许多金融应用都是随机问题。他们包含了大量的随机性和不确定性，因此这些问题都能够被随机控制理论很好的解决。这篇金融数学 代 写论文阐述了随机控制在金融中的运用。以金融实践为重点简要回顾了之前的研究并且深入了解最新的金融随机控制理论。文章利用蒙特卡洛模型进行估值分析，将随机控制技术应用到投资组合优化。为更加深入的研究随机控制，我们提供了一些可能的方向。

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Abstract

Many financial applications are stochastic problems in nature. They involve a significant amount of randomness and uncertainties, hence are well-handled by stochastic control theories. This study aims at demonstrating the utilization of stochastic control in finance. Specifically, a brief review of prior works provides some insight into the state-of-the-art of stochastic control theory, with a focus on the financial practice. A case study follows in which stochastic control techniques are applied to portfolio optimization. Monte Carlo simulations are used in analysis for estimation and approximation. In addition to the demonstration, the paper gives a discussion about more advanced topics in stochastic control, which can be potential directions for further study.

Keywords: stochastic control, portfolio optimization, Monte Carlo simulation

I. Introduction

Financial applications involve a considerate amount of uncertainties. Stochastic control, as a subfield of the traditional control theory, deals with uncertainties in the observations or the associated noise acquired from a dynamic system. It is thus a good candidate for handling problems with randomness, such as those typical decision-making problems under uncertainty in financial practice. Portfolio optimization, for example, is a typical one among this kind. Merton’s breakthrough work in the 1970s utilized the Hamilton-Jacobi-Bellman (HJB) equation in the computation of optimal trading strategy for a portfolio involving both risky and riskless assets and came up with fairly neat results, leading people to realize the applicability of stochastic modeling and control theory in the field of finance.

Among the various financial mathematics problems, portfolio optimization is the one that drawing the most attention due to its strong connection to the real practice. The very essence of finance is to manage wealth and gaining more value using the existing assets. A major financial activity is investment, in which investors expect the maximization of the portfolio value. By choosing the proportions of a variety of financial assets to be included in a portfolio, a decision-making process is completed, so a portfolio optimization problem may be treated as a decision-making problem as well. Most financial assets are risky in the sense that their values in future cannot be known without uncertainty. Only a couple of securities, such as the treasury bond, can be deemed as risk-free asset due to the credit of government. Therefore, which asset to select into a portfolio becomes a decision to make in a risky setting. A well-chosen portfolio can earn a fortune for the investor, while a poorly combined portfolio would cause disastrous loss. The implication is that, the selection criterion is the key to a well-performed portfolio. A considerate number of studies have been working on an effective selection criterion that lead to the maximum expected return under certain amount of risk, or the minimum uncertainty with a stable expected return. Modern portfolio theory, founded by Harry Markowitz in 1950s, brought the concept of “efficient portfolio”, which refers to a portfolio that maximizes the expected return contingent on a fixed amount of risk, which is measured by the standard deviation of the rate of return. With no doubt, even for such efficient portfolios there is a trade-off between the expected return and the risk. A higher expected rate of return always involves a higher potential risk. In the world of finance, there is no portfolio that can make a fortune with one hundred percent of certainty.

From a practical point of view, theoretical models are simplifications of the reality. Even the most complicated financial mathematics models cannot fully describe the actual practice. Taking the risk-free asset for an example, such asset is not really free from risk but are merely associated with much less risk than any other assets as a result of the government credit. Generally, real cases involve much more uncertainty than the theoretical description. An observation from practice may subject to various sources of noise and distortions, thus bringing in more complications and challenges to the analysis. It is beneficial to accommodate all the possible sources of risk in theoretical study, yet keep the model succinct and tractable. Stochastic control, after its first application to financial applications, have been further developed in a financial mathematical setting and found effective in modeling the decision-making problems from reality. Stochastic processes, especially Markov processes, provide relatively accurate descriptions to the practice of finance and economics. Many practical problems can be converted into stochastic control problems, which can be handled typically by dynamic programming principle. A drawback of this approach is that it results in non-linear partial differential equations (PDE), which are very intriguing but difficult to solve analytically. Such situation leads to the utilization of Monte Carlo simulation and numerical methods for approximated solutions. With proper implementation, numerical approximations can be good proxies for the analytical solutions and used as the guidance for practice.

