pricingandhedginginthefreightfutures market€¦ · industry-speciﬁc demand and economic growth...

Pricing and Hedging in the Freight Futures

Market

CCMR Discussion Paper 04-2012

Marcel Prokopczuk

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

Pricing and Hedging in the FreightFutures 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.

1

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 matchat every point in time.

6A survey of previous studies in the freight market in general is provided by Kavussanosand 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 asa 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 thereafterdue 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 ahedging 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 thefutures 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


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.


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 thefocus 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 employedin 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 + σξdz∗ξ (6)

Schwartz (1997):

dξ = κ(a∗ − ξ)dt + σξdz∗ξ (7)

Schwartz and Smith (2000):

dξ = a∗dt + σξdz∗ξ (8)

dχ = (−κχχ − λχ)dt + σχdz∗χ (9)

Korn (2005):

dξ = κξ(a∗ − ξ)dt + σξdz∗ξ (10)

dχ = (−κχχ − λχ)dt + σχdz∗χ, (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

F (T ) = eEQ[ln(ST )]+ 1

2V arQ[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 F̂ with maturity T̂ that cannot be

hedged by simply taking an offsetting position (e.g. because a contract with

maturity T̂ 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

h1∂F1

∂ξ=

∂F̂

∂ξ. (13)

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

h1∂F1

∂ξ+ h2

∂F2

∂ξ=

∂F̂

∂ξ(14)

h1∂F1

∂χ+ h2

∂F2

∂χ=

∂F̂

∂χ. (15)

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.


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


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, seealso 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., ξ̂t = E[ξt|ξt−1] and χ̂t = 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.


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.


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 repeatedthe 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.


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.


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

F (T ) = eEQ[ln(ST )]+ 1

2V arQ[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):

F (T ) = eξ0+aT+σ2

ξ

2T (17)

Schwartz (1997):

F (T ) = ee−κξT

ξ0+a∗(1−eκξT

)+σ2

ξ

4κξ(1−e

−2κξT)

(18)

Schwartz and Smith (2000):

F (T ) = ee−κξT

χ0+ξ0+a∗T−(1−e−κχ )λχ

κχ+A(T )

(19)

with A(T ) =σ2

χ

4κχ(1 − e−2κχT ) + 1

2σ2

ξT + (1 − eκχT )σξσχρ

κχ.

Korn (2005):

F (T ) = ee−κξT

ξ0+e−κχT χ0+a∗(1−e−κξT

)−(1−e−κχ )λχ

κχ+A(T )

(20)

with A(T ) =σ2

ξ

4κξ(1 − e−2κξT ) +

σ2χ

4κχ(1 − e−2κχT ) + (1 − e−(κξ+κχ)T )

σξσχρ

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 ofMoments, and Kalman filter maximum likelihood approaches to estimate latent factormodels to conclude that the Kalman filter is best.

21

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24

0

500

1000

1500

2000

2500

3000

3500

4000

Date

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

Mio

. U

SD

Trading Volume in the Dry Bulk Futures Market

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

Jan 2005 Jan 2006 Jan 2007 Jan 2008 Jan 20090

20

40

60

80

100

120

Date

US

D

Closest to Maturity Future

C4

C7

P2A

P3A

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

C4

words

words

Capesize

Richards Bay/Rotterdam, 150,000 mt dwt 10 per cent coal free in and out

and trimmed, scale load/25,000 mt Sundays holidays included discharge.

18 hours turn time at loading port and 12 hours at discharge port.

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.

Trading unit

Trading unit a a

a a 1000 mt

words

words

USD/Ton

words

words a a

a a Baltic

words

words

C7

words

words

Capesize

Bolivar/Rotterdam 150,000 mt dwt 10 per cent coal free in and

out trimmed, 50,000 mt Sundays holidays included loading/25,000 mt

Sundays holidays included discharge, 12 hours turn time at loading port

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.

Trading unit

Trading unit a a

a a 1000 mt

words

words

USD/Ton

words

words a a

a a Baltic

words

to be

P2A

words

to be a

Panamax

Basis a Baltic panamax 74,000 mt dwt not over 7 years of age, 89,000 cbm

grain, max loa 225m, draft 13.95m, 14.0 knots on 32mts fuel oil laden,

28mts fuel oil ballast and no diesel at sea, basis delivery Skaw-Gibraltar

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.

Trading unit

Trading unit a a

a a a Day

words

words

USD/Day

words

words a a

a a Baltic

words

words

words

P3A

words

words

words

Panamax

Basis a Baltic panamax 74,000 mt dwt not over 7 years of age, 89,000 cbm

grain, max loa 225m, draft 13.95m, 14.0 knots on 32mts fuel oil laden,

28 mts fuel oil ballast and no diesel at sea, for a trans Pacific round of

35/50 days either via Australia or Pacific (but not including short rounds

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.

