Improving the energy efficiency of the existing housing stock
given limited investment capacity
A real option approach in the Dutch housing sector.
Abstract
Improvements in the energy efficiency of the existing housing stock can have significant contributions to the necessary reduction in greenhouse gas emissions. Under current
investment capacity, the target set by the Dutch government (a nearly energy-neutral housing sector by 2050) is not achievable and therefore the investment capacity has to increase. This paper investigates the effect of an increase in investment capacity on the optimal investment path from an economic perspective using real option theory. This paper approaches the
investment program to improve the energy efficiency of the entire housing stock from a social planner perspective and finds that an increase in investment capacity results in higher
threshold levels of investment.
Keywords: energy efficiency, housing sector, real option, optimal investment path
1 1. Introduction
During the Paris Agreement of 2015, 196 Parties came together to set goals regarding
sustainable development in response to the global warming problem (see, e.g.,United Nations Framework Convention on Climate Change, 2015). Participating countries realized the necessity to reduce greenhouse gas emissions levels. The Paris Agreement required each country to communicate and maintain their post-2020 nationally determined contributions (NDCs).
Reduction of greenhouse gas emissions can be realized in several major sectors of the economy. It has been increasingly recognized that the housing sector can contribute to reductions in greenhouse gas emissions by improving the energy efficiency of the existing housing stock in the past decade (see, e.g., Sunikka, 2006). The Action Plan for Energy-efficient Housing in the UNECE Region, which was introduced in 2010 by the United Nations Economic Commission for Europe (UNECE), addresses the challenge of improving energy efficiency within the housing sector by providing a policy framework that sets out 12 goals over three different policy areas. The UNECE is aware, and mentions, that policy action is needed to encourage the transition towards a low-energy, and ultimately a zero-energy, housing sector (see, e.g., United Nations Economic Commission for Europe., 2010). The Dutch government has the ambition to move towards a nearly-energy neutral housing sector in 2050. To reach this goal, the energy efficiency of approximately seven million houses needs to be improved (see, e.g., Nijpels, 2018). Obviously, this investment cannot be completed within one investment decision and it takes multiple sequential investments to be realised. According to Aart van de Pal (Director Sustainable Energy at ECN) the capacity is limited to making 30- to 50 thousand houses nearly energy-neutral per year (see, e.g., Stichting Kennis Gebiedsontwikkeling, 2018). The improvement of the entire existing housing stock requires large investments which are irreversible. According to Van Hoek and Koning (2018), who published a report for the ‘Economisch Instituut voor de Bouw’, the average investment needed to make one house nearly energy-neutral is 30 thousand euro. The investment capacity needs to increase in order to meet the goals that are set by the Dutch government. Note that it takes multiple sequential investments to improve the energy
efficiency of the entire existing housing stock due to the limited investment capacity. This paper evaluates the investment program to improve the energy efficiency of the entire housing stock from an economic perspective, treating it as a sequence of multiple investment
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How does an increase in investment capacity affect the optimal path of investments for the investment program to improve the energy efficiency of the entire existing housing stock?
The investment program is approached from a social planner perspective. We assume the payoff realised upon completion of each investment to be uncertain. Moreover, the social planner has the option to postpone investment and the investments are characterized by large costs and irreversibility. Initially, the assumption is made that the investment costs are constant and equal for every investment. Nijpels (2018) mentions that improving the cost-effectiveness (e.g. reducing investment costs) is important to make the transition towards a nearly energy-neutral housing sector feasible. Therefore, this paper also aims to identify the optimal investment path under constant decreasing investment costs. This paper applies real option theory to evaluate the investment program given the characteristics of irreversibility, large costs of the investments and the uncertainty of the payoff. The expected payoffs are estimated using an explicit finite difference method which will be discussed in section 5 and Appendix A of this paper.
The decarbonisation of the built environment is a major topic among policymakers world-wide and harsh targets are set to realise this objective. This paper is interesting because it evaluates the improvements in energy efficiency from an economic perspective using real option theory, without focussing on the targets set by the government. There exists numerous papers that evaluate sequential investment decisions where it takes ‘time to complete’ using real option theory (see, e.g., Majd and Pincyck, 1987; Friedl, 2002). Most papers assume that the payoff from the investments is realised upon completion of the entire investment program, whereas this paper assumes that a one-time payoff is realised upon completion of each
investment stage. As far as I know, there is no other paper that evaluates sequential
investment decisions where it takes time to complete each investment stage and where a one-time payoff is realised upon completion of each investment stage, and put this into the context of improving the energy efficiency of the entire existing housing stock. In this way, this paper adds to the existing literature.
Using this novel method of real option theory, this paper finds that the threshold levels of the payoff that trigger investment are lower when the investment capacity is higher (e.g. shorter time to complete) for every investment stage except for the final stage. This result is
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approaches its final stages. Next to the payoff, the option to engage in the following
investment is acquired at each stage (except for the last stage). At earlier stages, more of these options are embedded within the option value, which drives the threshold level of the payoff that triggers investment down. Trigeorgis (1993) analyses the interactive characteristic of real options on the same underlying value, stating that future real options are part of the
underlying value of prior real options. Hence, the results obtained are in line with existing literature. When we identify the optimal investment path under the assumption of constant decreasing investment costs we find that the threshold level that triggers investment decreases at the beginning of the investment program, stays equal halfway the investment program and increases during the last stages (e.g. a smile pattern). Moreover, we observe that the optimal investment path (threshold levels) is lower and deviates between closer boundaries with constant decreasing investment costs. Finally, the optimal investment path is higher for larger volatility levels and lower for higher discount rates (expected return) keeping all else
constant. Both findings are in line with the characteristics of the optimal investment rule in real option theory discussed by Dixit and Pindyck (1994).
This paper is structured in the following way. Section 2 provides background information about the goals and regulation regarding the housing sector in the Netherlands. Section 3 will discuss existing literature within the field of real options, focussing on real options in general and existing literature closely related to the approach used in this paper. The model used to evaluate the option value of the investment decision is explained and introduced analytically in a simplified two-stage investment model setting in section 4. Afterwards, the numerical method is explained, the values of the variables incorporated in our model are discussed and the two-staged investment model is solved numerically in section 5. In section 6, the model will be extended to a multi-stage investment model and solved numerically, where the optimal path of investment under different investment capacities will be identified. Section 7 provides the analysis of the multi-stage investment model under the assumption of constant decreasing investment costs. Moreover, section 8 will provide a sensitivity analysis of the optimal investment path to different volatility levels of the payoff and discount rates. Finally, section 9 provides a conclusion and implications for policy-makers.
