# Investment decisions under uncertainty on LNG-powered vessels for environmental compliance

- Shun Chen
^{1}, - Shiyuan Zheng
^{1}Email author and - Qiang Zhang
^{1}

**Received: **1 November 2017

**Accepted: **18 April 2018

**Published: **25 April 2018

## Abstract

The shipping industry is investigating alternative fuels for ships, in order to comply with stricter emission requirements implemented by International Maritime Organization (IMO). Liquefied Natural Gas (LNG) is a promising alternative since it could reduce emissions substantially and offer potential fuel cost savings. But the investment in LNG fuelled vessels is currently facing a high degree of uncertainty, such as the differential between the prices of LNG and conventional maritime fuels, the availability of LNG and the reliability of its supply chain.

This paper makes an attempt to study the possibility of investing in LNG powered vessels under uncertainty. A deferral option model is proposed to quantify the value of flexibility for deferral based on multi-variables following specified stochastic processes. By exploiting the stochastic processes, it is possible to determine the value of deferral by solving a dynamic program using a least squares Monte Carlo simulation. The model is tested on an investment of a new chemical vessel with 19,000 dwt powered by LNG. Empirical analysis may suggest different investment strategies based on the probabilities of exercising an option and related option values each year. It indicates further that the attractiveness of LNG as ship fuel is dominated by a couple of parameters: difference of ship prices between a LNG powered vessel and a reference one, the price differential between LNG and conventional fuel prices, the share of the sailing time inside Emission Control Areas (ECAs), and the supply cost of LNG.

## Keywords

## Introduction

The shipping industry is a substantial emitter of air pollutants such as nitrogen oxides (NO_{x}), sulphur oxides (SO_{x}), carbon monoxide (CO) and carbon dioxide (CO2), because the vast majority (95%) of the world’s shipping fleet runs with engines powered by Intermediate Fuel Oil (IFO) for economic reasons (Cullinane and Bergqvist 2014). Though IFOs are cost effective, they contain high levels of asphalt, carbon residues, sulphur and metallic compounds and have high viscosity and low volatility properties as well.

_{x}and NO

_{x}, are subject to a series of step changes over the years, which are described in Tables 1 and 2.

Fuel sulphur contents: global and ECA limits

Date | Global limit [% mass] | Date | ECA limit [% mass] |
---|---|---|---|

Prior to 1/1 2010 | 4.5% | Prior to 1/7/2010 | 1.5% |

After 1/1 2010 | 3.5% | After 1/7/2010 | 1.0% |

After 1/1 2020 | 0.5% | After 1/1/2015 | 0.1% |

NO_{x} emission reduction programme

Tier | Date | NO | ||
---|---|---|---|---|

| 130 ≤ |
| ||

Tier I | 2000 | 17.0 | 45 × n | 9.8 |

Tier II | 2011 | 14.4 | 44 × n | 7.7 |

Tier III | 2016 | 3.4 | 9 × n | 1.96 |

_{x}and NO

_{x}emissions, suggested by Acciaro (2014). The first is the switch to higher-quality fuels, low in sulphur contents. The second is the use of exhaust gas cleaning systems, often referred to as maritime scrubbers, in combination with selective catalytic reduction to reduce nitrogen oxides for new vessels. And the third one is the use of vessels operating on LNG. Ship specifications, advantages and disadvantages of these three alternatives can be demonstrated in Table 4. It is further discussed in Acciaro (2014) that the use of scrubbers at sea is not well established due to the lack of detailed technical studies and accurate cost figures. Therefore, the alternative of installing exhaust gas cleaning systems is out of the scope and the focus is on distillate fuels (distillate oils include diesel fuels and fuel oils, e.g. Marine Gas Oil (MGO) and Intermediate Fuel Oil (IFO)) or on LNG.

