The stochastic optimal control of market regulation

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The Stochastic Optimal Control Of Market Regulation

Universiteit van Amsterdam Roetersstraat 11 1018 WB Amsterdam phone: +31-(0)6 460 800 54 e-mail: laurens.voogd@uva.nl homepage: http://www.uva.nl/feb/

Dissertation MSc Financial Econometrics

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The Stochastic Optimal Control

Of

Market Regulation

Universiteit van Amsterdam

Master Of Science

(Financial) Econometrics

Supervised By

Dr.ir. F.O.O. Wagener Prof. dr. H.P. Boswijk

Written By

Laurens Voogd

Born Palma de Mallorca, Spain

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Abstract

Financial markets can be characterized by actors which each have their own objec-tives. These objectives differ in terms of risk attitude of the actor and as a result in their short -or long term focus. Financial institutes, in contrast to authorities and society as a whole, typically think in short term objectives and are willing to engage in riskier investments than the authority considers (long term) responsible. During the 2008 financial crisis, these differences became painfully clear. While the financial crisis had several causes, is a generally accepted explanation that agents engaged into too risky endeavors which resulted in a sentiment of distrust amongst agents. To overcome those market frictions, the authority had to intervene and play a role as intermediary which came with great effort and cost. This research attempts to investigate how authorities and regulatory bodies can create a market environment which will prevent such market frictions as were experienced during the 2008 crisis. The research takes a (stochastic) optimal control approach in terms of solving the objectives of the actors in the market mathematically. To prevent those market frictions from ocuring, the authority can impose regulatory restrictions whenever the financial agent does no-longer comply with the pre-set regulatory requirements. The authority will evaluate this based on the level of the book-value and levered position of the financial agent. If these levels are considered (socially) unacceptable in terms of risk, will the authority intervene by limiting the behavior of the financial agent. Key findings of this research are the significant impact of regulatory require-ments and restrictions has on the economic performance of the authority agent and to a lesser extend on the economy as a whole. In the mathematical market, mar-ket intervention of the authority results in a (on average) 25% improvement on the authority’s reached objective and while doing so, improving the reaches objective of the economy as a whole by 1%. These gains result from restricting (excessive) risk taking of the financial agent. In the market model, the financial agent is will-ing to take a very risky financial position, with relatively low associated expected returns.The return the financial agent receives from taking on a riskier position has sharply diminishing returns to scale. This phenomena is the main cause of the improvements caused by market regulation.

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Abstract Abstract

Abreviations

ODE Ordinary differential equation. SDE Stochastic differential equation. PDE Partial differential equation.

HJB The Hamilton Jacobi Bellman equation. GCT Gershgorin circle theorem.

LLC Limited liable corporation. BTIS Backward time implicit scheme.

Nomenclature

Z(t) The state process (during the research we also refer to Z(t) as z for notational

convenience, or to zi as the discretized representation of Z(t)).

µ The drift rate of the stochastic diffusion process. ‡ The variance rate of the stochastic diffusion process. V(·, ·) The value-function.

V The value-function index, which will be used to represent the performance of the economy. Formally V := sDV(0, z)dz, here D denotes the problem

domain.

(u(t), c(t)) The leverage-rate and dividend controls of the firm.

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Chapter 1 Introduction

1.1 A brief overview

Over the last century financial market regulation has undergone several big changes. After the financial collapse in the U.S. that began in 1929, regulators decided to tighten the financial market regulation. The Glass-Steagall Act of 1933 and the Securities Act of 1933 were two of many acts that were passed by the United States Congress. The Securities Act of 1933 was enacted in order to ensure that the qual-ity of information about securities would improve. The Glass-Steagall Act of 1933 prohibited commercial banks from engaging in investment activities. The U.S. eco-nomic policy after the Great Depression was based on the theory created by John Maynard Keynes who stated that government regulation, in the form of either a re-duction in interest rates monetary- or government investment/spending fiscal policy will help create stable growth and high employment . During the early 1980?s the economic mainstream shifted toward the idea of efficient financial markets, which resulted in the deregulation of the financial markets. The efficient market hypothesis (EMH) , proposes that free financial markets will result in the most efficient alloca-tion of financial assets. This shift ultimately led to the repeal of the Glass-Steagall Act in 1999 by the Gramm-Leach-Bliley Act. The deregulation, was in addition accompanied by financial innovation. In the following decades financial markets be-came more and more characterized by complex and intransparant (derived) financial products. While it is impossible to attribute specific causes that led up to the 2008 financial crisis, economic mainstream does agree upon the fact that deregulation in combination with the intransparant nature of financial products took a great part in causing the 2008 financial crisis (M.Jickling,Causes of the 2008 financial crisis 2010, p.3).

