Investment Analysis of Low-Carbon Yard Cranes: Integrating Monte Carlo Simulation and Jump Diffusion Processes with a Hybrid American–European Real Options Approach

Abstract

In order to realize green and low-carbon transformation, some ports have explored the path of sustainable equipment upgrading by adjusting the energy structure of yard cranes in recent years. However, there are multiple uncertainties in the investment process of hydrogen-powered yard cranes, and the existing valuation methods fail to effectively deal with these dynamic changes and lack scientifically sound decision support tools. To address this problem, this study constructs a multi-factor real options model that integrates the dynamic uncertainties of hydrogen price, carbon price, and technology maturity. In this study, a geometric Brownian motion is used for hydrogen price simulation, a Markov chain model with jump diffusion term and stochastic volatility is used for carbon price simulation, and a learning curve method is used to quantify the evolution of technology maturity. Aiming at the long investment cycle of ports, a hybrid option strategy of “American and European” is designed, and the timing and scale of investment are dynamically optimized by Monte Carlo simulation and least squares regression. Based on the empirical analysis of Qingdao Port, the results show that the optimal investment plan for hydrogen-powered yard cranes project under the framework of a multi-factor option model is to use an American-type option to maintain moderate flexibility in the early stage, and to use a European-type option to lock in the return in the later stage. The study provides decision support for the green development of ports and enhances economic returns and carbon emission reduction benefits.

1. Introduction

As a vital component of international shipping and logistics, ports play a crucial role in facilitating global cargo flows and driving economic development [1]. A large number of ships gathered in ports would emit carbon dioxide, and intensive shipping routes and port operations also generate a large amount of carbon emissions. It has become an industry consensus to actively build green ecological ports and strengthen pollution prevention, energy saving, and emission reduction. Port enterprises have actively promoted the green transformation of terminals by adjusting the energy use system.

The low-carbon transformation of ports is a research hot spot in the field of green ports. Early research on energy use retrofitting in ports and cranes mainly focuses on oil-to-electricity conversion [2,3]. With the continuous development of clean energy, ports have shifted their perspectives from traditional electric energy to more efficient and environmentally friendly clean energy. They have also introduced solar energy, wind energy and other new energy technologies for transformation. Among them, the development of hydrogen energy has become an important strategic choice for accelerating energy transformation and cultivating new economic growth points [4].

In the context of green ports, the investment and construction of hydrogen-powered yard cranes will face many uncertainties from technological development, energy price fluctuations, carbon price fluctuations, changes in the macroeconomic environment, and emergencies. The many uncertainties increase the complexity of the investment environment and raise the risk of investment decisions made by port enterprises. Therefore, to ensure the project’s economic benefits and carbon reduction, hydrogen-powered yard crane investment decision-making issues require comprehensive and systematic study.

In existing research on port-related investments, traditional evaluation methods such as the net present value (NPV) approach are based on static assumptions [5] and thus fail to effectively capture the dynamic uncertainties [6]. Moreover, most current models tend to focus on a single factor [7], lacking an integrated perspective on the interaction among multiple sources of uncertainty. Although real options theory has been widely applied in sectors such as energy [8,9], it has rarely been adapted to reflect the unique characteristics of the port industry—such as high asset specificity and strong policy sensitivity—which has led to a disconnect between theoretical research on staged and flexible investment decision-making and practical needs. Empirically, although pilot projects involving hydrogen-powered cranes have been initiated in ports like Qingdao, there remains a lack of systematic quantification of their investment value and carbon reduction benefits.

This study addresses the existing research gap by developing a multifactor real options model. First, it innovatively integrates three categories of dynamic uncertainty—hydrogen prices, carbon prices, and technological maturity—into a unified analytical framework. Hydrogen prices are modeled using geometric Brownian motion; carbon prices are simulated through a Markov chain model incorporating both jump diffusion components and stochastic volatility; and technological maturity is quantified using a learning curve approach. This modeling framework overcomes the limitations of traditional static evaluation methods. Second, in view of the long investment horizon of port infrastructure, a hybrid American–European option strategy is proposed. Investment timing and scale are dynamically optimized through Monte Carlo simulation combined with least squares regression. On the empirical side, the hydrogen-powered yard crane project at Qingdao Port is used as a case study to construct and implement the multifactor simulation model. The results demonstrate the effectiveness of the proposed method in capturing dynamic uncertainties, as well as its practical feasibility and applicability in guiding complex, phased investment decisions and managing flexibility in real-world investment scenarios.

This study makes significant contributions on both theoretical and practical levels. Theoretically, it develops a dynamic investment decision-making model that simultaneously incorporates multiple sources of uncertainty, including hydrogen prices, carbon prices, and technological maturity, thereby extending the application of traditional real options approaches to the domain of green port infrastructure investment. By overcoming the limitations of single-factor modeling, the proposed model better reflects the complex interplay and evolutionary dynamics among multiple factors in green port projects, providing a robust theoretical foundation for infrastructure investment decision-making. Practically, the study introduces a flexible investment strategy that serves as a viable decision-support tool for port enterprises undertaking green equipment investments under conditions of high uncertainty.

The structure of this paper is as follows: Section 2 presents the literature review. Section 3 introduces the methodology for investment decision-making in the low-carbon transition of hydrogen-powered yard cranes. Section 4 outlines the results of the case study and conducts a sensitivity analysis. Section 5 provides a discussion of the findings, and Section 6 concludes the paper.

2. Literature Review

As an important node in the transit of land and sea transportation, emissions from port shipping account for about 3% of global greenhouse gas (GHG) emissions. The serious carbon emission problem of port shipping makes low-carbon transformation an important way to build green ports. Greenhouse gases from maritime transportation significantly impact climate change and affect air quality in coastal port cities [10]. It is predicted that maritime emissions will still account for 17% of global GHG emissions by 2050 with increasing energy efficiency and emission reduction efforts in ports. For this reason, the establishment of green ports and the optimization of port energy structure to reduce GHG emissions have become an inevitable trend in the future development of ports. The main energy supply of ports is currently undergoing a transformation from traditional energy to renewable and clean energy. The growing demand for carbon emission reduction at ports holds significant importance in addressing climate change. Low-carbon approaches and technologies within integrated energy systems of green ports have emerged as key drivers in advancing port decarbonization efforts [11].

2.1. Clean Energy Applications and Challenges in the Port Energy Transition

In order to build a clean and low-carbon energy system, some ports have applied clean energy including wind, solar, photovoltaic, and hydrogen energy to replace traditional fuels; at the same time, the implementation of intelligent projects such as oil-to-electricity conversion and shore power has enabled ports to systematize low-carbon development [12]. The specific selection and use of clean energy in ports still needs to be studied. For example, the application of wind energy in ports has the problems of unstable energy supply and environmental dependence; the harsh climate and strong sunlight exposure will lead to the decay of battery efficiency, which will reduce the service life of photovoltaic modules; the application of electricity as a driving energy source in ports has high development costs and investment risks, and the economic benefits are small, and the use of onshore power does not necessarily reduce the total emission of pollutants.

Compared with the above clean energy sources, hydrogen energy as a zero-carbon fuel has the advantages of environmental protection and economic affordability, which is the best choice for achieving carbon neutrality in ports. The environmental characteristics of hydrogen energy are reflected in the fact that its combustion does not produce greenhouse gases and ozone-depleting chemicals and does not produce acid rain components. Compared to electricity, the economic affordability of hydrogen energy is reflected in the fact that port cities are rich in industrial by-products of hydrogen resources, which can provide sufficient low-cost hydrogen. Compared to wind energy, the development of the hydrogen storage process allows for hydrogen to be stockpiled and transported in a safe and compact manner. From the point of view of energy conversion efficiency, the application of hydrogen to synthetic fuels is only 10–35% efficient, which is relatively inefficient compared to direct electrification scenarios that would require 2–14 times as much electricity [10].