Gas emission reduction resulting from operating with different emission control systems compared to the use of LNG

Abatement technology/measure | Emission reduction (%) | |||
---|---|---|---|---|

SO | NO | PM | CO | |

Basic internal engine modifications for 2 strokes, slow speed only | 0 | −20 | 0 | 0 |

Advanced internal engine modifications | 0 | −30 | 0 | 0 |

Direct water injection | 0 | −50 | 0 | 0 |

Humid air motors | 0 | −70 | 0 | 0 |

Exhaust gas recirculation + scrubbing | −93 | −35 | −63 | 0 |

Selective catalytic reduction (2.7% S residual oil fuel) | 0 | −90 | 0 | 0 |

Sea water scrubbing | − 75 | 0 | −25 | 0 |

Fuel switching (from 2.7% S to 1.5% S IFO) | −44 | 0 | −18 | 0 |

Fuel switching (from 2.7% > 1.5% S IFO) | − 81 | 0 | −20 | 0 |

Low S marine diesel (from 0.5% to > 0.1% S) | −80 | 0 | 0 | 0 |

Liquefied Natural Gas (LNG) | −90 | −80 | −100 | −20 |

Ship specifications, procs and cons of three emission abatement measures

Options | A | B | C |
---|---|---|---|

Oil powered ship with SCR | Oil powered ship with scrubber + SCR | LNG powered ship with SCR | |

Main engines and propulsion | |||

Engine type | Diesel | Diesel | Dual Fuel Diesel |

Propulsion system | Direct | Direct | Direct |

Auxiliary engine | |||

Engine type | Diesel | Diesel | Dual Fuel Gas |

Fuel (in ECA) | LSFO (Low Sulphur Fuel Oil) (0.1% S) | IFO | LNG, LSFO (0.1% S) |

Off ECA | LSFO (0.5% S) | LNG | |

SCR | Installed | Installed | Installed |

Scrubber | Not | Installed | Not |

Procs | -No or little modifications and investment needed -Well known and tested | -Lower fuel costs -Fuel available | -Potential CO2 reduction -Much lower maintenance |

Cons | -Higher fuel costs -Fuel availability uncertain -Wear and tear | -Higher OPEX, fuel consumption and maintenance costs -Ship design and stability challenges -Sludge management: need to create a logistics chain and adapt port infrastructure | -More space needed for the gas system on board -Bunkering points and associated logistics to be created -Safety aspects increase complexity of the supply chain, ship design and operations -Skilled and trained crew |

Among the technologies that are currently evaluated, the possibility for ships of switching to LNG as a main fuel has gained significant concerns during the last few years. The use of LNG as the fuel for vessels has potentially substantial advantages, since LNG allows for a significant reduction of NO_{x}, SO_{x}. It is quite attractive in particular for ECA compliance, as clearly demonstrated in Table 3. However, technical maturity, availability and costs appear to be the most critical issues for the success of LNG as a maritime fuel on a large scale (Acciaro et al. 2013).

The investment returns of two alternatives of using distillate oils or LNG are greatly dependent on the price differences between distillate oils and LNG. However, the future prices of LNG are largely unknown as an international market for natural gas does not exist and the future prices of IFO and MGO are largely volatile caused by global unstable economic and political situations. The uncertainty associated with price differentials among fuels could not be taken into consideration in a traditional discounted cash flow (DCF) analysis when making the economic evaluation of the ship investment. The traditional approach assumes that management, having made the decision to initiate a capital investment, will manage cash flows continuously as planned until the end of its pre-specified useful life. This approach ignores the ability of management to adapt or revise decisions in response to unexpected market developments (Lai and Trigeorgis 1995; Bendall and Manger 1988; Bendall and Stent 2007).

Real Options Analysis (ROA), in contrast, incorporates both the uncertainty inherent in the operating business environment and the ability to actively manage, control or alter, (albeit to switch inputs or technology, expand, abandon, defer, etc.) a project in response to the changing circumstances when new information becomes available (Trigeorgis 1991, 1993a, 1993b; Bendall and Stent 2007). There is a wide body of literature in the field of real options. Trigeorgis (1995, 1996) amongst others provide a good summary of the field. Till now the Real Option Analysis (ROA) has been applied successfully into the shipping industry to analyze managerial strategies, see Alizadeh and Nomikos (2009) for a review, including the decision to enter or exit a certain market (Dixit 1989, 1992); the extension of a time charter agreement (Bjerksund and Ekern 1995); the investment in a new service (Bendall and Stent 2003); the switch between dry and wet markets (Sødal et al. 2008, 2009); the investment in new vessels or portfolio of vessels (Hopp and Tsolakis 2004; Bendall and Stent 2003, 2005, 2007; Dikos 2008); or the change of flag for a vessel (Kavussanons and Tsekrekos 2011).