As a direct result of this global crisis, governments and regulators tightened regulation after two decades of deregulation. Many factors that caused the Great Recession were identified and regulated in order to prevent future crises. One factor that clearly contributed to the great recession was the excessive use of leverage. Commercial and investment banks had been using off balance sheet entities, which enabled them to evade capital control requirements. The commercial and investment banks who where involved in the creation of collateralized debt obligations used spe-cial purpose vehicles and commerspe-cial and investment banks who invested in CDOs created structured investment vehicles. As banks were required by Basel I rules to hold 8 percent of their capital against loans, the creation of those off balancesheet

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1.1. A BRIEF OVERVIEW CHAPTER 1. INTRODUCTION

vehicles, which were not required to hold any capital, was a way to evade capital adequacy rules. In this way banks were able to increase leverage substantially with borrowed funds.

Another contributing factor is the widely applied "fair value" accounting. Fair-value accounting involves reporting assets and liabilities on the balancesheet at fair value and recognizing changes in fair value as gains and losses in the income state-ment. Fair-value accounting can also be characterized as mark- to -market account-ing since it is common to set the fair-value equal to the market-value of the asset or liability (if there is a liquid market for the asset or liability). Critics argue that fair-value accounting exacerbated the severity of the 2007-09 financial crisis. The main allegations are that fair-value accounting contributes to excessive leverage in boom periods and leads to excessive eroded valuation in busts. The write-downs due to falling market prices deplete bank capital and set off a downward spiral, as banks are forced to sell assets at fire sale prices, which in turn can lead to conta-gion as fire-sales asset prices of one bank become relevant for other banks. These arguments are often taken at face value and evidence on problems created by fair-value accounting is rarely provided. Central bank and financial regulators tightened regulatory constraints, aiming to limit the sharp oscilations between levered boom periods and bust periods and ultimately preventing future reoccurrence of a 2007-09 financial crisis. One such action was the tightening of the leverage requirements to which financial intermediaries need to comply [Wayne R Landsman 2006]. It is widely agreed that high financial leverage, a high ratio of debt to the firm’s value, is a critical factor in the magnifying effects of financial crises. During the crisis asset value’s became highly uncertain which led to a decline in their value. As asset values decline, highly levered firms find their net worth sharply eroded and are forced to sell assets to avoid unacceptable risks of insolvency. But asset sales drive asset values down further, adversely impairing the balance sheets of other in-stitutions. These institutions in turn are forced to sell assets, creating a vicious cycle of balance sheet deterioration and asset sales. While the financial dynamics of such balance sheet adjustments have been widely discussed elsewhere, it is less well understood how this process affects macroeconomic outcomes, or that this pro-cess alone may generate an immediate and powerful international transmission of shocks. The financial positions of firms and the access to external finance of firms are crucial for the investment in and the development of an economy. This state-ment has become conventional wisdom in the finance literature. Unfortunately, the first strand omits to underline the financing constraints that are encountered by firms. The intransparant nature of financial markets due to financial innovation and the mark to market accounting applied by financial institutes result to a vulnerable and unstable system (M.Magnana, central bank and financial regulators tightened

regulatory constraints 2009, p.7). To strengthen bank capital requirements the

sec-ond Basel Accord and subsequently the third Basel Accord were developed. With the Basel III accord capital requirements, leverage ratios and liquidity requirements were adjusted in order to prevent future crises. According to the Basel III accord commercial and investments banks are no longer able to evade capital requirements with the use of off balance sheet entities such as SPV’s and SIV’s. A risk of all these regulations however is that they may stifle economic growth. Opponents of the Basel II and III accords are suggesting that this increased regulation will limit lending to the real economy and raise borrowing costs as banks will have to increase

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CHAPTER 1. INTRODUCTION 1.2. RESEARCH GOALS

the level of capital as prescribed in the Basel III accord.

