UNIVERSITY OF CALGARY. Crude Oil Hedging. An application of Currency Translated Options to Canada s Oil. Paul Obour A THESIS

Transcription

1 UNIVERSITY OF CALGARY Crude Oil Hedging An application of Currency Translated Options to Canada s Oil by Paul Obour A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS CALGARY, ALBERTA NOVEMBER, 2012 c Paul Obour 2012

2 Abstract This study presents a review of pricing and hedging currency translated options. It is intended for Canadian Oil producers seeking to mitigate their production and F/X risks. Currency translated options are options based upon a foreign asset but with a payout that occurs in another currency. Different types of currency translated options are covered: Flexos, Compos or joint options and Quantos. A special emphasis is placed on the comparison between the first and the third versions of this product since the latter is the only version that completely eliminates the currency risk to the commodity/equity investor. Any oil producer or consumer can diversify its risks by transforming its complete dependence on spot oil prices into a variety of exposures to forward, futures, and options markets. In the light of these transformations, we analyze the efficiency of linearly delta hedging with Flexos and Quantos and further examine the hedging implications if any. i

3 Acknowledgements It would not have been possible to write this thesis without the help and support of the kind people around me, to only some of whom it is possible to give particular mention here. First and foremost, my utmost gratitude to Dr. Antony F. Ware, whose patience and steadfast encouragement I have relied upon. Dr. Ware has been my inspiration as I hurdle all the obstacles in the completion of this research work. I would like to acknowledge the financial, academic and technical support from the Department of Mathematics & Statistics and its staff, partuicularly in the award of a Graduate Research Studentship that provided an additional financial support for this research. Special thanks to Dr. Anatoliy Swishchuk for partnering with PRMIA Calgary Chapter in organizing graduate research presentations to the Calgary Business community. My selection to present at the Final Event has had ripple benefits on my career. I would also like to thank my colleagues in the Financial Mathematics Lab, both past and present for sharing literature and invaluable assistance. Finally, I wish to express my love and gratitude to my beloved family; in particular my mum Theresa; my wife Ellen; for their understanding and endless love, through the duration of my studies in Calgary. ii

4 Table of Contents Abstract i Acknowledgements ii Table of Contents iii List of Tables v List of Figures vi List of Symbols vii 1 Introduction Hedging Instruments Purposes of Hedging Crude Oil Why Currency Translated Options? Currency Translated Options Flexible Exchange Rate Version - Flexos Composite or Joint Options - Compos Fixed Exchange Rate Version - Quantos Applications of Currency Translated Options Underlying Components Data Sets March 2009 Delivery June 2010 Delivery December 2011 Delivery Descriptive & Inferential Statistics March 2009 Delivery June 2010 Delivery December 2011 Delivery Estimation of Correlation Between WTI and USD-CAD F/X Log-returns Volatility Stationarity When WTI and F/X Returns are Normal But Not Jointly Normal Evidence of Joint Normality-Test of bivariate normality Using the Bivariate Copula Pricing Currency Translated Options The Black Model The Value of a Forward Contract, f t The Economy - Dynamics of the Domestic & Foreign Markets Pricing The Flexo Currency Translated Option Pricing the Compo Currency Translated Option Pricing the Quanto Currency Translated Option Hedging Currency Translated Options Linearly Delta Hedging Hedging the Flexo Currency Translated Options Hedging the Compo Currency Translated Options Hedging the Quanto Currency Translated Option iii

5 5.2 Computing the Profit and Loss (P&L) from a Hedging Strategy The Flexo P&L The Quanto P&L Exploring a Discrete - Time Delta Hedging Strategy Results Practical Issues Conclusions and Further work Conclusions Future Work Bibliography Appendix iv

6 List of Tables 2.1 Foreign stock/futures index contracts as defined by their payoffs at expiration Test Statistics for normality for WTI and F/X log-returns Estimated correlations between WTI and F/X log-returns Analysis of Mardia s asymmetry skewness and kurtosis for June 2010 contract Dependence of option prices on various parameters Base Case Value Parameters for Option Price Simulation Monte Carlo Simulation for Dynamic Delta Hedging P&L of WTI Call Option using Flexo Hegde Schemes P&L of WTI Call Option using Quanto Hegde Schemes v

7 List of Figures and Illustrations 3.1 March 2009 futures & CADTI prices used in historical testing June 2010 futures & CADTI prices used in historical testing December 2011 futures & CADTI prices used in historical testing Sample log-returns for December 2011 contract Comparison of daily log-returns of WTI and USD-CAD futures for March 2009 using a Normal Distribution and Quantile-Quantile plot to a histogram Comparison of daily log-returns of WTI and USD-CAD futures for June 2010 using a Normal Distribution and Quantile-Quantile plot to a histogram Comparison of daily log-returns of WTI and USD-CAD futures for December 2011 using a Normal Distribution and Quantile-Quantile plot to a histogram Year Volatilities of WTI Log-returns on December 2011 Contract Year Volatilities of USD-CAD F/X Log-returns on December 2011 Contract Plot of ACF of WTI log-returns Verifying autocorrelation in WTI log-returns for March 2009 Contract Scatter Plot of WTI and F/X futures to detect bivariate normality Quantile Plot of WTI and F/X June 2010 Contract Empirical Copula Function for WTI and F/X June 2010 Contract Efficiency of Delta Hedging Flexos using Monte Carlo Simulation Efficiency of Delta Hedging Quantos using Monte Carlo Simulation Daily Profit/Loss for delta hedging Flexos and Quantos vi