Until recently, the real options used for investment related to environmental compliance in shipping can be seen in Acciaro et al. (2013) and Acciaro (2014), which proposes a decision support model for ship owners based on the use of an option to defer in order to assess the viability of postponing investment in retrofitting vessels for environmental compliance. One of the main limitations in Acciaro et al. (2013) is that the price differential between LNG prices and distillates was fixed. Although the limitation was overcome in Acciaro (2014) by making the price differential change yearly, prices of LNG and distillates are just one sample of the fuel price distributions. If the sample path is changed, cost savings will be changed as well, and this may significantly influence the results.

In our paper, we propose a deferral option model where fuel prices are characterized by the mean-reverting geometric Brownian motion, and 300 sample paths are used to calculate the frequency of an investment opportunity. A dynamic program is designed to determine the value of the flexibility by making use of a least squares Monte Carlo simulation algorithm.

The main differences of our research with previous studies rest with three aspects. First, investment approaches incorporating uncertainty are abundant, but very little has dealt satisfactorily with non-stationary variables. In our study, fuel prices are characterized by non-stationary stochastic processes, because it seems not possible to make the prediction of these prices in the long term. Second, how the optimal time to invest changes with different paths and under different assumptions about market conditions is assessed. Third, various scenarios influencing the time to exercise option are discussed, providing support for shipping companies to make the investment decisions. The proposed model not only provides a tool to analyze the investment strategy under uncertainty, but it also provides an insight into the interdependency of variants influencing the investment in LNG powered vessels and will show the required flexibility in the decision-making for shipping companies.

This paper is structured as follows. The next section describes a proposed analytical modelling approach to measure the value of flexibility in deferring under non-stationary stochastic demand using real options solution methodologies. This is followed by a presentation of the data properties. We then proceed to present the empirical analysis. Some conclusions and directions for further research are addressed in the last section.

## Methodology

Uncertainty in real option analysis can be modelled as nonstationary stochastic processes, where the distribution may change over time. According to Dixit and Pindyck (1994), a stochastic process is defined as ‘a variable that evolves over time in a way that is at least in part random.’

Two common forms of such processes are the Geometric Brownian Motion with drift and the Ornstein-Uhlenbeck mean-reverting process. In relation to the LNG and distillate fuels markets, Geometric Brownian Motion (GBM) model, however, seems to have some drawbacks due to the fact that it does not capture the special characteristics such as mean reversion. Given a starting price, GBM would accept this as a normal event and would proceed randomly from there with no consideration of prior price levels and no greater probability of returning to the average price level. It allows that the price fluctuation could reach an unrealistic level when GBM is used to model spot prices. Mean reversion can be thought of a modification of the random walk, where price changes are not completely independent of one another but rather are related. Hence, prices of distillate fuels and LNG are supposed to follow mean-reverting processes in our study, which is bolstered by Li (2007), who argues that over time, the process of fuel prices tends to drift towards its long-term mean: such a process is called mean-reverting.

Where *P*_{o, t} means IFO prices at time t. \( {\overline{P}}_o \) is the mean of the variable *P*_{o, t}, *P*_{g, t} refers to LNG prices at time t, *dW* is an increment in a Wiener process and *η*_{
g
} and *σ*_{
g
} are parameters, \( {\overline{P}}_g \) denotes the mean of the variable *P*_{g, t}.*η*_{
o
} and *σ*_{
o
} are parameters.

The NPV is expanded on by incorporating the option premium, which represents the value from options that a static NPV analysis would not be able to capture. Benaroch and Kauffman (2000) present a clear comparison of this valuation versus a traditional discounted cash flow method. Chow and Regan (2011) conclude that many different kinds of real options can fit into the option premium in the framework provided by Trigeorgis. For example, Benaroch (2002) lists the many types of real options that can apply to information technology, including deferral, staging, scaling, abandonment, outsourcing, leasing, and compounding (interaction of multiple options).

Given an expiration time T and a project value based on one or more nonstationary stochastic processes, the problem of deciding the optimal time to exercise an option can be interpreted as an optimal stopping problem, according to Dixit and Pindyck (1994). An example of a real option defined as a stopping problem is the deferral option.

Where *u*_{
t
} is the decision to continue or stop the process at time *t*, *π*_{
t
} is the profit flow in the current time state as a function of *u*_{
t
} and state variable *x*_{
t
}, *ρ* is the discount rate for continuous compounding, *E*_{
t
} is the expectation at time *t*, and Φ_{
t
} is the value of the control problem. Equation (3) is commonly called the Bellman equation or the fundamental equation of optimality (Dixit and Pindyck 1994).