This paper is organized as follows. Section I and II consist of Part I of the study, which give a brief introduction and a thorough overview of the prior works on stochastic control and its applications in finance. Part II is an illustration of stochastic control problems, in which theoretical models are built with tuning parameters are given for computation. Monte Carlo simulations are also included as a device for problem solving. Part III of the work focuses on more advanced topics in stochastic control. It ends the paper with a summary of the latest promising developments in the field as well as suggestions for future work.

Apart from the optimization of trading strategy, asset pricing is also a critical part in financial activities. Precise pricing of financial assets is rather crucial in maintaining market efficiency. Over or under-estimating the value of the asset leads to opportunities of arbitrage, which are not supposed to exist in an efficient financial market. Asset pricing indeed involves a lot of randomness, so stochastic control is also intensively used for this subject. Merton (1973) [7] studied theory of rational option pricing. He lied the foundation for a rational theory of option pricing, based on which he extended the Black-Scholes model to weaker assumptions than those postulated such that the model fitted in more general situations. To be specific, the temptations of the extended version were: (i) the relatively weak condition underlying the derivation of the B-S type model could avoid dominance; (2) the terminal formula obtained was a function of “observable” variables; and (3) the model could be applied directly to the determination of the rational price of any kinds of options. This means that the scope of application was significantly broadened. According to the author, the model could be utilized in the pricing of various elements of the firm’s capital structure. Under the conditions when the Modigliani-Miller theorem held, one could use the entire value of the company as a baseline or benchmark and treated the individual securities within the capital structure as “options” or the so-called “contingent claims” on the company. The author suggested that a theory of the term structure of interest rates could be developed based on the proposed technique. In addition, the technique could be applied in the theory of speculative markets.

More recently, Galichon et al. [8] proposed a stochastic control approach to no-arbitrage bounds, given that the marginals were known. The study took the lookback options for a case study. The work was built with assumption of an investor allowed to dynamically trade the underlying asset and statically trade European call options for any possible trikes. The focus was superhedging under volatility. Typically, such a problem would be approached by Skorohod Embedding Problem (SEP), but this study provided a dual formulation that converted the superhedging problem into a continuous martingale optimal transportation problem. It was shown that the formulation enabled the recovery of previously known results about lookback options. The proposed method induced a new proof of the optimality of Azema-Yor solution of the SEP for a certain class of lookback options. The proposed approach, different from the SEP, was applicable to a wide class of exotics and also suitable for numerical approximations. In the case study, an optimal upper bound was given as █(U^μ (ξ)=inf┬(λ∈Λ_UC^μ )⁡{μ(λ)+u^λ (0,X_0,X_0 )}#(20) ) The main result provided a formulation of the robust superhedging problem based on the Kantorovich duality. It was argued that the derived duality result was advantageous in its generality. Also, similar to the SEP method, the solution of the dual formulation led to the associated optimal hedging strategy and the model in the worst scenario. The paper showed that the duality result derived was not only theoretically important, but also applicable to practice. The lookback option case study demonstrated the point. The author stressed that a major accomplishment of the study was that the proposed optimal transportation approach complemented the SEP approach.

References

[1] Merton, R. C. (1971). "Optimum consumption and portfolio rules in a continuous-time model". Journal of Economic Theory. 3 (4): 373–413. doi:10.1016/0022-0531(71)90038-X.

[2] R. C. MERTON, “Lifetime portfolio selection under uncertainty: the continuous-time Case”, Rev. Econ. Statist. LI (August, 1969), 247-257.

[3] Almgren, R. and N.Chriss, “Optimal execution of portfolio transactions”, J. Risk, 3:5, 39, 2001.

[4] Almgren, R., Optimal execution with nonlinear impact functions and trading-enhanced risk, Applied Mathematical Finance, 10, 1-18, 2003.

[5] R.Liu and J.Muhle-Karbe, “Portfolio Choice with Stochastic Investment Opportunities: a User’s Guide”, 2013.

[6] Gerhold, S., J.Muhle-Karbe and W.Schachermayer, “The Dual Optimizer for the Growth-Optimal Portfolio under Transaction Costs”, Finance and Stochastics, Vol. 17 (2013), No. 2, pp. 325-354.

[7] Merton, R.C., “Theory of rational option pricing”, The Bell Journal of Economics and Management Science, Vol. 4, No (Spring, 1973), pp. 141-183.

[8] Galichon, A., Henry-Labodere, P. and Touzi, N., “A Stochastic Control Approach to No-arbitrage Bounds Given Marginals, with an Application to Lookback Options”, The Annuals of Applied Probability, 2014, Vol. 24, No. 1, 312-336.

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