8 List of Symbols, Abbreviations and Nomenclature Symbol t T T S t Definition current time delivery time of forward contract or expiration time of option futures maturity time stock price at time t F WTI futures price in USD; F (t, T ) X 0 K f K d r d r f σ F σ X ρ XF q acf EWMA API WTI CADTI NYMEX N(d1) N(d2) guaranteed exchange rate in CAD per USD delivery or strike price in USD delivery or strike price in CAD Canadian riskless interest rate U.S. riskless interest rate estimated return volatility of WTI futures price in USD estimated return volatility of F/X futures - USD in CAD estimated correlation coefficient between returns X and F continuous dividend rate on index Auto-correlation function Exponentially Weighted Moving Average American Petroleum Institute West Texas Intermediate Canadian version of WTI: now WCS - Western Canada Select New York Merchantile Exchange, Inc proportion of the expected value of S t that comes from S > K the probability of the stock being greater than K at time t vii

9 Chapter 1 Introduction The dictionary definition of hedging - a means of protection against a loss - is too generic to describe the variety of purposes of modern hedging programs and it is too simple to reveal the wealth of choices a company has when developing its hedging program. Canada (13%) holds the world s third largest reserves after Venezuela (20%) and Saudi Arabia (18%) with over 174 billion barrels of proven reserves. Ranked as the sixth largest crude oil producing country in the world, over 99% of Canada s oil export are sent to the United States. Canada is the largest supplier of crude oil to the United States. In the international petroleum industry, crude oil products are traded on various oil bourses based on established chemical profiles, delivery locations, and financial terms. The chemical profiles specify important properties such as the oil s API gravity. The delivery locations are usually sea ports close to the oil fields from which the crude was obtained, and the pricing is usually quoted based on F.O.B. (free on board, without consideration of final delivery costs). The three most quoted oil products are North America s WTI crude, North Sea Brent crude, and the UAE Dubai crude, and their pricing is used as a barometer for the entire petroleum industry. Since July 26, 2010, NYMEX listed a Canadian Heavy Crude Oil (Net Energy) Index Futures contract (commodity code WCC) for trading on the NYMEX and for submission for clearing through the Chicago Merchantile Exchange Inc. (CME) ClearPort. The CME has a set of flexible clearing services open to over-the-counter (OTC) market participants to substantially mitigate counterparty risk and provide neutral settlement prices across asset classes with cash settled based on the Net Energy Canadian Daily Index (CDI) for WCS. Until the launch of the WCC, Canadian Oil producers used a synthetic underlying by converting the underlying WTI price from USD to CAD. 1

10 1.1 Hedging Instruments There are various mechanisms in the financial markets that the owner of a financial asset could use to hedge its value. Enumerated below are the subsets of strategies from the thousands of hedging strategies that producers and consumers of crude oil could deploy. I. Pure futures program - selling forwards consistently. II. Pure options program - selling or buying options consistently. The sale of certain options is highly speculative for those who are not naturally long crude oil; for a producer, however, it can be a useful hedging tool, assuming that management knows what it is doing. III. Option combination programs - the world of straddles, strangles and, the socalled costless collar. IV. Exotic options programs - Asian versus American options. The majority of hegdes are executed over the counter with Asian options. V. Various insurance programs by themselves or combination with derivatives. The oldest is insurance - a contract whereby one party undertakes to indemnify another against a loss. An organized futures market is another mechanism - an association of enterprises that buy and sell the future value of physical and financial commodities under specified rules. Options contracts are an offshoot of futures whereby one of the parties in the transaction sells the other the right to buy or sell a futures contract. Swaps and swaptions are outgrowths of futures and options. These hedging instruments have been traded extensively in the oil industry since 1979 and in the natural gas industry since Trade takes place on two organized exchanges - New York Mechantile Exchange (NYMEX) and Intercontinental Exchange (ICE) - and in 2

11 a bilateral OTC market where one party designs tailor-made instruments for another. The hedging instruments differ in how they apportion commodity price risk between the parties. A futures contract by itself commits a buyer and a seller to a given future price. If the settlement price is higher than the agreed price, one party has a hedging gain and the other has a hedging loss. In the options market, the buyer pays a premium for the right to buy or sell a futures contract at a given price. Since the strikes of the options on CADTI are in Canadian dollars, there is some protection for any gains made on the WTI options. The relationship between this strike price and the futures price determines whether the option will be exercised. If it is not exercised, the option seller keeps the premium, having delivered only the insurance. If it is exercised then, by definition, the option buyer has realized a hedging gain. 1.2 Purposes of Hedging Crude Oil The hedging objectives of an asset owner, like the crude oil producer, are bound to be different from those of a service buyer, like a refinery, which processes the crude oil. In the case of the producer, risk management issues vary from the least aggressive, using risk management instruments for catastrophic insurance, to the most aggressive, using risk management instruments to expand the asset base quickly. In an extreme case, a producer may even transform the company from an energy company trying to manage its own risks to a hedgemaster managing the risks of others as part of its core business, as BP, Koch, etc have done. The majority of crude oil and natural gas producers are passive hedgers whose main motivation is an untroublesome hedging program - perhaps nothing more complicated than taking a position based on the prompt, three, six, or 12-month strip of prices on the NYMEX. The prime directive of hedging programs is to reduce volatility of earnings (or costs). Volatility is defined in statistical terms as the annualized standard deviation of the natural log of the 3