2. Goals and regulation in the Dutch housing sector
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focusses on the improvements of the energy performance of the existing building stock and newly built buildings in Europe. According to the EPBD report, buildings account for 40 percent of the energy consumption within the European Union (see, e.g., The European Parliament and the Council of the European Union, 2010). Each Member State of the
European Union is required to comply with the EPBD and develop their policies based on the Directive.
Following the EPBD, the Dutch government developed their legislation regarding the energy efficiency of buildings, starting with the focus on the labelling of the energy performance of buildings from 2006 (see, e.g., Van Geel, 2006). Ed Nijpels, chairman of the ‘Klimaatberaad’, introduced the ‘Klimaatakkoord’ (hereafter KA) at year-end 2018 of which the main goals are to reduce carbon emissions by 49% by 2030 and 95% by 2050 compared to the carbon
emission levels in 1990. The Dutch government recognizes the importance of improvements in the energy efficiency of the housing sector to move towards a zero-carbon emission economy (see, e.g., Nijpels, 2018). Newly built houses need to comply with the Energy Performance of Building Directive (EPBD) guidelines set by the European Union, which require each Member State to set minimum energy performance requirements (see, e.g., The European Parliament and the Council of the European Union, 2010). Nijpels (2018) states that according to the ‘Gaswet’, which is a law that specifies the rules regarding the transport and delivery of gas, newly built houses are obliged to have no connection to gas since July 2018. Henceforth, all newly built houses are characterized by being nearly energy-neutral.
However, to reduce current greenhouse gas emission levels in the housing sector, the improvements in energy efficiency of the existing housing stock, rather than newly built houses, is most important. The improvements in energy efficiency of the existing housing stock is the most difficult task within the housing sector and the aim is to move towards a nearly zero-carbon emission housing sector in 2050. Regarding the built environment, the transformation and renovation of approximately 7 million houses and 1 million buildings is needed to realise the 95% carbon emission reduction by 2050. In order to realise the climate goals of 2030, approximately 1.5 million houses need to be renovated into nearly energy-neutral houses (see, e.g., Nijpels, 2018). According to Schellekens, Oei and Haffner (2019) and Nijpels (2018), improvement of the energy efficiency of 50 thousand houses per year in 2021, increasing to 200 thousand and 300 thousand houses in 2030 and 2050 respectively, is necessary to make a gradual energy transition feasible.
The improvements in energy efficiency can be realised in several ways including:
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utility costs are relatively small compared to the total cost of living, house owners do not prioritize energy efficiency when renovating their houses (see, e.g., Lulofs and Lettinga, 2003). According to Golubchikov and Deda (2012), a sound policy framework is needed in order for the transition towards a low-energy housing sector to be achievable. Nijpels (2018) states that the government aims to introduce numerous energy-efficient standards to which house-owners have to comply when renovating their houses. In doing so, tenants are protected against high energy costs and house-owners can attract subsidies when they renovate their houses in compliance with the standards. Moreover, the KA specifies several subsidies and policy instruments to address the problem regarding the unfeasibility of investments in improved energy efficiency. Examples mentioned in the KA is an investment subsidy in sustainable energy of one hundred million euro per year, a yearly increase in the energy taxes on gas and an increased tax reduction (see, e.g., Nijpels, 2018).
Next to the ambition and main goals of the government regarding the carbon emission reduction in the built environment and housing sector, Nijpels (2018) also specifies several requisites for the transition to be realised in the built environment. First of all, the ability to scale up sustainability investments is of importance in order to reduce the costs associated with improving the energy efficiency of houses. Especially innovative solutions are important to realise larger investment capacity (scaling up the volume) and decrease investment costs. As mentioned in the introduction, the current capacity is limited to making 30- to 50 thousand houses nearly energy-neutral per year (see, e.g., Stichting Kennis Gebiedsontwikkeling, 2018). This capacity needs to increase in order to meet the goals for 2030 and 2050. In addition, Schellekens, Oei and Haffner (2019) state that another main challenge is the lack of skilled installers within the labour market, which corresponds to the ability to scale up sustainability investments in the housing sector. Secondly, the financial sector is of great importance in order to finance the transition needed in the housing sector. For example, ABN AMRO Bank N.V., one of the top three banks in the Netherlands, provides a discount on mortgage interest rates for energy-efficient houses (see, e.g., ABN AMRO Bank N.V., 2019). Moreover, Nijpels (2018) mentions that a neighbourhood focussed approach is necessary regarding the improvement in energy-efficiency, which will take shape in 2021. Each
neighbourhood has different characteristics which require different solutions in improving the energy efficiency of the existing housing stock. Real-life examples proved that a
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capacity. This paper applies real option theory to the investment decision, since the
investments are characterized by irreversibility, large investment costs and the payoff of the investments are uncertain. In doing so, this paper aims to determine the optimal path of investments given the fact that there is a limited investment capacity and that multiple sequential investments are required to complete the investment program. In particular, this paper aims to determine the optimal path of investments under different (constant) investment capacities and volatilities, and how an increase in investment capacity affects the optimal path of investment. As mentioned briefly in this section, improvement in the cost effectiveness of the investments are important to make improvements in the energy-efficiency of houses feasible. Therefore, this paper also evaluates the optimal investment path under the assumption of constant decreasing investment costs.
3. Literature review: Real option theory
The orthodox discounted cash flow (DCF) approach to determine the net-present-value (NPV) of an investment is widely used by business and economics scholars to evaluate investment decisions. According to Parker (1968) one of the main challenges of using the DCF approach is to be able to forecasts and determine future cash flow schemes. Ross (1995) adds to this and argues that the NPV rule is somewhat irrelevant due to the optionality that exists in most investment decisions. Therefore, he states that all investment decisions should be treated as option pricing problems. Hull (2015) shares this view and argues that the problem of the classical NPV approach lies within the optionality embedded in investment decisions. Dixit and Pindyck (1994) mention three option-like characteristics whose interaction is of great importance to investment decisions, namely: irreversibility, uncertainty and timing.
The real option approach, based on option pricing theory, deals with the options embedded in investment opportunities. Berk, DeMarzo and Harford (2015) define a real option as the right to make a particular business decision, such as capital investment (which is relevant for our research). They mention the option to delay and the option to expand as common real options embedded in investment decisions. The optionality within investment decisions generally increase the NPV of the project when there is high uncertainty. In the remaining part of this section, literature on these common real options will be discussed. Furthermore, we will discuss literature on real options that is closely related to the analysis of this paper. 3.1.1 Option to delay investment
The option to delay refers to the timing option within an investment decision. When
investment opportunities are not immediately taken, the investment opportunity often remain in existence. The concept of ‘time value of money’, which suggest that there exists a
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theory provides other insights. Titman (1985) used real option theory to price vacant land and the option to invest (e.g. develop the land) and argues that postponing investment is valuable under uncertainty. Investing immediately could be seen as an opportunity cost, since you give up the opportunity to invest at a later time. Ingersoll and Ross (1992) also show that the value of the option to postpone investments increases with uncertainty. Boomsma, Meade and Fleten (2012) evaluate the difference in investment behaviour under different support schemes for investments in renewable energy. They use real option theory to account for the value of waiting arising from the ability to postpone the investment. They evaluate a Nordic case study of wind power investment project and find that, compared to renewable certificates, feed-in-tariffs support earlier investment.
