### Grant Program

Collaborative Research Grants Program 2020-2022

### Project Description

The goal of CISM is to engage mathematics directly in a range of other disciplines by widening the use and application of modern mathematical progress as effective research techniques among a variety of fields. The emphasis will be put on fostering scientific collaborations and interaction between the mathematical sciences and other disciplines as well as the industry.

The mathematical research in Kazakhstan has witnessed an enormous boost over the last years. In particular, it led to publications by young mathematicians in Kazakhstan in leading journals in pure and applied mathematics and in mathematical finance. Also, this led to the publication of high-level research monographs in leading scientific publishers (Birkhauser, CRC Press/Taylor and Francis) on several subjects leading to prestigious international prizes (see e.g. https://ffsb.espais.iec.cat/en/the-ferran-sunyer-i-balaguer-prize/ Ferran Sunyer i Balaguer Prize 2018). It would be an ideal moment for the Nazarbayev University to capitalise on these developments at this key moment and to consolidate the newly emerging internationally leading mathematical research in the form of its own interdisciplinary centre. The existing mathematical connections (especially to Massachusetts Institute of Technology, Imperial College London, in the top 10 of the world ranking of universities, and Al-Farabi Kazakh National University, top research university in Kazakhstan) would provide a great additional boost to the development and promotion of the centre.

Part of specific initial emphasis: In particular, in the following three years, a part of the emerging research deals with theory and applications of integral equations. As such, it allows one to effectively treat (both theoretically and numerically) a variety of problems appearing in a range of applications. One area of particular importance are models appearing in investments in risky projects under uncertainty (real options theory). Thus, a particular emphasis of the activities of the centre would be put on establishing and promoting links between mathematics and subjects related to optimal investment problems for Kazakhstan. This would lead to the affirming the leading position of the Nazarbayev University in sciences and their specific applications both in Kazakhstan and internationally.

Theory of integral equations is part of functional analysis, calculus, differential equations theory, operator theory, approximation theory, and of many other areas of pure and applied mathematics. Although centuries old, these subjects are under intense development, for use in pure and applied mathematics, mathematical modeling, engineering, physics as well as computer science. This stimulates continuous interest for researchers in these and related fields. In 2020-2022, the aim of this collaborative project is to study several modern aspects of the subject. The collaborative project has four main objectives in the fields of energy system modeling, pure mathematics, applied mathematics and algorithm development:

1. Investment theory: Develop models of energy systems that encompass energy generation from renewable and non-renewable sources, conversion, storage, transmission, and demand, and explicitly include the uncertainty and risk in these systems.

2. Pure mathematics: Study differential and integral equations arising in the above models. Publish obtained results in international high ranked mathematical journals.

3. Applied mathematics: Application of mathematical methods to the above models (e.g. using stochastic control and optimal stopping theories). Use analytical and numerical methods to analyse the proposed problems.

4. Algorithm and software development: Develop and implement efficient algorithms to solve numerically multi-dimensional optimal investment problems.

So, the main directions are: 1) understand the main sources of uncertainty for sustainable energy investment in KZ and formulate investment models that capture these; 2) develop (analytical or/and numerical) methods to solve these types of mathematical problems; 3) implement these methods and algorithms; 4) provide detailed analysis of the results.

In 2020-2022, we will study investments in exclusive projects with different cost structures, specifically a variable cost project (VCP) and a fixed cost project (FCP). We will derive the optimal investment policy taking account of the endogenous operating decisions for each underlying project. Choices between exclusive alternatives are prevalent in corporate decision-making. Common examples include choices between different production technologies (manufacturing), applications of a given technology (product choice), target firms (mergers and acquisitions), research programs (research and development) or advertising strategies (marketing). They are also quite common in individual decision-making, e.g., education choices, employment decisions, entertainment selections, etc. We mainly focus on situations where the underlying projects exhibit fundamental differences in cost structures, specifically in the mix between fixed and variable costs. An illustrative example is that of utilities who are nowadays faced with non-trivial choices between competing technologies when they invest in new power production units. On the one hand they can build plants using fossil fuels to generate power (e.g., coal- or gas-red plants). On the other hand they can use Green Energy - hereafter GE - (e.g., wind or solar plants). The distinguishing feature between these alternatives is their respective cost structures. In the former case, there are fixed costs and variable costs. Fixed costs consist of wages and maintenance costs, among others. Variable costs relate to the fuel used as an input in the production process. The price of fuel varies stochastically and is typically positively correlated with the electricity price, hence increases with it. Such a technology has lower operating leverage, depends less on demand and is less risky as a standalone project. In the latter case, the GE input is free and costs are independent of the price of electricity. Such a technology entails fixed costs, is more dependent on demand and has higher operating leverage. Hence, in essence, the choice between the two technologies is a choice between projects with lower and higher operating leverage. Motivated by this example, we study the general problematic of investing in exclusive alternatives consisting of a VCP and an FCP. The timing of the decision to invest is flexible. Moreover, each of the possible selections involves subsequent timing choices regarding operations. Some of the questions arising in this context are as follows. What is the optimal time to invest? Under what conditions is it optimal to invest in the FCP versus the VCP? What is the impact of operating leverage on these decisions? Applied to power generation projects, our model can be used to examine a host of issues. In particular, it can shed light on the relevance of subsidies and on the value of GE. Such issues are important not only for power operators but also for consumers and policy makers.