12 ratio of two successive prices. Companies that choose not to use financial market instruments for hedging submit themselves and their shareholders to the volatility of spot prices. Volatility reduction, therefore, is one of the standard measures of hedging effectiveness. 1.3 Why Currency Translated Options? This thesis seeks to examine options based upon a foreign asset where the payoff of the option occurs in another currency. The value of the contingent claim is affected by the comovement of the underlying asset price and the movement in some currency exchange rate. These products provide investors with the ability to invest in foreign assets without incurring foreign exchange risks. Thus, Canadian oil producers can hedge their underlying oil price and F/X exposure by using currency translated options (CTOs). We will then determine the hedging efficiencies of these classes of options. 4

13 Chapter 2 Currency Translated Options Investors and traders increasingly use derivatives on foreign indexes to obtain exposure to global markets. There are a variety of option styles available, each of which provides different degrees of exposure to the currency of the foreign index, see [13]. As a general overview, these products differ from standard options in that the value of the option is expressed in another currency. Currency Translated Options, CTOs, are pegged to foreign equities, with the strike price being in either foreign or domestic currency but with a payout that occurs in another currency at the exchange rate existing on expiration, see [18]. Currency translated options are offered with a variety of different features. There are four basic versions of these products; I. Foreign equity call struck in foreign currency, II. Foreign equity call struck in domestic currency, III. An equity linked foreign exchange call, and IV. A fixed exchange rate foreign equity call. Eric Reiner (1992), provided an analysis to the pricing and hedging of these products in the Black-Scholes setting. In this thesis we replace equity with commodity. We will thus consider this application to futures on commodities. 2.1 Flexible Exchange Rate Version - Flexos The simplest currency translated option is the flexible exchange rate version; which is simply an option on the foreign commodity with no protection against the movement in the currency. 5

14 Flexos are the first version of CTOs and are also referred to as Foreign Commodity Option Struck in Foreign Currency. The payoff for a call is C = max { X T t (F T t K f ), 0 }, (2.1) and a put is P = max { X T t (K f F T t ), 0 }, (2.2) where X T t is the value of the F/X futures contract at delivery, F T t is the value of the foreign commodity futures contract at t with the maturity of the futures at time T, where T T and K f is the strike price in foreign currency. The flexible element refers to the fact that the final payout of the option depends upon the final value of the exchange rate. 2.2 Composite or Joint Options - Compos The second and third versions of the product are called Composite or Joint Options. This is an option on an underlying, often a stock index, which is denominated in a second currency. These products offer more currency protection than the Flexos due to the fact that the strike price of the option is in the investor s home currency, but the value of the underlying asset remains in the foreign currency. The option value of the first Compos type depends on the product of the foreign commodity and the exchange rate at expiry. The call and put payoffs are defined as follows: C = max { (F T t X T t K d ), 0 } (2.3) where F T t and X T t P = max { (K d F T t X T t ), 0 } (2.4) are as defined earlier, and K d is the strike price in the investor s domestic currency. The second version is for the commodity investor trying to achieve some protection against exchange rate fluctuation. Canadian oil producers often use this version of currency translated option. 6

15 The third version is essentially the same product but expressed in terms of the currency market. This investor is a currency trader who is relating the payoff of the option to the performance of a foreign equity/commodity market. The value of this option is as a result of combining the currency option with the commodity futures contract to yield an F/X option - commodity linked foreign exchange option. The payoffs for a call and a put are respectively defined as: C = max { F T t (X T t K), 0 }, (2.5) P = max { F T t (K X T t ), 0 }, (2.6) where K is the strike of the exchange rate in domestic currency per unit of foreign currency. Ultimately, this product is a currency option but a minimum floor level has been established which is a function of the foreign commodity market, see [31]. In this product, the investor is exposed to the fall in the foreign commodity market but is protected against a fall in the currency market. For the Compo options, one version is the right to exchange the currency for a foreign commodity and the other version is the right to exchange the foreign commodity for the currency. They are an opposite reflection of each other like an image in a mirror. 2.3 Fixed Exchange Rate Version - Quantos The true Quanto (Quantity Adjusted Option - fourth version of CTO) is the fixed exchange rate version that assures that the final payoff of the option can be converted back to the investor s home currency at a guaranteed exchange rate. We can achieve a perfect hedge by using an instrument which explicitly allows a contingent, i.e; stochastic amount of foreign currency to be converted at a pre-specified exchange rate. Such quantity adjusting arrangements are generically termed Quantos. Quantos were originally developed in the late 1980 s to address the needs of investors who 7