1.2 Research goals

In the above overview we gave a brief summarization of the interactions between financial institutes and regulatory authorities over the past decades. Underlying to the interaction between financial institutes and regulatory agents is the, intrinsic difference between the objectives of financial institutes and those of regulatory bod-ies. The excessively levered financial positions of financial institutes result in both accelerated booms and busts. Since financial institutes are typically limited liable

corporations (LLC’s), the firm can in case of accelerated booms, "outperform" the

otherwise achieved result by leveraging its equity. In case of economic busts how-ever the financial actor is only limited liable for losses or even bankruptcy resulting from such economic busts. This allows the financial actor to ,without facing any consequences engage in even more risky investments. As a consequence will this excessive risk taking of the firm, lead to negative externalities for society as a whole. For instance once the firm reaches a point it can no longer immediately fulfill its financial obligations towards its creditors, the firms creditors will in turn need to postpone their own financial obligations towards their creditors and this cascading effect can lead to eroded asset values for all actors involved. Such social-costs can be externalized by financial institutes due to their limited liability and as a result, society as a whole, or in terms of this research, the authority agent will be held responsible for those negative externalities. The key question which this research attempts to answer is how can a regulatory authority influence a financial institute

to counterbalance the externalization of negative externalities in the objective of the financial institute? To answer this question, in the next chapter we will develop

a mathematical market model and give a description of the objective of both the financial agent (firm) and the authority agent. Once the model is described, we will then analyze the actions both agents can undertake in order to maximize their objectives and investigate on how those actions interact and influence the behavior of the agents.

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Chapter 2 The mathematical model

In the previous chapter we gave a broad overview of events leading up to the 2008 financial crisis. In this chapter we will develop a mathematical market model. The market contains two assets which we will discuss in the following section. Within this market, there are two active agents, a financial agent (firm) and an author-ity agent (authorauthor-ity). These two agents will be characterized by an agent specific objective and control-policies/instruments, with which they attempt to maximize their respective objective during a time period denoted by T. Below we attempt to develop a model in which the differences between the two agents are clearly stated and form a framework in which we can analyze the available strategies both agents can devise in order to maximize their own objective and how these strategies influ-ence the other agent. Ultimately we aim to answer the question how the authority agent can setup regulatory requirements that will yield an optimal outcome for the society in terms of the objective that the society (represented by the authority) will formulate.

2.1 The Market

We begin our model description by discussing the market which we will denote by M. The market contains two assets, an asset S(t) which represents a risky asset and an asset B(t) which is a risk-less asset. We model the price of the (risky asset) stock S(t) as the solution of the stochastic differential equation (SDE):

dS(t)

S(t) = µdt + ‡dW (t), (2.1)

where W (t) is a standard one-dimensional Brownian motion, and µ and ‡ > 0 are given constants. The constant µ denotes the expected log-return of the price of the stock. We can thus observe that the log-return is normally distributed with

dln(S(t)) ≥ N (µdt, ‡2dt). Thus ‡ denotes the variance rate of the log-return of the

risky asset S(t).

The risk-less asset B(t) available in the market solves the following ordinary

differential equation (ODE):

dB(t)

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2.2. THE FIRM CHAPTER 2. THE MATHEMATICAL MODEL

here, r represents the risk-free-rate and in accordance with portfolio theory we as-sume that µ > r. The assumption that µ > r is a result of the fact that risk-averse investors will require a risky asset to have a higher expected return than a risk-less asset. We finally assume that these two assets can be freely traded without any friction, such as transaction costs and that for the purpose of this research, the fi-nancial agent active in the market can always find a counterparty who is willing to buy or sell any amount of either one of the assets at the market price.

2.2 The firm

The firm active in the financial market M can invest in both S(t) and B(t). At time t, let X(t) be the investments in the bond, Y (t) the investments in equity, l(t) the rate of transfer from the bond holdings to the equity, m(t) the rate of opposite transfers and c(t) the rate of dividend. Since we assume no transaction costs when the firm refines its portfolio in terms of X(t) and Y (t), we are able to reduce l(t) and m(t) to a difference-process i.e. a(t) = l(t) ≠ m(t). We are able to make this assumption, since as discussed in the previous section, we assume that the firm can always find a counter party whom is willing to buy or sell any amount, of either asset in the market, at the currently revealing market price. Given the descriptions above, we can write X(t) and Y (t) to satisfy the following ODE and SDE:

dX(t) = rX(t)dt ≠ a(t)dt ≠ c(t)dt

dY(t) = Y (t)[µdt + ‡dW (t)] + a(t)dt. (2.3)

We will view the sum of the above two processes as the right-hand-side of the firms balance sheet and we will do so in order to rewrite the two processes, X(t) and Y (t) into one, book-value state-process of the firm which we will denote by Z(t). We can thus write:

Z(t) = X(t) + Y (t) = book-value state process of the firm at time t, u(t) = Y(t)

Z(t) = leverage-rate control of the firm at time t, c(t) = dividend control of the firm at time t,