Under the interaction of multiple systems in logistics and energy, the energy transition of key equipment in port operations has become one of the crucial pathways to achieving low-carbon port operations. Electrification, as a pivotal step in this transition, plays a significant role. For example, Ding et al. reported that by retrofitting diesel-powered rubber-tired gantry cranes (RTGs) in the yard with electric rail-mounted gantry cranes (RMGs), the carbon emissions of the terminal yard were reduced by approximately 17,630 tons compared to conventional terminals [13]. Building on the progress achieved through diesel-to-electric transitions, the shift from electric to hydrogen power has emerged as a key direction for further reducing carbon emissions and advancing energy transition in ports. Hydrogen fuel cells, as an efficient and clean power generation technology, are capable of reliably converting hydrogen into electricity to power port machinery and equipment. Pivetta et al. conducted a comprehensive analysis of 74 hydrogen projects worldwide, and through a combination of techno-economic data comparison and case study analysis, identified the use of hydrogen fuel cells to power port equipment as one of the primary pathways for hydrogen energy application in port settings [14]. Numerous ports are actively promoting the use of hydrogen-powered machinery and equipment. For example, the Port of Los Angeles and the Port of Long Beach in the United States have retrofitted their trucks with hydrogen power and deployed hydrogen fuel cell trucks; the Port of Qingdao applies hydrogen fuel cell trucks and develops hydrogen-powered automated railroad cranes; the Port of Singapore has developed and designed a hybrid system that combines hydrogen fuel cells and lithium batteries, which is applied to tire cranes. According to the port’s calculations, hydrogen-powered yard cranes can reduce 3.5 kg of carbon dioxide emissions and 0.11 kg of sulfur dioxide emissions for every TEU of containers operated. However, the large-scale application of hydrogen energy in ports is still constrained by low preparation efficiency, harsh storage conditions, limited level of technological maturity, and low return on investment, which are the key research directions for the next step.

At this stage, the hydrogen energy industry and carbon trading market are in the early stage of development, with high market volatility and uncertainty. Therefore, the complex investment environment for hydrogen-powered yard cranes to carry out large-scale replacement brings the risk of high investment costs and high technical demand. How to scientifically formulate investment decisions to accelerate enterprise transformation and upgrading is one of the hot spots of academic research.

2.2. Analysis of Uncertainties in Hydrogen-Powered Yard Cranes Investments

To minimize the adverse impact of uncertainty on the investment value assessment of hydrogen-powered yard crane projects, Hazar and Ulusoy emphasize the importance of accurately identifying the key sources of uncertainty within the project. They argue that these uncertainties must be systematically categorized and thoroughly analyzed to understand the potential impact of each factor on the project’s objectives [15]. Building on this foundation, Quitoras et al. further point out that these uncertainties not only affect the direct investment value of the project, but also exert complex influences on the overall cost and environmental benefits of the energy system. As a result, they can indirectly elevate the risk level associated with port investments [16]. Regarding the specific classification of uncertainty factors, studies by Nunes et al. and Xu et al. indicate that the investment process of hydrogen-powered yard crane projects involves a range of uncertainties. These include fluctuations in equipment investment costs, variability in energy conversion efficiency, uncertainty in hydrogen market prices, volatility in carbon trading market prices, the direction and intensity of macro-level policy regulation, and the dynamic evolution of technological maturity. These factors are interrelated and collectively contribute to the complex challenges faced in the assessment of project investment value [17,18].

Fluctuations in technological maturity introduce significant uncertainty into investment decisions for hydrogen-powered yard cranes at ports. Ampah et al. employed cost learning curves to quantitatively analyze how advancements in hydrogen production technologies impact hydrogen market prices. Their findings indicate that technological progress in hydrogen production leads to cost reductions, thereby creating opportunities for hydrogen energy to enter the market at scale and potentially replace high-carbon fossil fuels [19]. This aligns with the findings of Wang et al., who argue that reductions in technological costs can help trigger investment opportunities at an earlier stage. Their perspective further supports the view that technological uncertainty significantly influences the future cost trajectory of renewable energy and plays a critical role in shaping investment decisions [9].

Energy price volatility represents another major source of uncertainty affecting the value of hydrogen-powered yard crane projects, with its impact primarily manifesting in two key areas. First, fluctuations in the hydrogen energy market directly influence the operational energy costs of equipment. After conducting a bibliometric analysis to review the characteristics and trends in hydrogen production and storage research, Liu et al. found that the development of hydrogen energy still faces significant challenges, including high production costs and stringent storage requirements. They emphasized that the development of novel hydrogen storage materials has become a focal point in storage-related research [20]. Based on existing energy sources, Wen and Aziz, through data-driven modeling, further demonstrated that hydrogen prices and carbon taxes are critical drivers for the future development of the hydrogen economy. This underscores the complexity of hydrogen market price fluctuations and their potential impact on the investment value of related projects [21]. Second, volatility in the carbon market mainly affects the carbon reduction revenues of the equipment. In examining investment decision-making under uncertainty for coal-fired power plants transitioning to solar photovoltaic systems within an emissions trading scheme, Lin and Tan found that the introduction of carbon trading mechanisms increased the investment value of carbon capture, utilization, and storage (CCUS) projects while reducing the probability of investment failure. Although the mechanism was insufficient to trigger immediate investment under current economic conditions, its presence encouraged long-term investment, particularly in a low oil price environment [22]. This indicates that uncertainty in the carbon market also exerts a significant influence on project returns and should not be overlooked.

2.3. Applications, Limitations, and Improvements of Real Options Theory

For this type of investment decision problem with multiple uncertainties, the scholars’ research methods can be divided into three steps: identifying and quantitatively portraying the uncertainty factors affecting the investment strategy, selecting the investment strategy decision framework, and evaluating the investment strategy decision. In terms of the choice of investment decision-making framework, early investment analysis usually applies the net present value (NPV) method. But this method has the disadvantage of ignoring the value of real options and managerial flexibility in an uncertain environment, thus it cannot solve the problems of stochastic factors, uncertainty, and managerial flexibility involved in the investment [23]. To circumvent the drawbacks of the NPV approach, Black and Scholes proposed the Black–Scholes (B-S) option pricing formula [24]. In 1977, Myers and Turnbull, on the basis of the B-S option pricing model, proposed the concept of “real options” [25]. Since its development, the real options approach has become a widely used modeling method in investment decision research. Compared with the NPV method, the real options method is more suitable for R&D and transition stage investments. In projects with long economic life and high investment value, the application of real options method can effectively reduce the investment risk of investors. Among the existing studies, it is proposed that the real options approach has proved to be reliable in the transition assessment of emission reduction projects.

Wang et al. analyzed the real options value of photovoltaic power generation projects under the Renewable Portfolio Standard (RPS) policy framework and found that a staged investment strategy can significantly enhance project flexibility and value [9]. Becker et al. addressed the challenges of high risk and irreversibility in geothermal energy investment decisions. Through simulations of geothermal projects implemented in Brazil, their study demonstrated that real options theory can enhance the economic value of projects by increasing managerial flexibility and improving the identification of uncertainty factors [26]. Moreover, real options theory can be effectively applied to determine the optimal timing and scale of investment. In the context of geothermal resource investment in China, this approach has enabled investors to better cope with uncertainties and improve project profitability [22].

However, the real options model still has certain limitations in practical applications, mainly reflected in the idealization of the parameter and assumption conditions in the model. These limitations make it difficult for the model to estimate the value of the project investment that fits the actual investment situation. For this reason, many scholars have begun to explore the improvement of the real options method. Fedorov et al. integrated real options methods with decision analysis, employing the least squares Monte Carlo (LSM) algorithm to model and optimize uncertainty and managerial flexibility in staged project investments. This approach enables the capture of additional value under energy price volatility and resource uncertainty [27]. Similarly, Yu et al. used a nested compound real options model to evaluate the economic feasibility of retrofitting coal-fired power plants with carbon capture and storage (CCS) and carbon capture and utilization (CCU) technologies. Their findings suggest that under current high carbon prices, CCS investment is economically viable, while CCU remains uneconomical. However, a combined strategy could increase both the probability of investment and overall profitability [28]. This modeling approach is comparable to real options and dynamic programming frameworks used in other energy project investment studies. Nonetheless, it still faces challenges due to uncertainties in forecasting market prices, technological costs, and policy environments, indicating a need for further refinement.

