Risk Management of Currency Portfolios. Shirish Ranjit y. 15 April Abstract

Transcription

1 Risk Management of Currency Portfolios Shirish Ranjit y 15 April 1998 Abstract We formulate a model describing a currency portfolio, and then use a `Rigorous Global Search' method to nd the minimum market value of that portfolio over a given period of time with a given probability under certain assumptions. The fundamental problem of risk management is to measure risk of a portfolio. Risk managers have been exploring ways to quantify the characteristics of risks. The Bank of International Settlement (BIS) has approved the \Value at Risk" method of measuring risk as the industry standard. Thus, we base our mathematical model on the theory of the \Value at Risk." VaR is a single number that denes the maximum loss in the market value of a portfolio that can be expected over a given time period with a probability not exceeding a given number (such as one percent). The following models are formulated to nd VaR of portfolios: 1. A variance-covariance model sheds some light on the concept of VaR. 2. The Garman-Kohlhagen Model is used to value European style foreign exchange options. The paper walks through the model that was explored, and how it was solved using the optimizing software GlobSol. The problems were solved using GlobSol, not only nding the minimum value of the given portfolio containing any nite number of instruments, but also guaranteeing that the solution is the global minimum enclosed in tight bounds. The `Results' section demonstrates that we found guaranteed global minimum of the market value of a portfolio containing currency options, within a tight bound. This value is used to nd VaR of the portfolio. This work was supported in part by Sun Microsystems y Marquette University, P.O. Box 1881, Milwaukee, WI 53201{1881 1

2 Contents 1 Introduction Sources of Risk Hardware Requirements Mathematical Model Modeling the Value at Risk for a Portfolio One Underlying Market Factor Example More than One Underlying Market Factor Generalization Modeling the Value at Risk for an Option Portfolio Modeling the Value of a European Foreign Exchange Option VaR of an Option Portfolio The General Model The Model with More than One Risk Factor Methodology: Estimating Value at Risk Interval Arithmetic and GlobSol Interval Analysis Rigorous Global Solver (GlobSol) Why Is Interval Analysis an Appropriate Tool for Risk Managers? 25 4 Results 27 5 Conclusion 29 Appendix A Denitions 32 Appendix B Using GlobSol 34 The Fortran Program template The Required Syntax The Box Data le template The Conguration les Application I The Objective Function The Program File The Data File The User dened Data File The Result from GlobSol The Interpretation of the Result Makele Application II The Objective Function The Program File The Data File The Result from GlobSol The Interpretation of the Result

3 1 Introduction Risk management has always been a critical banking issue. Due to nancial innovations, the volatility in nancial markets has increased tremendously. After the break down of the Bretton Woods System in 1971, the volatility of exchange rates increased as nations changed their exchange rate system from xed exchange rates to oating exchange rates. The increase in the volatility of exchange rates exposed all the nancial managers to a greater uncertainty in achieving nancial objectives. Similarly, rapid advances in information technology increased proprietary trading activity and heightened the emphasis on money management. At the mean time, the growing complexity of nancial products has made it more diculty to measure the risks taken by nancial institutions, as accounting and disclosure rules have failed to keep pace with nancial innovation [14]. For banks with sizable currency portfolios, increased volatility of exchange values made the problem even more acute. Recent breathtaking losses in the derivatives market prompted research into objective methods of measuring a bank's or rm's risk exposure. Any methods developed to combat risk should be able to deal with multiple risk sources and their correlations. Those methods must also recognize the asymmetry of the distributions. Due to the complexities of the many risk factors and their interaction, a multidimensional approach to risk measurement is necessary [4]. A number of valuation methods emerged, including Value at Risk (VaR) which has become a de facto industry standard [14]. In 1993, the Group of Thirty released a study [13] that recommended VaR methods as the best methods to measure risk and capital adequacy. Since then, VaR methodology has become the industry standard. Though Value at Risk may be the accepted methodology, the industry has yet to agree on specic models. The Bank for International Settlements (BIS) allows banks to use their own internal VaR models. Under prespecied conditions, VaR provides a summary of the amount of risk, 3

4 given the risk characteristics of the portfolio. Value at Risk (VaR) is a measure of the maximum loss in the market value of a portfolio within a given time interval with a probability not exceeding a given number such as 99 percent. The Bank for International Settlements (BIS) suggests a 99% condence interval and a ten-day time interval in determining adequate bank capital. BIS also requires at least one year of historic data for estimating volatilities. The VaR approach yields a single estimate of potential loss that would be exceeded in value with only 1 percent probability. As the VaR methodology has gained international acceptance, it has become more important to banks as a risk management tool. Banks will be in a much better position to adjust their holdings to reduce the potential loss if they are able to accurately assess the maximum expected overnight loss exposure. This research uses rigorous global optimization tools to analyze VaR in a portfolio of currency derivatives. The paper is organized as follows. Section 2 is devoted to mathematical modeling. Section 3 describes the methodology, technology, and GlobSol software [7] used to solve the optimization problem. Section 4 demonstrates that we found guaranteed global minimum of the market value of a portfolio, which is used to nd VaR of the portfolio. Section 5 draws a conclusion and outlines directions for future research. Appendices A and B walk through the application of the GlobSol software to solve the two portfolio management problems. 1.1 Sources of Risk Risk factors are market characteristics whose change aects the market value of a portfolio. A portfolio is exposed to many types of risks. Market risks capture the change in the market value of a portfolio due to change in market conditions, such as: Foreign exchange risks: The change in the market value of a portfolio due to change in exchange rates, 4