(2.4)

as a characterization of the firm in terms of a state-process Z(t) and its available two controls (u(t), c(t)). The leverage-rate control of the firm u(t), again relies upon the assumption that the firm is able to find a counter party in the market, at any point in time, who is willing to buy or sell, any amount, of either X(t) or Y (t) without any friction such as transaction costs. This assumption requires that the portfolio of the firm is perfectly liquid and this is (especially in case the firm is in poor financial condition) not always true. In the next section, where we will discuss the authority agent, we will ensure a liquid portfolio of the firm by introducing " a firm backing authority". The authority will make liquid funds available to the firm whenever it is in poor financial condition and in need of liquidity. This assumption allows the firm to set a(t) and thus u(t), at any point in time, according to her preferences. We underline the importance of this assumption, the firm is able to (partly) externalize possible consequences of excessive risk taking or lack of managing liquidity, since it, in bust periods, can rely upon the provision of liquid funds by the authority

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CHAPTER 2. THE MATHEMATICAL MODEL 2.2. THE FIRM

agent. This is an oversimplification of events occurring during the 2008 financial crisis, we feel however that, it at least partly models the events during that pe-riod in which governments injected large amounts of liquidity into financial markets (M.R.Brunnermeier, Deciphering the Liquidity and Credit Crunch 2007 p.14). We can thus describe the firm via a book-value state process, which it can influence by using its two controls (u(t), c(t)):

dZ(t) = Z(t)[(r + u(t)(µ ≠ r))dt + u(t)‡dW(t)] ≠ c(t)dt,

Z(t) Ø 0. (2.5)

We have an additional restriction that Z(t) Ø 0. We assume that once Z(t) = 0, the firm will liquidate its assets and goes out of business (H. Mete Soner, Stochastic

Optimal Control In Finance p. 4). We underline that we will assume that the firm

will start the problem period to be investigated with an initial value Z(t0) = Z0

which at time t0 only consists of equity. Once the firm enters the problem period

it can use its leverage-rate control to lever its equity, but it is not allowed to raise additional equity by issuing new shares and it can only take on more debt. With this lever, it can of course increase its equity via natural growth. From the above, we can summarize the firm in terms of three variables, being a state-process Z(t) which represents the book-value of the firm at time t, and two control variables (u(t), c(t)), which denote the leverage-rate control and the dividend control with which the firm can influence the state-process at any point in time t. As we will discuss later, the authority will supply the firm with liquid funds in case it itself has liquidity problems. This assumption implies that the firm is able to continuously and limitlessly adjust the state of its book-value process via either the dividend - or the leverage rate control. The firm will apply those controls in order to maximize both its book-value growth as well as its dividend payments during the problem period. Before we proceed will we define the set of admissible controls A. We will assume that A := {(u(t), c(t)), u(t) Ø 0, c(t) Ø 0}. The assumption u(t) Ø 0 is a result of the reasoning that shorting equity in favor of a long position in a risk-less asset (u(t) < 0) is contradicting convex preference relations (J.Levin Choice

under uncertainty 2006, p.12). The firm would in such a case be able to increase

its expected return and at the same time decrease its risk on the expected return by increasing its leverage rate u(t). The dividend assumption that c(t) Ø 0 is solely based on the idea that we do not allow the equity holders to reinvest more funds into the firm. The firm will determine an optimal control policy (uú(t), cú(t)) œ A in

order to maximize its objective. Before we can elaborate on the specific formulation of the firms objective, we will first introduce a risk-premium.

Risk averse investors charge a risk-premium on more risky investments. Investors do this to compensate for the increase in uncertainty (A.Damodaran ,Equity Risk

Premiums (ERP): Determinants, Estimation and Implications ? The 2012 Edition

p.6). From Eq. 2.5, we observe that when the firm chooses its leverage-rate control to be a large positive value (highly levered), the book-value process will increase in uncertainty since the variance rate of the process is dependent on u(t). At this point, investors will charge a risk-premium to the firm whenever it wishes to take on more debt. To model this, we will use a proxy-cost of debt method, given by:

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2.2. THE FIRM CHAPTER 2. THE MATHEMATICAL MODEL

Here ” > 0. In other words, we believe that investors have an uniform assessment on the riskiness of lending funds to the firm, based upon the level of the leverage-rate of the firm u(t), for which they charge a risk-premium (rate) given by ﬁ(u(t)). The introduction of this risk premium yields the following cost of debt for the firm:

r(u(t)) = r + ”(u(t) ≠ 1)I(1,Œ)(u(t)). (2.7)