Our study relates to several branches of the literature. First, it contributes to the broad literature on growth options, operating leverage and firm value. The relation between operating leverage and risk is examined by Lev (1974). Real options models drawing the link between operating leverage and value premium can be found in Carlson, Fisher and Giammarino (2004), Zhang (2005) and Novy-Marx (2011). In contrast to this literature, our analysis studies the choice between exclusive projects with different leverage characteristics, specifically cost structures (fixed versus variable cost). It highlights, in particular, the role of the variable cost structure for the optimal timing and selection decisions, and the value of the project. Second, it also relates to the more specialized literature on power generation. Most of the literature in that area focuses on projects involving a given technology. Two exceptions are Decamps, Mariotti and Villeneuve (2006) and Siddiqui and Fleten (2010) who consider a choice between two exclusive alternatives. The first study develops a general method to deal with investments in the best of two technologies with values driven by a common underlying state variable. They show that the optimal investment region can consist of two intervals separated by a continuation interval. The second study adds the possibility of learning about the costs of one of the technologies after investing in it, but before deciding to deploy it. Assuming that the cost of deployment is incurred at the time of investment, they show that the model reduces to a single state variable problem as in Decamps, Mariotti and Villeneuve (2006). They proceed to examine the impact of learning on the optimal investment decision and the project valuation. The models considered in this project cannot be reduced to a single state variable. They involve two distinct underlying processes, for revenues and costs, as well as embedded (two-dimensional) optimal stopping time problems for the operating decisions of the underlying projects and the timing and selection decisions of the project manager. We will solve this two state variables compound optimal stopping problem in generality, assuming that the underlying processes follow geometric Brownian motions (GBM). In the application to power projects, the first state variable represents the price of electricity and the second one the price of gas. The GBM specification allows the spark spread to take negative values, when the price of gas adjusted by the heat rate exceeds the price of electricity, but also ensures that the price of gas cannot become negative. GBM, as a model of the long run behavior of the electricity price, is standard in the literature on power generation. Third, our research is also related to broader literatures on real options (RO) and American option pricing. Seminal contributions in the real option area highlight the values of flexibility and of waiting to invest (e.g., Brennan and Schwartz (1985), McDonald and Siegel (1985, 1986); see also Dixit and Pindyck (1994) and references therein). This literature relies on option pricing methods and typically uses partial differential equations to price projects. In contrast, we exploit a characterization of the value function based on the EIP representation, which is rooted in probabilistic/martingale methods (see Kim (1990) and Carr, Jarrow and Myneni (1992) for American options). We will also extend the literature on the valuation of multiasset American claims (e.g., Broadie and Detemple (1997)), by considering dual strike compound max-options and designing an implementable algorithm that solves systems of coupled integral equations of Fredholm type for optimal exercise boundaries. The compound structure of our problem and the presence of uncertain operating leverage modifies some of the properties characterizing optimal exercise decisions for max-options. The advantage of our algorithm compared to, e.g., PDE or Monte-Carlo methods, is the ability to tackle perpetual or long horizon problems, in addition to short horizon problems. This approach can help to address other challenging multi-dimensional stopping problems in real options and American option pricing applications.

All applicants have strong and complementary skills in the theory of differential and integral equations, probability theory, and numerical methods needed to address these problems.