*π*is the net present value of the total cost savings between a LNG fuelled vessel and a conventional one.

*f*(

*P*

_{o, j},

*Q*

_{o, j}) denotes annual operational costs and voyage costs dependent on the stochastic fuel oil prices at time t.

*f*(

*P*

_{g, j},

*Q*

_{g, j}) refers to annual operational costs and voyage costs dependent on the stochastic LNG prices at time t, Δ

*R*

_{o, g}means the difference of the residual values for two types, while Δ

*K*

_{o, g}is the difference of capital costs of two types of vessels.

Where \( {\pi}_{t_n} \) is the net present value of the total cost savings between a LNG fuelled vessel and a conventional one and it is a function dependent on the stochastic LNG prices, denoted as \( {P}_{g,{t}_{n+1}} \) and fuel oil prices, denoted as \( {P}_{o,{t}_{n+1}} \). *t*_{
n
} is a time state such that *t*_{n + 1} − *t*_{
n
} = Δ*t* and *n* = *T*/*∆t* is the final time horizon, but *n* = 0 is the initial time state being solved for.

In this example, the process is deferral and in each time state the problem is whether to continue deferring or to stop deferring and to invest immediately with a value of 0. The problem can be solved using differential equations if the state variable is an Ito process.

The deferral option in the investment of LNG vessels is a single option dynamic programming problem with an objective function that depends on the stochastic LNG and fuel oil prices. Because of the complex nature of this problem, it is not suitable to use finite difference methods to solve them, which may have a high computational cost. In this case, a relatively fast converging numerical option valuation method is necessary. This requirement rules out traditional Monte Carlo simulation methods as well. The Least Squares Monte Carlo simulation method (LSM) refined by Longstaff and Schwartz (2001) is therefore chosen for solving the option value, see Chow and Regan (2011). To apply this method, fuel oil prices *P*_{o, t} and LNG prices *P*_{g, t} need to be in a discrete format and the certain assumptions need to be made for the cost function to obtain the PV.

Where *η*_{
o
} is a generalized drift parameter and *σ*_{
o
} is a generalized diffusion parameter for oil-based prices; *η*_{
g
} is a generalized drift parameter and *σ*_{
g
} is a generalized diffusion parameter for LNG prices, and *W*_{Δt} can be simulated with a normal inverse function with \( \varepsilon \sqrt{\Delta t} \), where *ε* ∼ *N*(0, 1) Time series data of fuel prices would be necessary to estimate the parameters.

### The LSM solution algorithm

Because fuel prices are assumed to follow stochastic processes, numerical methods are necessary to solve realistic real option dynamic programming problems. Trigeorgis (1996) describes several common numerical methods developed to handle real option analysis. The methods can generally be categorized into three classes: finite difference, binomial lattice methods, or Monte Carlo simulation.

Chow and Regan (2011) point out that the finite difference method (Brennan and Schwartz 1977) is not very suitable for multiple options where the differential would be difficult to specify. The binomial lattice method (Cox et al. 1979) is not able to handle multidimensional variables very well because of its inherent method of simplifying probability distributions into branches. Traditional Monte Carlo simulation methods developed by Boyle (1977) are suitable in solving dynamic programming problems of multiple variables, while it has very high computational cost because at any time step, the backward dynamic program requires foreseeing the expected continuation function in the future time step. This complication resulted in the Least Squares Monte Carlo simulation method (LSM) refined by Longstaff and Schwartz (2001). LSM reduces the computational cost by using least squares regression at each intermediate step, based on the results of the following future time step along all the simulation paths (Chow and Regan 2011).

*ρ*, time horizon

*T*and a number of simulation paths

*P*.

- 1.
For each path

*ω*∈*P*and time state*t*_{ n }, 0 ≤*n*≤*T*/Δ*t*, generate simulated LNG and fuel oil prices using Eqs. (6)–(7). - 2.
For the known \( {P}_{t_0} \) and for each of the simulated \( {P}_{t_n}\left(\omega \right) \), 0 ≤

*n*≤*T*/*∆t*,*ω*∈*P*, evaluate an objective such as Eq.(4) to obtain \( {\pi}_{t_n} \) or the cost function valuation algorithm to obtain \( {\pi}_{t_n}\left({P}_{g,{t}_n},{P}_{o,{t}_n}\right) \). - 3.
Starting from