By introducing this risk-premium, we assume that investors will require the firm, in addition to the risk-free rate, to pay a risk premium whenever the firm is levered and which grows linearly as the firms leverage-rate increases. This premium is based upon the leverage-rate proxy, i.e., investors uniformly agree upon the fact that r(u(t)) is a valid proxy for the riskiness of lending funds to the firm. This increase in cost of debt aims to model the increase in (costly) default risk of the firm. The larger the leverage-rate of the firm, the greater will be the uncertainty of the book-value of the firm and thus its default-risk. While we, as will be discussed in the next section, already assume that the authority will provide liquid funds to the firm in case it needs them, we additionally assume that investors still feel the need to charge a (default) risk-premium since the they are not completely informed of "the firm backing authority" and thus expect to experience costs when retrieving their funds in case of a defaulting firm. Efficient market theory will argue that investors are able to hedge the risk associated with lending funds to the firm. As was experienced during the 2008 financial crisis however were commercial banks less willing to lend funds due to intransparant markets and an increase in uncertainty of available liquid funds and hence they required counter parties to pay risk-premiums (D.Gale, Liquidity Hoarding 2011,p.13). This assumption attempts to model the events occurring in the 2008 financial crisis in which commercial banks were hoarding liquidity and became less willing to loan out funds since they were increasingly uncertain of their own liquidity in the near future. For this reason they charged premiums to other parties who were in the market to borrow funds. This idea is also referred to as unequal borrowing/lending costs (R.A.Forsyth,Numerical Methods

for Controlled Hamilton-Jacobi-Bellman PDEs in Finance p7). We assume that the

"unequalness" between borrowing and lending costs differs between economic boom and bust periods and for this reason do we assume the problem period to be of finite time length and that the firm is operating in a bust period. Next, we need to make some additional assumptions on the preferences of the firm. We assume that the firms preferences are captured by the following utility-function U : [0, Œ] æ R+_,

given by:

U(c(t)) = c(t)“. (2.8)

Here 0 < “ < 1. The above utility function is commonly used in utility theory and satisfies both the property of non-satiation, which states that utility increases with book-value/dividend, i.e., that more book-value is preferred to less book-value, and that the firm is never satiated - it never has so much book-value that getting more would not be at least a little bit desirable, as well as the risk aversion property, which states that the utility function is concave or, in other words, that the marginal utility of book-value/dividend decreases as book-value/dividend increases.

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CHAPTER 2. THE MATHEMATICAL MODEL2.3. THE AUTHORITY AGENT

Given the above descriptions we can now formulate an objective which the firm will attempt to maximize. The objective functional is the expected discounted utility derived from dividend plus the utility derived from the remaining book-value of the firm at the end of the problem period:

Jf = E 5 e≠—·U(Z(·)) + ⁄ · t0 e ≠—t_U_(c(t))dt6_, · = inf{t0 Æ t < T, Z(t) = 0} · T, s.t.

dZ(t) = Z(t)[(r(u(t)) + u(t)(µ ≠ r(u(t))))dt + u(t)‡dW(t)] ≠ c(t)dt, Z(t0) = Z0.

(2.9)

Here — > 0 denotes discount-rate and Z0 the initial-value of the book-value of the

firm, at the beginning of the problem period. The stopping-time · captures the fact that the firm will liquidate its balance sheet whenever it reaches the point Z(t) = 0 and if this state is never reached, we assume the firm to be active for a finite amount of time denoted by T . In other words, is the firm attempting to maximize the growth on equity for its shareholders, which is captures in the terminal book-value of the firm and while attempting to establish as much growth as possible, it does also attempt to pay as much dividend as possible and it derives (discounted) utility from both items, as shown below:

Jf = E[ _e≠—·_U_(Z(·))

¸ ˚˙ ˝

derived utility from book-value (equity) growth

+ ⁄ ·

t0

e≠—tU(c(t))dt

¸ ˚˙ ˝

utility derived from dividend

].

(2.10)

Formally, it will determine an optimal control policy (uú(t), cú(t)) œ A in order to

maximize its objective functional. We introduce the value-function V (·, ·) as the maximized objective functional V (t, Z(t)) = sup_{(u(t),c(t))œA}Jf. In the next chapter

we will develop the procedure used by the firm to determine this optimal control-policy (uú(t), cú(t)).

