Pricing and Hedging in the Freight Futures Market

CCMR Discussion Paper 04-2012

Marcel Prokopczuk Pricing and Hedging in the Freight Futures Market

Marcel Prokopczuk∗

April 2010

Abstract In this article, we consider the pricing and hedging of single route dry bulk freight futures contracts traded on the International Maritime Exchange. Thus far, this relatively young market has received almost no academic attention. In contrast to many other commodity markets, freight services are non-storable, making a simple cost-of-carry valuation impossible. We empirically compare the pricing and hedging accuracy of a variety of continuous-time futures pricing models. Our results show that the inclusion of a second stochastic factor signiﬁcantly improves the pricing and hedging accuracy. Overall, the results indicate that the Schwartz and Smith (2000) two-factor model provides the best performance.

JEL classiﬁcation: G13, C50, Q40

Keywords: Freight Futures, Hedging, Shipping Derivatives, Imarex

∗ICMA Centre, Henley Business School, University of Reading, Whiteknights, Reading, RG6 6BA, United Kingdom. e-mail:[email protected]. Telephone: +44-118- 378-4389. Fax: +44-118-931-4741.

Electronic copy available at: http://ssrn.com/abstract=1565551 I Introduction

In a globalised world, efficient goods transport from continent to continent is an increasingly indispensable economic growth factor. More than 95 % of world trade (in volume) is carried by marine vessels. Transport volume has risen worldwide from 2800 Mio tons in 1986 to 4700 in 2005.1 The world relies on fleets of ships with a cargo carrying capacity of 960 million deadweight tons (dwt)2 to carry every conceivable type of product. In fact, from grain to crude oil, iron ore to chemicals, seaborne trade amounted to more than 7.1 billion tonnes in 2005.3 The market for freight rates is complex. Market participants include shipowners, operators and charterers, all of whom are exposed to significant price risk. Reasons for changes in freight rates are manifold. From a long-term perspective, vessel supply determines the supply curve of shipping services. Thus, information on vessel availability, production and scrapping has a direct influence on equilibrium price levels. Demand for shipping services is closely linked to the demand for the commodities that require transportation. The more goods used in industrial production, the greater the demand for delivery services. Thus, the equilibrium freight rate level responds to changes in industry-specific demand and economic growth in general. In the short run, the cost of operating a vessel greatly fluctuates with the cost of shipping fuel, also known as ‘bunker’, which comprises up to 30 % of operating costs. Naturally, bunker prices are closely linked to crude oil prices and, therefore, freight rates react to changes in oil price levels. Weather may also strongly influence short-term changes in demand for shipping services; a harvest that occurs earlier than expected in one region of the world increases the need for transportation immediately, as many products cannot be stored indefinitely.

1See Kavussanos and Visvikis (2006a). 2Deadweight ton (dwt) is a measurement to express the weight of cargo, fuel, stores, passengers and crew carried by a ship when loaded to its maximum. 3See the Baltic Exchange Web site: www.balticexchange.com.

Electronic copy available at: http://ssrn.com/abstract=1565551 From a financial perspective, freight rates are typically considered part of the commodity market.4 However, freight rates exhibit some characteristics that distinguish them from most other markets. In contrast to all major traded commodities freight rates can be considered costs of services, not products. Therefore, they are essentially not storable, a property that makes simple cost-of-carry valuations of futures contracts for freight rates impossible.5 Further, the freight rate spot market shows a high degree of volatility, which causes significant risks for shipowners and charterers alike, creating a substantial hedging demand. Consequently, as for other commodities, futures markets have been established to meet this demand. As a result, the availability of futures in the market creates the need for appropriate valuation models. In the present paper, we study the pricing and hedging performance of four different pricing models for dry bulk freight futures traded on the newly established International Maritime Exchange (Imarex). By doing so, we contribute to the literature in various ways. First, to the best of our knowledge, we are the first to study the pricing and hedging performance of no-arbitrage pricing models in the freight derivatives market as previous research, such as Kavussanos and Nomikos (1999), Haigh (2000), Kavussanos and Nomikos (2003), Kavussanos and Visvikis (2004), and Batchelor et al. (2007), focuses mainly on econometric aspects of the forward freight market.6 Second, we are also first to analyse these pricing models with respect to their hedging performance, an aspect that may be considered even more important than pricing for market participants. Third, extant literature only analyses index futures and forward freight agreements that are traded over-the-counter. In 2001, the Imarex launched the first futures contracts written on individual

4See Geman (2008). 5The only other major nonstorable commodity is electricity. However, in contrast to electricity, freight rate prices behave less erratically as supply and demand need not match at every point in time. 6A survey of previous studies in the freight market in general is provided by Kavussanos and Visvikis (2006b).