Zeng and Chen integrated game theory with real options to develop a game options approach, which was used to examine the impact of incentive policies for energy storage systems on the development of microgrids [29]. This hybrid methodology not only retains the strengths of traditional real options models but also extends their applicability and flexibility by incorporating complementary theoretical frameworks, providing a more effective analytical tool for addressing complex investment decisions. Arin and Ozbayoglu, in response to the pricing biases of the traditional Black–Scholes (BS) model—particularly in the valuation of deep in-the-money or out-of-the-money options—proposed a hybrid computational intelligence model based on deep learning [30]. By leveraging deep neural networks (DNNs), their model significantly improved pricing accuracy, especially under conditions of high market volatility and for tail options, achieving a 94.5% reduction in mean squared error (MSE) compared to the BS model. Despite its impressive performance on specific datasets, the model’s generalizability to other option types or market conditions remains to be fully validated. Moreover, aspects such as computational complexity, real-time responsiveness, interpretability, and integration with dynamic market factors still require further development.

Building upon traditional real options theory, recent studies have extended the framework into multifactor options models by incorporating a broader range of project-specific information to more comprehensively analyze uncertainties and managerial flexibility. In terms of model enhancement, researchers have improved stochastic processes or simultaneously adjusted parameter settings and the underlying stochastic structure to better reflect real-world complexities. For instance, Zhao et al. introduced a jump diffusion process into the option pricing model, allowing the model to account for sudden market shocks and better capture the impact of abrupt fluctuations in uncertain factors [31]. Building on this, D’Amico et al. proposed a Markov chain-based compound real options valuation framework for multi-stage R&D projects. By addressing correlated success probabilities and the stochastic duration of development phases, their model offers a structured approach to describe sequential decision stages and outcomes, thereby expanding the applicability of real options theory to more intricate R&D investment contexts [32]. Moreover, Cortazar et al. developed advanced multifactor models, including a two-factor model with stochastic volatility and jump structures, and a four-factor model incorporating stochastic holding costs and stochastic volatility. These models yield optimal decision strategies under general Markovian dynamics, offering refined analytical tools for investment decision-making in highly complex and uncertain environments [33]. Collectively, these studies provide valuable insights into the enhancement of real options models, enabling them to more effectively address the complexity and uncertainty inherent in real-world investment projects.

3. Methodology

3.1. Simulation of Uncertainty Factors

Hydrogen price, carbon price, and technology maturity were chosen as the main factors in modeling the uncertainties, as they directly affect the investment costs, carbon reduction benefits, and technology risks of hydrogen-powered yard cranes. Hydrogen price determines the operating cost of hydrogen-powered yard cranes, and fluctuations will directly affect the expenditure per unit of time, which in turn affects the economics of the project. Carbon price, on the other hand, determines the size of carbon emission reduction benefits, and fluctuations not only reflect policy risks, but also relate to whether carbon emission reduction benefits can be converted into economic gains. Technology maturity affects equipment investment costs and equipment reliability. As technology advances, lower equipment costs and greater system integration help improve investment returns and reduce risks. Changes in hydrogen price, carbon price, and technology maturity together determine project feasibility and economic benefits, and modeling them as variables helps to fully reflect the complexity and dynamics of hydrogen-powered yard cranes investment.

3.1.1. Hydrogen Price

Hydrogen prices are characterized by volatility, a downward trend, and significant uncertainty. These prices are influenced by a variety of factors, including upstream hydrogen production costs, midstream storage and transportation expenses, and downstream infrastructure and operational costs of hydrogen refueling stations. Although technological advancements and economies of scale are contributing to a gradual decline in hydrogen prices, the early-stage nature of market development continues to introduce considerable uncertainty [34,35]. The geometric Brownian motion (GBM) model assumes that asset prices follow a lognormal distribution and describes their stochastic evolution through a drift term and a volatility term. This model has been widely used in asset price modeling, option pricing, and risk management due to its mathematical tractability and ability to capture random fluctuations [36,37]. The continuity and lognormal distribution properties of GBM make it particularly suitable for modeling the stochastic behavior of hydrogen prices, especially in scenarios where price changes are relatively continuous and influenced by random factors [38]. Therefore, this study adopts the GBM model to simulate the dynamic evolution of hydrogen prices, as shown in Equations (1) and (2):

$${dh}_p=h_{\mu }\cdot h_{p,t}\cdot d_t+h_{\sigma }\cdot h_{p,t}\cdot {dW}_t$$

(1)

$$d{W}_t={\epsilon }_t\cdot \sqrt{d_t}$$

(2)

$h_p$ denotes the hydrogen price; ${dh}_p$ denotes the change in hydrogen price, representing its instantaneous variation over the time interval $d_t$; $h_{\mu }$ denotes the drift rate of the hydrogen price, determining the steady rate at which the price increases or decreases over time; $h_{p,t}$ denotes the momentary $t$ hydrogen price; $d_t$ denotes the time step used in simulating the continuous evolution of hydrogen price, capturing the gradual and continuous component of price changes over time; $h_{\sigma }$ denotes the volatility of the hydrogen price, which determines the magnitude of its random fluctuations and reflects the extent to which the price is influenced by market uncertainty; ${dW}_t$ denotes the increment of a Wiener process (Brownian motion), used to simulate the impact of random market fluctuations on the hydrogen price; ${\epsilon }_t$ is a random variable that follows a standard normal distribution, used to simulate the unpredictable stochastic fluctuations in hydrogen price changes; $\sqrt{d_t}$ denotes the square root of the time increment ${\mathrm{d}}_{\mathrm{t}}$, used to calculate the stochastic component of the process, ensuring that the magnitude of random fluctuations decreases as the time step becomes smaller.

3.1.2. Carbon Price

The volatility of carbon prices will cause a large uncertainty impact on the production cost of enterprises or projects. The price theory [39], regression model [40], geometric Brownian motion [41], Markov process with jump diffusion term, and stochastic volatility have been used to describe the change rule of carbon price in the current research.

The carbon price jump phenomenon in the investment decision problem of hydrogen-powered yard cranes mainly comes from the impact of sudden events on the carbon price. Specifically, the emergence of unexpected events such as new coronavirus infection shock, geopolitical conflicts, and extreme weather will bring sudden impacts on the carbon emissions trading market, which will cause the carbon price to be disturbed and fluctuate in a short period of time [42]. Discontinuous changes in carbon price will have a greater impact on the cost and revenue of the hydrogen-powered yard crane construction project, which in turn affects the investment value of the project. Therefore, it is necessary to introduce the jump diffusion process in the simulation of carbon price to explain the impact of unexpected events on the investment value of the project.

The stochastic fluctuation phenomenon in the investment decision problem of the hydrogen power yard cranes mainly comes from the influence of changes in the external market environment on the carbon price. Due to the obvious future nature of the carbon price, the carbon market will be affected by the changes in the economic situation and the fluctuation of the market environment. In order to cope with such changes, scholars introduced stochastic volatility, adjusting the volatility from a constant to a time-varying value, so as to modify the real options model [32,33].

A Markov chain model with jump diffusion components and stochastic volatility is capable of addressing complex problems in financial derivative pricing, portfolio optimization, and risk management [43]. By simultaneously capturing the jump behavior and volatility randomness of carbon prices, the model effectively accounts for nonlinear fluctuations, volatility spillover effects, and policy sensitivity in carbon price dynamics. This provides a more accurate framework for risk assessment and price forecasting in carbon markets [44].

