate and the foreign exchange rates that we work with, exist and are unique. This is explained by Filipovic [7] and characterized by Theorem 4.1below. By putting special importance on this, it

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allows us to calculate an expected rate for a specific foreign exchange rate at any future time 𝑡 up until time 𝑇 . The intention of finding such a future value is to calculate the return on the portfolio. Within this return calculation, it is also possible to consider the risk associated. This is explained in Chapter 6. Of course there are many parameters and economic changes that may not be accounted for within a chosen SDE, and due to the difficulty already involved with estimating derivative prices based on a function of said foreign exchange rates or proxy assets, the accuracy of the final estimated foreign exchange rate may not be in line with its true value in the future.

Theorem 4.1 (Existence & Uniqueness). Let 𝑡 ∈ [0, 𝑇 ] with 𝑡0 = 0 and denote the

sequence of random variables (𝑋𝑡) which satisfies the SDE in Definition 4.8 with an

initial, real value 𝑋₀ = 𝑥 ∈ R. By Filipovic in [7], if the SDE satisfies the Lipschitz condition defined by

|𝜇(𝑥, 𝑡) − 𝜇(𝑧, 𝑡)| + |𝜎(𝑥, 𝑡) − 𝜎(𝑧, 𝑡)| < 𝐶 |𝑥 − 𝑧|

and the Linear Growth condition given by

|𝜇(𝑥, 𝑡)| + |𝜎(𝑥, 𝑡)| ≤ 𝐶(︀1 + |𝑥| )︀

where 𝐶 is some suitably chosen constant, provided that 𝑋₀ is independent of the Brow-nian Motion 𝑊𝑖 for every 𝑖 ∈ [0, 𝑇 ], then for every timespace initial point denoted (𝑥, 𝑡0) ∈ R𝑛×[0, 𝑇 ], there exists a unique solution 𝑋 = 𝑋(𝑥,𝑡0) of Equation (4.2.1). Since all currencies, including Frontier currencies, and proxy assets, considered in this paper are observable, even if it is difficult to estimate the true parameters, the fact is that these parameters exist. Such parameters may change over time, and they are seen as random processes themselves. However, they are considered finite whilst the currencies and proxy asset derivatives are in use. For processes such as the short rate, it is not possible to observe the spot rate. So the implied interest rate gathered from assets such a zero-coupon government bonds or money market rates may need to be used as an estimation.

Finding the Solution to a Simple SDE. We consider Equation (4.2.1), where we take 𝜇(𝑋𝑡, 𝑡) = 𝜇 and 𝜎(𝑋𝑡, 𝑡) = 𝜎. Thus the SDE simplifies to a linear one of constant, finite parameters. Thus the unique solution is found by integrating and rearranging as

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follows: ∫︁ 𝑇 0 𝑑𝑋𝑡= ∫︁ 𝑇 0 𝜇𝑑𝑡+ ∫︁ 𝑇 0 𝜎𝑑𝑊𝑡, =⇒ 𝑋𝑇 − 𝑋0 = 𝜇(𝑇 − 0) + 𝜎(𝑊𝑇 − 𝑊0) =⇒ 𝑋𝑇 = 𝑋0+ 𝜇𝑇 + 𝜎𝑊𝑇.

Theorem 4.2(Markov Property). Following from Definition4.8, any strong and unique solution(𝑋𝑡)𝑡∈[0,𝑇 ] satisfying the SDE 𝑑𝑋𝑡as described in Equation (4.2.1) satisfies the

Markov property denoted

E [︁ 𝑓(𝑋𝑥0 𝑡 ) | ℱ𝑠]︁ = E[︁𝑓(𝑋𝑡−𝑠𝑢 ) ]︁ 𝑢=𝑋𝑠𝑥0

where 𝑓 is a measurable, bounded function from R𝑛→ R and 𝑠, 𝑡 ∈ [0, 𝑇 ] for all 𝑠 ≤ 𝑡. The Markov property is described by Bayer [16] and is one of the strongest and most useful assumptions we have in the modelling of these stochastic processes. It simplifies the movement of the process from a present state into one at a future time point. It will be mentioned again in Chapter7 under the Monte Carlo Method for simulations. Remark. A heuristic approach to questioning the distribution of a 𝛿-time change of a stochastic process is based on its differential equation. If we base this on a small time change of 𝛿, the calculation of 𝑋𝑡+𝛿−𝑋𝑡is normally distributed with a mean of 𝜇(𝑋𝑡, 𝑡)𝛿 and a variance of 𝜎(𝑋𝑡, 𝑡)2𝛿. The 𝛿-change in 𝑋 is independent of the historical changes in 𝑋. Both these assumptions hold because of the fact that increments of a Brownian Motion are independent and normally distributed as described in Definition 3.2. This Remark will be reintroduced as an assumption when choosing a correlation method in Section5.5.