2 routes of the freight market. Thus, our paper contributes to the empirical literature as we are the first to study this relatively new market. As stated, the non-storability of freight services makes a simple cost-of- carry valuation of futures contracts impossible. Thus, one must employ an alternate pricing approach/model when valuing freight futures contracts. In this paper, we focus our attention on affine continuous-time models of the spot price dynamics that have been successfully employed for other commodity markets. The affine continuous-time framework has the substantial advantage of allowing closed-form valuation for the freight futures contracts. More specifically, we consider the one-factor price dynamics and pricing models proposed by Black (1976) and Schwartz (1997), as well as the two-factor models developed by Schwartz and Smith (2000) and Korn (2005). All four models are estimated employing a Kalman filter-based approach for the four main dry bulk futures traded at the Imarex. We then study the pricing and hedging performance for each model. The two-factor models are found to significantly improve the pricing accuracy for the considered futures contracts. Moreover, the Schwartz and Smith (2000) model yields the best results when considering the hedging performance. The remainder of the paper is structured as follows. Section II describes the bulk freight market as well as the data used in our empirical study. Section III describes the pricing models considered and outlines their estimation, while Section IV presents the empirical results of our study. Section V concludes.

II Market and Data Description

A. The Dry Bulk Freight Market

Seaborne trade can roughly be divided into ﬁve groups: dry bulk, oil tanker, container, gas tanker and other. In this paper, we study the dry bulk futures market. Overall, dry bulk commodities represent about 38 % of all seaborne trade and are thus, a major segment of the entire market. Dry bulk vessels are

3 classified according to their size; principally Handysize, Handymax, Panamax, and Capesize. The first two are of minor importance for the derivatives market, as they primarily carry minor bulks (see below) and thus, represent a small market share. With about 70,000 dwt, Panamax are the largest ships that will fit through the Panama Canal and mainly carry grain and coal from America and Australia to Europe and Asia. Capesize vessels are typically above 150,000 dwt and round Cape Horn to travel between the Pacific and the Atlantic Ocean as they do not fit through the Panama Canal.7 Dry bulk commodities are typically divided into two distinct categories: major and minor bulks. Major bulks comprise two thirds of the dry bulk sector and include iron ore (27 %), coal (26 %) and grain (14 %). Due to their major role in the world economy (iron ore is the raw material for steel production, coal is used to produce electricity and for steel production) they are shipped by larger vessels, such as the Capesize and Panamax. Minor bulks comprise steel, steel products, fertilisers, sugar, cement, non-ferrous metal ores, salt, etc. These are shipped primarily in smaller vessels such as Handymax and Handysize ships. The primary methods of vessel charters include: bareboat, time, and voyage charters. In a bareboat charter, owners rent their vessels in return for monthly payments. The owner has no responsibilities for crewing, victualling, etc. This type of charter is of minor importance to the markets considered in this paper. Time charter shipowners run their vessels under instructions of charterers and have no responsibilities for the commercial management. The charterer pays the shipowner a per-day fee plus costs, such as fuel, port fees, etc. By contrast, the charterer in a voyage charter pays the owner a per ton fee to carry freight from Point A to Point B while the shipowner is responsible for every detail of the operation and bears all costs.8 To provide price information to market participants, the Baltic Exchange

7See the Imarex Web site (www.Imarex.com) and Kavussanos and Visvikis (2006a). 8See Gray (1986), who compares these two charter types by hiring a car (time charter) versus hiring a taxi (voyage charter).

4 began calculating the Baltic Freight Index (BFI) in 1985, which is computed as the average assessment by a panel of ship brokers around the world for a basket of 11 dry cargo routes. Today, the Baltic Exchange provides more than 40 daily single-route assessments as well as ﬁve indices, namely the: Baltic Dry Index (BDI), Baltic Capesize Index (BCI), Baltic Panamax Index (BPI), Baltic Supramax Index (BSI), Baltic Handysize Index (BHSI) and Baltic International Tanker Routes (BITR).9

B. The Imarex Freight Futures Market

The world’s first global freight futures contract was launched in May 1985 by the Baltic Exchange through the Baltic International Freight Futures Exchange (BIFFEX). This contract was written on the Baltic Freight Index (BFI) and was also the world’s first futures contract written on a service rather than an asset in the typical sense.10 The main objective for creating this futures contract was to provide a hedging device for the shipping industry in response to strong price fluctuations on the spot market.11 However, comprised of an average of 11 routes, there was limited hedging effectiveness as many market participants were only exposed to a subset of the constituent routes. Thus, the cross-hedge quality of this futures contract was extremely limited, which resulted in low trading interest and consequently, the termination of the contract in April 2002.12 The low hedging quality of the BIFFEX futures leaded to the formalisation of an over-the-counter (OTC) freight forward market in 1992 that enabled market participants to trade single routes of the Baltic Exchange freight segment, although with the usual disadvantages of OTC contracts, i.e., credit

9See the Baltic Exchange Web site (www.balticexchange.com) for more details, such as a description of all routes and the panel members. 10In 1999, the BFI was replaced by the BPI. 11See Haigh (2000), who notes that at the same time, a similar contract began trading at the Bermuda-based International Futures Exchange, but ceased trading shortly thereafter due to insuﬃcient trading volume. 12See Kavussanos and Nomikos (2000) and Kavussanos and Nomikos (2003).