3.1.2. Option to expand
Investments opportunities are often not constrained to a one-time investment. Investment opportunities can be regarded as compound options, which are options on options. Siddiqui and Maribu (2009) use real option theory to evaluate the real option to invest in a gas-fired distributed generation unit where the natural gas price is uncertain. After the investment in the gas-fired distributed generation unit, the option arises to expand and invest in additional capacity and/or to invest in a heat exchanger. They find that the real option value increases with uncertainty and hence a sequential investment is preferred. Oppositional, when uncertainty is low a direct investment in the gas-fired distributed generation and additional capacity or heat exchanger is preferred. Pindyck (1988) does not look at two sequential
options to expand but uses real option theory to observe the option of continuous expansion of the capacity of a monopolistic firm given that the future demand is stochastic and uncertain. Pindyck (1988) finds that investment in capacity expansion occurs in spurts, only when demand increases and rises above historical levels. When demand decreases, no investment in capacity expansion occurs. In fact, these papers are both characterized by sequential
investment decisions, which is related to the analysis in this paper and will be covered in the next section.
3.2. Sequential investment
In this paper we evaluate a staged investments program to improve the energy efficiency of the entire existing housing stock in The Netherlands. Our problem can be regarded to as a compound option, where investing at one stage gives the option to invest in the following stage and each stage takes time to complete. There exists a broad range of literature that investigate similar sequential investment decisions using real option theory.
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outlays proceed continuously and each dollar spent represents a new investment stage. The payoff derived from the investment is contingent upon completion of the entire program. They observe that the value of the investment program increases as the uncertainty in the payoff increases, given the fact that one can postpone the next investment outlay (e.g. postpone construction). Moreover, they find that when the opportunity costs of the time to build is large, a decrease in time to build (higher construction flexibility) decreases the threshold level at which it pays to invest, whereas the construction flexibility has less
pronounced effects on this threshold level when the opportunity costs are rather low. Next to that, Majd and Pindyck (1987) show that higher construction flexibility increases the value of the program.
Friedl (2002) investigates the value of the option to build a firm in a sequential investment setting with time to build. His model is almost identical to that of Pindyck and Majd (1987) given that his model also takes into account that investment outlays within the building stage can only proceed at a limited rate and that the payoff from the investment is realised after completion of the whole project (the building phase). Friedl (2002) adds the option to suspend operation in the operating phase to the model of Pindyck and Majd (1987). He plots the threshold level to invest to the expected drift rate of the (stochastic) payoff for different levels of the time to build. The change in the threshold level with the expected drift rate is more pronounced when the time to build is longer (lower maximum investment rate). Moreover, the threshold level for investment is higher in case of the lower maximum investment rate
compared to the model with the higher maximum investment rate, except when the expected drift rate is close to zero.
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A third model that is similar to the one used in this paper is that of Bar-Ilan and Strange (1998). They evaluate a two-stage sequential investment model where the investment at each stage takes time to complete. The payoff is realised after completion of the second
investment. Moreover, they can suspend between the first and second investment stage. One of the things that they investigate is the effect of different lags (time to complete) on the investment. In their model, the investment cost of the second stage is adjusted so that the NPV of the project cost constant. They find that it does not take substantial high lags in order for the threshold level of the first investment to be lower than the threshold level of the second investment when suspension is possible. The threshold level of the first investment is always (substantially) higher than that of the second investment when suspension between stages is not possible, since there is no additional option value that is given up when investing in the second stage. When there is no time to complete (investment lag) they find that the threshold level of the first investment is higher than that of the second in both the cases when the option to suspend between stages is present and when there is no option to suspend, implying that the second investment is immediately triggered when the first investment is triggered.
Another paper worth mentioning is that of Trigeorgis (1993). He evaluates an sequential investment setting where each investment outlay presents a different real option, being the option to engage in the first investment, the option to abandon at the next investment, the option to contract and expand during the subsequent investments during the building stage. In the operating stage, when the project generates its cash flow, the manager has the real option to switch to an alternative. The value of the project is assumed to be stochastic and uncertain in his model. Trigeorgis (1993) analyses the interactive characteristic of real options on the same underlying value, stating that future real options are part of the underlying value of prior real options. He finds that the degree of interaction between options is larger for options of the same type and increases, which makes the options less additive, when time to maturity
differences are larger and when options are deeper in the money. This paper gives us an understanding about the nature and characteristics regarding the interaction of sequential real options which is useful to keep in mind for our own analysis.
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to the evaluation of the option value under different extraction rates in the paper of Brennan and Schwartz (1985). Similarities between the model used in the paper of Bar-Ilan and Strange (1998) and this paper is the fact that we include that each investment stage requires time to complete before one can engage in the next investment and we aim to identify the differences in threshold levels for each investment stage for different time to complete. Next to that, this paper’s model incorporates the option to postpone the next investment stage (real option). This paper’s model deviates from the fact that each investment outlay generates a payoff realised upon completion of each stage instead of generating the first payoff after the completion of the whole sequence of investments. Moreover, this paper assumes constant investment costs at each stage during the entire investment program. The next section illustrates our model analytically in a two-stage investment model.
4. Model
Consider, from a social planner perspective, that the goal is to improve the energy efficiency of the entire existing housing stock. This investment cannot be completed at once given limited capacity (e.g. number of installers) and therefore has to occur in multiple stages. Therefore, the investment decision can be regarded as a compound option. In other words, when the first investment to improve a portion of the existing housing stock is being made, the option to invest in the improvement of the next portion arises and so on. We assume that it takes ‘time to complete’ each stage of investments and that the portion of existing housing stock that is completed within each stage is completed simultaneously (e.g. at the same time). We evaluate a model where the investments (𝐼) are assumed to be identical in size and time to complete. Moreover, in contrast to the literature described in section 3.2, one-time payoffs (𝑃) are realised upon the completion of each investment stage. In this model, the payoff of
improvements in energy efficiency is considered to be uncertain and evolves stochastically, following a Geometric Brownian Motion (GBM) process:
𝑑𝑃 = µ𝑃𝑑𝑡 + 𝜎𝑃𝑑𝑧 (1)
where 𝑃 denotes the one-time payoff, µ denotes the drift rate (e.g. growth rate of the payoff), σ denotes the volatility (uncertainty) of the payoff and dz is the increment of the standard Wiener process. For simplicity, we solely incorporate the payoff into our model instead of multiple stochastic variables that affect the payoff, such as the electricity and gas price
(through avoided costs) and the social benefits that arise from the improved energy efficiency of the existing housing stock. However, note that the payoff is contingent upon these
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the investments to be irreversible.