5 Interest rate risks: The change in the market value of a portfolio due to change in domestic or foreign interest rates, Volatility risk: The risk brought about by a change in the volatility of a spot exchange rate. 1.2 Hardware Requirements The required hardware is a machine (such as a Sun SPARC, or other UNIX box, or a PC) with f90 (Fortran 90) compiler. The speed of the solution depends on the speed of the machine. 2 Mathematical Model The section is subdivided into three main sections. The rst section (2.1) develops the general concept of Value at Risk (VaR) using variance-covariance models or a correlation matrix. The second section (2.2) describes the Garman-Kohlhagen model for valuing a currency option portfolio. The third section (2.3) outlines the model with more than one risk factor. 2.1 Modeling the Value at Risk for a Portfolio In this section, VaR is rst dened. Then a simple model for a portfolio is introduced to clarify the concept. The simple model is used to generalize VaR. Value at Risk (VaR) is a number that relates the potential loss in a portfolio to the probability of the loss. Under prespecied conditions and the given risk characteristics of the portfolio, VaR provides a useful summary of the amount at risk [4]. VaR is dened as the maximum decline in the portfolio market value that may occur within a given time interval (such as a day) with a probability not exceeding a given number (such as 99%). The VaR may be represented as: Prob(P VaR) = ; 5

6 where P is the change in portfolio value and VaR is the value at risk with a probability level, such as 0:01 for 99% probability. Hence, VaR is considered to be a one sided probability i.e. the value of the portfolio moves in an adverse direction only. In the subsequent sub-sections, the generalization of VaR is developed mathematically models using the covariance matrix One Underlying Market Factor A simple VaR model having one underlying market factor such as US$ to British Pounds, is developed in this section. Let F the factor sensitivity which incorporates the amount of holding of each currency, V the portfolio value, V the change in portfolio value, y the underlying market factor, such as interest rates, y a marginal change in the underlying market factor. The factor sensitivity is the change in the portfolio value due to a marginal change in the underlying market factor F = V=y; which can be rewritten as V = F y: If the change in the market factor is allowed to vary within a 99% condence interval, then V is the maximum loss in the market value that would be expected to incur with a 99 percent probability over night. Therefore, the volatility of the value of the portfolio is a function of the volatility of the underlying market factor 2 v = F 2 2 y: The variance (volatility) is calculated as second moment of the probability distribution function that is sample variance. If there is only one underlying market factor, then the Value at Risk for a 99% condence interval, = 2:33, can be expressed as follows: VaR =? F y =? v : 6

7 2.1.2 Example Suppose the portfolio consists of a short position in the Australian dollar (A$) and a short position in the British pound($), with the portfolio denominated in the US Dollars($). Let the annual volatility ( A$) of the A$ be 16:83% and that of the $ ($) be 8:41%, with 252 trading days annually. Let the correlation coecient of the two currencies equal 0:5. Finally, assume the factor sensitivity of the A$ to be $5000 and the factor sensitivity of the $ to be $7500. The factor sensitivity of each foreign exchange rate incorporates the total holding of that foreign exchange in the portfolio. The portfolio has $ worth of Australian dollars, and $ worth of $. Therefore, the total value of the portfolio is $1; 250; 000. The spot prices are 1:633$=$ and 0:6687$=A$ The daily volatility of the A$ is A$ d, and the daily volatility of $ is d $. The daily volatilities are calculated as follows: d A$ = A$ (252) 1=2 = 1:06%, and for the $, d $ = $ (252) 1=2 = 0:53%. The value at risk is calculated as follows. VaR 2 = [(2:33)(1:06)5000] 2 +[(2:33)(0:53)7500] 2 +2(0:5)(2:33) 2 (1:06)(0:53)(5000)(7500): VaR 2 = $ :8 Thus, VaR= $18779:01 is the maximum loss the portfolio can incur at the 99% con- dence level. In other words, there is only 1% probability that the maximum loss in the market value of the portfolio will exceed $18779:01. Thus, the portfolio can loose 1:50% of the total market value More than One Underlying Market Factor When there are more than one market factors, the problem becomes more complex, as changes in the market factors may be correlated. For two market factors, the change in market value of the portfolio is V = F y y + F z z; (1) 7