5 risks and transaction costs. In 2001, the newly established, Oslo-based International Maritime Ex- change (Imarex) began to offer freight futures contracts written on single routes. In 2006, the Imarex merged with NOS Clearing, which had been providing clearing services for the Imarex since inception. In addition to dry-bulk and tanker futures, Asian-style options were introduced in 2006 and 2005, respectively.13 Trading via the Imarex is conducted anonymously and is based on a hybrid exchange concept that combines direct electronic execution and voice brokerage. Trading is possible either as a direct member of the Imarex or through a General Clearing Member (GCM). Currently, the Imarex has over 200 direct members, representing a mix of oil companies, shipowners, traders and financial companies. The GCM include 15 major financial institutions, such as Goldman Sachs and JP Morgan. According to Imarex, the exchange offers the most liquid electronic marketplace in the freight market.14 Total nominal trade volume amounted to US$ 8,575 million in 2009.15 The dry bulk futures segment studied in this paper is the largest segment of the Imarex. It amounts to almost 50 % of trading volume for the period under study. Figure 1 displays the trading volume from 2005 to 2009 in which one can observe strong growth beginning in the first quarter of 2006 until the third quarter of 2008, ending with a deep plunge that is most likely a consequence of the present economic crisis.16

[FIGURE 1 ABOUT HERE]

In this paper, we consider the four available single-route dry bulk futures traded on the Imarex. These include two Capesize voyage charter routes:

13In addition to freight derivatives, Imarex oﬀers a bunker fuel contract to provide a hedging mechanism for vessel operating costs. 14See the Imarex Web page: www.Imarex.com. 15Excluding December 2009, which was not yet available at time of writing. 16Interestingly, the trading volume in options increased from 2008 to 2009. Thus, the recently introduced options market has most likely taken away some of the liquidity in the futures market.

6 C4 (Richards Bay to Rotterdam) and C7 (Bolivar to Rotterdam), and two Panamax time charter routes, P2A (Gibraltar to Far East) and P3A (Paciﬁc round). Table 1 provides summary information for the four considered underlyings. For Routes C4 and C7 (P2A and P3A), 12 (6) consecutive monthly maturities are traded.17 The settlement price is the arithmetic average of the spot prices over the delivery period, which includes the last seven business days in the delivery month to reduce the possibility of market manipulation. Trading ends on the last trading day prior to the delivery period.

[TABLE 1 ABOUT HERE]

C. Data

We sample weekly prices of the ﬁrst six closest to maturity futures for the following four routes: C4, C7, P2A and P3A. Each time series is constructed by rolling monthly to the next following futures contract once the last trading day is reached. The sample period covers ﬁve years and extends from January 2005 to December 2009, yielding 262 observations per contract that correspond to 1,572 observations per route. All data was downloaded from Bloomberg. Figure 2 displays the time series of the four closest to maturity futures. Note that we scale the prices of P2A and P3A by 10−3 to facilitate the presentation. One can observe a gradual increase of futures prices from mid-2005 with a sharp decline that follows at the end of 2008.

[FIGURE 2 ABOUT HERE]

Table 2 reports summary statistics of the weekly log return of the closest to maturity futures and the six-month ahead futures for the four considered contracts. All contracts exhibit a negative mean return during the sample period. The standard deviation can be considered rather high, compared with other commodities like crude oil or copper, which might be a consequence of the non-storability of freight services.

17In addition, three consecutive yearly contracts are traded for C4 and C7.

7 By visual inspection of the time series in Figure 2, it is not clear whether or not the futures prices are stationary. Thus, we conduct the Philipps-Perron test of a unit root which cannot be rejected at the price level. At the return level, all test statistics are signiﬁcant at the 1 % level, indicating stationarity of the log return series. Comparing the volatilities of the contracts across maturity provides evidence for the Samuelson eﬀect, i.e., increasing volatility for decreasing futures horizons.18

[TABLE 2 ABOUT HERE]

Figure 2 clearly shows that the considered futures move together to some extent. To quantify the joint movement, we compute the correlation matrix of the closest to maturity and the six-month ahead log return series, which are reported in Table 3. It can be observed that although the correlation among the series is strong, it is far from perfect. The highest dependence can be observed between the two voyage charter Routes C4 and C7.19 The moderate level of correlation also highlights the need for single-route futures as a futures contract written on a basket of routes, such as the BIFFEX future discussed in the previous section, certainly oﬀers only a poor hedging performance for speciﬁc regions of the world.

[TABLE 3 ABOUT HERE]

III Futures Pricing and Hedging

In this section, we brieﬂy introduce the four considered price dynamics. We then provide futures pricing formulas, discuss hedging and outline the estimation approach.

18See Samuelson (1965). 19In addition to correlation at the return level, the considered futures contracts might also be co-integrated at the price level. As the dependence structure among futures is not the focus of this paper, we do not investigate this issue further.