Therefore, in this study, the Markov chain with jump diffusion term and stochastic volatility is used to portray the changing law of the carbon price, as shown in Equation (3):

$$d{c}_p=D_t+V_t+J_t$$

(3)

$c_p$ denotes a carbon price; $d{c}_p$ denotes the change in carbon price, representing its instantaneous variation over the time interval $d_t$; specifically, this change comprises three components: the drift term $D_t$, the volatility term $V_t$, and the jump term $J_t$, as defined in Equations (4), (5) and (7).

$$D_t=c_{\mu }\cdot c_{p,t}\cdot d_t$$

(4)

$D_t$ denotes the drift term, representing the product of the drift rate, the instantaneous carbon price, and the time increment. Specifically, $c_{\mu }$ denotes the drift rate of the carbon price, determining the average trend of price movement and reflecting the steady rate at which the price increases or decreases over time; $c_{p,t}$ denotes the momentary $t$ carbon price; $d_t$ denotes the change in time at moment $t$.

$${V_t=c}_{\sigma }\cdot \sqrt{v_t}\cdot c_{p,t}\cdot {dW}_t$$

(5)

$V_t$ denotes the volatility term, which captures the extent of variation in price fluctuations and reflects how market uncertainty evolves over time. Specifically, $c_{\sigma }$ denotes the volatility of the carbon price, which determines the magnitude of its random fluctuations and reflects the extent to which the price is influenced by market uncertainty; $\sqrt{v_t}$ denotes the magnitude or intensity of the stochastic volatility, reflecting the instantaneous variation in the amplitude of carbon price fluctuations. It is used to modulate the strength of the random shocks from Brownian motion on the carbon price; $c_{p,t}$ denotes the momentary $t$ carbon price; ${dW}_t$ denotes a Brownian motion, used to simulate the impact of random market fluctuations on the carbon price.

In Equation (5), the equation for the change in stochastic volatility is shown in Equation (6):

$$d{v}_t=k\cdot {(\theta -v}_t)\cdot d_t+\eta \cdot \sqrt{v_t}\cdot {dZ}_t$$

(6)

$d{v}_t$ denotes the change in volatility $v_t$ over the time interval $d_t$; $k$ denotes the mean reversion rate of volatility, representing the speed at which the stochastic volatility reverts to its long-term average level; $\theta$ denotes the long-term average of volatility, representing the average level toward which market volatility tends to revert in the absence of random shocks or fluctuations; $v_t$ denotes stochastic volatility, which describes the variability in the magnitude of price fluctuations and indicates that market uncertainty changes over time; $d_t$ denotes the infinitesimal change in time at moment $t$; $\eta$ denotes the volatility of volatility, describing the extent to which the volatility itself varies over time and reflecting the potential intensity of fluctuations in market volatility; $\sqrt{v_t}$ denotes the magnitude or intensity of stochastic volatility, reflecting the instantaneous variation in the amplitude of carbon price fluctuations, and is used to modulate the strength of random shocks from Brownian motion on the carbon price; $d{Z}_t$ is a standard Brownian motion used to describe random fluctuations in continuous time. It is correlated with the Brownian motion $d{W}_t$ with a correlation coefficient $\rho$, which indicates the degree of correlation between the two processes. Specifically, if $\rho =1$, the two processes are perfectly positively correlated; if $\rho =0$, they are independent; and if $\rho =-1$, they are perfectly negatively correlated.

$${J_t=c}_{p,t}\cdot d{J}_t$$

(7)

$J_t$ denotes the jump term, which simulates abrupt, discontinuous, and significant fluctuations in carbon price caused by unexpected market events. It represents a jump process that follows a compound distribution consisting of a Poisson process and a normal distribution.

$d{J}_t$ denotes the change in the jump process at time $t$, used to capture the total variation caused by sudden jump events. The equation of variation in the jumping process is shown in Equation (8):

$$d{J}_t=\sum _{i=1}^{N_t}{Y}_i$$

(8)

$N_t$ denotes the Poisson process representing the number of jump events per unit time, describing the frequency at which these events occur. The jump counts exhibit Markov chain properties, with jump intensity $\lambda$ indicating the average number of jump events per unit time; $Y_i$ denotes the price change caused by each jump event, which follows a normal distribution with a mean of ${\mu }_J$ and a standard deviation of ${\sigma }_J$. Here, ${\mu }_J$ denotes the average jump amplitude, while ${\sigma }_J$ captures the volatility or uncertainty of the jump amplitude, reflecting the degree of dispersion in the magnitude of the jumps.

The carbon price simulation methodology in this study introduces a Markov chain model Equation (7) with jump diffusion term ($d{J}_t$) and stochastic volatility ($d{v}_t$) based on the traditional geometric Brownian motion model (${dW}_t$). This improvement enhances the adaptability of the model to the volatility of the carbon market and is able to more accurately characterize the changes in carbon prices under high volatility and policy-driven conditions. With the jump diffusion term, the model is able to simulate abrupt changes in carbon prices, while the stochastic volatility term makes the simulation results more reflective of market uncertainty. The improved model is able to cope with the complexity of carbon price changes more effectively than the traditional geometric Brownian motion model, providing support for carbon-related investment decisions.

3.1.3. Technology Maturity

Hydrogen-powered yard crane equipment is a new energy device in the current stage of its life cycle, with a small application scale and low degree of standardization. As the technological maturity of hydrogen-powered yard crane equipment increases, the investment cost of the equipment will decrease. The studies of Penisa et al. and Yao et al. show that learning curves can effectively portray the changes in the investment cost of this type of renewable energy-related technology and equipment [45,46]. Therefore, the learning curve method is used to portray the investment cost of each piece of equipment [47], as shown in Equation (9):

$$E_{c,x}=a\cdot x^{-b}$$

(9)

$E_{cx}$ represents the investment cost of the $x$ unit; $\mathrm{a}$ represents the investment cost of the port’s first hydrogen-powered yard crane unit; and $b$ represents the percentage decrease in the investment cost of the unit that occurs as the technological maturity of the unit increases.

3.2. Cost–Benefit Model

At each stage of the hydrogen-powered yard crane construction project, every decision made by the port leads to changes in both costs and returns. The cost of the hydrogen power yard crane equipment construction project ($T_C$) is expressed as the sum of the annual hydrogen energy consumption cost ($H_{cy}$), the annual periodic maintenance cost ($M_{cy}$), and the investment cost of the hydrogen power yard crane equipment ($E_c$), as shown in Equation (10):

$$T_C=H_{cy}+M_{cy}{+E}_c$$

(10)

• The annual hydrogen energy consumption cost ($H_{cy}$) refers to the total yearly cost of hydrogen energy consumption, which varies according to the hydrogen price and the total number of hydrogen-powered yard crane units. It consists of the product of the number of hydrogen-powered yard cranes invested in ($X$), the number of TEUs of containers to be handled by each hydrogen-powered yard crane per year ($Q$), the amount of hydrogen used by the hydrogen-powered yard cranes for each TEU handled ($h_q$), and the price of hydrogen ($h_p$), as shown in Equation (11):

$$H_{cy}=X\cdot Q\cdot h_q\cdot h_p$$

(11)

• The annual periodic maintenance cost ($M_{cy}$) refers to the yearly maintenance and servicing expenses for hydrogen-powered yard crane equipment, which vary with the total number of units deployed. It consists of the product of the number of hydrogen-powered yard cranes invested in ($X$) and the annual maintenance cost ($m_c$), as shown in Equation (12):

$$M_{cy}=X\cdot m_c$$

(12)

• The annual hydrogen power yard crane equipment investment cost ($E_c$) refers to the total one-time expenditure for the procurement and installation of the equipment, which varies with the number of units. It consists of the product of the number of hydrogen power yard cranes ($X$) invested in construction and the annual investment cost per hydrogen power yard cranes ($E_{c,x}$), as shown in Equation (13):

$$E_c=X·E_{c,x}$$

(13)

The benefit ($T_R$) from the hydrogen power yard crane construction project consists of the sum of annual cost savings ($S_r$), annual carbon emission reduction benefit ($C_r$), and relevant government subsidies ($G$), as shown in Equation (14):

$$T_R=S_r+C_r+G$$

(14)

• The annual savings ($S_r$) is the sum of the annual electrical energy cost savings ($E_{rt}$) and annual maintenance cost savings ($M_{rt}$) that the port can realize after completing the equipment transition relative to conventional yard cranes, given that the port has the same number of hydrogen yard cranes and conventional yard cranes, as shown in Equation (15):

$$S_r=E_{rt}+M_{rt}$$

(15)

Following the hydrogen-powered retrofit of traditional yard cranes, the original equipment is decommissioned and replaced by new hydrogen-powered units, thereby eliminating all electricity consumption and associated electricity costs incurred during the operation of the traditional cranes. Given that yard cranes are among the primary electricity consumers within port areas [48], their deactivation results in the complete removal of their electricity demand, leading to a reduction in the port’s overall electricity expenditure [49]. The portion of electricity costs saved as a result of the shutdown of traditional yard cranes constitutes the achievable annual electricity consumption savings.