8 where the second market factor is denoted by z and its factor sensitivity is F z. The variance of the market value of the portfolio is 2 v = (F y) 2 2 y + (F z) 2 2 z + (F y)(f z ) yz y z ; (2) where F y the factor sensitivity of market factor y, which incorporates the amount of holding in y, F z the factor sensitivity of market factor z, which incorporates the amount of holding in z, 2 y the variance of y, 2 z the variance of z, yz the correlation coecient of y and z. The change in the portfolio value resulting from standard deviation changes in the underlying market factors is VaR 2 = ( F y ) 2 2 y + ( F z) 2 2 z + ( F y)( F z ) yz y z = 2 2 v : Thus, the Value at Risk is VaR =? v ; (3) where v is given by Equation (2) Generalization The Value at Risk is VaR =? v ; where v is the standard deviation of the market value of the portfolio. The VaR can be written in the general form nx nx VaR 2 = F ij ij ( i )( j ); (4) i=1 j=1 8

9 where n number of factors in the portfolio i, j 1::n 2:33 for 99% condence interval F ij factor sensitivity of i and j currency which incorporates the amount of holding of each currency ij the correlation coecient for i 6= j 1, if i = j i, j historical standard deviation of the i th and j th currency respectively. Equation (4) denes the VaR model for n market factors. For example, the market factor can be the amount of holding in a currency. Then, equation (4) is the VaR for the portfolio containing n currencies. The above model uses a covariance matrix of the portfolio to nd VaR. The VaR, in general, is the standard deviation of the market value of a portfolio times the, the number associated with a probability such as = 2:33 for 99% probability. 2.2 Modeling the Value at Risk for an Option Portfolio Options are examples of derivative securities. That is, their value is derived from some other asset, such as stocks, bonds, or, as in our case, a foreign currency(a currency other than the universal currency). Options are derived from real, tangible assets, but are merely contracts. There is a writer for every buyer of an option. There are two types of options in the currency market. One is a call, and the other is a put. A currency call option is the right, but not the obligation, to buy a sum of foreign exchange at a xed exchange rate, called the strike price, at a specied point in the future, called the expiration date. Similarly, a currency put option is the right, but not the obligation, to sell a sum of foreign exchange at a xed strike price at a specied point in the future, called the expiration date. In the options market, the seller is often called the writer of an option. The options can be classied into two styles depending on how they can be exercised. If an option can be exercised on or before the expiration date, it is called an American option. If an option can be exercised only on the expiration date, it is 9

10 called a European option. The buyer of a call or a put option, is said to be long in those options. The seller of a call or a put option, is said to be short in those options. The currency option market is probably the world's largest option market. Though most of the trading occurs in the private or over-the-counter market, the Bank of International Settlements (BIS) survey data estimated the daily volume of currency option trading at the daily value of $22 billion. The currency option market represents a small portion of the total foreign exchange trading, which is estimated more than $1 trillion dollars a day. The major money centers are London, New York, and Tokyo. In the following sections, the Garman-Kohlhagen model will be explored in detail [11]. The Garman-Kohlhagen model values foreign exchange options. First, the portfolio of foreign exchange options is valued at the spot prices. Then, the global minimum value of the portfolio is found by allowing the spot prices to change by standard deviations, in our case = 2:33 representing a 99% condence interval. The VaR of the portfolio is the dierence between the market value at the spot prices and the minimum market value when the spot prices are allowed to vary within standard deviations Modeling the Value of a European Foreign Exchange Option Of the four basic types of foreign exchange trade (spot, forward, future, and option), the option contract is arguably the most complex. Therefore, we will present a pricing model for a European style option. There are a number of option pricing models, each having its roots in the Black- Scholes Model. The Garman-Kohlhagen (G-K) model is derived from the Black-Scholes model for valuing European style foreign exchange options. The G-K model has three main assumptions, ([2] and [11]). 1. G-K assumes that there are no taxes, transactions costs, or restrictions on short 10

11 selling of either options or of currencies. All traders in both capital and foreign exchange markets are price takers, that is, no one economic agent has market power. This provides frictionless market conditions. 2. The foreign and domestic interest rates, are riskless and constant over the term of the option's life. The interest rates are expressed as continuously compounded rates. Incorporation of interest rate is the major modication G-K makes to the Black-Scholes model. 3. The G-K model assumes that the spot exchange rate S has lognormal distribution with constant standard deviation. The resulting G-K pricing model for a call option is V C = Se?r f t N(h)? Ee?r dt N h? v p t ; (5) and the model for the put option is V P =?Se?r f t N(?h) + Ee?r dt N v p t? h ; (6) where S spot rate expressed as domestic to foreign currency units E exercise or strike price of the option r d risk-free return in the domestic currency r f risk-free return in the foreign currency N cumulative normal distribution v annual volatility t time remaining to expiration of the option, in years h = ln S E + r d? r f + v2 2 t 1 v A call or put Delta () is dened as a measure of an instrument's price sensitivity to a unit or marginal, change in the underlying instrument. For example, an option delta is a measure of the change in the option's price due to a change in the underlying spot rate. The call delta for this model is pt : C = V c S = e?r f t N(h); (7) 11