8 A. Price Dynamics

One of the ﬁrst stochastic commodity pricing models proposed in the literature is the model of Black (1976). Black assumes that the log spot price follows an arithmetic Brownian motion. Precisely, let ξ = ln(S), then

dξ = adt + σξdzξ, (1)

where zξ is a standard Brownian motion, a the drift and σξ > 0 the volatility of the process. Although Black mainly considers the pricing of options on storable commodities, we can employ the same spot price dynamics in the context of futures pricing on non-storable assets. The second one-factor spot price dynamics we consider was proposed by Schwartz (1997). In contrast to the dynamics proposed by Black (1976), Schwartz acknowledged that many commodity prices show strong signs of mean reversion. Thus, he proposed to model the log spot price as an Ornstein-Uhlenbeck process:

dξ = κξ(a − ξ)dt + σξdzξ. (2)

The speed of mean-reversion toward the long-run equilibrium price is governed by the parameter κξ > 0 while the long-term equilibrium log price itself is characterized by a. Empirical research has shown that, in many commodity markets, a second stochastic factor improves the pricing and/or hedging of futures contracts.20 Therefore, we also consider the two-factor model developed by Schwartz and Smith (2000).21 The short-term/long-term model of Schwartz and Smith (2000) is a latent

20See, e.g., Schwartz (1997). 21Note that this model is mathematically equivalent to the two factor model considered in Schwartz (1997). The model of Schwartz and Smith (2000) has been successfully employed in the literature for a variety of commodity futures markets; see e.g., Schwartz and Smith (2000) for crude oil, Manoliu and Tompaidis (2002) for natural gas, Lucia and Schwartz (2002) for electricity, and Sorensen (2002) for corn, soybean, and wheat markets.

9 factor model equivalent to the model previously developed by Gibson and Schwartz (1990). The advantage of the latent factor formulation is given by the fact that the stochastic factors are more ‘orthogonal’, which enhances empirical estimation. The economic intuition of their model can be described as follows: the log spot price is assumed to be the sum of two stochastic factors, i.e. ln(S) = ξ + χ. The ﬁrst factor ξ represents the long-term equilibrium of the considered commodity log spot price and follows an arithmetic Brownian motion

dξ = adt + σξdzξ, (3) reﬂecting the uncertainty with respect to the long-run equilibrium price level. The second stochastic factor χ is assumed to capture short-term deviations from this equilibrium price. As these deviations should not prevail forever, χ is assumed to follow an Ornstein-Uhlenbeck process reverting to zero

dχ = −κχχdt + σχdzχ. (4)

The coefficient κχ > 0 characterises the degree of mean-reversion, i.e., the rate at which short-term deviations will disappear. The half-life of these shocks can be computed as ln(2)/κχ. zξ and zχ are standard Brownian motions correlated with correlation coefficient ρ. The Schwartz and Smith (2000) model assumes that the long-run equilibrium price level follows a non-stationary process. Korn (2005) argued that this assumption might lead to inferior model performance if the price equilibrium actually follows a stationary process. Therefore, Korn proposed replacing the first stochastic factor of the Schwartz and Smith (2000) model by an Ornstein-Uhlenbeck process. The dynamics of (3) changes to

dξ = κξ(a − ξ)dt + σξdzξ. (5)

Thus, the log spot price becomes the sum of two mean reverting processes.

10 A natural next step would be to increase the number of stochastic factors to three. We did so and considered three-factor models in the spirit of Cortazar and Naranjo (2006). However, the estimation yielded unstable, ﬂuctuating parameter estimates that indicated an overparametrization. Therefore, we refrain from reporting these results to remain focused on the most interesting aspects of our study.

B. Futures Pricing

Following Cox and Ross (1976) and Harrison and Kreps (1979) the no-arbitrage price of a future with maturity T is given as the risk-neutral expectation of the spot price at T . Therefore, one needs to rewrite the model under the risk-neutral measure in order to derive futures pricing formulas. Let λξ and

λχ denote the market prices of risk of the state variables ξ and χ, respectively. These parameters represent the compensation a risk-averse agent requires per unit of risk for each of the two risk factors.22 The risk-neutral versions of the considered processes are then given as follows.

Black (1976): ∗ ∗ dξ = a dt + σξdzξ (6)

Schwartz (1997): ∗ ∗ dξ = κ(a − ξ)dt + σξdzξ (7)

Schwartz and Smith (2000):

∗ ∗ dξ = a dt + σξdzξ (8) ∗ dχ = (−κχχ − λχ)dt + σχdzχ (9)

Korn (2005):

∗ ∗ dξ = κξ(a − ξ)dt + σξdzξ (10) ∗ dχ = (−κχχ − λχ)dt + σχdzχ, (11)

22Note that we implicitly assume the market prices of risk being constant through time.

11 ∗ with the risk-neutral drift rate denoted by a = a − λξ; zξ and zχ are again two (correlated) Brownian motions. The risk-neutral versions of the model dynamics allow us to obtain the no-arbitrage futures price as

Q 1 Q F (T )= eE [ln(ST )]+ 2 V ar [ln(ST )]. (12)

The risk-neutral expectation and variance of the log spot price at T depend on the underlying spot price dynamics. We provide the exact futures pricing formulas of the four models in the Appendix.