The annual cost of electricity for conventional yard cranes ($E_{rt}$) consists of the product of the number of hydrogen-powered yard cranes invested in ($X$), the number of TEUs of containers to be handled annually by each conventional yard crane ($Q$), and the cost of electricity for each TEU handled by the conventional yard cranes ($e_{ct}$), as shown in Equation (16):

$$E_{rt}=X\cdot Q\cdot e_{ct}$$

(16)

Following the hydrogen-powered retrofit of traditional yard cranes, the original equipment is decommissioned and replaced by new hydrogen-powered units, thereby eliminating the need for maintenance activities and associated expenditures required during the operation of the conventional equipment. Given the high operational intensity and significant depreciation rate of yard cranes [50], their maintenance demands are typically frequent, making the cost-saving effect of equipment replacement particularly notable. The portion of costs saved through the substitution of traditional yard cranes with hydrogen-powered ones thus constitutes the achievable annual maintenance cost savings.

The annual maintenance cost of a conventional yard cranes ($M_{rt}$) consists of the product of the number of hydrogen-powered yard cranes units invested in construction ($X$) and the annual maintenance cost per conventional yard cranes ($m_{ct}$), as shown in Equation (17):

$$M_{rt}=X\cdot m_{ct}$$

(17)

• The annual carbon emission reduction benefit ($C_r$) refers to the economic gains generated from the reduction in carbon emissions during the operation of hydrogen-powered yard crane equipment. This benefit varies with both the carbon price and the total number of deployed units. It consists of the product of the number of hydrogen-powered yard crane equipment invested in ($X$), the number of TEUs of containers to be handled by each yard crane per year ($Q$), the carbon emissions reduced by each TEU handled by the hydrogen-powered yard cranes ($c_{ne}$) and the carbon price ($c_p$), as shown in Equation (18):

$$C_r=X\cdot Q\cdot c_{ne}\cdot c_p$$

(18)

• The government-related subsidy ($G$) consists of the product of the number of hydrogen-powered yard crane equipment invested in the construction ($X$) and the government subsidy for the construction of each hydrogen-powered yard cranes ($I_g$), as shown in Equation (19):

$$G=X\cdot I_g$$

(19)

Integrating the cost module and the revenue module, analyzing from the perspective of annual cash flow, the cash inflow includes annual cost savings ($S_r$), annual carbon emission reduction benefits ($C_r$) and government-related subsidies ($G$), and the cash outflow includes annual hydrogen energy consumption costs ($H_{cy}$), annual regular maintenance costs ($M_{cy}$), and equipment investment costs ($E_c$). Therefore, the net return that can be expected from the investment at stage $t$ is as follows:

$$N_t=S_r+C_r+G-\left(H_{cy}+M_{cy}+E_c\right)$$

(20)

In this case, the hydrogen price obeys a geometric Brownian motion, and the carbon price obeys a Markov process with a jump diffusion term and stochastic volatility, both of which are accounted for as variables in the formula. Changes in equipment investment costs with technology maturity can be converted to a constant value by accumulating the cost of individual equipment. As shown in Figure 1.

3.3. Real Options Model Solution Methods

On the basis of the cost–benefit model constructed in the previous section, in order to effectively deal with the impact of uncertainty factors on investment decision-making, it is necessary to introduce dynamic analytical tools to systematically analyze the timing of investment and the structure of returns. In this context, this study proposes the concept of “optimality” as the core logic support for subsequent model construction and solution.

In this study, “optimality” refers to identifying the most favorable investment strategy for hydrogen-powered yard crane construction projects by comprehensively accounting for uncertainties such as hydrogen price, carbon price, and technological maturity. Through the development of a multifactor real options model, the goal is to assist ports in formulating an investment plan that achieves the best possible balance between economic returns and carbon emission reductions [51,52]. This optimal strategy not only maximizes the investment value of the project but also ensures a high level of cumulative carbon reduction, thereby supporting the port’s green and low-carbon transition while safeguarding the economic interests of port enterprises. Specifically, optimality is reflected in determining the best investment timing and scale for hydrogen-powered yard crane projects through a scientifically grounded decision-making approach, enabling ports to effectively manage the risks and challenges posed by uncertain factors.

Under this framework, the performance of different option strategies is modeled to systematically identify investment paths that strike the best balance between economics and low-carbon performance, ensuring that investment options are quantifiably “optimal”.

Monte Carlo simulation is used to solve real options in this study as a whole. For different real options scenarios, this study uses least squares Monte Carlo simulation to solve the portion of American-like real options and standard Monte Carlo simulation to solve the portion of European-style real options.

For the least squares Monte Carlo simulation method for American-like real options, as shown in Figure 2.

• Set and enter the following parameters: the number of simulation paths, i.e., the number of simulations $\mathrm{Z}$; the investment time node contained in each simulation path, i.e., the investment decision time node $\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}$.
• Use the geometric Brownian motion, Markov chain with jump diffusion term and stochastic volatility, and learning curve methods from Section 3.1 to simulate the price of carbon (Equations (3)–(8)), the price of hydrogen (Equations (1) and (2)), and the price of the investment in equipment (Equation (9)).
• Calculate the return on investment for each phase of the investment using the cost–benefit model in Section 3.2 (Equation (20)).
• Based on the return on investment for each stage of investment in the previous step (3), perform a least squares regression on the moments of investment other than the final stage, and fit a model to estimate the future expected value (Hold Value) for each stage.

$$Y_t=A{{·N}_{t-1}}^2+B·N_{t-1}+C$$

(21)

$Y_t$ denotes the predicted value of investment in hydrogen-powered yard cranes at stage $t;$ $N_{t-1}$ denotes the net return on investment in hydrogen-powered yard cranes at stage $t-1$; $A$ is the coefficient of the quadratic term in the regression model; $B$ is the coefficient of the linear term in the regression model; $C$ is the constant term in the regression model.

• Start backtracking from the last investment stage, if Exercise Valued $N_t$ > 0, invest, if Exercise Valued $N_t$ < 0, start backtracking.

Exercise Valued $N_t$ denotes the option exercise value of the hydrogen-powered yard crane investment at stage $t$.

• At stage $t$, compare Exercise Valued $N_t$ with Hold Value $Y_t$, invest if Exercise Valued is higher, and continue backtracking if Hold Value is higher.

$$IF{N}_t>Y_t,V_t=N_t;ELSE{continuestage}_{t-1}$$

(22)

Exercise Valued $N_t$ denotes the option exercise value of the hydrogen-powered yard crane investment at stage $t$.

Hold Value $Y_t$ denotes the option holding value of the hydrogen-powered yard crane investment at stage $t$.

$V_t$ denotes the value of the decision chosen for the hydrogen-powered yard crane investment at stage $t$.