12 and the put delta is P = V p S =?er f t N(?h): (8) Gamma (?) is a measure of the change in delta due to a marginal change in the underlying instrument. For example, the option Gamma is a measure of the change in the option delta due to a marginal change in the underlying spot rate. Gamma is the same for a call and put option where N 0 is the normal distribution.? = 2 V c h i S 2 = e?r t f N 0 1 (h) Sv p t ; (9) Delta, Gamma, and price volatility are assumed constant in calculating the maximum loss, in other words VaR, for a given trading period. We also assume that the no arbitrage rule holds. Figure 1 is a portfolio containing one put option. The x axis represents the spot price S1 which is allowed to change from 1:4 to 1:8. The y axis shows the changes in portfolio value due to change in the spot price. This shows that a put is valuable when the spot price is lower than the strike price (1:64). The graph is nearly linear except near the strike price. The minimum value of a put lies at the upper bound of the spot price VaR of an Option Portfolio Suppose we have a portfolio consisting of only one British pound call option, and our universal currency is the US dollar for this example. Suppose the unknown spot rate is S($=$), the option strike is E($=$) = 1:64, and there is one month to expiration. During that month, we assume the British pound Eurocurrency rate is r f % = 0:0687 and the U.S. dollar Eurocurrency rate is r d % = 0:0556. We will assume that the annual volatility of the dollar/pound spot be v% = 12% from the historical data. The spot rate will be allowed to vary = 2:33 standard deviations, that is creating 12

13 S1-0.1 Figure 1: A put option a 99% condence interval, from the current value S0. Thus the problem is dened as min V C = Se?r f t N(h)? Ee?r dt N h? v p t ; (10) such that S0? S S0 + ; where is the daily volatility of the spot price for calculating the VaR over a one day period. The solution to this problem V is the minimum market value of the portfolio. The dierence between the current market value, that is the portfolio valued at the current spot price (S0), and the minimum market value is the maximum loss the portfolio may incur with the 99% probability over a one day period. Though we create 99% condence interval for the spot price S, this may not give exactly a 99% condence interval for the global minimum value. However, for the purpose of exposition in this paper, we assume that we found the minimum value with 99% probability 1. 1 Due to the properties of interval arithmetic, 99% condence intervals in the independent variables do not translate into 99% probability for the minimum value of the portfolio, and hence for the VaR. However, it is possible to create 100% condence interval by taking the interval in which all the data fall. This will give 100% probability for VaR. 13

14 The minimum market value for the given interval is $1:35. This is the guaranteed global minimum value of portfolio holding one call option on British Pounds. Now the VaR is calculated as the current market portfolio value $1:85 minus the minimum market value of the portfolio $1:35. Thus, VaR = $1:85? $1:35 = $0:50. This shows that almost 27% of the current portfolio value can be lost if adverse conditions hold true. Figure 2 is a portfolio containing a call option. The x axis represents the spot price S1, which is allowed to change from 1:4 to 1:8. The y axis shows the changes in portfolio value due to changes in the spot price. This is the graphical representation of the optimization problem of equation (10). This shows that a call option is more valuable if the spot price move above the strike price (1:64). The graph is nearly linear except near the strike price. The minimum value of a put lies at the lower bound of the spot price S Figure 2: A call option 14

15 2.2.3 The General Model Assuming there are no correlations among foreign exchange rates, the model (10) can be extended to a portfolio containing n = nc + np number of options, nc calls and np puts, as follows. V Ci = S i e?r f t N(h)? Ee?r dt N V P i =?S i e?r f t N(?h) + Ee?r dt N h? v p t ; and v p t? h : The objective function is given by min such that Xnc! np X m Ci V Ci + m P i V P i i=1 i=1 (11) S i0? i S i S i0 + i where S i the unknown (future) spot rate of currency i which is allowed to vary within a 99% of condence interval and is expressed as universal currency to units of currency i S i0 the spot rate at time 0 of currency i expressed as universal currency to units of currency i i the daily volatility of the spot price of currency i 2.33 for 99% probability m Ci is the amount of holding in ith currency on a call option m P i is the amount of holding in ith currency on a put option. The minimum value of the objective function (11) of a portfolio containing options is found allowing all the spot prices to move independently standard deviations. The VaR of the portfolio is the dierence between the minimum market value of the portfolio in the interval [S l ; S u ], where S l lower bound of the spot price and S u is the upper bound of the spot price, and the market value of the portfolio at the current spot prices(s i0). A graph of the market value of a portfolio containing a call option in a currency S1, and another call option in another currency S2, is given in Figure 3. The spot prices of both currencies are allowed to vary from 0 to 1:5 on the x and y axes. The choice 15

16 of the interval of the spot rates is purely arbitrary as it would be informative to see how the function behaves in the wide interval. This will tell us whether the function is smooth and continuous. The gure tells how the market value of the portfolio changes as the spot rates are changed from 0 to 1:5. The combination of calls (a graph of call option is in Figure 2) and puts (a graph of a put option is in Figure 1) in a portfolio can produce a complicated surface, even though the surface may show monotonicity in some intervals. The `dip' in the graph may very well be the global minimum of the function for this interval. Hence, at higher dimension, when puts and calls are linearly combined, the surface may have many `dips' due to the curves in the graph of a put and a call, as in Figures 1 and 2. Thus, when a portfolio contains puts and calls with appropriate hedging, the summation of those instruments produces a surface with many `dips' as in an egg carton. This is due to the shapes of a call option as in Figure 2 and a put option as in Figure 1. Therefore, the combination of calls and puts in a portfolio can produce a complicated surface S S Figure 3: Two call option in two currencies 16