C. Hedging

In practice, hedging open positions is as least as important as pricing. Linear contracts, such as futures, are typically delta hedged.23 In theory, we can obtain an instantaneous risk-free position by building a delta neutral portfolio. Consider an open forward position Fˆ with maturity Tˆ that cannot be hedged by simply taking an oﬀsetting position (e.g. because a contract with maturity Tˆ is not available). As the number of contracts needed to delta hedge this position is equal to the number of risk factors, we need one other contract

F1 with maturity T1 in the one factor models framework. The hedge ratio h1 can be computed as ∂F ∂Fˆ h 1 = . (13) 1 ∂ξ ∂ξ In the two factor models framework, we need a second contract to build up the delta hedge. The hedge ratios h1 and h2 are obtained by solving the following system of equations

∂F ∂F ∂Fˆ h 1 + h 2 = (14) 1 ∂ξ 2 ∂ξ ∂ξ ∂F ∂F ∂Fˆ h 1 + h 2 = . (15) 1 ∂χ 2 ∂χ ∂χ

23See also Schwartz (1997) or Korn (2005).

12 D. Estimation

In order to study the pricing and hedging performance of the four models considered, we split our data set in two. The first four years of data are used for estimation and in-sample evaluation. The last year of data is then used for out-of-sample analyses. We estimate all four models employing a Kalman filter-based maximum likelihood approach.24 As the latent state variables are all (jointly) normally distributed this technique is the standard approach for estimating the considered models. This is mainly due to the fact that, for a linear Gaussian model, the Kalman filter approach is optimal in the least-squared sense, exploring time-series and cross-sectional properties of the data simultaneously. An alternative approach is to follow a purely maximum likelihood method in the spirit of Chen and Scott (1993). This approach, relying on an inversion of the measurement equation, i.e., the futures pricing formula, has the substantive disadvantage that one must make an assumption as to which futures contracts are observed without error.25 More details on the Kalman filter-based estimation of the models are outlined in the Appendix.

IV Empirical Results

A. Parameter Estimates

Table 4 provides the parameter estimates for the four models considered and the four routes: C4, C7, P2A, and P3A. The ﬁrst item of note is that the mean reversion parameters κξ and κχ are all estimated signiﬁcantly; the same is true for the volatility parameters σξ and σχ. On the other hand, the drift parameters for the non-stationary models (Black (1976) and Schwartz and Smith (2000)), as well as all risk premia are only estimated with low precision.

24See, e.g., Harvey (1989) on Kalman ﬁltering. 25One has to choose one (two) contract(s), as the one-factor (two-factor) model consists of one (two) risk factor(s).

13 This result was also observed by Schwartz and Smith (2000) and Korn (2005) when calibrating their models to crude oil futures prices. The correlation parameters are relatively close to zero in the Schwartz and Smith (2000) model, whereas these parameters take strongly negative values in the Korn (2005) model. Comparing the estimated parameter values across underlyings, one observes that the estimates for the Routes C4 and C7 are very similar. In contrast, the estimated values for the Routes P2A and P3A show higher volatilities and diﬀerent degrees of factor correlations.

[TABLE 4 ABOUT HERE]

A first possibility for judging model performances is to consider their in- sample fit, measured by the score of the likelihood function. Note that most model variants are not nested in each other, and, thus, a formal likelihood ratio test cannot be conducted. However, when inspecting the respective log likelihood values provided in Table 4, one can see that the Black (1976) model provides the worst fit to the data, while the Schwartz (1997) model improves the fit to some extent. The greatest improvement can be observed for the Schwartz and Smith (2000) model. Although having one parameter less than the Korn (2005) model, it provides an almost-equal fit for the Routes C4 and C7 and an even better fit for Routes P2A and P3A.

B. Pricing Performance

We ﬁrst compare the pricing accuracy of the four considered models based on the in-sample period 2005 – 2008. Table 5 reports the root mean squared errors in monetary terms (RMSE) and the relative root mean squared errors in percentage terms (RRMSE) for each considered contract as well as an overall error for each route and model.26

[TABLE 5 ABOUT HERE]

26Note that the reported error terms are based on log prices. Analysing the models’ performance based on log prices is most appropriate as these were used for estimation, see also Schwartz (1997).