Sequential investment problems within real option theory are solved working backwards (see, e.g., Siddiqui and Maribu, 2008; Dixit and Pindyck, 1994). The backwards approach will also be applied in this paper. The remainder of this section will illustrate our model analytically in a two-stage investment model.
4.1. Two-stage investment model
Figure 1
This figure gives a visual representation of the two-stage investment model.
Consider an investment program to improve the energy efficiency of the entire existing housing stock. We assume that the total required investment is divided into two separate and identical investments. Figure 1 provides a visualisation of the two-stage investment model. As mentioned in previous section, the payoff realised after each investment is uncertain and follows GBM (see equation 1).
At the start, which will be referred to as stage 0 in our model, no investments have been made and the social planner has the option to decide whether to invest amount (𝐼) or wait. When the first investment is made, it takes time to complete the first investment, which will be denoted by 𝑇. After completion, the first investment generates a one-time payoff (𝑃). At this point (stage 1), we have the option to either engage in the second investment to further improve the energy efficiency of the remaining housing stock, or to postpone the second investment. Once the second investment has been made, it (again) takes time to complete the investment. After completion of the second (and final) investment, the payoff is obtained and there is no investment decision left (stage 2), since the energy efficiency of the entire housing stock is improved at this point.
4.1.1. Stage 2: Completion of the total investment
In stage 2 the total investment has been realised and the energy efficiency of the entire existing housing stock has been improved. As the entire existing housing stock is energy-neutral, no further investments decisions are to be made at this point and hence there is no remaining option value. The one-time payoff is received upon completion (e.g. after the ‘time to complete’) of the second (and final) investment. Since no further options to invest are
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obtained, this payoff represents the value that you get from the investment. This value is denoted as 𝑃 and is equal to:
𝑃 = 𝑃 (2)
4.1.2. Stage 1: The second investment decision
In stage 1, the decision has to be made whether to engage in the second (and final) investment and pay the second investment outlay 𝐼, or to postpone this investment (wait). As mentioned in section 4, the investments take the same ‘time to complete’. Recall that the payoff is
uncertain and evolves stochastically in continuous time, which is also the case during the time to complete. Therefore, we derive the expected payoff, and hence the expected value, that we get when we decide to engage in the second investment. The expected value that we get when we decide to engage in the second investment equals:
𝑃 = 𝑃 (3)
where denotes the expectation. As mentioned, we have to account for the fact that the payoff evolves stochastically during the time to complete and follows the GBM process (see equation 1). Moreover, we have to discount the expected payoff by the appropriate discount rate over the time to complete to get the expected value at the moment that we decide to engage in the second investment. The discount rate is denoted as 𝑟 in our model. Hence, the expected value at the decision-making moment equals:
𝑃 = 𝑃𝑒− 𝑟−𝜇 𝑇 (4)
However, there is also the option to wait and not engage in the second investment. It might be interesting to engage in the second investment at a later moment in time, since the payoff is uncertain and evolves stochastically. When the payoff is sufficiently high, one would directly engage in the second investment. Dixit and Pindyck (1994) show that the option value to invest satisfies the Bellman equation, which is the following partial differential equation (PDE) in our model in continuous time:
𝑟 𝑃 = µ𝑃𝜕𝐹
𝜕𝑃+ σ 𝑃
𝜕2𝐹
𝜕𝑃2 (5)
Note that we disregard the one-time payoffs that are realised by earlier investments. In other words, we have the real option to postpone investment or to invest the outlay 𝐼 and get 𝑃 . The Bellman equation enables us to compute the option value of postponing the second investment, which will be denotes as 𝑃 . The general solution to the Bellman equation is (see, e.g., Dixit and Pindyck, 1994):
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Where 𝛽 and 𝛽 equal,
𝛽 = −µ− 1 2𝜎 2 𝜎2 + √(µ− 1 2𝜎2) 2 + 𝑟𝜎2 𝜎2 > 1 (7) 𝛽 = −µ− 1 2𝜎 2 𝜎2 − √(µ− 12𝜎2)2+ 𝑟𝜎2 𝜎2 < 0 (8)
Note that we have two unknown constants and . We can impose boundary conditions to this equation by evaluating the option value at both boundaries where 𝑃 equals zero or 𝑃 equals the threshold level that triggers investment. Intuitively, when the payoff equals zero, one would never invest as the value also equals zero in this case. In order for this to be true, we require to be equal to zero. Hence, the second part of equation 6 is eliminated and the option value becomes:
𝑃 = 𝑃𝛽1 (9)
At this point, we computed the value of what we get if we decide to engage in the second investment, which is 𝑃 . Next to that, we identified the value before the second investment and after completion of the first investment ( 𝑃 ), which is the real option to invest. We can use these expression to identify the threshold level for the payoff, which we will denote as 𝑃∗, at which we are willing to pay investment outlay 𝐼 and engage in the second
investment. In order to find this threshold we have to impose the value matching (VMC) and smooth pasting condition (SPC) (see, e.g., Pindyck and Dixit, 1994):
VMC: 𝑃∗ = 𝑃∗ - 𝐼 (10)
SPC: 𝜕𝐹1
𝜕𝑃 =
𝜕𝐸 𝑉2
𝜕𝑃 (11)
Since 𝑃∗ and 𝑃 are non-linear we also need the smooth pasting condition to hold
to ensure optimality in the threshold level 𝑃∗. The smooth pasting condition takes the
first-order derivatives of the value functions with respect to the payoff (𝑃). We can substitute equation 4 and 9 into both conditions to obtain:
VMC: 𝑃∗𝛽1 = 𝑃∗𝑒− 𝑟−µ 𝑇− 𝐼 (12)
SPC: 𝛽 𝑃∗𝛽1− = 𝑒− 𝑟−µ 𝑇 (13)
Note that 𝐼 is not a function of 𝑃 and therefore drops out on the right-hand side of the smooth pasting condition. Combining the value matching and smooth pasting condition enables us to calculate the optimal threshold level for the payoff that triggers investment (𝑃∗):
𝑃∗ = 𝛽1
𝛽1 − 𝐼𝑒
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We can substitute 𝑃∗ into the value matching condition and solve this analytically in order to find the corresponding value of the constant :
=𝑃∗𝑒− 𝑟−µ 𝑇
𝑃∗𝛽1 −
𝐼
𝑃∗𝛽1 (15)
When the payoff is larger than the threshold level (𝑃 > 𝑃∗) we are willing to pay the outlay 𝐼 and engage in the second (and final) investment. When the payoff is smaller than the
threshold level (𝑃 < 𝑃∗), we rather wait for the expected payoff to increase before we engage in the second investment. Note that 𝑃 and 𝑃 - 𝐼 are equal for 𝑃 = 𝑃∗.