• After determining the investment time of this path through backtracking, calculate the present value of these real options to obtain the real options investment value of this path.

$$V_{t,discount}=e^{-t·r}·V_t$$

(23)

$V_{t,discount}$ denotes the discounted decision value of the hydrogen-powered yard crane investment at stage $t$. $e$ is the base of the natural logarithm,$t$ represents the time span from the current stage to the investment decision stage, and $r$ is the discount rate.

• Repeat the least squares Monte Carlo simulation and compute the average of real options across all paths.

$$Optionvalue={\displaystyle \frac{1}{N}}\sum _{i=1}^N{V}_{ti,discount}$$

(24)

$Optionvalue$ refers to the final value of the option, that is, the average present value of the option across all simulated paths. $V_{ti,discount}$ denotes the discounted option value at time $ti$ along the $i$-th simulation path.

For the standard Monte Carlo simulation of European real options, as shown in Figure 3:

• Set and enter the following parameters: the number of simulation paths, i.e., the number of simulations $\mathrm{Z}$; the investment time nodes contained in each simulation path, i.e., the investment decision time node $\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}$;
• Use the geometric Brownian motion, Markov chain with jump diffusion term and stochastic volatility, and learning curve methods in Section 3.1 to simulate the price of carbon (Equations (3)–(8)), the price of hydrogen (Equations (1) and (2)), and the price of the equipment investment (Equation (9)).
• Calculate the return on investment for each phase of the investment using the cost–benefit model in Section 3.2 (Equation (20)).
• Calculate the value of the real options from the first stage onwards.

$$V_1=Max(N_1,0)$$

(25)

$V_1$ denotes the option value at stage 1; $N_1$ denotes the net return on investment in hydrogen-powered yard cranes at stage 1.

• If the current real options value is greater than zero, i.e., investment is made, the calculation of the European real options value for the next period continues. If the current real options value is equal to zero, i.e., no investment is made, the path is terminated.

$$IF{V}_{t-1}>0,V_t=Max\left(N_t,0\right),ELSE{V}_t=0$$

(26)

$V_t$ denotes the option value at stage $t$; $N_t$ denotes the net return on investment in hydrogen-powered yard cranes at stage $t$.

• Calculate the discounted value of the real options value for all stages and sum to obtain the value of the real options investment for the path.

$$V_{ti,discount}=\sum _{i=1}^N{e^{-t·r}·V}_{ti}$$

(27)

$V_{ti,discount}$ denotes the discounted option value at time $t$ along the $i$-th simulation path; $V_{ti}$ denotes the option value at time $t$ along the $i$-th simulation path before discounting; $e$ is the base of the natural logarithm; $t$ denotes the time span from the current stage to the investment decision stage for hydrogen-powered yard cranes; $r$ is the discount rate.

• Repeat the Monte Carlo simulation and compute the average of real options across all paths.

$$Optionvalue={\displaystyle \frac{1}{N}}\sum _{i=1}^N{V}_{ti}$$

(28)

• $Optionvalue$ refers to the final value of the option, that is the average present value of the option across all simulated paths.

4. Case Study Applications

Qingdao Port is the sixth largest container port in the world, with container throughput exceeding 30 million TEU in 2023. Qingdao Port continues phase II of a fully automated container terminal, in a continuous effort to promote the construction of green ports and the application of clean energy. Qingdao Port carried out the construction project of hydrogen-powered yard cranes in 2021, deploying six hydrogen-powered yard cranes in the automated terminal’s pilot application. In order to verify the feasibility of the model, this study chooses Qingdao Port Automated Terminal as the study case, and the parameter settings of the Qingdao Port Hydrogen Power Yard Cranes Multi-Factor Investment Decision Model are shown in

Table 1. Multi-factor real options model parameters.

Parameter	Description	Value	Unit
$a$	Investment cost for the first hydrogen-powered yard crane	2.4	Million RMB
$Q$	Number of standard containers to be handled per yard crane per year	40,000	TEU
$h_q$	Hydrogen consumption for handling one standard container with a hydrogen-powered yard crane	0.2	kg/TEU
$m_c$	Annual maintenance cost for each hydrogen-powered yard crane	10,000	RMB/unit
$e_{ct}$	Electricity cost for handling one standard container with a traditional yard crane	6	RMB/TEU
$m_{ct}$	Annual maintenance cost for each traditional yard crane	64,000	RMB/Unit
$c_{ne}$	Carbon emission reduction per standard container handled by hydrogen-powered yard crane	2.1	kg/TEU
$T$	Project duration	15	Year
$r$	Discount rate	0.06	——
$L$	Service life of hydrogen-powered yard crane equipment	15	Year
$I_g$	Government subsidy for construction of each hydrogen-powered yard crane	500,000
$h_p$	Initial hydrogen price	30	RMB/kg
$h_{\mu }$	Drift rate of hydrogen price	−0.03	——
$h_{\sigma }$	Fluctuation rate of hydrogen price	0.03	——
$c_p$	Initial carbon price	48	RMB/ton
$c_{\mu }$	Drift rate of carbon price	0.038	——
$c_{\sigma }$	Fluctuation rate of carbon price	0.035	——
$k$	Volatility mean reversion rate	1.0	——
$\theta$	Long-term average of volatility	0.02	——
$\eta$	Volatility of volatilities	0.05	——
$\rho$	Relevance of the Brownian motion	0.25	——
$\lambda$	Jumping intensity	0.05	——
${\mu }_J$	Mean value of jumps	0	——
${\sigma }_J$	Standard deviation of jump amplitude	0.1	——
$b$	Proportion of equipment cost reduction as technology matures	0.1	——

:

In this study, the starting moment of the phased investment program of the hydrogen-powered yard crane construction project is selected as 2021, and the time interval of phased investment is set as 5 years. The results of the probability prediction of the phased construction process of the hydrogen-powered yard cranes in Qingdao Port and the calculation of the desired phased construction process are shown in

Table 2. Probability prediction results of the phased construction process of the hydrogen-powered yard cranes.

Construction Process	2021–2025	2026–2030	2031–2035
6	15%	5%	1%
19	80%	40%	10%
38	4%	55%	80%
76	1%	5%	9%
Expected construction process	18	32	40

:

4.1. Design of Real Options Program for Yard Crane Investment

Considering the adoption of the dual yard crane loading and unloading process in Qingdao Port Automated Container Terminal yard, the renovation of two yard cranes will be carried out at the same time. Therefore, it is determined that the stage-by-stage construction process of hydrogen-powered yard cranes is as follows: completing partial renovation (18 units), completing further renovation (32 units), and becoming the main equipment type in the port (40 units). Starting in 2021, the port conducts three decision periods, 2021–2025, 2026–2030, and 2031–2035, resulting in four investment decision scenarios, as shown in

Table 3. Real options scheme design table.

	2021–2025	2026–2030	2031–2035	Maximum Number of Investment Units	Investment Strategy
Project 1	2021–2035 12 American Real Options			12	Conservative
Project 2	2021–2025 12 European Call Real Options	2026–2035 14 American Call Real Options		26	Cautious up front, flexible later
Project 3	2021–2030 12 American Call Real Options		2031–2035 14 European Call Real Options	26	Moderate flexibility at the front end, stable investment at the back end
Project 4	2021–2025 12 European Call Real Options	2026–2030 14 European Call Real Options	2031–2035 8 European Call Real Options	34	Aggressive expansionist

:

Each of the four scenarios in the table utilizes a different combination of real options to accommodate different investment strategies and market expectations. Scenario 1 uses a single class of American call real options and chooses the right time to invest in a fixed 12 units during the period 2021–2035, which is a conservative investment emphasizing a reduction in the possible negative impacts of uncertainty. Scenario 2 uses European call real options (12 units in 2021–2025) first, then switches to American call real options (14 units in 2026–2035), realizing a progressive investment strategy of caution in the early stage and flexible adjustment in the later stage, with a total investment size of 26 units. This program is optimistic about hydrogen energy yard crane investment on the whole, so it considers making an initial 12-unit investment, but it is more concerned with the impact of subsequent uncertainty, so it designs American-like real options to ensure investment flexibility. Scenario 3 selects early American-like call real options (12 units in 2021–2030) and later European-style call real options (14 units in 2031–2035) to maintain moderate flexibility in the early stage and lock in gains in the later stage, again for 26 units of investment. Scenario 4 adopts full European-style call real options, gradually increasing the investment in three stages (12 units → 14 units → 8 units), with a total investment of up to 34 units, which belongs to the aggressive expansion type of program and is suitable for the large-scale investment program in the event that the price of hydrogen and carbon are both conducive to investment.