14 Comparing the pricing accuracy across routes first, one observes that the RMSE are smallest for Routes C4 and C7, followed by P2A and P3A. However, due to different price quotations, the voyage charter Routes C4 and C7 trade at substantially lower price levels than the time charter Routes P2A and P3A. Thus, it is more appropriate to consider the RRMSE when comparing errors across routes. Overall relative pricing errors are smallest for P2A with an RRMSE between 0.25 % and 0.54 %, followed by P3A with an RRMSE between 0.32 % and 0.88 %, C7 with an RRMSE between 0.61 % and 2.56 %, and C4 with an RRMSE between 0.77 % and 2.66 %. Comparing the pricing errors across maturities, we can observe substantially higher values at the short end of the futures curve, i.e., for the one-month contracts. Overall, when comparing the in-sample pricing accuracy across models, we yield two major conclusions. First, the two-factor models substantially improve the in-sample pricing performance. This is especially true for the voyage charter Routes C4 and C7, where the RMSE and RRMSE are reduced by a factor of three. Within the one-factor models, the mean-reverting Schwartz (1997) model slightly outperforms the non-stationary Black (1976) dynamics. Second, the in-sample pricing accuracy of the Schwartz and Smith (2000) and the Korn (2005) models are very similar. Although the Schwartz and Smith (2000) model provides slightly lower errors in most cases, the differences are small. We now turn to the out-of-sample pricing comparison. Given the parameters estimated using data from January 2005 to December 2008, we value all futures contracts traded between January 2009 and December 2009. All parameters except for the latent state variables are held constant. For the state variables ξt and χt we employ one-week ahead forecasts based on the

Kalman ﬁlter, i.e., ξˆt = E[ξt|ξt−1] andχ ˆt = E[χt|χt−1]. This yields a truly out-of-sample framework for one-week ahead pricing forecasts. The results (RMSE and RRMSE) are presented in Table 6.

[TABLE 6 ABOUT HERE]

15 Firstly, we can observe that, not surprisingly, the out-of-sample pricing errors are substantially higher than for the in-sample case. However, the ranking across routes remains identical. The lowest RRMSE of 1.18 % to 1.33 % is observed for the Route P2A, followed by Route P3A with 1.78 % to 2.31 %, Route C7 with 3.46 % to 4.17 %, and Route C4 with 3.65 % to 4.40 %. Comparing pricing accuracy across models shows that the benefits of employing a two-factor model are lesser compared to the in-sample case, but still clearly observable. The ranking of the two-factor models reverses, however, as for the in-sample case, the results are quite similar. To analyse the statistical significance of the previous results, we perform a series of tests. We first calculate the weekly out-of-sample pricing error of each route by calculating the RMSE across maturity, obtaining a time series of RMSE for each route and model. We then compute the difference of these RMSE and test whether the mean difference is significantly different from zero by employing a standard t-test. To account for possible autocorrelation and heteroscedasticity, robust standard errors are computed employing the method of Newey and West (1987). To ensure the robustness of our results with respect to possible outliers, we further compute the median difference between each weekly model error. The median is then tested for significance by employing the non-parametric sign test developed by Wilcoxon (1945). The results of these tests are reported in Table 7. To keep the presentation manageable, the table contains a matrix of relative error reductions and significance levels organised as follows: the upper triangular of each matrix contains the reduction in mean RMSE when moving from the model with the lower number of parameters to the model with the higher number of parameters. For example, considering Route C4, the reduction in mean RMSE when moving from the Black (1976) model to the Schwartz (1997) model is 4.17 %. When moving from the Black (1976) model to the Schwartz and Smith (2000) model, the reduction yields 17.70 %. Moving from the Schwartz (1997) model to the Schwartz and Smith (2000) model reduces the mean RMSE by

16 14.11 %, and so on. The lower triangle contains reductions in median RMSE, again when moving from the model with fewer parameters to the one with a greater number of parameters. Thus, for example, the Korn (2005) model yields a 11.66 % lower median RMSE than the Schwartz 97 model. The results presented in Table 7 confirm the significance of the previous findings; the difference between the two one-factor models is insignificant. Both two-factor models improve the out-of-sample pricing significantly. The size of the mean error reduction is between 13.45 % and 21.90 % while the median error is reduced by 7.91 % to 19.16 %. Comparing the two-factor models shows that the Korn (2005) model slightly improves pricing compared to the Schwartz and Smith (2000) model. Differences are, however, small, and only significant for the voyage charter Routes C4 and C7.

[TABLE 7 ABOUT HERE]

C. Hedging

To investigate the hedging performance of the models, we perform the following analysis. On each observation date in the out-of-sample period, we construct a portfolio to delta hedge the four-months ahead future.27 Hedge ratios are computed as described in Section III.C. To remain comparable, we use the two-months ahead future for the one-factor models, and one- and three-months ahead futures for the two-factor models. The portfolio of two, respectively three, futures contracts is initially delta neutral and remains constant for one week. As it is free of cost to build the portfolio, a perfect hedge should result in zero change of wealth. Thus, the hedging error can be measured as change in value of the futures portfolio. Table 8 provides mean hedging errors (ME), standard deviations of hedging errors (SD), mean absolute hedging errors (MAE), and root mean squared hedging errors (RMSE). As positive and negative hedging errors cancel over

27The choice of the four months contract is to some extent arbitrary. We have repeated the analysis for other maturities yielding qualitatively similar results.

17 time, MAE and RMSE are typically preferred for judging the performance of a hedging strategy. As for pricing accuracy, the model based on the Black (1976) dynamics performs worst in all instances. The Schwartz (1997) model already yields clear improvements. Not unexpectedly, the two-factor models further improve the hedging accuracy. However, in contrast to the pricing results the Schwartz and Smith (2000) model outperforms the Korn (2005) model. For all routes, the Schwartz and Smith (2000) model yields the smallest values for every considered error metric.