4.1.3. Stage 0: The first investment decision
As mentioned earlier, stage 0 refers to the situation where no investment has been made yet and the social planner needs to decide whether to engage in the first investment or to postpone investment. Similar to the second investment decision, a one-time payoff (𝑃) is received upon completion of the first investment. However, in this case the option to engage in the second investment ( 𝑃 ) is also obtained from the first investment. This gives us the following value after completion of the first investment, which will be denotes as 𝑃 :
𝑃 = 𝑃 + 𝑃 (16)
Similarly to the procedures in section 4.1.1 and 4.1.2, the one-time payoff and the option to engage in the second investment are obtained upon completion of the first investment stage. Since the payoff is uncertain and stochastic, we need to derive the expected (discounted) payoff over the time to complete. Note that the option to engage in the second investment is a function of the payoff. Therefore expectations over the time to complete regarding this option value also need to be derived. This yields the following value that is received at the moment we decide to engage in the first investment:
𝑃 = 𝑃 + 𝑃 (17)
Henceforth, the decision has to be made whether to pay investment outlay 𝐼 and receive 𝑃 or to postpone the first investment, which will be denoted as 𝑃 . Similar to previous section, we can compute the value of the option to engage in the first investment ( 𝑃 ) using the Bellman equation (see equation 5). Similar to previous section, the Bellman equation has the following general solution:
𝑃 = 𝐺 𝑃𝛽1+ 𝐺 𝑃𝛽2 (18)
Note that the only difference is the notation of the constants to keep things uncluttered. We can eliminate 𝐺 by imposing the boundary conditions similarly to what we did in section 4.1.2. In order to find the threshold level 𝑃∗ at which we are willing to engage in the first
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previous section. At the decision making moment to engage in the first investment, we have the option to either wait and invest at a later moment in time ( 𝑃 ) or to pay investment outlay 𝐼 and receive 𝑃 . The VMC and SPC are expressed as follows:
VMC: 𝐺 𝑃∗𝛽1 = 𝑒− 𝑟−µ 𝑇 𝑃∗+ 𝑃∗𝛽1 − 𝐼 (19)
SPC: 𝛽 𝐺 𝑃∗𝛽1− = 𝑒− 𝑟−µ 𝑇 1 + 𝛽 𝑃∗𝛽1− (20)
Solving the analytical value of the threshold level 𝑃∗ and constant 𝐺 is difficult at this stage and will not be done in this paper. This paper provides a numerical programme to evaluate the investment program in a two-stage investment model and extends the analysis to a multi-stage investment model. Section 5 explains the numerical method used in this paper, discusses the values of the variables used in the model and provides a numerical solution to the two-stage investment program. The remainder of this paper extends the investment program to a multi-stage investment program, where we are interested in finding the optimal investment path under different investment capacities and decreasing investment costs. Moreover, we perform a sensitivity analysis with respect to the volatility and discount rate.
5. Numerical application
The previous section illustrated our model analytically in the context of a two-stage investment model. Realistically, the improvement of the entire housing stock’s energy efficiency is an ongoing process and takes more than two investments in order to be realised. Extending our model to a multi-stage investment makes it even more difficult to provide an analytical solution for the threshold levels of investment at each stage. Henceforth, the remainder of this paper will provide a numerical solution to the investment program in the two-stage investment model and multi-stage investment model, where we are interested in finding the optimal path of investments (e.g. threshold levels of the payoff that trigger investment at each stage) under different levels of investment capacity. As mentioned in the introduction, the investments are characterized by irreversibility and the payoffs that are realised upon completion of each investment stage are uncertain and evolve stochastically according to GBM. In this paper, we derive the expectations of future payoffs using a
16
partial differential equation (see equation 5). Using the explicit finite difference method is equivalent to approximating the stochastic process of the payoff by a discrete trinomial tree process (see, e.g., Clewlow and Strickland, 1998). This method replaces the partial
differences with finite differences, which simplifies the partial differential equation and therefore allows us to carry out the numerical analysis and identify the optimal investment path. The procedure for the explicit finite difference method is shown in Appendix A. In order to provide a numerical solution, a code is written in Python which is shown in Appendix B. The real option (the decision whether to invest) can be approached as an American option with an infinite maturity. The investment program in a multi-stage
investment model basically resembles a compound option. Before providing the results, we first discuss the values of the variables that are used in the model.
5.1.1. Payoff and investment
As mentioned throughout the paper, the payoffs realised from the investments are stochastic and move according to Geometric Brownian Motion (see equation 1). Note that the payoff of improving the energy efficiency of houses is contingent upon the electricity and gas price through avoided costs. According to Woldring (2019), the average Dutch utility bill consists for 64.4% of gas costs and 35.6% of electricity costs as of January 1st of 2019. In order to find drift rate and volatility of the payoff, data on the gas and electricity prices (including and excluding taxes and levies) for Dutch households has been extracted from Eurostat. The data obtained consists of bi-annual electricity and gas prices for Dutch households over a nine-year period from 2010 up until and including 2018. We solely looked into the yearly change in gas and electricity prices to deal with seasonality. The drift rate is obtained by computing the mean yearly change in the gas and electricity price (including taxes and levies) and combine these by taking the weighted average using the weights according to Woldring (2019). In doing so, the drift rate equals 1.50%. The volatility of the payoff is calculated using the general formula for the calculation of the standard deviation of a portfolio:
𝜎𝑝 = √𝑤𝑔𝜎𝑔 + 𝑤𝑒𝜎𝑒 + 𝑤𝑔𝑤𝑒𝜎𝑔𝜎𝑒𝜌 (21)
where 𝜎𝑝 is the volatility of the payoff, 𝜎𝑔 and 𝜎𝑒 are the volatility of the growth in the gas and electricity price including taxes and levies respectively, 𝑤𝑔 and 𝑤𝑒 denote the weights of gas and electricity in the Dutch utility bill and 𝜌 denote the correlation between the gas and electricity growth (excluding taxes and levies). The volatility of the payoff equals
approximately 6%1. This paper also evaluates the optimal path of investment (threshold levels that trigger investment) under different levels of volatility in section 8.1.