2.3 The Authority Agent

The authority attempts to maximize the objective, which is formulated by society as a whole. While society benefits from efficient financial markets, which will help allocate scarce resources efficiently. is society, in contrast to the firm, also influenced by negative externalities resulting from excessive risk taking by the firm. In the previous section we disccussed the firm and stated that the firm consist of a perfectly liquid portfolio. This assumption is a simplification and not realistic. Typically when the firm has a book-value 0 Æ Z(t) Æ 1, we will assume that its portfolio is

not perfectly liquid and at this point the authority has to intervene by supplying the

firm with liquid funds to ensure the firm to have a perfectly liquid portfolio. As was shown during the 2008 financial crisis, financial institutes are relatively illiquid and cannot immediately satisfy their financial obligations towards its creditors. This phenomena has a cascading effect since its creditors in turn are faced with a decline in its liquidity and will be forced to (fire-sale) its asets in order to satisfy their

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2.3. THE AUTHORITY AGENTCHAPTER 2. THE MATHEMATICAL MODEL

financial obligations. Ultimately this will lead to eroded asset values and thus to a cost for society. The objective of the authority is to setup regulatory measures

that will allow for efficient financial markets but will also take into account that excessive risk taking of financial institutes will result in costly interventions for the authority by supplying liquid funds to firms which are in poor financial condition. In

our model, we define "poor financial condition" to be a book-value 0 Æ Z(t) Æ 1. In such a state we assume that the authority feels obliged to supply the firm with liquid funds, because that it experiences the firm to be too-big-to-fail and it considers a bankruptcy of the firm to be socially irresponsible since it will have an cascading, negative effect on employment and efficiency of financial markets.

Whenever the authority has to intervene by supplying the firm with liquid funds, we will assume that it can do so in accordance to their terms and conditions. Once the firm reaches a state 0 Æ Z(t) Æ 1, the authority will supply the firm with liquid funds and during such time, the authority can define leverage-rate requirements and

dividend requirements with which the firm has to comply in order for them to receive

those liquid funds. In other words, the authority requires the firm to meet a state

requirement given by Z(t) œ R where R = [1, Œ). Whenever the firm does not

comply with this state-requirement, the authority intervenes by supplying the firm with liquid funds and during this period it has instruments with which it can restrict the control policies of the firm.

(i) Dividend restriction: The authority can limit or even temporary terminate the possible dividend control of the firm. Whenever the firm complies to the state-process requirements determined by the authority, its available dividend controls are given by the set C := {c(t), c(t) Ø 0}. Once the firm no-longer satisfies those state-process requirements (Z(t) /œ R), the regulator is granted authority by law to intervene. The dividend-restriction can be applied in such a way that it further limits the set of admissible consumption control such that Cr := {c(t), c(t) = ﬂ}. Introducing ﬂ will limit or even temporary terminate

the dividend control of the firm until a point at which the firm again complies with the state-process requirements. In order words, will the authority "take over" the dividend policy of the firm and impose a strategy with which the firm has to comply. The instrument that the authority can apply is thus the level of fl. The authority is free to decide on the level of fl but in contrast to the firm does it has to decide on the optimal level of fl at the start of the problem period and is not allowed to alter this level during the problem period. (ii) Leverage restriction:The authority, as a second action, can force the firm

to adjust the setup of its portfolio. This control will lead to a more stable book-value process of the firm. Similar to the dividend restriction above, is the regulator granted authority to intervene in the leverage control of the firm and alter the set of admissible leverage controls U := {u(t), 0 Æ u(t) < Œ} to Ur := {u(t), u(t) = ¯u}. Again, the authority will enforce a leverage-rate

policy upon the firm and the firm is no-longer able to control its leverage-rate authonnomously.

(iii) Dividend-leverage restriction:The authority can apply both actions to-gether, so limiting dividend and forcing a shift in the portfolio setup of the firm.

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CHAPTER 2. THE MATHEMATICAL MODEL2.3. THE AUTHORITY AGENT

The authority, in contrast to the firm, is limited in the timing of its instruments (¯u, ﬂ). It can only set the levels of the instruments at the beginning of the problem period. Below we outline the interaction between the authority and the firm and at which points in time they are able to declare their optimal strategy.

t= t0 Start At the start of the problem period the authority will declare the

regula-tory requirements given by Z(t) œ R, which is given by R := [1, Œ). Next to the requirements, will the authority also declare the control restrictions (¯u, ﬂ) whenever the firm does not comply to the requirements during the problem period.

t œ (t0, ·) During Once the period has started, the authority can no longer alter its

instruments ﬂ, ¯u.The firm is the only remaining agent who can (costless) alter its control strategy, instantaneously during the problem period. We assume that the authority has full insight into the controls of the firm and its book-value process at any time t.

t= · · T Termination At this point the problem period terminates. This can either

happen at the time T = 1 or at a point t = · which is defined as the infenum of either T or the point in time which the firm reaches a book-value equal to zero. At this point neither agent can apply any more controls and can only determine the value of their objective Jf and Ja respectively.

