A Jump-Diffusion Model of Shipping Freight Rate

LI Xuying, GU Xianbin School of Economics and Management

Shanghai Maritime University Shanghai, China

e-mail: xyli@shmtu.edu.cn, guxianbinmail@126.com

Abstract—To draw a model that can describe shipping freight rate fluctuation precisely has never been so important since the ship investment evaluated under the framework of real options. We set a general double exponential jump-diffusion model, which generates a leptokurtic distribution. Compared with geometric Brownian motion model, the general double exponential jump diffusion model is more appropriately in matching the key features of shipping freight rate returns.

Keywords-double exponetial jump diffusion ; geometric Brownian motion; MCMC; Cramer-Von Mises criterion; shipping freight rate

I. INTRODUCTION In recent years, real options, which introduces the

principle of options into general investment and management and therefore provides more alternatives for decision-makers, has been used in shipping investment decisions.

Shipping freight rate model is not only an indispensable part but also a core element in applying real options in shipping investment decisions. A precise model can help getting a right choice.

Li(2007)[1] researched the behavior patterns of shipping freight rate fluctuations and set a geometric Brownian motion model for shipping freight rate. In this model, the shipping freight rate fluctuation is a stochastic process with fixed drift rate and variance. However, the true world is not as simple as the model described. Gu and Li(2009)[2] analyzed the shipping freight rate of past years and found that the distribution of the shipping freight rate returns has leptokurtic feature, which conflicts with the geometric Brownian motion model.

Kou(2002)[3] proposed a double exponential jump diffusion (DEJD) model, which can generate distributions with high peak and fat tail feature. Liu(2008)[4] adjusted Kou’s model by generalizing the double exponential distribution. With an added parameter, the general double exponential jump-diffusion(GDEJD) model is able to reflect the over-reactions in markets.

In this paper, Li’s shipping freight rate model is modified. Instead of geometric Brownian motion, the shipping freight rate is defined as a jump-diffusion process. By introducing the GDEJD, the new model is capable of measuring the degree of over-reactions in the shipping market as well as matching the high peak and fat tail feature of true returns’ distribution

II. THE MODEL Shipping freight rate under jump-diffusion processes can

be written as:

( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−++= ∑

=

tN

iittttt JdPdwPdtPdP

11σμ (1)

where

tP - Shipping freight rate at time t μ - The instantaneous expected return

2σ - The instantaneous volatility of the asset’s return conditional on that the Poisson jump event does not occurred

tw - A Wiener process

( )tN - A Poisson process with rate λ { }iJ - A sequence of independent and identically

distributed non-negative random variables; And, ( )JY ln= has a double exponential distribution

with probability density function:

{ } { }1 2( ) ( )

1 2( ) 0 ( ) 0

1 2

( )

1, 0

y yY y yf y p e I q e Iη κ η κ

κ κη η

η η

− − −− ≥ − <= +

> >(2)

where , 0p q ≥ 1p q+ = ,represent the probabilities of upward and downward jumps. The parameter κ denotes the critical position of the jump direction (i.e. the degree of over-reaction). If the jump size is bigger thanκ , the jump will be consider as an upward one, vice versa. The more different between κ and zero, the more irrational the market is.

Being differ from the geometric Brownian motion，the jump diffusion process is a mixture of both continuous diffusion path and discontinuous jump path. So it can explain the volatility caused by the impacts of the unpredictable information.

Solving the stochastic differential equation in (1), gives the dynamics of the shipping freight rate:

( )

∏=

⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

tN

iitt JwtPP

1

2

0 2exp σσμ (3)

by where, 101 =∏ =i ；

2009 Second International Conference on Future Information Technology and Management Engineering

978-0-7695-3880-8/09 $26.00 © 2009 IEEE

DOI 10.1109/FITME.2009.83

313

III. PARAMETER ESTIMATION The unknown parameters are μ ,σ ,the ratio of Poisson

process ( λ ) and the parameters of the jump size ( κηη ,,, 21p ). MCMC (Markov Chain Monte Carlo) methods give a relative simple way to compute them.