4.2. Analysis of Investment Decision Results

4.2.1. Uncertainty Factor Simulation Results

The multi-factor real options model needs to take the uncertainty factor simulation results as model inputs; therefore, it is necessary to simulate the change rule of uncertainty factors first.

• Hydrogen price simulation

According to (Equations (1) and (2)), combined with the current stage of hydrogen industry construction in Qingdao Port and the table parameter settings, this study simulates the changing law of hydrogen price for hydrogen-powered yard crane use. In total, 10,000 simulations of hydrogen prices in 2021–2050 are conducted. All simulation data are visualized using box-and-line diagrams, as shown in Figure 4.

The industrial by-production of hydrogen around Qingdao Port as the source of energy supply for hydrogen-powered yard cranes in 2021–2025 reduces fuel costs significantly, and the cost of hydrogen energy consumption in the operating cost of hydrogen-powered yard cranes at this stage is negligible. In the “Several Measures on Promoting the High-Quality Development of Green and Low-Carbon Industries” issued by NDRC, it is pointed out that the price of hydrogen will be lowered to 30 RMB/kg after 2024. So, this study will set the price of hydrogen energy for hydrogen-powered yard cranes from 2025 to 30 RMB/kg. After 2025, the cost of hydrogen energy for hydrogen-powered yard cranes will gradually decrease with the development of the hydrogen industry in Qingdao. By 2050, the average hydrogen price will reach 14.02 RMB/kg, the maximum value will reach 23.77 RMB/kg, and the minimum value will reach 8.4 RMB/kg.

• Carbon price simulation

China’s carbon emissions trading opened in July 2021, with an opening price of 48 RMB/ton. According to the “2021 China Carbon Price Survey Report”, the expected carbon price survey results show that the average carbon price in the national carbon market in 2022 is expected to be 49 RMB/ton. Based on the above information, the parameter settings, equation, and the change rule of carbon price is simulated. As shown in Figure 5, the initial value of carbon price is set to 48 RMB/ton at the starting point of the construction project in 2021. During the simulation period of the investment decision, the Chinese government strengthens the regulation of the carbon market, which makes the carbon quota decrease and the price of CERs increase, resulting in a large increase in the carbon price, and the carbon price reaches an average of 140 RMB/ton in 2050, with the maximum value reaching 268.8 RMB/ton and the minimum value reaching 75.1 RMB/ton.

• Simulation of changes in equipment investment costs

As a large port handling equipment, the initial investment cost of hydrogen-powered yard cranes is high. In this study, the learning curve is used to calculate the initial investment cost change of the hydrogen-powered yard cranes, and the simulation results are shown in Figure 6. The cost of the hydrogen-powered yard crane equipment decreases as the technology matures, resulting in a change in the overall investment cost of the project. At the beginning of the project period, the investment cost of a single hydrogen-powered yard crane is RMB 2.4 million. With the increase in technology maturity, the investment cost of a single hydrogen-powered yard crane can be reduced to RMB 1.78 million when retrofitting to 20 units.

4.2.2. Analysis of Investment Decision-Making Results

The simulation results of hydrogen price, carbon price, and equipment investment cost were used as input variables and substituted into the multi-factor real options model of the hydrogen-powered yard crane construction project for solving. The least squares Monte Carlo simulation method is used to simulate the four investment decision real options. Each of the four investment real options is simulated 10,000 times by Python 3.13.1 programming to eliminate the error caused by randomness. Eventually, the project investment values of the investment decisions are calculated and shown in Figure 7. In the case scenarios, the project investment values from high to low are Scenario 3, Scenario 4, Scenario 2, and Scenario 1, and the carbon emission reductions are Scenario 3, Scenario 4, Scenario 2, and Scenario 1, respectively, from high to low. The real options with the highest carbon emission reduction are selected while ensuring the maximization of port revenue. Therefore, it is determined that under the multi-factor real options model, Scenario 3 is the optimal investment decision real option for the hydrogen-powered yard crane construction project.

Finally, the optimal investment timing and investment scale of the hydrogen-powered yard crane construction project of Qingdao Port Automated Terminal under the multi-factor real options model of hydrogen-powered yard crane investment decision is: with the use of the American-like real options in the early stage (12 units in 2021–2030), and the use of the European real options in the later stage (14 units in 2031–2035), and the project investment value is RMB 8,734,400 and the cumulative carbon emission reduction is 32,117 tons. These results suggest that the initial stage of the project investment requires more flexibility considering the uncertain factors. Then, at a later stage of the project, the continuous investment of the hydrogen yard crane could be considered.

4.3. Sensitivity Analysis

This study also conducts further simulations to accurately assess the impact of the fluctuation of the three types of uncertainty factors on the optimal investment decision plan and project investment value, including hydrogen price, carbon price, and technological maturity. It is necessary to carry out a sensitivity analysis of the main uncertainty factors in the multifactorial real options model. In this study, we analyze the sensitivity of the drift rate and volatility of hydrogen price, the drift rate and volatility of carbon price, and the learning curve parameters, respectively.

4.3.1. Hydrogen Price Drift Rate and Volatility Rate

In order to assess the impact of hydrogen price drift rate parameter changes on the investment decision of the hydrogen-powered yard cranes project, on the basis of the hydrogen price drift rate of −0.03 set in the previous study, the range of values will be expanded to −0.01 to −0.05, and then on the basis of the hydrogen price volatility of 0.03, the range of values will be adjusted to 0.01 to 0.05. Figure 8 and Figure 9 shows the value of the project investment in each investment decision scheme with the hydrogen price drift rate and volatility rate parameter changes.

As shown in Figure 8, in the multi-factor real options model, when the hydrogen price drift rate is −0.01, the project investment value of investment plan 1, investment plan 2, and investment plan 4 are lower, and the optimal investment program is still investment plan 3. As the hydrogen price drift rate decreases to −0.04, the project investment value of Scenario 4 exceeds that of real Scenario 3, and Scenario 4 becomes the optimal investment plan. As shown in Figure 9, under the five scenarios of hydrogen price volatility from 0.01 to 0.05, the project investment value of Scenario 3 is the largest, and at this time, the optimal investment plan is Scenario 3.

The reason is that under the multi-factor real options model, the hydrogen price drift rate determines the long-term expectation of the hydrogen price. When the hydrogen price drift rate gradually changes from −0.05 to −0.01, it indicates that the expected downward trend of the hydrogen price is slowing down, and the energy cost of hydrogen in the future cash flow prediction will be affected by the slowing down of the decline in the hydrogen price. The change in hydrogen price volatility from 0.05 to 0.01 indicates that the uncertainty and market volatility of the hydrogen price has decreased significantly, which reduces the “volatility premium” of the real options. Therefore, changes in hydrogen price drift and volatility can bring investment flexibility to the port hydrogen-powered yard crane construction project and have a significant impact on the outcome of the project investment decision.

4.3.2. Carbon Price Drift Rate and Volatility Rate

In order to assess the impact of carbon price drift rate and volatility parameter changes on the investment decision, the range of values of drift rate is expanded to 0.018 to 0.058, and the range of values of volatility is expanded to 0.015 to 0.055 on the basis of the carbon price drift rate and volatility set in the previous section. Figure 10 and Figure 11 show the changes in the project’s investment value with the carbon price drift rate and volatility parameter changes in the various investment decision scenarios.