Trading unit

Trading unit

Trading unit a a

a a a Day

words

words

words

USD/Day

words

words

words a a

a a Baltic

27

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 PPreturn

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 CorrelationsThis 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 EstimatesThis 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

Black 76 Schwartz 97 SS 2000 Korn 2005 Black 76 Schwartz 97 SS 2000 Korn 2005

κξ - 0.2830*** 3.0194*** 0.8017*** κξ - 0.2638*** 2.7654*** 0.7470***

κχ - - - 1.6660*** κχ - - - 1.5203***

a -0.2062 2.2596*** -0.1015 3.0518*** a -0.2010 2.2328*** -0.0803 3.5702***

a∗ -0.2748 1.9958*** -0.2777 4.0011*** a∗ -0.2854 1.8796*** -0.2976 4.2374***

λχ - - -0.2661 0.0768 λχ - - -0.2866 -0.5794

σξ 0.5109*** 0.5497*** 0.4446*** 1.2091*** σξ 0.4679*** 0.5064*** 0.4179*** 1.1736***

σχ - - 0.5637*** 1.2247*** σχ - - 0.5187*** 1.2032***

ρ - - -0.0367 -0.8543*** ρ - - -0.0821 -0.8714***

ξ0 3.0459*** 3.0432*** 2.8243*** 2.9810*** ξ0 3.0220*** 3.0205*** 2.7744*** 3.1708***

χ0 - - 0.3531*** 0.6098 χ0 - - 0.3972*** 0.8612*

LL 2992.1 3073.5 3675.2 3679.8 LL 3071.0 3152.6 3874.2 3878.2

Route P2A Route P3A


κξ - 0.1882*** 2.1339*** 0.4641*** κξ - 0.2140*** 3.9719*** 0.1018***

κχ - - - 1.2128*** κχ - - - 2.9352***

a -0.3331 8.9127*** -0.1302 9.9977*** a -0.3921 8.4040*** -0.2876 6.1302***

a∗ 0.3501 8.5831*** -0.5056 9.7090*** a∗ -0.3717 8.4965*** -0.4192 8.1302***

λχ - - -0.6127** 0.5982 λχ - - -0.4656 1.7878

σξ 0.6167*** 0.6371*** 0.6159*** 1.3443*** σξ 0.6691*** 0.7146*** 0.6204*** 0.6733***

σχ - - 0.5997*** 1.3028*** σχ - - 0.7458*** 0.9781***

ρ - - -0.3450*** -0.8691*** ρ - - 0.0011 -0.2375*

ξ0 10.7056*** 10.7048*** 10.5035*** 10.9141*** ξ0 10.5389*** 10.5500*** 10.3743*** 10.8388***

χ0 - - 0.2115 0.0403 χ0 - - 0.2231* -0.8681***

LL 2989.5 3068.0 3658.7 3656.6 LL 2780.4 2832.5 3476.3 3441.3

30

Table 5: In-Sample Pricing Errors

This table reports in-sample (estimation period January 2005 to December 2008) root

mean squared errors (RMSE) and relative root mean squared errors (RRMSE) for

each maturity and for the entire samples across maturities and time (All).