*n*=*T*/*∆t*, use LSM to solve the backward dynamic program Eq. (5).① If

*n*=*T*/*∆t*, let \( {\Phi}_{t_n}\left({P}_{g,{t}_n}\left(\omega \right),{P}_{o,{t}_n}\left(\omega \right)\right)=\max \left(0,{\pi}_{t_n}\left({P}_{g,{t}_n},{P}_{o,{t}_n}\right)\right) \).② Keep track of the optimal decision with the variable

*θ*(*ω*,*t*_{ n }); if \( {\Phi}_{t_n}\left({P}_{g,{t}_n}\left(\omega \right),{P}_{o,{t}_n}\left(\omega \right)\right)>0 \), then that sample path is “in the money” and is assigned a value of*θ*(*ω*,*t*_{ n }) = 1. Otherwise, let*θ*(*ω*,*t*_{ n }) = 0.③ Let

*n*=*n*− 1. If*n*= 0, go to step 4.④ Estimate \( {\widehat{\Phi}}_{t_{n+1}} \) using least squares regression with Hermite polynomial series, although other polynomials that form an orthonormal basis such as Laguerre series can also be used:

*i*+ 1) th polynomial can be represented recursively as

Where the *β*_{
i
} coefficients are estimated using least squares. The *x* values are \( {\pi}_{t_n}\left({P}_{g,{t}_n}\left(\omega \right),{P}_{o,{t}_n}\left(\omega \right)\right)\left|{}_{\pi_{t_n>0}}\right. \) at each simulation path at the particular time interval.

⑤ Use the estimate \( {\widehat{\Phi}}_{t_{n+1}} \) to solve Eq. (5). If the optimal decision is to wait because \( {\left(1+\rho \right)}^{-\Delta t}\cdot E\left[{\widehat{\Phi}}_{t_{n+1}}\right]>{\pi}_{t_n}\left({P}_{g,{t}_n}\left(\omega \right),{P}_{o,{t}_n}\left(\omega \right)\right) \), then the option on that simulation path is still in the money and *θ*(*ω*, *t*_{
n
}) = 1.

*n*> 0, go to step ③. Otherwise, the value obtained from Eq. (5) is the value of option.

- 4.
If \( {\Phi}_{t_0}>0 \), then the option is worth keeping. If \( {\left(1+\rho \right)}^{-\Delta t}E\left[{\widehat{\Phi}}_{t_1}\right]>{\pi}_{t_0}\left({P}_{g,{t}_0},{P}_{o,{t}_0}\right)\Big)>0 \), then the best strategy is to defer the option. If \( {\Phi}_{t_0}>0 \) and \( {\left(1+\rho \right)}^{-\Delta t}E\left[{\widehat{\Phi}}_{t_1}\right]\le {\pi}_{t_0}\left({P}_{g,{t}_0},{P}_{o,{t}_0}\right)\Big) \) then the best strategy is to invest immediately.

- 5.
The algorithm may need to be returned at increasing values of ∏ until the option value stops increasing beyond some tolerance.

## Empirical analysis

The real options analysis with LSM solution algorithm is applied to the investment strategy in shipping. As introduced in the first section, the most realistic alternatives are three if the focus is to reduce SO_{x} and NO_{x} emissions: the switch to higher-quality fuels, the use of exhaust gas cleaning systems, in combination with selective catalytic reduction to reduce nitrogen oxides for new vessels; and the use of vessels operating on LNG. Our research focuses on distillate fuels or LNG for a new chemical ship for the environmental compliance. The alternative of installing exhaust gas emission reduction devices, such as scrubbers, is beyond the scope of this paper, as explained in Section 1. Before presenting the model results, figures and information used in this paper will be explained at first.