1 The correlation equals 0.9396, the weights for the gas and electricity are 0.644 and 0.356
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As mentioned previously, the average investment costs needed to make a house nearly energy-neutral is 30 thousand euro according to Van Hoek and Koning (2018). In order to reach the goals for 2050, the energy efficiency of approximately seven million houses needs to be improved (see, e.g., Schellekens, Oei, and Haffner 2019). This means that a total investment of approximately 210 billion euro is needed to complete the entire investment program. Improving the cost effectiveness is important to make the investments in energy-efficiency feasible. Improving the cost effectiveness can be realised through, for example, innovative solutions. Therefore, this paper also evaluates the optimal path of investment under constant decreasing investment costs.
5.1.2. Other variables
Next to the payoff and investment, we need measures for the discount rate, the investment capacity, the time to complete and the time to maturity of the real option. Current market conditions in the Netherlands are characterized by low interest rates. The risk free rate, based on ten year Dutch government bonds, was approximately 0% on May 31st of 2019 (see, e.g., IEX, 2019). Hermelink and De Jager (2015) wrote a report on the appropriate discount rates for energy and climate modelling to evaluate energy efficiency measures. They mentioned that the appropriate social discount rate2 that should be used in the Netherlands is 5,5% consisting of a risk free rate of 2.5% and a risk premium of 3%. Similarly, Werkgroep Discontovoet (2015) wrote a report regarding appropriate discount rates, focussing on different policy areas in the Netherlands. They recommend using a flat term structure and a discount rate of 3% regarding the valuation of investments within the policy area ‘CO2’. This discount rate is based on a risk free rate of 0% and a risk premium of 3%. Taking the reports mentioned above into account and the current Dutch risk free rate of approximately 0%, this paper uses a discount rate of 3% (based on a 3% risk premium) to discount the payoffs. Note that the discount factor represents the required return of investing.
As mentioned in the introduction, the current capacity is limited to making 30- to 50 thousand houses nearly energy-neutral per year (see, e.g., Stichting Kennis Gebiedsontwikkeling, 2018). In our model we assume constant investment capacity. If the capacity stays constant at the current level, it takes approximately 140 to 230 years to improve the energy efficiency of the entire housing stock. In order to account for an increasing investment capacity in the upcoming decades, our base case assumes a constant investment capacity of improving the energy efficiency of 100 thousand houses per year. Moreover, to answer the research question mentioned in the introduction, this paper also identifies the optimal path of investment under different levels of investment capacity. The time to complete is contingent upon the capacity and number of investment stages in our model. We evaluate the investment program under different maximum investment capacities and therefore the time to complete will vary. For
2 Hermelink and de Jager (2015) define the social discount rate as the discount rate that reflects the
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each model the time to complete will be specified.
Recall from this section that a real option resembles an American option with infinite time to maturity. The time to maturity is set such that extending the time to maturity is negligible. In this paper, the time to maturity of the real option is set at 50 years.
5.2. Results two-stage investment model
The option value and the threshold level of the payoff at each stage that triggers investment in a two-stage investment program is obtained in this section. In the next section, the model will be extended to a multi-stage investment model and evaluated under different investment capacities.
When the energy efficiency of the entire existing housing stock can be improved in two investment stages, this means that 105 billion euro will be invested at each stage for the improvement of 3.5 million houses (7 million in total). When the investment capacity is limited to improving the energy efficiency of 100 thousand houses per year, each investment stage takes 35 year to complete (𝑇 = 35). Similar to the analytical approach, we discuss the results working backwards, starting with the second investment stage. Figure 2 illustrates the option value and threshold level of investment for the payoff (𝑃∗) at the second investment
stage.
Figure 2
This figure shows the relationship between the option value and the payoff for the second investment stage. The dotted line illustrates the value of the option to invest for different payoffs and the solid line
illustrates the value of what we get when we engage in the second investment. 𝑃∗ indicates the threshold level of the payoff that triggers the second investment.
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value at this point is equal to 128,25 billion euro. For payoffs smaller than the threshold level, the value of the option to postpone investment ( 𝑃 ) is higher than the value what we get if we engage in the investment ( 𝑃 − 𝐼). Therefore, no investment will take place when 𝑃 < 𝑃∗. Figure 3 illustrates the option value and the threshold level for the payoff 𝑃∗ for the first investment.
Figure 3
This figure shows the relationship between the option value and the payoff for the first investment stage. The dotted line illustrates the value of the option to invest for different payoffs and the solid line
illustrates the value of what we get when we engage in the second investment. 𝑃∗ indicates the threshold level of the payoff that triggers the first investment.
Again, the threshold level is identified at the point where the option value to postpone investment ( 𝑃 ) is equal to the value what we get from investing minus the investment costs ( 𝑃 − 𝐼). If we compare the graphs for the second and first investment, we find that the option value at the threshold level 𝑃∗ is significantly higher for the first investment compared to the second investment. The threshold level of the payoff for the first investment is equal to 330,45 billion euro, which is approximately equal to 94,414 euro if we express this in the payoff per house. Note that for both the first and second investment the threshold level of the payoff is significantly high such that improvements in the energy efficiency in a two-stage investment model will likely not happen. At the threshold level for the payoff, the option value equals 169,68 billion euro. The threshold level of the payoff for the first
investment is lower since the option to engage in the second investment is obtained next to the payoff of the first investment. Recall that there remains no option to engage in the next
20 6. A multi-stage investment model under different investment capacities
This section provides the optimal path of investments (e.g. the threshold level of investment at each stage) under the assumption that the investment program takes one hundred stages to complete. The model could be extended to one where each investment stage represents the improvement of energy efficiency of a few houses and approximates continuous investment. However, this is beyond the scope of this paper, whereas this paper aims to identify the effect of increased investment capacity on the optimal investment path.
When one hundred (identical) stages are needed to complete the investment program, the energy efficiency of 70 thousand houses will be improved at each stage. The required investment outlay per stage is 2.1 billion. As stated throughout the paper, the main objective of this research is to find the effect of an increased investment capacity on the optimal path of investment. Remember from section 5.1.2 that our base case assumes a constant investment capacity of improving the energy efficiency of 100 thousand houses per year. Additionally, this section will identify the optimal investment path when the capacity of improving the energy efficiency is 200 thousand and 400 thousand houses per year. Note that the time to complete parameter (𝑇) in our model is contingent upon the investment capacity and number of investment stages. When the investment capacity is 100 thousand houses per year, each stages takes 0.7 years to complete (𝑇 = 0.7). When the investment capacity is 200 thousand and 400 thousand houses per year, the time to complete (𝑇) is 0.35 years and 0.175 years respectively. The values of the remaining parameters are unchanged3. Figure 4 shows the findings in a multi-stage setting and shows the threshold level of investment per investment stage in case of different investment capacities.