[TABLE 8 ABOUT HERE]

Similar as for the pricing errors, we test the reduction of hedging errors on statistical significance. Table 9 reports the mean and median of the relative reduction in absolute hedging errors following the same logic as Table 7. It can be seen that the differences are of substantial size; the reductions in mean and median RMSE are up to 50 %. The Schwartz and Smith (2000) model dominates all other models in this regard. This is especially true for the time charter Routes P2A and P3A, where the model significantly outperforms the Korn (2005) model by 4 % to 10 %. Overall, we conclude that the two-factor models substantially outperform the one-factor models in every respect. Both two-factor models seem to perform equally well with respect to pricing; however, the Schwartz and Smith (2000) model clearly yields superior hedging results.

[TABLE 9 ABOUT HERE]

V Conclusion

In this paper, we empirically study the pricing and hedging of dry bulk futures contracts traded on the International Maritime Exchange. As the non-storability of freight services makes a simple cost-of-carry valuation impossible, we implement four diﬀerent continuous-time no-arbitrage pricing

18 models, which are empirically tested and compared with respect to their pricing accuracy and hedging effectiveness. We find that the two-factor models significantly outperform the one-factor models in every aspect. Within the class of two-factor models, we find similar performance with respect to pricing accuracy. With respect to hedging, we find the Schwartz and Smith (2000) model to yield better results than that proposed by Korn (2005). Therefore, we recommend the model of Schwartz and Smith (2000) be used when considering pricing and hedging for freight futures contracts.

19 VI Appendix

A. Futures Pricing Formulas

The general futures pricing formula is given by

Q 1 Q F (T )= eE [ln(ST )]+ 2 V ar [ln(ST )], (16) where the variance under the risk-neutral measure is identical to the variance under the real measure. As the ﬁrst two moments of the considered stochastic processes are well known, it is straight forward to obtain the respective futures formula. These are given as:

Black (1976): 2 σξ F (T )= eξ0+aT + 2 T (17)

Schwartz (1997):

2 − σ −2 κξT ∗ − κξT ξ − κξT e ξ0+a (1 e )+ 4 (1 e ) F (T )= e κξ (18)

Schwartz and Smith (2000):

−κ T ∗ −κ λχ e ξ χ0+ξ0+a T −(1−e χ ) +A(T ) F (T )= e κχ (19)

2 σ − σ σ ρ with A(T )= χ (1 − e 2κχT )+ 1 σ2T + (1 − eκχT ) ξ χ . 4κχ 2 ξ κχ

Korn (2005):

−κ T −κ T ∗ −κ T −κ λχ e ξ ξ0+e χ χ0+a (1−e ξ )−(1−e χ ) +A(T ) F (T )= e κχ (20)

2 2 σ − σ − − σ σ ρ with A(T )= ξ (1 − e 2κξT )+ χ (1 − e 2κχT )+(1 − e (κξ+κχ)T ) ξ χ . 4κξ 4κχ κξ+κχ

20 B. Kalman Filter Estimation Outline

The Kalman ﬁlter is a recursive procedure to calculate estimates of a latent state vector. In addition, if the parameters of the data-generating processes are unknown, they can be estimated using a maximum likelihood approach. This estimation method is the standard approach for estimating Gaussian latent factor models.28 To follow this approach, one must consider a discretised version of the model to be estimated. The evolution of the state variables is called the transition equation and can be written as

xt = c + Qxt−1 + ωt, (21)

′ where xt = ξt for the one-factor models and xt = [ξt, χt] for the two-factor models. c is a N x 1 vector, Q is a N x N matrix, and ωt a N x 1 vector of serially uncorrelated disturbances, where N = 1 for the one-factor models and N = 2 for the two-factor models. The relationship between state variables and observations is described by the measurement equation which has the general form of

yt = d + Zxt + εt. (22)

The 6 x 1 vector yt contains the observed log prices of the six available futures contracts at date t, d is a 6 x 1 vector, Z a 6 x N matrix, and εt a 6 x 1 vector of serially uncorrelated disturbances. Details on the Kalman ﬁlter recursion can be found in Chapter 3 of Harvey (1989).

28Duffee and Stanton (2004) compare pure maximum likelihood, Efficient Method of Moments, and Kalman filter maximum likelihood approaches to estimate latent factor models to conclude that the Kalman filter is best.

21 References

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22 Harvey, A. (1989). Forecasting, structural time series models and the Kalman ﬁlter. Cambridge University Press.

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24 Trading Volume in the Dry Bulk Futures Market 4000

3500

3000

2500

2000

Mio. USD 1500

1000

500

0 05:Q1 05:Q2 05:Q3 05:Q4 06:Q1 06:Q2 06:Q3 06:Q4 07:Q1 07:Q2 07:Q3 07:Q4 08:Q1 08:Q2 08:Q3 08:Q4 09:Q1 09:Q2 09:Q3 09:Q4 Date

Figure 1: Freight Futures Trading Volume

This ﬁgure displays the trading volume (in Mio USD) in the Imarex dry bulk futures market on an quarterly basis over the 2005 to 2009 period. The data was obtained from the Imarex Web page: www.Imarex.com.