16 wanted to participate in the explosive rally in the Japanese stock market without being exposed to movements in the Yen. This type of option is a guaranteed exchange rate foreign commodity option and it is the only type that completely eliminates any risk in the exchange rate, see [14]. The payoffs for a Quanto call and put are as follows: C = max { X 0 (F T t K f ), 0 } = max { (F T t X 0 K d ), 0 }, (2.7) P = max { X 0 (K f F T t ), 0 } = max { (K d F T t X 0 ), 0 }, (2.8) where X 0 is the multiplier that determines how many dollars the contract pays per foreign index point. X 0 is defined at the beginning of the contract. Unfortunately, there is no simple buy and hold strategy that guarantees a payoff of X 0 F T CAD dollars at expiration. The reason is that the final CAD value of USD at expiration is unknown. The only strategy for replicating the Quanto payoff in CAD is a dynamic one, and invloves creating a synthetic fund portfolio of traded securities such that it always has the CAD value X 0 F, an exposure of X 0 to the CAD, and zero exposure to the USD. 2.4 Applications of Currency Translated Options These options have become increasingly popular as investors desire exposure to foreign assets without assuming the foreign exchange risk. Most of the demand is for bond and stock index options. A CTO can in theory exist for any asset or liability denominated in a currency other than that in which it is usually traded. One example of such a structure which is offered on an organised exchange is the Chicago Merchantile Exchange s Nikkei 225 stock index contract, which uses the nominal price of the yen-denominated index applied to a US Dollar notional principal. Fund managers in the United States and Europe has shown much interest for these products. However, one of the major problems when investing in other countries has been the currency 8

17 risk. Derivative products such as exchange traded options and futures have made the process easier by only tying up a relatively small amount of the capital that is the initial margin in the case of the futures or premium in the case of the option, see [24]. However, when either the option become in-the-money or futures prices change (which has cash flow impacts on the margin account), the investor will face currency risk. The Currency Translated Options have been developed to eliminate the currency risks associated with foreign investments. The following table summarizes the payoffs and the properties of the various contracts. CTO Forward payoff in CAD Call Option payoff in CAD Comments Flexos X T t (F T t K f ) max { X T t (F T t K f ), 0 } - sign of payoff/moneyness is independent of X T t - magnitude of payoff in CAD depends upon X T t Compos X T t F T t K d max { X T t F T t K d, 0 } - sign of payoff/moneyness depends upon X T t - magnitude of payoff in CAD depends upon X T t Quantos X 0 (F T t K f ) max { X 0 (F T t K f ), 0 } - sign of payoff/moneyness is independent of X T t - magnitude of payoff in CAD is independent of X T t Table 2.1: Foreign stock/futures index contracts as defined by their payoffs at expiration 9

18 Chapter 3 Underlying Components 3.1 Data Sets The data sets chosen for this analysis seek to illustrate very different market dynamics by examining time periods with different economic conditions. The data for WTI futures contracts are the daily settled prices reported by NYMEX, and the F/X futures data is the Bank of Canada noon rate with delivery for the same expiry. Added to the historical WTI and F/X data, strikes set in Canadian Dollars, are established on CADTI with the final ending price of the synthetic CADTI underlier. The Canadian version of WTI futures contract (CADTI) is a synthetic futures contract, that is only traded over-the-counter. It shares the same delivery port and oil grade as WTI. The only difference has got to do with the currency involved in the contract. Why do Canadian oil producers use this product? One reason could be due to the illiquid nature of some of the positions in the Canadian market which merits the need for this synthetic product. This new contract is achieved by taking the product of the WTI futures price in USD with the corresponding exchange rate to get the new price in CAD, see [12]. Several different data sets were used in this research. We compute the pricing and hedging of this product by historically testing three different data sets at different economic periods. We will use CADTI prices in the pricing and hedging of Flexible Exchange Rate version of CTOs (Flexos). Unlike Quantos, Flexos require the exchange rate dynamics through out the life of the contract. A thorough description of the statistical analysis of these data sets are presented in the next section. 10

19 3.1.1 March 2009 Delivery There are 376 observations in this data set. It ranges from August 22, 2007 to the option s expiry on February 20, 2009 with a final CADTI price of $ Figure 3.1 illiustrates the price paths of WTI and F/X futures. (a) WTI futures price (b) F/X futures price (c) CADTI prices Figure 3.1: March 2009 futures & CADTI prices used in historical testing. This data set spans with the peak of oil prices in 2008 just before the financial meltdown which lasted for the second period of The sharp decrease in oil prices reflected in the relationship between the USD-CAD for the same period. We estimated the correlation coefficient between WTI and USD/CAD F/X futures prices to be WTI and CADTI prices seem to synchronize in their movement. 11

20 3.1.2 June 2010 Delivery This contract runs from November 24, 2008 to its expiry on May 20, 2010 with a final CADTI price of $ This is the smallest data set (with 371 prices) in the series of data considered. The price path of WTI and F/X futures are illustrated in Figure 3.2. This data set begins right after the global financial crisis of 2008 and represents the gradual recovery of oil prices and the fall of the US dollar. (a) WTI futures price (b) F/X futures price (c) CADTI prices Figure 3.2: June 2010 futures & CADTI prices used in historical testing. The plots in Figure 3.2 shows a strong co-movement between the respective prices. The futures prices appear to increase with time. The correlation coefficeint between the WTI and USD/CAD futures prices was estimated to be