Let tΔ be a small time interval, the discretized version of (1) is

( ) ( )( )2

( )

( ) 1

12( ) ( ) 1 exp 1

( ) ( ) N t t

ii N t

t W t t W tP t P t t

P t P t Y

μ σ σ

+Δ

= +

⎧ ⎫⎛ ⎞− Δ + +Δ −⎜ ⎟⎪ ⎪⎝ ⎠Δ +Δ ⎪ ⎪= − = −⎨ ⎬⎪ ⎪+⎪ ⎪⎩ ⎭∑

(4)

If tΔ is small enough, (4) can be approximated to

∑Δ+

+=

+Δ+Δ=Δ )(

1)()()( ttN

tNiit YZtt

tPtP σμ (5)

Where

tZ is an standard normal , ( )JY ln= , and, under the assumption of a Poisson process, the probability of having one jump in time interval ( ]ttt Δ+, is tΔλ ,and that of having more than one jump is ( )tΔο .Therefore, we have

⎩⎨⎧

Δ−=Δ=

=∑Δ+

+= tptpY

Y ittN

tNii λ

λ1,0

,)(

1)( (6)

Let tB be the Bernoulli random variable with

kBP t == )1( , kBP t −== 1)0( , by where tk Δ≡ λ .

Combining the prior results, tr as the return of shipping freight rate in tΔ , can be written as

tttt YBZtttPtPr +Δ+Δ=Δ≡ σμ)()(

(7)

Consequently, the conditional distribution of tr is

( )tYBtNr ttYBt ttΔ+Δ 2

,,, ,~| σμσμ (8) According to Hammersly-Clifford theorem and (8), the

posterior distributions of the unknown parameters are all available. Let

1 2( , , . . . , )nX r r r= , ),,,,,,( 21 κηησμθ kp= ,

),...,,( 21 nBBBB = , ),...,,( 21 nYYYY = , the distributions are as fellows,

( ) ( ) ( )μθμσμ fYBXfXYBf X ,,,|,,,| ∝ ( ) ( ) ( )σθμσσ fYBXfXYBf X ,,,|,,,| ∝ ( ) ( ) ( ) ( )pfpfpYfYBpf Y |,,,|,,,,| 2121 κκηηκηη ∝ ( ) ( ) ( )12121 ,,,|,,,,| ηκηηκηη fpYfYBpf Y∝ ( ) ( ) ( )22112 ,,,|,,,,| ηκηηκηη fpYfYBpf Y∝ ( ) ( ) ( ) ( )κκκηηηηκ fpfpYfYBpf Y |,,,|,,,,| 2121 ∝

( ) ( ) ( )| |f k B Binomial B k f k∝ ( ) ( ) ( )θθθ |11,,|,,|1 ==∝= ttttXt BfBYXfXYBf ( ) ( ) ( )θθθ |00,,|,,|0 ==∝= ttttXt BfBYXfXYBf ( ) ( ) ( )θθθ |,,|,,| tYtttXt YfYBXfXBYf ∝

where ( )|f i i - The conditional distributions of the parameters

( )|Binomial i i - The Binomial distribution

( )|Yf i i -Represents (2)

( ) ( )2

2

1| , , exp22

t t tX t t t

r t B Yf X B Y

ttμ

θσπ σ

⎡ ⎤− Δ −= −⎢ ⎥

ΔΔ ⎢ ⎥⎣ ⎦ By using slice Gibbs sampler[5], the MCMC estimation can be done effectively. Gu(2009)[6] gives the details.

IV. COMPARISON WITH GEOMETRIC BROWNIAN MOTION When valuing the real options of ship investment, the

more realistic the shipping freight rate model is the more precise will the assessment be. For this reason, we set “correctness of simulating the shipping market’s risk” as a criterion when arguing a model.

A simple way to realize this standard is to generate a sample by the model and then compare with the actual data.

A. Cramer-Von Mises Criterion Cramer-Von Mises Criterion is used for judging whether

two independent samples come from the same distribution. The null hypothesis is that two independent samples

Mxxx ,,, 21 and Nyyy ,,, 21 come from the same (unknown) distribution.

Define T as

( ) ( )4 1

6U MNT

NM N M M N−= −

+ + (9)

Where

( ) ( )22

1 1

N M

i ji j

U N r i M s j= =

= − + −∑ ∑ ’ (10)

ii sr , -The rank of each sample; Mixing two samples and computing the T value of

every combination can get the distribution of T value under the null hypothesis. Then the p value of null hypothesis is given. Anderson(1962)[7]and Burr(1963)[8]discussed it in detail.

B. Data and Estimaton Baltic Supramax Index(BSI) published by Baltic

Exchange is a vane of dry bulk shipping market. The dataset used here is BSI from 2005/7/1 to 2009/2/23, which obtained form Clarkson Research1 and includes 189 observations. All the computations in this paper were done with R. 1 http://www.clarksons.net

314

The parameters of the double exponential jump diffusion model are estimated by MCMC method. The prior distributions are

)25,0(~ Nμ , )05.0,5(~ IGσ , )4,8(~ Betap ,)40,2(~ betak , )25,0(~ Nκ , )1,2(~1 paretoη ,

)2(~ 22 χη .