First of all, Figure 4 shows that the threshold level of the payoff that triggers investment increases (or stay equal) for every stage that brings us closer to the completion of the investment program. This relationship holds under all three levels of investment capacity. Overall we observe a positive relation between the threshold level of investment and the number of investment stages that have been completed. Reasoning behind this is that there are more options embedded in the option value of the investment program at earlier stages
(compound option). Therefore the threshold level that triggers investment is lower at earlier stages. This result is exactly what we would expect given what we learned from the paper of Trigeorgis (1993), who described the interactive nature of sequential options on the same underlying value.
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Figure 4
This figure presents the optimal investment path in a multi-stage investment model under different levels of investment capacity. The black line represents the base case where the investment capacity is
100.000 houses per year. The blue line represents the optimal investment path when the investment capacity is 200.000 houses per year. The green line represent the optimal investment path when the
investment capacity is 400.000 houses per year.
When comparing the results to that of the two-stage investment model described in section 5.2, we observe that the threshold level of investment per house for the first investment stage is significantly lower in the multi-stage investment model for all three investment capacities. The same reasoning of embedded (compound) options associated with the larger number of investment stages holds for this finding.
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investment. As mentioned earlier, the targets set by the government can only be realised when the investment capacity increases. However, with increased investment capacity, the threshold levels that trigger investment are higher at each stage, which might imply that it actually takes longer to complete the entire investment program (since we need higher payoff levels).
Hence, there exists a trade-off for increasing the investment capacity between approaching the investment program with the targets set and approaching it from an economic perspective. When the final investment stages are approached, the threshold levels of investment converge to each other and are approximately equal. Note that for the last stage the threshold level of investment is the highest in case of lower investment capacity. Moreover, the final investment stages require significantly high payoffs (approximately 67.000 per house) in order for the investment to be triggered regardless the investment capacity. Implications are that it might be unprofitable to improve the energy efficiency of the entire existing housing stock.
7. A multi-stage investment model with constant decreasing investment costs
The assumption that investment costs are equal at every investment stage is not realistic. Nijpels (2018) states that innovations are important for the investment costs to decline and making the improvement of energy efficiency feasible. Therefore, we are interested in
determining the optimal investment path under declining investment costs. In this section, we evaluate the optimal investment path in the multi-stage investment model (one hundred stages) under the assumption that the investment costs decline with ten million euro at each stage, with all other variables being equal to our base case4. Figure 5 presents our findings. Figure 5 shows a different optimal investment path than the investment model with constant investment costs (see section 6). Whereas the initial model showed a positive relationship between the threshold level of investment and the number of stages, Figure 5 shows a ‘smile’ pattern for the optimal path of investment. The threshold level of investment decreases during the first part of the investment program, stagnates (is equal) halfway through the investment program and increases during the final stages of the investment program. Moreover, we observe that the threshold level of the payoff per house ranges between approximately 33,460 and 36,172 euro during the entire investment program, showing a lower and more constant optimal investment path compared to the case of constant investment costs. Obviously, we cannot say for sure that the investment costs will decline at every investment stage with ten million euro. However, these findings strongly suggest that a policy scheme aimed at the improvement of the cost effectiveness (innovation within the housing sector) improves the feasibility of the investments and triggers investment at a lower payoff levels even for the first investment stage, where the investment costs are 2.1 billion in both models. Keep in mind that
4 The first investment equals 2.1 billion, the volatility equals 6%, the discount rate equals 3%, the drift
23
this is from the perspective of a social planner. Households might want to wait until
investment costs decline given the fact that the improvement in energy efficiency is a single investment decision where no further options are obtained when they decide to engage in the investment. From their perspective, we can better look at the last investment stage, where the households have the option to postpone investment but no further options are obtained upon investing. Because the investment costs are lower in the model with constant decreasing investment costs compared to our initial investment model (in the last investment stage), the threshold level of investment is lower at the last investment stage.
Figure 5
The optimal investment path in the multi-stage investment model under constant decreasing investment costs. The y-axis shows the threshold level of investment per house (x 1000) and the x-axis
shows the investment stages. The investment costs decrease with ten million euros per stage.
8. Sensitivity Analysis
8.1. A multi-stage investment model under different volatilities
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Figure 6
This figure illustrates the difference in the optimal investment path for different levels of uncertainty. The black line represents the optimal investment path when the volatility is 6%, the blue line represents the optimal investment path when the volatility is 12% and the green line represents the
optimal investment path when the volatility is 20%.
As expected, the figure shows that the threshold level of investment is higher when the volatility is higher at every investment stage. Due to higher volatility the option value to postpone investment increases and therefore we require a higher payoff in order to engage in the investment and give up the option to postpone investment. Note that the difference is less pronounced at the earlier investment stages than at the final investment stages. In the last section we already identified the high threshold level at the final investment stages, which could make it difficult for the investment program to be entirely completed. When the
volatility is 20%, the threshold level per house that triggers investment exceeds 100.000 euro at the final investment stages. Implications of these results are that the likelihood that energy efficiency of the entire housing stock will be improved is lower when the volatility of the payoff is larger.
8.2. A multi-stage investment model under different discount factors
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Figure 7
This figure illustrates the difference in the optimal investment path for different discount rates. The
black line represents the optimal investment path in the base case where the discount rate is 3%. The blue line represents the optimal investment path when the discount rate is 5% and the green line
represents the optimal investment path when the discount rate is 7%.
Figure 7 shows the findings that a higher discount rate lowers the threshold level of
investment. Note that the discount rate represents the required rate of return for investors. The difference between the discount rate and the drift rate of the payoff (growth rate) can be seen as the opportunity costs of keeping the option alive (see, e.g. Dixit and Pyndick , 1994). If the required rate of return increases and the drift rate (growth rate) of the payoff stays equal, the opportunity costs of keeping the option alive increases. Therefore, a higher discount rate (keeping all else constant) reduces the threshold level of investment.
9. Conclusion and implications
26
path from an economic perspective, ignoring the targets set by the Dutch government.