The objective of the authority is to develop regulatory instruments that both allow for efficient financial markets and as well as prevent the firm from excessive risk taking since this will result in costly interventions. The cost of supplying the firm with liquid funds whenever the firm does not meet the regulatory state-requirement

Z(t) /œ R is captured in the following cost function:

Ÿ(Z(t)) =„[(1 ≠ u(t))Z(t)]2I_{[0,1)◊(1,Œ)}(Z(t), u(t)). (2.11)

Here „ > 0. The authority is assumed to be required to provide the firm with a fraction of its outstanding debt in liquid funds (debt of firm = (1 ≠ u(t))Z(t)). The funds that the authority has to make available to the firm once it no-longer meets the requirement Z(t) œ R can not be used in other parts of the economy, which the authority considers to be costly since it dampens economic growth and development. With this cost function in mind, the authority can formulate the objective function it will attempt to maximize as follows:

Ja = E 5 e≠—·U(Z(·)) + ⁄ · t0 e≠—tU(cú(t)) ≠ Ÿ(Z(t))]dt 6 = Júf _≠⁄ · t0 e≠—tŸ(Z(t))dt, · = inf{t0 Æ t < T, Z(t) = 0} · T, s.t. dZ(t) = Z(t)[(r(uú(t)) + uú(t)(µ ≠ r(uú(t))))dt + uú(t)‡dW (t)] ≠ cú(t)dt, Z(t0) = Z0, (uú_{(t), c}ú_{(t)) = argsup} (u(t),c(t))œAr Jf. (2.12)

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2.3. THE AUTHORITY AGENTCHAPTER 2. THE MATHEMATICAL MODEL

Here — > 0 is the discount rate. In other words the authority can only influence the set ¯u and ﬂ in order to limit the set of admissible controls of the firm A and convert it to Ar. While doing so it will assume that the firm will attempt to maximize

the firms objective functional Jf, given it is limited to A

r. Here we have denoted

the maximized objective of the firm, given Ar, by Júf. The authority attempts to

decide on ¯u and ﬂ in order to allow for efficient financial markets and while doing so minimize the costs it will encounter in case it has to supply the firm with liquidity.

Ja= E[e≠—·U(Z(·)) +

⁄ · t0 e

≠—t_U_(cú_(t))dt

¸ ˚˙ ˝

efficient financial market

≠

⁄ · t0 e

≠—t_Ÿ_(Z(t))dt

¸ ˚˙ ˝

costly market interventions

]

(2.13)

In words: the authority will attempt to design instruments in which it can set regulatory requirements such that it will both allow for a efficient financial market as well as prevent costly interventions whenever the firm no-longer can comply with set state-requirements. It can do this by restricting the firms set of admissible controls. In the next chapter we will discuss how the authority will attempt to maximize its objective.

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Chapter 3 Methodology

In the previous chapter we setup a model in which two agents, the firm and the authority can interact and by doing so they each attempt to maximize their own objective. In this chapter we will develop procedures that the agents will use respec-tively, in order to find their respective optimal control policy / optimal instruments. First we will start discussing the optimization problem of the firm. The firm will solve its optimization problem by solving the Hamilton Jacobi Bellman equation

(HJB) . After briefly discussing the theory behind the HJB equation and the

nu-merical method which the firm will apply to solve the PDE, we will proceed by discussing the Newton Raphson method, which the authority will apply to find its optimal instruments.

3.1 The Hamilton Jacobi Bellman equation.

In the previous chapter we formulated both the objective functional of the firm Jf

and its state-process Z(t). In this chapter we will use a generalized notation for notational convenience. First let us define an objective functional given by :

Jf = E 5 ›(·, Z(·)) + ⁄ · t0 f(t, Z(t), ‹(t))dt 6 , · = inf{t0 Æ t < T, Z(t) = 0} · T, s.t. dZ(t) = –(t, Z(t), ‹(t))dt + Â(t, Z(t), ‹(t))dW (t), Z(t0) = Z0. (3.1)

Here, we denote ‹(t) to be a generalized (vector) of control policies available to the firm. Again we like to underline that we have adjusted the notation of the above problem purely for notational convenience and that the underlying problem is identical to the one as described in Eq.2.9. With the above dynamic optimization problem, can we define a value-function V (·, ·). The value-function is defined as the maximized objective functional with respect to the admissible control policies

‹(t) œ A, given by:

V(t, Z(t)) = sup

‹(t)œA

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3.1. THE HAMILTON JACOBI BELLMAN EQUATION.CHAPTER 3. METHODOLOGY

For an initial state (t0, z0) œ T ◊ (0, Œ) we say that ‹ú(t) œ A is an optimal control

if V (t, Z(t)) = Jf(t, Z(t), ‹ú(t)). Given the notion of the value-function, is the firm

attempting to derive a PDE for V (·, ·) which allows the firm to determine its optimal control law at any point in time and state space.. In order for the firm to be able to do so, we assume the following:

(i) There exists an optimal control law ‹ú(t).