### Data description

Ship Parameters for two types of chemical ships

Options | Unit | Traditional Type | New Type |
---|---|---|---|

Oil ship with SCR | LNG fuelled ship with SCR | ||

Main engines and propulsion | |||

Engine speed | – | Low | Low |

Engine type | – | Diesel | Dual Fuel Diesel |

Propulsion system | – | Direct | Direct |

Auxiliary engine | |||

Engine speed | – | Medium | Medium |

Engine type | – | Diesel | Dual Fuel gas |

Fuel (in ECA) | – | LSFO (0.1% S) | LNG, LSFO |

Off ECA | – | MGO (0.5% S) | LNG |

SCR | Installed | Installed | |

Deadweight | DWT | 19,000 | 19,000 |

Speed | Knot | 13.3 | 13.3 |

Days at sea | Days | 35 | 35 |

Days at ports | Days | 13 | 13 |

Fuel consumption at sea | Tonnes/day | 23.7 | 20.88 |

Fuel consumption at port(MGO) | Tonnes/day | 9.4 | 3.12 |

The study makes an attempt to solve the probability of each option of investing in a LNG powered vessel between 2016 and 2020. Costs are estimated of a LNG powered vessel in comparison to a reference vessel burning maritime fuel oils. For a conventional ship, 0.1% LSFO (low sulphur fuel oil) is used inside an ECA. Off the ECA, fuel oils not exceeding 3.5% sulphur could be used by 2020, after which the global sulphur cap will be reduced to 0.5%. For the LNG powered ship, dual-fuel low speed diesel engine is installed as main engine. LNG is the main fuel and LSFO is the pilot fuel, so SCR is required to comply with Tier III. At the voyage of Houston-Ulsan, it is estimated that one ship operates roughly 25% within the ECA annually.

### Investment deferral option

Uncertainty of an investment in a LNG powered ship lies in prices of oil-based fuels and LNG, each of which could evolve as a mean-reverting geometric Brownian motion, as explained in Section 1. The deferral option model can be solved in several steps as follows.

### Step 1: To simulate fuel prices and estimate the net present values of total costs

*η*

_{ o },

*η*

_{ ghh },

*η*

_{ gjp }, \( \overline {P_o} \), \( \overline {P_{ghh}} \), \( \overline {P_{gjp}} \),

*σ*

_{ o },

*σ*

_{ ghh }and

*σ*

_{ gjp }are required to be estimated. First, the sample data are used to estimate the parameters

*a*

_{ o },

*b*

_{ o },

*a*

_{ ghh },

*a*

_{ gjp },

*b*

_{ ghh }and

*b*

_{ gjp }in the following equations by OLS:

Where *σ*_{ε, o}, *σ*_{ε, ghh} and *σ*_{ε, gjp} are the standard deviations of regression equations, respectively. *P*_{
ghh
} refers to LNG prices at Henry Hub, \( {\overline{P}}_{ghh} \) denotes the mean of the variable *P*_{
ghh
}. *P*_{
gjp
} refers to LNG prices at Japan, \( {\overline{P}}_{gjp} \) denotes the mean of the variable *P*_{
gjp
}.

Where *P*_{mgo, t} refers to MGO price, *P*_{o, t} means IFO price at time t, *α* is the parameter and *β* the intercept.

*a*

_{ o },

*b*

_{ o },

*a*

_{ ghh },

*a*

_{ gjp },

*b*

_{ ghh }and

*b*

_{ gjp }and calculate parameters of regression equations. Results are presented in Table 6.

Parameters | Estimates | Parameters | Estimates | Parameters | Estimates | |
---|---|---|---|---|---|---|

IFO |
| 0.002456 | \( {\overline{P}}_o \) | 5.5638 |
| 0.0377 |

MGO |
| 0.894557 |
| 1.1109 | – | – |

LNG |
| 0.01616 | \( {\overline{P}}_{ghh} \) | 1.5653 |
| 0.04939 |

LNG |
| 0.01376 | \( {\overline{P}}_{gjp} \) | 2.1945 |
| 0.05562 |

The net present values of total costs each period for 300 sample paths for two types of vessels (unit:$)

Sample path j | 1–20 | 2–21 | 3–22 | 4–23 | 5–24 |

1 | 93,652,148 | 94,619,235 | 96,308,355 | 97,813,552 | 99,229,020 |

2 | 87,457,616 | 86,617,530 | 85,429,349 | 84,239,065 | 83,983,471 |

3 | 91,875,435 | 92,990,715 | 94,714,786 | 96,839,855 | 99,295,831 |

114,598,621 | 116,251,992 | 117,713,689 | 118,678,309 | 119,689,962 | |

298 | 93,650,277 | 93,710,972 | 93,995,208 | 94,113,740 | 94,653,258 |

299 | 93,532,797 | 94,041,005 | 94,933,723 | 95,451,127 | 95,576,234 |

300 | 82,098,887 | 82,644,984 | 83,956,705 | 85,022,315 | 85,737,473 |

Part 2: The net present values of total costs each period for 300 sample paths for a LNG powered vessel | |||||