25 Closest to Maturity Future 120 C4 C7 P2A P3A 100

60 USD

0 Jan 2005 Jan 2006 Jan 2007 Jan 2008 Jan 2009 Date

Figure 2: Closest to Maturity Futures

This ﬁgure displays the historical time series of the closest to maturity futures C4, C7, P2A and P3A. Note that P2A and P3A are scaled by 10−3 to facilitate presentation.

26 Table 1: Dry Bulk Routes Descriptions This table provides descriptions of the underlying dry bulk freight routes of the Imarex freight futures C4, C7, P2A and P3A. Source: Baltic Exchange.

Trading unit Price Settlement Route Vessel Description (LOT Size) quotation Index words words Richards Bay/Rotterdam, 150,000 mt dwt 10 per cent coal free in and out Trading unit words words words words and trimmed, scale load/25,000 mt Sundays holidays included discharge. Trading unit a a words words a a C4 Capesize 18 hours turn time at loading port and 12 hours at discharge port. a a 1000 mt USD/Ton a a Baltic Laydays 25 days forward from date of index, cancelling 40 days forward from date of index. Vessel’s age max. 15 years. Freight based on metric tonnes. 3.75 per cent total commission. words words Bolivar/Rotterdam 150,000 mt dwt 10 per cent coal free in and Trading unit words words words words out trimmed, 50,000 mt Sundays holidays included loading/25,000 mt Trading unit a a words words a a C7 Capesize Sundays holidays included discharge, 12 hours turn time at loading port a a 1000 mt USD/Ton a a Baltic

27 and 12 hours turn time at discharge port. Laydays 20 days forward from date of index, cancelling maximum 35 days forward from date of index. Vessel’s age maximum 15 years. 3.75 pct total commission. words words Basis a Baltic panamax 74,000 mt dwt not over 7 years of age, 89,000 cbm Trading unit words words to be to be a grain, max loa 225m, draft 13.95m, 14.0 knots on 32mts fuel oil laden, Trading unit a a words words a a P2A Panamax 28mts fuel oil ballast and no diesel at sea, basis delivery Skaw-Gibraltar aaa Day USD/Day a a Baltic range, for a trip to the Far East, redelivery Taiwan-Japan range, duration 60/65 days. Loading 15-20 days ahead in the loading area. Cargo basis grain, ore, coal, or similar. words words Basis a Baltic panamax 74,000 mt dwt not over 7 years of age, 89,000 cbm Trading unit words words words words grain, max loa 225m, draft 13.95m, 14.0 knots on 32mts fuel oil laden, Trading unit words words words words 28 mts fuel oil ballast and no diesel at sea, for a trans Pacific round of Trading unit a a words words a a P3A Panamax 35/50 days either via Australia or Pacific (but not including short rounds aaa Day USD/Day a a Baltic such as Vostochny/Japan), delivery and redelivery Japan/South Korea range. Loading 15-20 days ahead in the loading area. Cargo basis grain, ore, coal or similar. 3.75 per cent total commission. Table 2: Futures Returns Summary Statistics This table provides summary statistics for the weekly log returns of the four Imarex freight futures under study that are closest to maturity: C4(1), C7(1) P2A(1), and P3A(1) and six months to maturity: C4(6), C7(6), P2A(6) and P3A(6). The mean µ and the standard deviation σ are annualized. PPlevel and PPreturn denote the test statistic of the Phillips-Perron test for a unit root. The asterisk * indicates significance at the 1 % level.

µ σ PPlevel P Preturn C4(1) -0.0690 0.7008 -0.79 -15.33* C7(1) -0.0375 0.6203 -0.69 -12.42* P2A(1) -0.0070 0.7145 -0.59 -13.72* P3A(1) -0.0756 1.0626 -0.85 -13.55*

C4(6) -0.0550 0.4740 -0.65 -14.47* C7(6) -0.0279 0.4492 -0.56 -13.34* P2A(6) -0.0235 0.8078 -0.81 -17.92* P3A(6) -0.0848 0.7143 -0.84 -15.18*

28 Table 3: Returns Correlations This table reports correlations of the weekly log returns of the four freight futures under study: C4, C7, P2A and P3A. The upper triangle displays the correlations of one-month ahead futures returns while the lower triangle displays the correlations of the six-months ahead futures.

C4 C7 P2A P3A C4 - 0.8516 0.6096 0.5266 C7 0.8691 - 0.6811 0.5710 P2A 0.6051 0.6174 - 0.7682 P3A 0.5396 0.5874 0.3669 -

29 Table 4: Parameter Estimates This table reports the estimated parameters for the four freight futures considered. All parameters are estimated by the Kalman ﬁlter maximum likelihood approach using weekly data over the 2005 to 2008 period. LL denotes the log likelihood score. * denotes statistical signiﬁcance at the 10 %, ** at the 5 %, and *** at the 1 % level.

Route C4 Route C7