Route C4

RMSE RRMSE


1 0.1197 0.1142 0.0387 0.0410 1 5.32% 4.81% 1.45% 1.62%

2 0.0710 0.0677 0.0087 0.0086 2 2.88% 2.62% 0.32% 0.32%

3 0.0366 0.0342 0.0201 0.0196 3 1.37% 1.24% 0.71% 0.69%

4 0.0004 0.0027 0.0168 0.0171 4 0.01% 0.09% 0.59% 0.60%

5 0.0279 0.0244 0.0057 0.0057 5 1.06% 0.90% 0.20% 0.20%

6 0.0443 0.0359 0.0193 0.0183 6 1.70% 1.36% 0.67% 0.65%

All 0.0625 0.0587 0.0211 0.0216 All 2.66% 2.39% 0.77% 0.82%

Route C7

RMSE RRMSE


1 0.1138 0.1089 0.0283 0.0315 1 5.13% 4.67% 1.07% 1.26%

2 0.0660 0.0634 0.0058 0.0054 2 2.69% 2.46% 0.22% 0.21%

3 0.0335 0.0317 0.0148 0.0143 3 1.30% 1.19% 0.54% 0.52%

4 0.0000 0.0000 0.0128 0.0132 4 0.00% 0.00% 0.46% 0.47%

5 0.0264 0.0246 0.0063 0.0062 5 1.03% 0.94% 0.22% 0.22%

6 0.0444 0.0372 0.0212 0.0195 6 1.71% 1.43% 0.73% 0.68%

All 0.0593 0.0561 0.0169 0.0174 All 2.56% 2.32% 0.61% 0.66%

Route P2A

RMSE RRMSE


1 0.0763 0.0945 0.0290 0.0306 1 0.80% 0.98% 0.29% 0.30%

2 0.0345 0.0570 0.0057 0.0054 2 0.35% 0.58% 0.06% 0.05%

3 0.0000 0.0292 0.0115 0.0114 3 0.00% 0.29% 0.11% 0.11%

4 0.0307 0.0000 0.0076 0.0081 4 0.31% 0.00% 0.07% 0.08%

5 0.0495 0.0245 0.0134 0.0136 5 0.51% 0.24% 0.13% 0.13%

6 0.0825 0.0605 0.0535 0.0524 6 0.82% 0.59% 0.51% 0.50%

All 0.0536 0.0537 0.0262 0.0261 All 0.54% 0.55% 0.25% 0.25%

Route P3A

RMSE RRMSE


1 0.1606 0.1538 0.0565 0.0603 1 1.79% 1.68% 0.60% 0.64%

2 0.0799 0.0755 0.0085 0.0073 2 0.87% 0.80% 0.09% 0.07%

3 0.0357 0.0336 0.0128 0.0130 3 0.38% 0.35% 0.13% 0.13%

4 0.0000 0.0000 0.0048 0.0053 4 0.00% 0.00% 0.05% 0.05%

5 0.0273 0.0261 0.0136 0.0137 5 0.29% 0.27% 0.14% 0.14%

6 0.0663 0.0636 0.0461 0.0448 6 0.67% 0.63% 0.45% 0.44%

All 0.0802 0.0766 0.0310 0.0319 All 0.88% 0.82% 0.32% 0.33%

31

Table 6: Out-of-Sample Pricing Errors

This table reports out-of-sample (January 2009 to December 2009) root mean squared

errors (RMSE) and relative root mean squared errors (RRMSE) for each maturity

and across maturities and time (All).

