Application of stochastic differential equations and real option theory in investment decision problems

Stochastic differential equations;Investments--Decision making--Mathematical models;Real options (Finance)--Mathematical models

Chavanasporn, Walailuck ; Ewald, Christian-Oliver ; Ewald, Christian-Oliver

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Stochastic differential equations;    Investments--Decision making--Mathematical models;    Real options (Finance)--Mathematical models;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/1691/WalailuckChavanaspornPhDThesis.pdf?sequence=3&isAllowed=y

来源: DR-NTU

【 摘 要 】

This thesis contains a discussion of four problems arising from the application ofstochastic differential equations and real option theory to investment decision problems in a continuous-time framework. It is based on four papers written jointly withthe author’s supervisor.In the first problem, we study an evolutionary stock market model in a continuous-time framework where uncertainty in dividends is produced by a single Wiener process.The model is an adaptation to a continuous-time framework of a discrete evolutionarystock market model developed by Evstigneev, Hens and Schenk-Hoppé (2006). Weconsider the case of fix-mix strategies and derive the stochastic differential equationswhich determine the evolution of the wealth processes of the various market players.The wealth dynamics for various initial set-ups of the market are simulated.In the second problem, we apply an entry-exit model in real option theory to studyconcessionary agreements between a private company and a state government torun a privatised business or project. The private company can choose the time toenter into the agreement and can also choose the time to exit the agreement if theproject becomes unprofitable. An early termination of the agreement by the companymight mean that it has to pay a penalty fee to the government. Optimal times forthe company to enter and exit the agreement are calculated. The dynamics of theproject are assumed to follow either a geometric mean reversion process or geometricBrownian motion. A comparative analysis is provided. Particular emphasis is givento the role of uncertainty and how uncertainty affects the average time that theconcessionary agreement is active. The effect of uncertainty is studied by using MonteCarlo simulation.In the third problem, we study numerical methods for solving stochastic optimalcontrol problems which are linear in the control. In particular, we investigate methodsbased on spline functions for solving the two-point boundary value problems thatarise from the method of dynamic programming. In the general case, where onlythe value function and its first derivative are guaranteed to be continuous, piecewisequadratic polynomials are used in the solution. However, under certain conditions,the continuity of the second derivative is also guaranteed. In this case, piecewisecubic polynomials are used in the solution. We show how the computational timeand memory requirements of the solution algorithm can be improved by effectivelyreducing the dimension of the problem. Numerical examples which demonstrate theeffectiveness of our method are provided.Lastly, we study the situation where, by partial privatisation, a government givesa private company the opportunity to invest in a government-owned business. Afterpayment of an initial instalment cost, the private company’s investments are assumedto be flexible within a range [0, k] while the investment in the business continues. Wemodel the problem in a real option framework and use a geometric mean reversionprocess to describe the dynamics of the business. We use the method of dynamicprogramming to determine the optimal time for the private company to enter andpay the initial instalment cost as well as the optimal dynamic investment strategythat it follows afterwards. Since an analytic solution cannot be obtained for thedynamic programming equations, we use quadratic splines to obtain a numericalsolution. Finally we determine the optimal degree of privatisation in our model fromthe perspective of the government.