1 | 79,517,962 | 80,917,733 | 81,922,366 | 82,876,357 | 84,012,918 |

2 | 80,967,571 | 82,311,144 | 83,878,890 | 85,459,745 | 86,618,243 |

3 | 81,442,731 | 82,768,563 | 83,555,735 | 84,776,658 | 86,053,819 |

83,744,186 | 84,999,982 | 86,230,016 | 87,056,316 | 87,897,931 | |

298 | 84,168,935 | 85,602,135 | 86,892,395 | 88,276,463 | 89,917,868 |

299 | 79,497,737 | 80,667,512 | 82,373,912 | 84,099,198 | 85,495,554 |

300 | 79,918,498 | 81,261,929 | 82,257,027 | 83,598,094 | 84,725,689 |

### Step 2: to determine the optimal stopping time by using LSM program

In order to use the LSM algorithm, the option needs to be exercised in discrete times in the interval [0, T]. In this research, the time to maturity is 2020, the investment option could be exercised within the period of 2016 to 2020, so intervals (n) are 6 and the length (Δ*t*) is equal to 1. The underlying variables (\( {P}_{g,{t}_n} \) and \( {P}_{o,{t}_n} \)) are then simulated with 300 paths.

Where \( {\pi}_{t_n} \) is the net present value of the total cost savings between a LNG fuelled vessel and a conventional one and it is a function dependent on the stochastic LNG prices, denoted as \( {P}_{g,{t}_{n+1}} \) and fuel oil prices, denoted as \( {P}_{o,{t}_{n+1}} \). *t*_{
n
} is a time state such that *t*_{n + 1} − *t*_{
n
} = Δ*t* = 1 and *n* = *T*/Δ*t* = 6 is the final time horizon, but *n* = 0 is the initial time state being solved for.

Since LSM will be used to solve the backward dynamic program, let’s start with Year 5, i.e. the year of 2020. In Year 5, the shipowner has the possibility of investing in a new technology or deferring the investment decision. Because \( {\Phi}_{t_n}\left({P}_{g,{t}_n}\left(\omega \right),{P}_{o,{t}_n}\left(\omega \right)\right)=\max \left(0,{\pi}_{t_n}\left({P}_{g,{t}_n},{P}_{o,{t}_n}\right)\right) \), Keep track of the optimal decision with the variable *θ*(*ω*, *t*_{5}); if \( {\Phi}_{t_5}\left({P}_{g,{t}_5}\left(\omega \right),{P}_{o,{t}_5}\left(\omega \right)\right)>0 \) (it means the present value of total costs over 20 years of a ship burning oil-based fuels is larger than that of a LNG powered one), then that sample path is “in the money” and is assigned a value of *θ*(*ω*, *t*_{5}) = 1. Otherwise, *θ*(*ω*, *t*_{5}) = 0. We continue to go to Year 4 and do the same work till Year 1.

Simulation results of 300 sample paths in the case of ECA 25%^{a}

Sample path j | The optimal year to exercise | Option value ($) | Discount rate | Present value of savings ($) |
---|---|---|---|---|

1 | 1 | 18,889,317 | 1 | 18,889,317 |

2 | 2 | 8,481,576 | 1.26248 | 6,718,203 |

3 | 0 | 0 | 0 | 0 |

4 | 2 | 7,095,164 | 1.26248 | 5,620,021 |

5 | 0 | 0 | 0 | 0 |

298 | 1 | 15,268,176 | 1 | 15,268,176 |

299 | 1 | 20,818,910 | 1 | 20,818,910 |

300 | 0 | 0 | 0 | 0 |

Simulated option value: | 10,066,289 |

Sample distribution of time to exercise option for each year in the case of ECA 25%

Year | 2016 | 2017 | 2018 | 2019 | 2020 | No exercise | Option Value |
---|---|---|---|---|---|---|---|

Frequency | 233 | 9 | 1 | 1 | 2 | 54 | $100,266,289 |

Percent | 77.7% | 3% | 0.3% | 0.3% | 0.7% | 18% |

### Step 3: to consider some scenarios

The similar findings can also be detected in the other two cases of ECA 50%, i.e. a vessel operates 50% of 1 year in an ECA and ECA100%, i.e. a vessel operates the whole year in an ECA. It is clearly seen that if a vessel spends more time sailing in an ECA and the ship is cheaper, this option would be more attractive. The results suggest that an immediate investment is more attractive and deferring the investment tends to be less valuable in general compared to the first case. It is interesting to find that if a vessel using LNG spends the whole year sailing in an ECA and the ship value is only 10% higher than that of a reference one, the result prefers an immediate investment in 2016, and the probability of no investment during a 5-year period is shrunk to 5.3%.