17 2.3 The Model with More than One Risk Factor In option model (10), the domestic and foreign interest rates and volatility are allowed to vary within = 2:33 standard deviations, i.e. within a 99% condence interval. The purpose of this variation of the model is to examine the eects of changes in risk factors on the market value of an option. A time series of both domestic and foreign interest rates is obtained to calculate the variance of each interest rate. The volatility of the forward rate is used to keep the problem simple and manageable. Here, we take either one call option or one put option for optimization. The problem can be formulated as follows p V Ci = S i e?r f it N(h)? Ee?r dit N h? v i t ; or p V P i =?S i e?r f it N(?h) + Ee?r dit N v i t? h : The objective function is either minimize the value of a call or minimize the value of a put as follows: min (m Ci V Ci ) ; or min (m P i V P i ) such that S i0? i S i S i0 + i r f0? rf r f i r f0 + rf r d0? rd r di r d0 + rd v i0? v v i v i0 + v 17

18 where S i the unknown (future) spot rate of currency i expressed as universal currency to units of currency i S i0 the spot rate at time 0 of currency i expressed as universal currency to units of currency i r f0 the foreign interest rate at time 0 r f i the unknown (future) foreign interest rate r d0 the domestic interest rate at time 0 r di the unknown (future) domestic interest rate v i0 the volatility of the forward rate at time 0 v i the unknown (future) volatility of forward rate i the daily volatility of currency i rf the daily volatility of foreign interest rates rd the daily volatility of domestic interest rates v the daily volatility of volatility of forward rate m Ci is the amount of holding in ith currency on a call option m P i is the amount of holding in ith currency on a put option. The dierence between the minimum market value of the option in the intervals [S l ; S u ], [r dl ; r du ], [r f l ; r f u ], [v l ; v u ] and the market value of the option at the current spot price, the current domestic and foreign interest rates and the current volatility, is the VaR of the option. The solution to this problem is V. A graph of the value of a portfolio containing a call option is given in Figure 4. The spot price S1 and the foreign interest rate r f are allowed to vary on the y and x axes respectively. The spot price is allowed to change from 0 to 2, while the foreign interest rate is allowed to change from 0 to 30%. The choice of the interval of the spot rate and the interest rate are purely arbitrary in order to see how the function behaves in the wide interval. This provides us information on whether the function is smooth and continuous. The graph shows that the surface descends steeply and then it becomes at. When r d and are also allowed to vary, then the shape of the objective function may become complicated. When more risk factors are allowed to vary, the surface may be at or may have steep slope. The minimum value of the portfolio very well be in the interval at which the function is at. For this type of surface, optimization methods that use interval analysis are appropriate. 18

19 rf S Figure 4: The value of portfolio, the spot price, and the foreign interest rate 3 Methodology: Estimating Value at Risk The total market value of the portfolio is evaluated in light of several risk factors. Some risk factors are allowed to vary within = 2:33 standard deviations creating 99% condence intervals for each factor, while other factors are assumed constant. For example, in the option model spot exchange rates are allowed to vary, while volatility of the spot exchange rate is assumed constant. The volatility of the spot rate is found using historical data. The BIS requires use of at least one year historical data in estimating volatilities. The objective is to determine the maximum overnight loss a bank could face on the portfolio of trades with a probability not exceeding a given number, assuming that the portfolio is illiquid overnight. Though we create 99% condence intervals for risk factors, this does not give a 99% condence interval for the calculated minimum value. However, for the purpose of exposition in this paper, we assume that we nd the minimum value with 99% probability. The maximum loss is determined for the portfolio closing position by allowing 19

20 various risk factors to change within specied ranges, and re-evaluating the market value of the portfolio. As the number of risk factors and the number of currencies represented in the portfolio increases, the theoretical shape of the portfolio frontier, i.e. the function dening the portfolio, becomes more complicated than that shown in Figures 3 and 4. Thus, the calculation of the maximum loss becomes quite dicult. Therefore, it becomes important to guarantee that a global minimum is found, as a local minimum may leave a great deal of risk hidden. For example, Swiss Bank Corporation divides the condence interval in each variable at 15 discrete points. Then they calculate the value of their portfolio using VaR models. The process assumes that the market value of the portfolio is monotonically decreasing in the interval. However, from Figure 3, it is seen that this may not be true. Hence, the Bank is exposed to hidden risk between two discrete points. For example, if the portfolio in Figure 3 is valued at 0:5 and 0:7 in S2, we fail to capture the dip around 0:6. This exposes the Bank to hidden risk with possible disastrous consequences. This problem does not arise when using interval arithmetic methods. It is not only the mathematical model that simplies the problem of nding the global minimum, but also the method for solving the problem. We use a global search method based on interval arithmetic. The interval method solves an optimization problem of the following type. Let V : S R n! R be a continuous, dierentiable function dening the total market value of a portfolio. Consider the global optimization problem, min V (s); s2s possibly with linear or nonlinear equality or inequality constraints g(s). Using interval arithmetic not only the global optimum is found but also the solution is validated within tight bounds for the minimizer S and/or the optimum value V. 20