Route C4

RMSE RRMSE


1 0.1627 0.1610 0.1253 0.1206 1 6.59% 6.17% 4.72% 4.50%

2 0.1042 0.1030 0.1002 0.0998 2 4.30% 4.12% 4.01% 3.98%

3 0.0845 0.0852 0.0886 0.0895 3 3.49% 3.49% 3.63% 3.66%

4 0.0819 0.0815 0.0821 0.0831 4 3.36% 3.35% 3.38% 3.41%

5 0.0882 0.0819 0.0762 0.0766 5 3.63% 3.36% 3.12% 3.15%

6 0.1023 0.0870 0.0755 0.0732 6 4.24% 3.58% 3.09% 3.01%

All 0.1076 0.1038 0.0930 0.0919 All 4.40% 4.13% 3.70% 3.65%

Route C7

RMSE RRMSE


1 0.1610 0.1587 0.1236 0.1203 1 6.06% 5.70% 4.36% 4.19%

2 0.1057 0.1052 0.1009 0.1006 2 4.03% 3.90% 3.76% 3.74%

3 0.0859 0.0871 0.0908 0.0918 3 3.26% 3.29% 3.45% 3.48%

4 0.0827 0.0825 0.0846 0.0856 4 3.17% 3.17% 3.23% 3.27%

5 0.0893 0.0841 0.0780 0.0780 5 3.49% 3.27% 2.98% 2.99%

6 0.1080 0.0972 0.0776 0.0751 6 4.30% 3.82% 3.00% 2.91%

All 0.1088 0.1058 0.0939 0.0932 All 4.17% 3.95% 3.50% 3.46%

Route P2A

RMSE RRMSE


1 0.1752 0.2085 0.1432 0.1412 1 1.74% 2.08% 1.41% 1.39%

2 0.1197 0.1404 0.1201 0.1199 2 1.18% 1.39% 1.19% 1.19%

3 0.1047 0.1071 0.1073 0.1081 3 1.04% 1.07% 1.07% 1.08%

4 0.1120 0.0987 0.1011 0.1017 4 1.13% 0.99% 1.01% 1.02%

5 0.1197 0.0973 0.0992 0.0993 5 1.22% 0.98% 0.99% 0.99%

6 0.1506 0.1325 0.1312 0.1326 6 1.55% 1.37% 1.34% 1.36%

All 0.1326 0.1363 0.1181 0.1182 All 1.33% 1.37% 1.18% 1.18%

Route P3A

RMSE RRMSE


1 0.3495 0.3429 0.2403 0.2352 1 4.23% 4.10% 2.76% 2.75%

2 0.1953 0.1955 0.1701 0.1647 2 2.13% 2.11% 1.81% 1.76%

3 0.1523 0.1538 0.1573 0.1560 3 1.64% 1.65% 1.68% 1.67%

4 0.1315 0.1307 0.1344 0.1325 4 1.39% 1.38% 1.42% 1.40%

5 0.1376 0.1318 0.1268 0.1244 5 1.46% 1.40% 1.34% 1.32%

6 0.1622 0.1475 0.1325 0.1265 6 1.71% 1.56% 1.39% 1.33%

All 0.2025 0.1982 0.1649 0.1612 All 2.31% 2.25% 1.80% 1.78%

32

Table 7: Reduction of Out-of-Sample Pricing Errors

This table reports mean and median reductions of RMSE (calculated across maturity)

between models. The upper triangle of each matrix reports the average relative

reduction (mean) while the lower triangle reports the relative reduction of the median

errors.

Route C4 Route C7


Black 76 - 4.17% 17.70%*** 19.18%*** Black 76 - 3.70% 20.87%*** 21.90%***

Schwartz 97 -0.39% - 14.11%*** 15.66%*** Schwartz 97 -1.63% - 17.82%*** 18.89%***

SS 2000 12.65%*** 11.90%*** - 1.80%*** SS 2000 17.12%*** 15.92%*** - 1.30%***

Korn 2005 12.51%*** 11.66%*** 1.69%*** - Korn 2005 17.98%*** 18.00%*** 0.67%*** -

Route P2A Route P3A


Black 76 - -4.77% 13.45%** 13.23%** Black 76 - -1.23% 16.18%* 17.97%**

Schwartz 97 -9.15%* - 17.40%*** 17.18%*** Schwartz 97 -6.01% - 17.20%** 18.97%**

SS 2000 8.15%*** 16.41%*** - -0.26% SS 2000 3.52%* 10.05%*** - 2.15%

Korn 2005 7.91%*** 17.40%*** 0.20% - Korn 2005 13.75%*** 19.16%*** 14.25% -

33

Table 8: Hedging 4 Months Future

This table reports error statistics of the four-months future delta hedge. ME reports

the mean hedging error, whereas SD provides the standard deviation of hedging errors.

MAE reports the mean absolute hedging error, RMSE the root mean squared hedging

error.

Route C4 Route C7


ME -0.0323 -0.0483 0.0162 0.0204 ME -0.0305 -0.0457 0.0180 0.0222

SD 0.4938 0.4002 0.2564 0.2728 SD 0.4580 0.3681 0.2690 0.2861

MAE 0.3012 0.2646 0.1790 0.1899 MAE 0.2778 0.2459 0.1678 0.1745

RMSE 0.4900 0.3992 0.2544 0.2709 RMSE 0.4545 0.3673 0.2670 0.2842

Route P2A Route P3A


ME -93.6 -111.5 31.7 39.1 ME -67.5 -123.4 13.0 25.0

SD 997.0 803.3 491.3 513.2 SD 663.3 646.2 341.6 350.4

MAE 706.7 585.4 380.7 402.9 MAE 468.7 438.4 231.3 240.7

RMSE 991.6 803.1 487.5 509.7 RMSE 660.2 651.6 338.5 347.8

34

Table 9: Reduction of Hedging Errors

This table reports mean and median reductions of mean absolute hedging errors

between models. The upper triangle of each matrix reports the average relative

reduction (mean) while the lower triangle reports the relative reduction of the median

errors.

Route C4 Route C7


Black 76 - 12.16%* 40.57%** 36.96%** Black 76 - 11.50% 39.60%* 37.20%*

Schwartz 97 4.97%* - 32.35%* 28.24%* Schwartz 97 2.68% - 31.75%* 29.05%

SS 2000 14.63%** 23.33%* - -6.07% SS 2000 36.05%*** 27.20%** - -3.96%

Korn 2005 22.25%* 13.10% -3.15%* - Korn 2005 28.27%** 27.09%** -1.23% -

Route P2A Route P3A


Black 76 - 17.16%*** 46.14%*** 42.99%*** Black 76 - 6.45% 50.64%*** 48.64%***

Schwartz 97 14.13%*** - 34.98%*** 31.18%** Schwartz 97 8.91% - 47.23%** 45.10%**

SS 2000 39.11%*** 20.07%** - -5.84%*** SS 2000 50.80%*** 32.95%*** - -4.04%*

Korn 2005 42.62%*** 15.64%* -6.20%*** - Korn 2005 44.50%*** 30.80%*** -9.76%*** -

35

pricingandhedginginthefreightfutures market€¦ · industry-speciﬁc demand and economic growth...

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