The mathematical research in Kazakhstan has witnessed an enormous boost over the last years. In particular, it led to publications by young mathematicians in Kazakhstan in leading journals in pure and applied mathematics and in mathematical finance. Also, this led to the publication of high-level research monographs in leading scientific publishers (Birkhauser, CRC Press/Taylor and Francis) on several subjects leading to prestigious international prizes (see e.g. https://ffsb.espais.iec.cat/en/the-ferran-sunyer-i-balaguer-prize/ Ferran Sunyer i Balaguer Prize 2018). It would be an ideal moment for the Nazarbayev University to capitalise on these developments at this key moment and to consolidate the newly emerging internationally leading mathematical research in the form of its own interdisciplinary centre. The existing mathematical connections (especially to Massachusetts Institute of Technology, Imperial College London, in the top 10 of the world ranking of universities, and Al-Farabi Kazakh National University, top research university in Kazakhstan) would provide a great additional boost to the development and promotion of the centre.

Part of specific initial emphasis: In particular, in the following three years, a part of the emerging research deals with theory and applications of integral equations. As such, it allows one to effectively treat (both theoretically and numerically) a variety of problems appearing in a range of applications. One area of particular importance are models appearing in investments in risky projects under uncertainty (real options theory). Thus, a particular emphasis of the activities of the centre would be put on establishing and promoting links between mathematics and subjects related to optimal investment problems for Kazakhstan. This would lead to the affirming the leading position of the Nazarbayev University in sciences and their specific applications both in Kazakhstan and internationally.

Theory of integral equations is part of functional analysis, calculus, differential equations theory, operator theory, approximation theory, and of many other areas of pure and applied mathematics. Although centuries old, these subjects are under intense development, for use in pure and applied mathematics, mathematical modeling, engineering, physics as well as computer science. This stimulates continuous interest for researchers in these and related fields. In 2020-2022, the aim of this collaborative project is to study several modern aspects of the subject. The collaborative project has four main objectives in the fields of energy system modeling, pure mathematics, applied mathematics and algorithm development:

1. Investment theory: Develop models of energy systems that encompass energy generation from renewable and non-renewable sources, conversion, storage, transmission, and demand, and explicitly include the uncertainty and risk in these systems.

2. Pure mathematics: Study differential and integral equations arising in the above models. Publish obtained results in international high ranked mathematical journals.

3. Applied mathematics: Application of mathematical methods to the above models (e.g. using stochastic control and optimal stopping theories). Use analytical and numerical methods to analyse the proposed problems.

4. Algorithm and software development: Develop and implement efficient algorithms to solve numerically multi-dimensional optimal investment problems.

So, the main directions are: 1) understand the main sources of uncertainty for sustainable energy investment in KZ and formulate investment models that capture these; 2) develop (analytical or/and numerical) methods to solve these types of mathematical problems; 3) implement these methods and algorithms; 4) provide detailed analysis of the results.

In 2020-2022, we will study investments in exclusive projects with different cost structures, specifically a variable cost project (VCP) and a fixed cost project (FCP). We will derive the optimal investment policy taking account of the endogenous operating decisions for each underlying project. Choices between exclusive alternatives are prevalent in corporate decision-making. Common examples include choices between different production technologies (manufacturing), applications of a given technology (product choice), target firms (mergers and acquisitions), research programs (research and development) or advertising strategies (marketing). They are also quite common in individual decision-making, e.g., education choices, employment decisions, entertainment selections, etc. We mainly focus on situations where the underlying projects exhibit fundamental differences in cost structures, specifically in the mix between fixed and variable costs. An illustrative example is that of utilities who are nowadays faced with non-trivial choices between competing technologies when they invest in new power production units. On the one hand they can build plants using fossil fuels to generate power (e.g., coal- or gas-red plants). On the other hand they can use Green Energy - hereafter GE - (e.g., wind or solar plants). The distinguishing feature between these alternatives is their respective cost structures. In the former case, there are fixed costs and variable costs. Fixed costs consist of wages and maintenance costs, among others. Variable costs relate to the fuel used as an input in the production process. The price of fuel varies stochastically and is typically positively correlated with the electricity price, hence increases with it. Such a technology has lower operating leverage, depends less on demand and is less risky as a standalone project. In the latter case, the GE input is free and costs are independent of the price of electricity. Such a technology entails fixed costs, is more dependent on demand and has higher operating leverage. Hence, in essence, the choice between the two technologies is a choice between projects with lower and higher operating leverage. Motivated by this example, we study the general problematic of investing in exclusive alternatives consisting of a VCP and an FCP. The timing of the decision to invest is flexible. Moreover, each of the possible selections involves subsequent timing choices regarding operations. Some of the questions arising in this context are as follows. What is the optimal time to invest? Under what conditions is it optimal to invest in the FCP versus the VCP? What is the impact of operating leverage on these decisions? Applied to power generation projects, our model can be used to examine a host of issues. In particular, it can shed light on the relevance of subsidies and on the value of GE. Such issues are important not only for power operators but also for consumers and policy makers.