4.2.1 Short Rate Model

For the duration of this thesis, we have considered a finite time horizon [𝑡0, 𝑇] for our hedging purposes. To characterize the stochastic differential equation, we will need to

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consider a drift and volatility term for each. We proceed by considering

𝑑𝑟_𝑡ℎ = (𝑏 + 𝛽𝑟ℎ_𝑡)𝑑𝑡 + 𝜎_𝑡ℎ𝑑𝑊𝑡 (4.2.2)

𝑑𝑟_𝑡𝑑= 𝜇𝑑_𝑡𝑑𝑡+ 𝜎𝑑𝑑𝑊_𝑡* (4.2.3)

where 𝑟ℎ _{is the short rate for the hard or primary currency which is USD and 𝑟}𝑑 _{is the} short rate for the domestic or foreign currency. It is further assumed that both BMs, 𝑊𝑡 and 𝑊*

𝑡, within the SDEs are independent and under the risk neutral measure P*of the market. We have chosen the well known Vasiçek model for the primary currency (USD) short rate which allows negative interest rates which are a possibility in the modern financial world as is currently seen with the the Euro Interbank Offer Rate (EURIBOR). The EURIBOR is defined by “the average interest rate at which European Banks offer short-term unsecured lending between themselves”. Since the Vasiçek model has a mean reverting drift, the deduction is that the rate will eventually tend back to its mean rate after a period/s of fluctuations. We chose to use the well known Ho-Lee Model for the foreign (secondary) currency since in Frontier markets, there is very little historic or present market information. Hence the possibility of a changing 𝜇(𝑡) based on time is a more efficient proposition.

Lemma 4.2 (Ho-Lee Model). The well known Ho-Lee short rate is defined by

𝑑𝑟𝑑_𝑡 = 𝜇𝑑_𝑡𝑑𝑡+ 𝜎𝑑𝑑𝑊_𝑡* whose explicit unique solution is

𝑟𝑑_𝑡 = 𝑟𝑑₀+

∫︁ 𝑡 0

𝜇𝑑_𝑠𝑑𝑠+ 𝜎𝑑𝑊_𝑡*.

Proof of Lemma 4.2. To solve for the local currency short rate equation 𝑟_𝑡𝑑, we start

with

𝑑𝑟𝑑_𝑡 = 𝜇𝑑_𝑡𝑑𝑡+ 𝜎𝑑𝑑𝑊_𝑡*

We make note that the right hand side of the SDE is not dependent on the short rate, which allows us to smoothly integrate both sides without the use of a specifically chosen

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function: ∫︁ 𝑡 0 𝑑𝑟_𝑠𝑑= ∫︁ 𝑡 0 𝜇𝑑_𝑠𝑑𝑠+ 𝜎𝑑 ∫︁ 𝑡 0 𝑑𝑊_𝑠* =⇒ 𝑟𝑑_𝑡 = 𝑟𝑑₀+ ∫︁ 𝑡 0 𝜇𝑑_𝑠𝑑𝑠+ 𝜎𝑑𝑊_𝑡*.

Lemma 4.3 (Vasiçek Model). The well known Vasiçek short rate is defined by

𝑑𝑟_𝑡ℎ = (𝑏 + 𝛽𝑟ℎ_𝑡)𝑑𝑡 + 𝜎_𝑡ℎ𝑑𝑊𝑡 (4.2.4)

whose explicit unique solution is

𝑟_𝑡ℎ = 𝑟₀ℎ𝑒𝛽𝑡+ 𝑏 𝛽[𝑒 −𝛽𝑡₋_{1] + 𝜎𝑒}𝛽𝑡 ∫︁ 𝑡 0 𝑒−𝛽𝑠𝑑𝑊𝑠.

Proof of Lemma 4.3. To solve for the primary currency short rate equation, the Vasicek

model, we start with the SDE shown in Equation (4.2.4). For the efficiency of the proof, we denote that 𝑟ℎ _{= 𝑟. We know that we need to use a suitable function and then use} Ito’s Formula given in AppendixBwhich simplifies to:

𝑑𝑓(𝑟, 𝑡) =(︁𝑓(𝑟, 𝑡) 𝑑𝑡 + 𝜇 𝑓(𝑟, 𝑡) 𝑑𝑟 + 1 2𝜎2 𝑑2𝑓(𝑟, 𝑡) 𝑑𝑟2 )︁ 𝑑𝑡+ 𝜎𝑑𝑓(𝑟, 𝑡) 𝑑𝑟 𝑑𝑊𝑡

where 𝜇 = (𝑏+𝛽𝑟). We consider the function 𝑓(𝑟, 𝑡) = 𝑟 ·𝑒−𝛽_{, we calculate the following} derivatives 𝑑𝑓 𝑑𝑡 = −𝛽 𝑟𝑒 −𝛽𝑡_, 𝑑𝑓 𝑑𝑟 = 𝑒 −𝛽𝑡 _& 𝑑2𝑓 𝑑𝑟2 = 𝑑𝑓 𝑑𝑟𝑒 −𝛽𝑡 _{= 0} Now using Ito’s Formula, we have

𝑑𝑓(𝑟, 𝑡) = 𝑑(︀𝑟 · 𝑒𝛽𝑡)︀

=(︁− 𝛽𝑟𝑒−𝛽𝑡+ (𝑏 + 𝛽𝑟)𝑒−𝛽𝑡+ 0)︁𝑑𝑡+ 𝜎(︀𝑒−𝛽𝑡)︀𝑑𝑊𝑡 = 𝑏𝑒−𝛽𝑡_𝑑𝑡_{+ 𝜎𝑒}−𝛽𝑡_𝑑𝑊

Application of Proxy Hedging in an Illiquid Market

University of Amsterdam

MSc Stochastics and Financial Mathematics

An Application of Proxy Hedging in

Emerging Markets

Abstract

Preface

Contents

1

Introduction & Preliminaries

2

Financial Markets

2.1

Foreign Exchange Market

2.2

Derivative Markets

2.3

Market Efficiency

2.4

Discounting Future Cash Flows

2.5

Portfolio Construction

2.6

Downside Risk Strategies

3

Modelling

3.1

Motivation & Preliminaries

3.2

Portfolio Theory

Forwards & Contingent Claims

3.3

Completeness & Risk-Neutral Measures

4

Stochastic Processes

4.1

Martingales, Variance and Covariance

4.2

Stochastic Differential Equations