Table I. shows the estimation results of the jump diffusion model.

TABLE I. ESTIMATION OF JUMP DIFFUSION MODEL

Parameters μ σ k p 1η 2η κ

0.5994 0.1874 0.4319 0.9101 10.1002 3.1762 -0.0994

Parameter k tells that there are almost 20 jumps happened every year, and the p implies that about 90% of the jumps are upward. The shipping market is a bit irrational because of 0.0994κ = − , and it can be inferred that some speculators consider the rapid drops, which is smaller than 9.94%, as the signals of long position.

A geometric Brownian motion shipping freight rate model is

( )20 exp / 2t tP P t wμ σ σ⎡ ⎤= − +⎣ ⎦ . (11)

Estimation of its parameters is relatively easy, because the return of shipping freight rate belongs to a normal with mean ( )2 / 2 tμ σ− Δ and variance 2 tσ Δ while the shipping freight rate is a geometric Brownian motion. Table II. gives the results of the model.

TABLE II. ESTIMATION OF GEOMETRIC BROWNIAN MOTION MODEL

Parameters μ σ

0.0768 0.6164

C. Results Putting all estimated parameters into the models

respectively and simulating two independent samples, then the T values can be computed with actual BSI data. The p values of each sample are shown in Table III.

TABLE III. RESULTS OF CRAMER-VON MISES CRITERION

Probability of null hypothesis

Models

Jump diffusion Geometric Brownian

p 0.5344 0.0038

Obviously, Cramer-Von Mises Criterion refused the null hypothesis that the sample from geometric Brownian motion model and the actual BSI come from a same distribution. However, the test does not reject that the sample generated by general double exponential jump-diffusion model and the real data generated is from a same distribution.

Fig1. is the distributions of the three dataset. The black solid line is return distribution of BSI, the blue dash line is

the return distribution generated by jump diffusion model and the red dot line is that of geometric Brownian model.

Figure 1. Three distributions

Evidently, the double exponential jump diffusion model fits the BSI better than geometric Brownian motion.

V. CONCLUSION The jump-diffusion model contains a jump part, which

can explain the violent wave of shipping freight rate when some unexpected news impacts the market.

The paper presented a general double exponential jump- diffusion model for shipping freight rate. The model’s parameters are estimated by MCMC method.

The upper of Fig2.displays the volatility of BSI weekly returns, the lower parts is the jumps picked out by the GDEJD model. The graph indicates that the GDEJD model is close to the real world.

Figure 2. Jump of BSI

The jump-diffusion model is supposed to be more realistic and better than geometric Brownian motion model in simulating the market’s risk. By applying Cramer-Von Mises Criterion, the intuition is proved.

ACKNOWLEDGMENT The research is supported by Innovation Program of

Shanghai Municipal Education Commission (10ZZ101), the Foundation of Science (2009164) and the Key Discipline (XR0101) of Shanghai Maritime University.

REFERENCES

315

[1] LI Yaoding, “Ship investment under uncertainty,” Dissertation,Shanghai Maritime University,2007.

[2] Gu Xianbin, LI Xuying,“Empirical analysis on long memory property of Baltic dry index”,Journal of Shanghai Maritime University,Vol.30 No.1, Mar. 2009,pp.40-44.

[3] S.G.Kou, “A jump-diffusion model for option pricing”,Management Science, Vol.48, No. 8, August 2002,pp.1086-1101

[4] LIU Xiaoshu, “Empirical comparison of three double-exponential-jump-diffusion models,”South China Journal of Economics, Vol.2,2008,pp.64-72.

[5] Deepak K. Agarwal and Alan E. Gelfand, “Slice gibbs sampling for simulation based fitting of spatial data models,”, Statist. Comp. Vol.15, 2005,pp:61–69.

[6] Gu Xianbin, “An application of real option method in ship investment under jump-diffusion process,”Disseration, Shanghai Maritime University,2009.

[7] T.W.Anderson, “On the distribution of the two-sample cramer-von mises criterion,” The Annals of Mathematical Statistic, Vol.33,pp:1148-1159,1962

[8] E.J.Burr, “Distribution of the two-sample cramer-vom-mises criterion for small equal samples,” The Annals of Mathematical Statistic, Vol.34,pp:95-101,1963

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