21 3.1.3 December 2011 Delivery This is the largest (with 403 observations) data set in our analysis and ranges from May 17, 2010 to November 18, 2011, approximately 2 years data, with a final CADTI price of $ Figure 3.3 shows the price path of WTI and F/X futures. (a) WTI futures price (b) F/X futures price (c) CADTI prices Figure 3.3: December 2011 futures & CADTI prices used in historical testing. Surprisingly, the ripple effect of the 2008 global financial meltdown is evident in this data set as prices of oil begin to appreciate from the second quarter of 2010 through to the second quarter of This same period also witnessed the decline of the US dollar. We estimated the correlation coefficient between WTI and F/X prices to be The next section explores the nature of the distribution for the various data sets considered. 13

22 3.2 Descriptive & Inferential Statistics Since financial markets data are often stochastic and continuous in nature, we will imploy the use of continuous-time stochastic processes in modelling them. From the data sets described above, we found the highest correlation coefficient between the WTI and USD-CAD F/X futures prices to be If S(t 1) and S(t) are two consecutive futures prices for a series, the log return r(t) (also called the continuously compounded return) is defined as: [ ] S(t) r(t) = ln = ln[s(t)] ln[s(t 1)] (3.1) S(t 1) The advantage of looking at log returns of a series is that one can see relative changes in the variable and compare directly with other variables whose values may have very different base values. Figure 3.4: Sample log-returns for December 2011 contract. This is the difference between the natural log of the assets price at time t and the natural log of its price at the previous step in time. We determine the features of the underlying dynamics and the kind of distribution to be used for it. Thus, we start by looking at the distribution of returns of each of the processes under study. 14

23 The normal distribution has been widely used to model log-returns of financial data, which is the assumption when using a Geometric Brownian motion driven model. However, statistical analysis of the distribution of individual asset returns frequently finds fat tails, skewness, and other non-normal features, see [10] and [30]. We therefore test the normality assumption of the asset returns before looking elsewhere for other distributions by generating histograms from the data sets and superimposing each histogram with the normal density curve. The normal distribution is a symmetric distribution with a single peak and an expected kurtosis of 3. Since we are comparing two data sets, it is often desirable to know if the assumption of a common distribution is justified. If so, then location and scale estimators can pool both data sets to obtain estimates of the common location and scale. If two samples do differ, it is also useful to gain some understanding of the differences. To verify this assumption, we do Quantile-Quantile plot on each data set. The quantile-quantile (Q Q) plot is a graphical technique for determining if two data sets come from populations with a common distribution. A Q Q plot is a plot of the quantiles of the first data set against the quantiles of the second data set. More abstractly, given two cumulative probability distribution functions G and H, with associated quantile functions: q = G 1 and q = H 1 (the inverse function of the CDF is the quantile function), the QQ plot draws the qth quantile of G against the qth quantile of H for a range of values of q. Thus, the QQ plot is a parametric curve indexed over [0, 1] with values in the real plane R 2. If two distributions being compared are similar, the points in the Q-Q plot will approximately lie on the line y = x. If the distributions are linearly related, the points in the Q Q plot will approximately lie on a line, but not necessarily on the line y = x. Since Q Q plots compare distributions, there is no need for the values to be observed as pairs, as in scatterplots. 15

24 3.2.1 March 2009 Delivery Figure: 3.5 shows that the returns on both the WTI and F/X will be appropriate for the normal distribution. However, the USD-CAD F/X returns have an excess kurtosis and a negative skew due to the decline of the USD-CAD exchange rate during the period under study. (a) Hist of WTI log-returns (b) Q-Q plot of WTI log-returns (c) Hist of USD-CAD F/X log-returns (d) Q-Q plot of USD-CAD F/X log-returns Figure 3.5: Comparison of daily log-returns of WTI and USD-CAD futures for March 2009 using a Normal Distribution and Quantile-Quantile plot to a histogram. 16

25 3.2.2 June 2010 Delivery From Figure 3.6 we see that both returns seem to have come from a normal distribution even though the WTI returns show a negative skew. This could be due to the fluctuations in the oil prices from the inception of the contract. These fluctuations were as a result of the aftermath effect of the global financial meltdown. (a) Hist of WTI log-returns (b) Q-Q plot of WTI log-returns (c) Hist of USD-CAD F/X log-returns (d) Q-Q plot of USD-CAD F/X log-returns Figure 3.6: Comparison of daily log-returns of WTI and USD-CAD futures for June 2010 using a Normal Distribution and Quantile-Quantile plot to a histogram. 17

26 3.2.3 December 2011 Delivery Though the sample size in this contract is quite large, there seem to be some outliers in both return distribution. Figure 3.7 illustrates the distributions for the WTI and F/X futures with both returns showing evidence of log-normality behaviour. Both returns have excess kurtosis. The WTI returns exhibit negative skewness. (a) Hist of WTI log-returns (b) Q-Q plot of WTI log-returns (c) Hist of USD-CAD F/X log-returns (d) Q-Q plot of USD-CAD F/X log-returns Figure 3.7: Comparison of daily log-returns of WTI and USD-CAD futures for December 2011 using a Normal Distribution and Quantile-Quantile plot to a histogram. By observing the various histograms and QQ-plots for the data sets, we will now consider an alternative stronger normality tests that test the null hypothesis that a sample x 1,..., x n 18