As seen in Figure 10 and Figure 11, the carbon price drift rate and volatility changes have little impact on the project investment value, and the optimal investment decision scheme is still Scheme 3. As the carbon price drift rate increases, the carbon price rises, and the reduction in carbon emissions benefits the low carbon transition, making the project investment value show an upward trend. In the multi-factor real options model, as seen in Figure 10, when the carbon price drift rate rises from 0.018 to 0.058, the project investment value grows from RMB 2,419,595.97 to RMB 2,829,121.40, and the project investment value goes up by 16.9%. When the carbon price volatility increases from 0.015 to 0.055, as seen in Figure 11, the value of the project decreases by RMB 9916.94, and there is no “volatility premium”. The reason is that the Markov chain of jump diffusion term and stochastic volatility is introduced in the carbon price simulation. The jump diffusion term may reduce the real options value in the case of extremely high volatility, and the uncertainty of stochastic volatility increases the difficulty of the price prediction, so the introduction of the jump diffusion term and stochastic volatility has an irregular effect on the project value.

4.3.3. Learning Curve Parameters

Since the development of hydrogen-powered yard cranes is in the early stages of the life cycle, the learning curve parameter b takes a small range of values, with 0.1 to 0.4 being a reasonable interval [34]. The larger the learning curve parameter b, the stronger the learning effect of the equipment technology and the lower the investment cost of a single unit. When the value of learning curve parameter b is taken from 0.1 to 0.4, the results of project investment value calculation for each investment program are shown in Figure 12.

The value of project investment under Scenarios 1 to 4 is positively correlated with parameter b. With the enhancement of the technology learning effect, the value of project investment in each investment scenario grows significantly. When parameter $b$ changes from 0.1 to 0.4, the investment values of Scenarios 1 to 4 are positive, and the average growth rate of the maximum investment value is 122%.

Since the hydrogen-powered yard crane construction project is characterized by a high investment amount, the initial investment cost of hydrogen-powered yard crane equipment is the most important cost of the project, so the change in the initial investment cost of hydrogen-powered yard crane equipment has a more significant impact on the investment decision of the project compared to the parameters of hydrogen price and carbon price. The sensitivity analysis results show that when the technical maturity of hydrogen-powered yard crane equipment is improved faster, the financial pressure of port investment and construction decreases, and the port will be more inclined to complete more hydrogen-powered yard crane equipment renovation in the decision-making period under the background of the green port.

5. Discussion

The optimal investment decision strategy developed in this study integrates the characteristics of both American-style and European-style real options to effectively address market uncertainties across different investment phases. In the early investment phase (2021–2030), hydrogen technologies remain immature, and significant volatility in hydrogen and carbon prices creates a highly uncertain investment environment. To manage this uncertainty, an American-style real options approach is adopted, allowing investors the flexibility to choose the optimal timing for exercising the investment option within the investment window. This dynamic flexibility enables adaptation to evolving market and policy conditions and helps mitigate risks associated with premature or suboptimal investment decisions. In contrast, during the later investment phase (2031–2035), the hydrogen industry is expected to reach a higher level of technological maturity, with a more stable market and policy environment. Under these conditions, a European-style real options strategy is employed, where investment decisions are made at the option’s expiration. This approach facilitates greater certainty in determining future investment scale and returns, while avoiding frequent adjustments in response to short-term market fluctuations. By combining the flexibility of the American-style approach in the early stage with the certainty of the European-style strategy in the later stage, the proposed investment plan reduces early-stage investment risks while ensuring the stability and predictability of returns in the later phase. This hybrid strategy ultimately maximizes the project’s overall investment value and cumulative carbon reduction benefits.

Based on the results of the sensitivity analysis, it is evident that when the hydrogen price drift rate decreases progressively from −0.01 to −0.05, the economic returns of all projects exhibit a significant upward trend. Notably, the total revenue of Project 4 increases from approximately RMB 2 million to nearly RMB 18 million, indicating a particularly substantial gain. A lower hydrogen price drift rate implies reduced volatility, and a more stable hydrogen price environment helps mitigate investment uncertainty, thereby enhancing both the predictability and profitability of investment decisions. In contrast, when the hydrogen price volatility ranges from 0.01 to 0.05, the economic returns across projects remain largely stable. For instance, Project 4 maintains a consistent revenue of around RMB 8 million throughout this range, suggesting that short-term fluctuations in hydrogen price have a relatively limited impact on the project’s economic performance. Moreover, as the carbon price drift rate increases from 0.018 to 0.058, project revenues also show a steady upward trend. Taking Project 4 as an example, total revenue rises from approximately RMB 8 million to over RMB 10 million, indicating that long-term trends in carbon pricing have a more pronounced influence on project profitability. In contrast, increasing the carbon price volatility from 0.015 to 0.055 does not lead to significant changes in economic returns, suggesting that short-term volatility is insufficient to meaningfully affect long-term investment decisions. Additionally, as the learning curve parameter $b$ increases from 0.1 to 0.4, the total revenue of all investment scenarios rises significantly. In particular, the revenue of Project 4 increases from approximately RMB 7 million to over RMB 20 million, further confirming the critical role of scaling up equipment investment and enhancing technological advancement in boosting long-term project profitability. In summary, when making investment decisions for hydrogen-powered yard crane projects, firms should focus primarily on the long-term trends of hydrogen and carbon prices and determine appropriate scales of equipment and technological investment to effectively ensure the maximization of investment returns.

Due to the comprehensive consideration of the variables already in the model and the limitations of the availability of existing market data. In this study, we primarily focus on three major sources of uncertainty: hydrogen price, carbon price, and technological maturity. Future research could further expand the scope of uncertainty factors by incorporating additional variables such as government policies (e.g., subsidy schemes, carbon taxation), fuel cell efficiency, and maintenance costs. Government subsidies for hydrogen energy, changes in carbon trading regulations, and the introduction of environmental policies may all affect hydrogen prices and project returns. Technological advancements and improvements in fuel cell efficiency may influence the long-term operational costs of the project. Additionally, since hydrogen technology is still in its early stages of development, equipment maintenance costs may evolve as technological maturity increases and large-scale deployment advances. Evaluating the impacts of these additional factors on the economic feasibility of hydrogen-powered yard cranes would provide more accurate guidance for investment decision-making and enhance the model’s relevance in real-world applications.

Due to limitations in data acquisition, this study was not able to include different types of ports in the study for a full comparative analysis. Significant differences exist among ports in terms of scale, energy structure, policy environment, and equipment infrastructure, all of which may exert varying influences on investment decisions. Therefore, future research could conduct comparative analyses by selecting different types of ports—such as inland ports and energy ports—or by incorporating a larger, multi-port dataset to test the applicability and robustness of the proposed model across diverse port environments. This would enhance the generalizability and practical value of the research findings, offering a broader empirical foundation to support investment decision-making in the context of port decarbonization.

6. Conclusions

To achieve an optimal balance between economic returns and carbon reduction benefits in hydrogen-powered yard crane projects, this study develops an investment decision-making model based on real options theory, incorporating multiple sources of uncertainty, including hydrogen price, carbon price, and technological maturity. A case study of Qingdao Port is conducted to demonstrate the model’s adaptability and feasibility in the context of green port development. The model design integrates features of both European and American options: in the early stages, an American-style strategy is adopted to enhance investment flexibility, while in the later stages, a European-style approach is employed to ensure the stability of configuration decisions, effectively aligning with the phased nature of port equipment investment. The study focuses on the investment decision challenges arising from fluctuations in hydrogen prices, carbon pricing dynamics, and technological maturity uncertainties during the green port transition. It proposes an analytical framework capable of dynamically responding to changes in uncertainty. The results indicate that the proposed model performs well in identifying investment value, evaluating carbon reduction benefits, and supporting the formulation of phased investment strategies. It enhances the scientific rigor and operability of green equipment investment decisions, providing both theoretical support and practical guidance for the orderly implementation of hydrogen-powered yard cranes in green port settings, and offering an effective solution framework for low-carbon investment decision-making under complex and uncertain conditions.