21 3.1 Interval Arithmetic and GlobSol In this section, interval arithmetic is dened, and we explain how it is used in GlobSol to solve optimization problems. The last subsection argues why GlobSol is an eective tool for risk managers Interval Analysis What is interval arithmetic? In mathematics, real numbers are combined using real arithmetic. Interval arithmetic is a generalization of real numbers. In interval arithmetic, interval numbers are combined using interval arithmetic. Similar to real arithmetic, interval numbers replace real numbers, and interval arithmetic replaces real arithmetic [6]. Boldface or brackets denote interval quantities, for example x := [0:5; 1:0] = [x l ; x u ]. Upper case letters denote vectors and matrices. For example, let x = [1; 3] and y = [6; 9] be two intervals of real numbers. Addition of two intervals works in the following way. 1. Take the smallest numbers from intervals x and y and add those two numbers: = Take the largest numbers from intervals x and y and add those two numbers: = The resulting interval from z = x + y = [7; 12]. 4. In a practical implementation, we must round the lower bound downward and the upper bound upward to guarantee containment. The resulting interval z contains all the possible sums of the numbers from intervals x and y. The addition of one number from x and another number from y cannot lie outside interval z. Other operations on the intervals work similarly. 21

22 In general, if x = [x l ; x u ] is an interval with lower bound x l and upper bound x u and y = [y l ; y u ], the elementary operations for interval arithmetic obey x op y = fx op y j x 2 x and y 2 yg for op 2 f+;?; ; g: The most important reason this arithmetic is used in solving minimization problems is the Fundamental Theorem of Interval Analysis. Theorem 1 (Fundamental Theorem of Interval Analysis [9]) If a rational function is evaluated using interval arithmetic instead of oating-point arithmetic, the resulting interval is guaranteed to enclose the range of function values. A consequence is that a global minimum can never be discarded. Surveys of interval analysis appear in Global Optimization Using Interval Analysis [6] or in Rigorous Global Search: Continuous Problems [7]. The following is an example of how the rigorous global search method discards areas where minima could not possibly lie. The goal of this exercise is to show how interval arithmetic computations can guarantee to enclose the range of a function. If f : R! R, f(x) := ff(x) : x 2 xg. An interval extension f : IR! IR is f(x) := ff(x) : x 2 xg, for all intervals x IR. To illustrate interval arithmetic, let f : R 2! R be the Rosenbrock function f(x; y) := 100? y? x (1? x) 2. Consider the interval vector X := (x; y) := ([0:7; 1]; [0:0; 0:3]) [1]. Then for x 2 x and y 2 y, f(x; y) = 100 y? x (1? x) = 100 y? x (1? x) 2 [0:0; 0:3]? [0:7; 1] (1? [0:7; 1]) 2 = 100 ([0:0; 0:3]? [0:49; 1]) 2 + [0:0; 0:3] 2 = 100 [?1:0;?0:19] 2 + [0:0; 0:09] = [3:61; 100:09]: 22

23 Here, we have used exact interval arithmetic. The implementation uses outwardly directed rounding to preserve containment in the presence of machine round-o errors. The point is that we have computed guaranteed lower and upper bounds for the range of f over the box [0:7; 1][0:0; 0:3] using only the operations of interval arithmetic. Since f(0; 0) = 1, we can immediately state rigorously that the box [0:7; 1] [0:0; 0:3] cannot contain a global minimum of the Rosenbrock function, and exclude it from further consideration. Here, we simply picked the point (0; 0) because it is easy to evaluate the function exactly at that point. In practice, a signicant part of the algorithm tries to pick good candidates for global minima. GlobSol also uses more sophisticated techniques to achieve tighter range bounds. Throughout the algorithm, interval arithmetic is used to give easily computable, not necessarily tight, range bounds for desired quantities Rigorous Global Solver (GlobSol) GlobSol is a software package for rigorous global optimization that uses interval analysis to compute guaranteed bounds for the range of a function. GlobSol uses interval optimization algorithms along with automatic dierentiation. GlobSol has been applied successfully in a wide variety of problems spanning many dierent industries. Various organizations such as GE, NASA, and engineering companies are using the method in their optimization problems. The optimization software is described in Rigorous Global Search: Continuous Problems [7]. GlobSol uses many dierent approaches in nding global minima. Here, we give a general overview of the technique. GlobSol computes bounds on the minimum value V of the objective function V at the point or points S. If the objective function V does not have an unique minimum, then all the global minima are enclosed in tight bounds. GlobSol works in the following way. 23