27 comes from a normally distributed population. Two of such tests are the Shapiro -Wilk and the Lilliefors tests. Definition The Shapiro - Wilk test is an analysis of variance test for normality. By considering the sample observed returns {r(t 1 ),..., r(t n )}, we order them from smallest to largest, r (1) r (2) r (n), and let r (i) denote the ith order statistic. Again from the observed returns sample size n, obtain the coefficients a 1, a 2,..., a k where k is approximately (n + 1)/2 if n is odd and n/2 otherwise. The test statistic is: W = [ k i=1 a i(r (n i+1) r (i) ) n i=1 (r( i) r) 2 ] 2 where the coefficients a i are the critical values of the sampling distribution given by (a 1,..., a n ) = m V 1 (m V 1 V 1 m) 1/2 where m = (m 1,..., m n ) and m 1,..., m n are the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution, and V is the covariance matrix of those order statistics. This test is often called the W test. If the p-value is less than the chosen alpha level, then the null hypothesis is rejected (i.e. one concludes the data are not from a normally distributed population) else we do not reject the null hypothesis. Definition The Lilliefors test evaluates the hypothesis that the sample has a normal distribution with unspecified mean and variance against the alternative hypothesis that the sample does not have a normal distribution. The main difference from the well-known Kolmogorov-Smirnov test (K-S test) is in the assumption about the mean and standard deviation of the normal distribution. The K-S test assumes the mean and standard deviation of 19

28 the population normal distribution are known; Lilliefors test does not make this assumption. The test statistic is computed from the Z i s, defined by Z i = (x i x)/s, i = 1,..., n. Let S(x) be the empirical distribution function based on the Z i s. The Lilliefors test statitic T L is defined by T L = sup F (x) S(x) where F (x) is the standard normal distribution function. Results of the test for normality performed on the three data sets are shown in Table 3.1. The table contains the Test Statistic of each of these tests conducted. We rejected the null hypothesis that the samples are from a normally distributed population in the Lilliefors test since in each instance the test statistic was greater than the critical value of 0.05 while we do no reject the null hypothesis in the intance of the Shapiro-Wilk test. Thus, both WTI and F/X returns could be modelled using a log-normal distribution. Test Data Set Mar 09 Jun 10 Dec 11 WTI Lilliefors F/X WTI Shapiro-Wilk F/X Table 3.1: Test Statistics for normality for WTI and F/X log-returns. We will model the dynamics of the Flexo and Quanto type currency translated options since much work has been done on the Compos, see [20]. Thus the main focus will be to determine the hedging efficiencies of these two currency translated options. Again, we can model both the WTI and F/X futures contracts with a Gaussian driven stochastic process Estimation of Correlation Between WTI and USD-CAD F/X Log-returns From the analysis of the data sets in the previous sections, we saw that correlation (ρ) does exist between the WTI and USD/CAD futures as well as their log-returns. But the 20

29 correlation that this pair exhibits is a real dynamic one, ever changing with global economic cycles. Since we are looking at two different assets, it is useful for us to determine the impact of correlation on the model to be used in pricing and hedging them. The most familiar measure of dependence between two quantities is the Pearson productmoment correlation coefficient, or Pearson s correlation. It is obtained by dividing the covariance of the two variables by the product of their standard deviations as follows: ρ(r 1, r 2 ) = corr(r 1, r 2 ) = Cov(r 1, r 2 ) σ 1 σ 2 = E[r 1 E[r 1 ]]E[r 2 E[r 2 ]] σ 1 σ 2 where E is the expected value operator. We will assign higher weights to the more recent data since the behaviour of assets are based on current economic indicators, see [8]. We will use exponentially weighted moving average (EWMA) to assign weights between the WTI and F/X futures data. Definition In an EWMA, the weights decrease exponentially so that recent data points are much more important than older data points. The weighting for each older datum point decreases exponentially, never reaching zero. The univariate form of the model is: σ 2 t = λσ 2 t 1 + (1 λ)r 2 t 1 (3.2) where the weight λ is assumed to be known - often set at 0.94 for daily returns. We estimated the actual correlation parameters from historical testing to prevent errors which might be due to risks in correlation. Table 3.2 illiustrates the results from estimation of correlations for each of the contracts. Mar 09 Jun 10 Dec 11 Non-Weighted EWMA (λ = 0.94) Table 3.2: Estimated correlations between WTI and F/X log-returns. 21

30 3.3 Volatility Volatility estimation is of central importance to risk management and pricing. While for a simulated process with known drift and volatility the procedure for assessing estimator performance is straightforward, the same is not true for real market processes where both drift and volatility are unobservable, see [3], [15], [18]. The classical standard deviation measure, σ, of the log-returns, r(t i ), derived from equation: (3.1) ( σ = 1 n n 2 1 r(t i ) n 1 2 r(t i )) (3.3) n(n 1) i=1 i=1 It is important to consider how estimators perform for processes in which volatility is not constant. However, in the Black-Scholes setting of constant volatility on the log-normally distributed returns, the volatility is scaled by time. This is estimated as: σ = σ tn t 1. Figure 3.8: 5 - Year Volatilities of WTI Log-returns on December 2011 Contract 22