Our study relates to several branches of the literature. First, it contributes to the broad literature on growth options, operating leverage and firm value. The relation between operating leverage and risk is examined by Lev (1974). Real options models drawing the link between operating leverage and value premium can be found in Carlson, Fisher and Giammarino (2004), Zhang (2005) and Novy-Marx (2011). In contrast to this literature, our analysis studies the choice between exclusive projects with different leverage characteristics, specifically cost structures (fixed versus variable cost). It highlights, in particular, the role of the variable cost structure for the optimal timing and selection decisions, and the value of the project. Second, it also relates to the more specialized literature on power generation. Most of the literature in that area focuses on projects involving a given technology. Two exceptions are Decamps, Mariotti and Villeneuve (2006) and Siddiqui and Fleten (2010) who consider a choice between two exclusive alternatives. The first study develops a general method to deal with investments in the best of two technologies with values driven by a common underlying state variable. They show that the optimal investment region can consist of two intervals separated by a continuation interval. The second study adds the possibility of learning about the costs of one of the technologies after investing in it, but before deciding to deploy it. Assuming that the cost of deployment is incurred at the time of investment, they show that the model reduces to a single state variable problem as in Decamps, Mariotti and Villeneuve (2006). They proceed to examine the impact of learning on the optimal investment decision and the project valuation. The models considered in this project cannot be reduced to a single state variable. They involve two distinct underlying processes, for revenues and costs, as well as embedded (two-dimensional) optimal stopping time problems for the operating decisions of the underlying projects and the timing and selection decisions of the project manager. We will solve this two state variables compound optimal stopping problem in generality, assuming that the underlying processes follow geometric Brownian motions (GBM). In the application to power projects, the first state variable represents the price of electricity and the second one the price of gas. The GBM specification allows the spark spread to take negative values, when the price of gas adjusted by the heat rate exceeds the price of electricity, but also ensures that the price of gas cannot become negative. GBM, as a model of the long run behavior of the electricity price, is standard in the literature on power generation. Third, our research is also related to broader literatures on real options (RO) and American option pricing. Seminal contributions in the real option area highlight the values of flexibility and of waiting to invest (e.g., Brennan and Schwartz (1985), McDonald and Siegel (1985, 1986); see also Dixit and Pindyck (1994) and references therein). This literature relies on option pricing methods and typically uses partial differential equations to price projects. In contrast, we exploit a characterization of the value function based on the EIP representation, which is rooted in probabilistic/martingale methods (see Kim (1990) and Carr, Jarrow and Myneni (1992) for American options). We will also extend the literature on the valuation of multiasset American claims (e.g., Broadie and Detemple (1997)), by considering dual strike compound max-options and designing an implementable algorithm that solves systems of coupled integral equations of Fredholm type for optimal exercise boundaries. The compound structure of our problem and the presence of uncertain operating leverage modifies some of the properties characterizing optimal exercise decisions for max-options. The advantage of our algorithm compared to, e.g., PDE or Monte-Carlo methods, is the ability to tackle perpetual or long horizon problems, in addition to short horizon problems. This approach can help to address other challenging multi-dimensional stopping problems in real options and American option pricing applications.

All applicants have strong and complementary skills in the theory of differential and integral equations, probability theory, and numerical methods needed to address these problems.

Short title | CISM |
---|---|

Status | Active |

Effective start/end date | 1/1/20 → 12/31/22 |

### Fingerprint

Mathematics

Variable cost

Operating leverage

Fixed costs

State variable

Kazakhstan

Integral equations

Optimal investment

Cost structure

Gas

Real options

Costs

Differential equations

Electricity

Power generation

Numerical methods

Geometric Brownian motion

Electricity price

Exercise

Uncertainty

### Keywords

- Investment theory
- theory of integral equations
- energy system modeling
- numerical simulation
- finance