24 1. Dene a search region (box) where the global minimum is sought. 2. Partition the search region and discard areas which cannot possibly contain the global minimum. There are various methods for discarding areas (boxes). A few methods will be discussed below. 3. Try to prove there is a local minimum in a small box. 4. Iterate step 2 until a suciently small bound is obtained for the minimum or minima of the objective function V. First of all, the initial search region is partitioned in smaller boxes which are kept in a list L. Then, various methods are used to either throw the boxes away or shrink it to smaller ones. Generally, a solution is sought in a single box. The method for discarding boxes are as follows. Obtain various possible candidates for minimum. Then evaluate the objective function V and nd the smallest V. The V is an upper bound for V. Now, boxes with V > V can be discarded as they cannot possibly contain the minimum. Discard those boxes where the gradient g of the objective function V is nonzero. Since the minimum occurs at g = 0, the boxes that do not contain the gradient g = 0 can be discarded. One of the condition for minimization is that in the neighborhood of minimum, the function is convex. Hence, the boxes where V is not convex can be discarded. An Interval Newton method is used to verify existence of a unique local minimum in a box as well as to tighten the bounds when possible. The above methods are used repeatedly until the global minimum for the objective function V is found with a tight bound or minima are found for V with tight bounds. 24

25 3.1.3 Why Is Interval Analysis an Appropriate Tool for Risk Managers? Interval analysis provides an appropriate method to solve risk management problems because most of the risk management problems in nance involve valuing a portfolio by allowing some risk factors to change within an interval. In this case, the risk factors for foreign exchange options are allowed to change within 99% condence interval, and the portfolio is minimized within that interval. Since the problem involves nding the minimum value of a function dening a portfolio in an interval, interval analysis is the appropriate method to solve risk management problems. The major reason to use the method and hence GlobSol software is that it guarantees the solution to be the global minimum within a tight bound. In risk management, it is imperative that we nd the global minimum, not the local minimum. If we found just the local minimum, there may be great deal of hidden loss which in time may lead a bank or a nancial institution to fail. For example, Swiss Bank values their portfolio at 15 points within 99% condence interval of each risk factor. It could be very well that the objective function may `dip' below between two chosen points. For example, if the portfolio in Figure 3 is valued at 0:5 and 0:7 in S2, we fail to capture the dip around 0:6. Hence, the risk manager fail to capture the possible global minimum. Thus, GlobSol will be a useful tool for a risk managers on Wall Street. Risk managers are dealing with complicated functions which may have many minima on a at surface (e.g. Figure 4). For example, a function dening a portfolio may look like an egg carton, or a nearly at minimum when the portfolio contains combination of calls and puts as in Figures 1 and 2. GlobSol is highly eective in not only solving these types of problems but guarantees that the global minimum has been found within a tight bound. These types of problems are very hard and virtually impossible to solve using real arithmetic. Many mathematicians stated that such a guarantee was impossible. Their argument was that optimization algorithms can sam- 25

26 ple the objective function and perhaps some of its derivatives at only a nite number of distinct points. Hence, there is no guarantee that a function to be minimized does not dip to smaller value between sample points [6]. Therefore, no algorithms based on real arithmetic can guarantee the global minimum. However, GlobSol uses interval methods that do not depend on the sampling at points and then evaluating the function as real arithmetic does. As it depends on throwing out regions that do not contain global minima, it is not possible to skip any \dips" that are candidates for global minima. Even if the solver could not nd the exact minimum, the tight bounds guarantee that the global minimum is enclosed. Hence, the solution is guaranteed to contain any global minimum. What is the advantage to risk mangers? The important advantage is that we have the solution to our objective function which is very hard to solve using real arithmetic. The solution is guaranteed to be a global minimum which is a very important fact for the risk managers. Lastly, we get a bound for solutions. This bound tells that the global minimum cannot be less than the lower bound. This is another important fact for the risk managers because, by profession, risk managers' job is to estimate the minimum value of the portfolio. The lower bound of the solution tells a risk manager that it is absolutely not possible for the portfolio to have value less than the lower bound under the given conditions. So even though a 99% condence is used to meet BIS standard, the lower bound of the solution will give a greater condence level than 99% condence level. Hence, the lower bound of the solution has a greater importance to risk managers as it is the minimum value the portfolio can have, given the risk factors. For further readings in interval analysis, refer to the World-Wide-Web site The web site is a collection of links to many sites that do research on interval analysis. 26

27 4 Results In this section, we demonstrate that we found guaranteed global minimum of the market value of a portfolio, and hence VaR of the portfolio with a 99% probability. We use a portfolio containing two calls and two puts as an example explained in detail in Appendix B Application I. The portfolio contains a call and a put on British Pounds (BP) and a call and a put on German Marks(DM). The universal currency is US $. The portfolio has 100 of each call and put. The strike price for BP is $1:64=$, while the strike price for DM is $0:535=DM. The domestic interest rate is 5:56%, and the foreign interest rates are 6:87% for UK, and 3:09% for Germany. The maturity date is 30 days. The historical volatility of BP is 12%, while the historical volatility of DM is 14%. GlobSol shows that the minimum portfolio value occurs at exchange rates of one British Pound equal to $1:641 and one German Mark equal to $0:5338. The minimum market value of the portfolio is $6:19. GlobSol guarantees that this is the minimum value that the portfolio can incur within the given conditions. The market value of the portfolio at the spot price one British Pound equal to $1:633 and one German Mark equal to $0:545 is equal to $6:75. The maximum loss is $6:75? $6:19 = $0:56. We can be sure that the loss will be under $0:56 with 99% probability. The loss is about 8:3% of the current market portfolio value. The VaR tells the manager of a nancial institution that the company will loose 8:3% of their the current market value if the adverse conditions hold true. Hence, the VaR sets a standard on how much a company is willing to risk. If the risk in the portfolio is not according to company guidelines, then investors may need to hedge some risks in their portfolio to reduce the risk in the portfolio. Similarly, for the second model which is explained in detail in Application II, the portfolio contains a call option on British Pounds. The spot price, domestic interest rate, foreign interest rate, and the volatility of the forward rate are allowed to move 27