31 Stochastic volatility models are one approach to resolve a shortcoming of the Black-Scholes model; that the underlying volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlying security. By assuming that the volatility of the underlying price is a stochastic process rather than a constant, it becomes possible to model derivatives more accurately Figure 3.9: 2 - Year Volatilities of USD-CAD F/X Log-returns on December 2011 Contract The crude oil price exhibits a high degree of volatility which varies significantly over time. Such characteristics imply that the oil market is a promising area for testing volatility models. Unfortunately, we shall not treat this in this work. A direct measurement of volatility is thus difficult in practice. Since we assume the market is efficient, it provides us with proper option premiums. This feature forms the concept of implied volatility. 23

32 Definition Implied Volatility, ι is the volatility, for which the Black-Scholes price equals the market price V BS (t, S t, K, T ; ι) = V MK. (3.4) Note, that the put - call parity implies that puts and calls with the same strike have identical implied volatilities. Implied volatility can be thought of as a consensus among the market participants about the future level of volatility - assuming a fair allocation of information, as well as a same model used by all market participants for pricing options. A concept closely related to implied volatility is smile effect - volatility obtained from market prices is often U - shaped, having its minimum near - the - money, often defined as an interval, for which 0.95 m Deviation of implied volatility from a constant Black-Scholes volatility can be viewed as the risk premium payable to the holder of the short position, which indirectly implies volatility to be fungible. Many researchers have found out that volatility is mean-reverting. A unique implied volatility given the Black-Scholes price can be found with numerical procedures (such as Newton-Raphson used by Matlab), since C BS σ = Λ > 0. This legitimates a market standard to quote prices in terms of implied volatilities. Most of the time implied volatility is larger than historical. Implied volatility increases with time to maturity. 24

33 3.4 Stationarity Definition A stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time or space. Let {X t } be a stochastic process and let F X (x t1+τ,..., x tk+τ ) represent the cumulative distribution function of the joint distribution of {X t } at times t 1+τ,..., t k+τ. Then, {X t } is said to be stationary if, for all k, and for all t 1,..., t k, F X (x t1+τ,..., x tk+τ ) = F X (x t1,..., x tk ). (3.5) Since τ does not affect F X (.), F X is not a function of time. A stationary process has the property that the mean, variance and autocorrelation structure do not change over time. The most important property of a stationary process is that the auto-correlation function (acf) depends on lag alone and does not change with the time at which the function was calculated, see [27]. A weakly stationary process has a constant mean and acf (and therefore variance). A truly stationary (or strongly stationary) process has all higher-order moments constant including the variance and mean. Definition Autocorrelation refers to the correlation of a time series with its own past and future values. Autocorrelation is also sometimes called lagged correlation or serial correlation. Given measurements, X 1, X 2,..., X N at time t 1,..., t N, the lag k autocorrelation function is define as: ρ k = N k i=1 (X i X)(X i+k X) N i=1 (X i X) 2 (3.6) 25

34 Figure 3.10: Plot of ACF of WTI log-returns Autocorrelation is a correlation coefficient. However, instead of correlation between two different variables, the correlation is between two values of the same variable at times t i and t i+k. We can also determine the behaviour of the joint distribution by plotting a scatter diagram using the pairs (X i, X i+k ), which is also known as the lag plot. When the autocorrelation is used to detect non-randomness, it is usually only the first (lag 1) autocorrelation that is of interest. There should be no clear pattern among the lagged returns. Larger financial time series data sets often exhibit some dynamics that can be attributed to the exogenous factors that have different characteristics over different time intervals. A conclusion of stationarity is reasonable when one considers the nature of the tracking process that is occurring. Stationarity exists if the mean and variance of the data remains constant. 26

35 Figure 3.11: Verifying autocorrelation in WTI log-returns for March 2009 Contract The average returns for both WTI and USD-CAD F/X are similar and the volatility of the returns of each asset increases towards the contract expiry. Results from Figure 3.11 indicates that we have not identified the presence of autocorrelation in the data sets. Fundamentally, the pattern of the ACF s and the pattern of the PACF s are used to identify which model might be a good starting model. If there are more significant ACF s than significant PACF s then an AR model is suggested as the ACF is dominant. If the converse is true where the PACF is dominant then an MA model might be appropriate. The order of the model is suggested by the number of significant values in the subordinate. In our case, the absence of autocorrelation in the returns suggests that we can go ahead and model in the Gaussian way. 27

36 3.5 When WTI and F/X Returns are Normal But Not Jointly Normal Evidence of Joint Normality-Test of bivariate normality Normality on each of the variables separately is a necessary, but not sufficient, condition for multivariate normality to hold. The bivariate normal distribution was fundamental in the development of simple regression and correlation. Another property of a multivariate normal distribution imply that all pairs of variables must be bivariate normal. Bivariate normality, for correlated variables, implies that the scatterplots for each pair of variables will be elliptical; the higher the correlation, the thinner the ellipse. So, as a partial check on multivariate normality, one could verify univariate normality in every single variable and then obtain the scatterplots for pairs of variables and see if they are approximately elliptical. Figure 3.12 illustrates a semi-elliptical behaviour of June 2010 contract on WTI and F/X futures. Figure 3.12: Scatter Plot of WTI and F/X futures to detect bivariate normality 28