It may be claimed that the model suggests an investment option in LNG depending on three parameters: the price differentials between LNG and oil-based fuels, the difference of new-building prices between a new ship burning LNG and a conventional one burning traditional maritime fuels, and the share of operation within ECAs.

For the cases of ECA 25 and ECA 50%, the results prefer waiting and no investment within the 5-year period. For the case of ECA 100%, only 12.3% of the time to exercise option would lie within the first year, and the result still suggests deferring the investment to the future after 2020 at the probability of 68.7%.

In general, deferring the investment appears to be more valuable as a strategy if LNG prices at Japan are used in the model. This could be explained by higher mean values and bigger standardised deviation of the simulated LNG prices at Japan. In other words, LNG prices at Japan are significantly higher than those from Henry Hub in the USA. The bigger variability and mean values of LNG prices affect substantially the total savings obtainable between the operation of two types of vessels.

The results indicate that there is no single answer to what is the best optimal time of making an investment in a LNG fuelled ship. The best timing would be dependent on the amount of time spent sailing in ECAs, the price differential of ship values between two types of vessels, the supply cost related to the consumption tax and bunkering cost, and LNG prices at different stations.

## Conclusions

The investment in LNG fuelled vessels is currently facing a high degree of uncertainty, such as the differential between the prices of LNG and conventional maritime fuels, the availability of LNG and the reliability of its supply chain. The discounted cash flow techniques cannot incorporate the flexibility to respond to new information and to defer the investment, therefore, real options analysis is favoured by academia to accommodate flexibility in the investment decision so that the valuation of a project can reflect operating and strategic adaptability.

The real options literature describes methods for quantifying the value of flexibility to execute such a strategy. Until recently, Chow and Regan (2011) have proposed an option methodology for network designs that feature multidimensional and stochastic variables. To date it has been the first time such an option methodology has been considered in deferring an investment in a LNG powered new ship related to environmental compliance in shipping. The proposed deferring option model is a numerical option evaluation method using the LSM algorithm to solve a backward dynamic program.

The prices of maritime fuels are assumed to follow the mean-reverting geometric Brownian motion, which are contained in the deferral option model. The model is solved following LSM algorithm proposed by Chow and Regan (2011). Based on the mean-reverting geometric Brownian motion equations, 300 sample paths of simulated weekly fuel prices over the period of 2015 to 2040 can be generated. Once fuel prices are given, the savings of the total costs for two types of chemical vessels can be computed. The deferral option model proposed in this paper can be solved by simulation within Matlab and the distribution of time to exercise option and the call option values are obtained for each path.

Results indicate that the attractiveness of LNG as the ship fuel compared to the use of distillate fuels is dominated by several parameters, including the price difference between LNG and IFO; the share of operation inside ECAs, the differential of new ship values between a LNG powered ship and a reference one and the supply cost.

The model to predict costs for LNG systems on board chemical vessels offers extensive possibilities to study additional variants. Options include different vessel sizes, fuel consumption of vessels, route profiles, and other LNG related operational costs.

In our study, we only examine the possibility of investing in a LNG powered vessel to comply with emission requirements of SO_{x} and NO_{x}. While the contribution of LNG as the ship fuel to CO_{2} reduction sparks off considerable debate and it will be left for further study.

## Declarations

### Acknowledgements

We thank the editor and anonymous referees for their valuable comments and suggestions. We are also grateful for comments from participants at the 14th World Conference on Transport Research in Shanghai, China, July 2016.

### Funding

This paper is funded by Shanghai Natural Science Foundation (13ZR1418600).

### Authors’ contributions

The first author is responsible for the acquisition and the analysis of data, making the empirical analysis, drafting the article and revising it based on comments from other co-authors. The corresponding author SY.Z contributed to this paper by designing the code for LSM (Least Square Monte Carlo Simulation) algorithm in the programming language and testing it within Matlab. The third co-author QZ is responsible for the literature review concerning the abatement technologies of meeting emission requirements, and policies aiming to improve the ocean environment. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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