28 within 2:33 standard deviations (99% condence intervals) from their current values. The intervals of those variables are in the `Data File' section in Appendix B. The minimum value of the portfolio is found using GlobSol. The minimum occurs at spot price of one British Pound equal to $1:604, at the foreign interest rate 0:069, at domestic interest rate of 0:0548 and at the volatility of 11:75%. The minimum market value of the portfolio is $0:989. We guarantee that the global minimum is $0:989 for the given conditions. The market value of the portfolio at Spot Strike Domestic interest rate Foreign interest rate Time to expiration 30 Volatility 0.12 Number of options 100 is $1:049. Thus, the VaR is given by $1:049?$0:989 = $0:060. We can be sure that the loss will be under $0:06 with 99% probability. The loss is about 5:72% of the current market portfolio value ($1:049). If the company is not willing to loose 5:72% of the market value of the portfolio, then their investors can use various hedging methods to reduce the risks. The minimum value of the portfolio occurs at the lower bound of the spot price, the upper bound of foreign interest rate, the lower bound of domestic interest rate and the lower bound of the volatility. This veries the model of call option as the value of call option is minimum at the lower bound of spot price, the dierence between interest rates (r d? r f ) is the largest negative number, and the volatility is the smallest. Hence, the minimum value of the call option obtained from running GlobSol is dependable. In conclusion, we not only found the optimum values for function dening portfolios 28

29 guaranteeing that we found the global minimum, hence the VaR of the portfolio. 5 Conclusion Risk management has always been a critical banking issue. Risk managers have been trying to quantify the risk in complex portfolio. The market characteristics that aect the market value of a portfolio are called risk factors. Therefore, our major goal is to identify the risk factors to develop an appropriate mathematical model to measure risk of a portfolio of nancial products, specially portfolios containing currency options. There are many dierent methods to measure risk of a portfolio containing currency options. \Value at Risk", developed in early 1990, has been the norm in measuring the risk of a portfolio. Since, it has been approved by the Bank of International Settlement (BIS), the mathematical model is based on the theory of the \Value at Risk." Value at Risk is nothing but the maximum possible loss in the market value of a portfolio over a given time period with a given probability. It is the amount of currency such as dollar, that the total loss might exceed within a certain time period with a certain probability, for a portfolio [4]. The objective is to calculate total value of the portfolio in light of several risk factors; and then determine the maximum overnight (or over a given time period) loss that could incur on that portfolio of products, assuming that the portfolio is illiquid. The maximum loss is determined for the portfolio closing position by allowing the risk factors such as exchange rate or interest rate to change within a specied ranges. As the number of risk factors increases, and as the number of currencies represented in the portfolio increases, the mathematical shape of the portfolio becomes more complex (e.g. Figures 3 and 4). Hence, the calculation of the maximum loss becomes quite challenging. It is very important to nd the global maximum in estimating the risk of a portfolio. In mathematical jargon, risk managers are interested in nding not the local minimum but the global minimum of the mathematical function dening the 29

30 portfolio. Thus, it is very important to guarantee that the minimum found is the global minimum. The uncertainty in the global minimum will produce a doubtful nancial decision and hence possible nancial disaster. The claim is that not only the maximum loss of the given portfolio containing any nite number of any currency options is found, but also guarantee that it is the global minimum which will be enclosed in tight bounds. Future Research: The assumptions of the models can be relaxed such that the correlation between currencies are incorporated. The model could be improvement by incorporating volatility forecasting models into G-K model. A possible extension will be nding optimum hedge in the portfolio after nding VaR. The problems can be model in light of interval arithmetics. Acknowledgment I would like to thank Dr. Joe Daniels, Dr. Peter Toumano, and Dr. George Corliss for helpful comments and their assistance in making this paper. References [1] George F. Corliss, Chenyi Hu, R. Baker Kearfott, and G. William Walster. Rigorous global search { executive summary. Technical Report No. 442, Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, Wisc., April Global Solutions Working Note 1. [2] David F. Derosa. Options on Foreign Exchange. Probus Publishing Company, Chicago, [3] Darrell Due and Jun Pan. An overview of value at risk. The Journal of Derivatives, pages 7{48, Spring [4] Giord Fong and Oldrich A. Vasicek. A multidimensional framework for risk analysis. Financial Analysts Journal, pages 51{57, July/August