Using this novel method of real option theory, this paper finds that the threshold levels of the payoff that trigger investment (e.g. optimal investment path) are lower when the investment capacity is higher (e.g. shorter time to complete) for every investment stage except for the final stage. This result is contradictory with the findings of Majd and Pindyck (1987) and Friedl (2002). Both find that a decrease in time to build decreases the threshold level of investment. Bar-Ilan and Strange (1998) observed lower threshold levels for longer
investment lags (time to complete) except for low and significantly high lags. Note that the model used in this paper is different than the models used in the papers mentioned above, whereas this paper assumes that a one-time payoff is received upon completion of each investment stage and the investment costs are equal for each investment stage. When the time to complete is longer, the benefits of the payoff being stochastic are larger and hence the threshold level that triggers investment is lower. Moreover, this paper finds that the threshold level of investment is lower at earlier stages of the investment program and increases when the investment program approaches its final stages. Note that next to the payoff, the option to engage in the following investment is acquired at each stage (except for the last stage). At earlier stages, more of these options are embedded within the option value, which drives the threshold level of the payoff that triggers investment down. Trigeorgis (1993) analyses the interactive characteristic of real options on the same underlying value, stating that future real options are part of the underlying value of prior real options. Hence, the results obtained are in line with existing literature.
Nijpels (2018) mentions that the improvements in cost-effectiveness (e.g. lower investment cost) is necessary to make the improvements in energy efficiency of the existing housing stock feasible. Henceforth, this paper also focussed on the effect of constant decreasing investment costs on the optimal investment path. The findings show that the threshold level that triggers investment decreases at the beginning of the investment program, stays equal halfway the investment program and increases during the last stages (e.g. a smile pattern) under the assumption of constant decreasing investment costs. Moreover, we observe that the optimal investment path (threshold levels) is lower and deviates between closer boundaries with constant decreasing investment costs. Finally, the optimal investment path is higher for larger volatility levels and lower for higher discount rates (expected return) keeping all else constant. Both findings are in line with the characteristics of the optimal investment rule in real option theory discussed by Dixit and Pindyck (1994).
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Appendix A: Explicit Finite Difference method
The payoff received upon completion of each investment stage is stochastic and uncertain and follows a Geometric Brownian Motion (GBM) process:
𝑑𝑃 = µ𝑃𝑑𝑡 + 𝜎𝑃𝑑𝑧 (1A)
It is more convenient to work in terms of 𝑥 = 𝑙𝑛 𝑃 , which transforms the stochastic differential equation (equation 1A) to:
𝑑𝑥 = 𝜈𝑑𝑡 + 𝜎𝑑𝑧 (2A)
𝜈 = µ − 1
2𝜎
2 (3A)
When we work in terms of the logarithm of 𝑃, applying Itô’s lemma gives the following partial differential equation (PDE):
31
The explicit finite difference method is equivalent to approximating the diffusion process of the payoff by a discrete trinomial process. By using the explicit finite difference method, we can transform the trinomial tree into a rectangular grid. In order to compute the option values of the call option at the top node and lowers node in the grid we have to impose the following boundary conditions:
𝜕𝐹
𝜕𝑃= 1 for large 𝑃 (5A)
𝜕𝐹
𝜕𝑃= 0 for small 𝑃 (6A)
In order to obtain the rectangular grid, we imagine that time and space are divided up into discrete intervals (∆𝑡 and ∆𝑥). ∆𝑥 cannot be chosen independently of ∆𝑡. It is numerically most
efficient if
∆𝑥 = 𝜎√3∆𝑡. (7A)
The explicit finite difference method simplifies the PDE by replacing the partial differences in equation 4A with finite differences. The explicit finite difference method replaces𝜕𝐹
𝜕𝑡 with forward
differences and replaces 𝜕𝐹
𝜕𝑥 and 𝜕2𝐹
𝜕𝑥2 with central differences:
𝜕𝐹 𝜕𝑡 = 𝑖+1, 𝑗− 𝑖, 𝑗 ∆𝑡 (8A) 𝜕𝐹 𝜕𝑥 = 𝑖+1, 𝑗+1− 𝑖+1, 𝑗−1 2∆𝑥 (9A) 𝜕2𝐹 𝜕𝑥2 = 𝑖+1, 𝑗+1− 2 𝑖+1, 𝑗+ 𝑖+1, 𝑗−1 ∆𝑥2 (10A)
substituting equation 8A-10A into the PDE (see equation 4A) yield the following expression: −𝐹𝑖+1, 𝑗− 𝐹𝑖, 𝑗 ∆𝑡 = 𝜎 𝐹𝑖+1, 𝑗+1− 𝐹𝑖+1, 𝑗+ 𝐹𝑖+1, 𝑗−1 ∆𝑥2 + 𝜈 𝐹𝑖+1, 𝑗+1− 𝐹𝑖+1, 𝑗−1 ∆𝑥 − 𝑟 𝑖+ , 𝑗 (11A)
Which can be rewritten as:
𝑖, 𝑗 = 𝑝𝑢 𝑖+ , 𝑗+ + 𝑝𝑚 𝑖+ , 𝑗 + 𝑝𝑑 𝑖+ , 𝑗− (12A)
where 𝑝𝑢, 𝑝𝑚 and 𝑝𝑑 are the probabilities that the payoff goes up by ∆𝑥, stays the same, or goes down by ∆𝑥 respectively, over a small time interval ∆𝑡. Note that the probabilities need to add up to one:
𝑝𝑢+ 𝑝𝑚+ 𝑝𝑑 = 1 (13A)
Solving the equations gives us the following expressions for 𝑝𝑢, 𝑝𝑚 and 𝑝𝑑:
32
𝑝𝑑 = ∆𝑡 ( 𝜎2
∆𝑥2−
𝜈
∆𝑥) (16A)
Note that 𝑝𝑢, 𝑝𝑚 and 𝑝𝑑 are used to determine the expected payoff (diffusion process) at each time interval.
B: Python programming code (model) # European call option code
def tri_eur_cal(P, T, sig, mu, r, n, nj, I, V_ame): dt = T/n nu = mu - 0.5*sig**2 dx = sig*sqrt(3*dt) edx = exp(dx) mu = r - div pu = 0.5 * dt * ((sig/dx)**2 + nu/dx) pm = 1 - dt * (sig/dx)**2 - r * dt pd = 0.5 * dt * ((sig/dx)**2 - nu/dx) Pt = P*exp(-nj*dx)*edx**np.linspace(0, 2*nj, 2*nj+1) strike = I*np.ones(2*nj+1) payoff= np.maximum(0,(Pt+V_ame)-strike) c1 = np.zeros([(n+1),(2*nj+1)]) c1[n,:]=payoff[:] for i in range(n, 0, -1): c1[i-1,1:2*nj]=pu*c1[i,2:2*nj+1]+pm*c1[i,1:2*nj]+pd*c1[i,0:2*nj-1] c1[i-1,0] = c1[i-1,1]
c1[i-1,2*nj] = c1[i-1,2*nj-1] + ((Pt+V_ame)[2*nj]-(Pt+V_ame)[2*nj-1]) return(c1[0,:])
Eur_cont_value = tri_eur_cal(P, t, sig, mu, r, n, nj, I, V_ame)
# American call option code