37 We perform a further test of multivariate normality by using the Mardia s test (1974). This calculates Mardia s multivariate skewness and kurtosis coefficients as well as their corresponding statistical test. For large sample size, the multivariate skewness is asymptotically distributed as a Chi-square random variable; here it is corrected for small sample size. Results from this test are presented in Table 3.3. Multivariate Coefficient Statistic df p Skewness Skewness corrected for sample small size Kurtosis Table 3.3: Analysis of Mardia s asymmetry skewness and kurtosis for June 2010 contract With a given significance level of 0.05, the multivariate skewness (corrected for small sample) and kurtosis results are significant. Although the data set seem to have some outliers (that can be modified), there is a slight evidence of bivariate normality in the June 2010 contract. Figure 3.13: Quantile Plot of WTI and F/X June 2010 Contract 29

38 We then estimate the Mahalanobis distance of a multivariate vector with mean vector µ and covariance matrix S from: D M (v) = (v µ) T S 1 (v µ), (3.7) where v = [X 1, X 2 ]. We plot this distance against quantiles of a Chi-square distribution as shown in Figure 3.13 above. This graph is a powerful way to assess for multivariate normality. As in any normality plot, we wish to see a straight 45 line that ensures multivariate normality Using the Bivariate Copula We can get a whole variety of different behaviours by considering the bivariate copula that is associated with (X, Y ) via Sklar s theorem. If we use the Gaussian copula, then we get (X, Y ) are jointly normal, and so Z = X + Y is normally distributed. If the copula is not the Gaussian copula, then X and Y are each still marginally distributed as normals, but are not jointly normal and so the sum will not be normally distributed, in general. Copulas enable us to imbed the marginal distributions extracted from vertical spreads in the options markets in a multivariate pricing kernel. Copulas allow us not only to separate the impact on the joint distribution of the marginals and the association structure, but also to exploit non-parametric measures of the latter. A possible strategy to address the problem of dependency under non-normality is to separate the two issues, i.e. working with non-gaussian marginal probability distributions and using some techniques to combine these distributions in a bivariate setting. The main advantage of the copula approach to pricing is to write the bivariate pricing kernel as a function of univariate pricing functions, see [21]. 30

39 Definition A 2-dimensional copula is a real function C: [0, 1] 2 [0, 1] with the following properties: I. u [0, 1], C(0, u) = C(u, 0) = 0, II. u [0, 1], C(u, 1) = u and C(1, u) = u. III. (u 1, u 2 ), (v 1, v 2 ) [0, 1] [0, 1] with u 1 v 1 and u 2 v 2 : C(u 2, v 2 ) C(u 1, u 2 ) C(u 2, v 1 ) + C(u 1, v 1 ) 0 Thus a copula is a multivariate distribution with support in [0, 1] n and with uniform marginals. As such, it can represent the joint distribution function of two standard uniform random variables U 1, U 2 : C(u 1, u 2 ) = P r (U 1 u 1, U 2 u 2 ) (3.8) We can use this feature in order to re-write via copulas the joint distribution function of two (even non-uniform) random variables. The most interesting fact about copulas in this sense is Sklars theorem. Figure 3.14 below illustrates the behaviour of the WTI and F/X contracts using an Empirical Bivariate Copula function. This is evidential that returns of these contracts can be modeled using copulas. (a) Scatterplot of the empirical copula (b) Scatterplot of the Gaussian copula Figure 3.14: Empirical Copula Function for WTI and F/X June 2010 Contract. 31

40 Theorem Sklar s theorem Let F (x, y) be a joint distribution function with continuous marginals F 1 (x) and F 2 (y). Then there exists a unique copula C such that F (x, y) = C(F 1 (x), F 2 (y)) Conversely, if C is a copula and F 1 (x), F 2 (y) are continuous univariate distributions, F (x, y) = C(F 1 (x), F 2 (y)) is a joint distribution function with marginals F 1 (x), F 2 (y). The theorem suggests then to represent the multiplicity of joint distributions consistent with given marginals through copulas. Three specific copulas are worth mentioning: the product copula, the minimum and the maximum copulas. Families of copulas which encompass all of these copulas are called comprehensive. As for the first, the copula representation of a distribution F degenerates into the so-called product copula, C(v, z) = vz, if and only if X and Y are independent. As for the others, they derive from the well-known Fréchet-Hoeffding result in probability theory, stating that every joint distribution function is constrained between the bounds: max {F 1 (x) + F 2 (y) 1, 0} F (x, y) min {F 1 (x), F 2 (y)} (3.9) As a consequence of Sklar s theorem, the Fréchet-Hoeffding bounds exist for copulas too: max {v + z 1, 0} C(u, v) min {v, z} In correspondence of the extreme copula bounds, there is perfect positive and negative dependence between the variables, and every variable can be obtained as a deterministic function of the other (see Embrechts, McNeil and Straumann, 1999 for a proof). Let us define the generalized inverse of a distribution function y = F 2 (x), as F 1 2 (y) = inf {t R : F 2 (t) y, 0 < y < 1} 32