(ii) The value function V (·, ·) is regular in the sense that V (·, ·) œ C1,2_.

(iii) A number of limiting procedures as shown in the Appendix A.4 can be justified. The HJB is a partial differential equation (PDE) which most often has the

value-function of a dynamic optimization problem, as its unique solution (L. Smears, Hamilton-Jacobi-Bellman Equations Analysis and Numerical Analysis). The HJB

that describes the dynamic optimization problem of the firm is given by (see Ap-pendix for derivation):

≠ˆV_ˆt(t, Z(t)) = sup ‹(t)œA [f(t, Z(t), ‹(t)) +–(t, Z(t), ‹(t))ˆV ˆZ(t, Z(t)) +Â(t, Z(t), ‹(t))ˆ2V ˆZ2(t, Z(t)) D , V(T, z) = ›(T, z), V(t, 0) = 0. (3.3)

By introducing a differential operator of second order: L‹(t)V = –(t, Z(t), ‹(t)) ˆ ˆZV(t, Z(t)) + Â(t, Z(t), ‹(t)) ˆ2 ˆZ2V(t, Z(t)), (3.4) we obtain: ≠ˆV ˆtt(t, Z(t)) = sup‹(t)œA Ë f(t, Z(t), ‹(t)) + L‹(t)VÈ, V(T, z) = ›(T, z), V(t, 0) = 0. (3.5)

Before proceeding we make the following assumptions:

Assumption I: (Properties of the HJB.) We make the assumption that the

coefficients –, Â, f are continuous functions of (t, Z(t), ‹(t)), with Â Ø 0 and that –, Â and f are bounded on 0 Æ Z(t) Æ zf (where zf is the boundary value of

the Z(t)-domain). Since we restrict ourselves to a finite computational domain

0 Æ Z(t) Æ zf, we avoid difficulties associated with coefficients that grow with Z(t)

as Z(t) æ Œ. We also assume that the set of admissible controls ‹(t) œ A are compact (i.e. a closed, bounded set).

During our analysis will we ensure bounded values of –, Â and f by restricting the control variables (u(t), c(t)) to have an upper-bound, denoted by umax = 1000, cmax =

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CHAPTER 3. METHODOLOGY3.1. THE HAMILTON JACOBI BELLMAN EQUATION.

1000. The upper-bounds are large relative to the computational domain given by (t, Z(t)) œ [0, 1] ◊ [0, 1] and chosen such that they are unlikely to be reached during our computations. Given the above assumptions on the HJB equation, we can con-sider the RHS of the above HJB equation as a static optimization problem foreach (t, Z(t)) œ T ◊ [0, zf], can we find ‹ú(t) that solves the first order condition (FOC)

d d‹(t)

Ë

f(t, Z(t), ‹(t)) + L‹(t)_VÈ= 0. _(3.6)

which enables us to formulate ‹ú(t) in terms of derivatives w.r.t. Z(t) of V (·, ·).

Finally can we define the HJB equation purely in terms that are only dependent on

t and Z(t), given by:

≠ˆV ˆt(t, Z(t)) = Ë f(t, Z(t), ‹ú(t)) + L‹ú(t)VÈ, V(T, z) = ›(T, z), V(t, 0) = 0. (3.7)

Given enough regularity, the value-function satisfies the above PDE plus bound-ary conditions (Tomas Bjork,Stochastic optimal control with appllications in finance 2010, p33). While in few cases there can be found an analytical solution to the value-function, that solves the HJB, is it in our case more convenient to determine

V(·, ·) numerically. In the next section we will discuss the method we will use to do

so.

3.1.1 Numerical Solution Of The Hamilton Jacobi Bellman

Equation

Our goal is to approximate a solution to the above described HJB, i.e., to find a function (or some discrete approximation to this function) which satisfies the above given relationship between various of its derivatives on some given region of space and time, along with some boundary conditions along the edges of this domain. In general this is a difficult problem and only rarely can an analytic formula be found for the solution. A finite difference method proceeds by replacing the derivatives in the partial differential equations by finite difference approximations. This gives a large algebraic system of equations to be solved. For notational convenience, we will (temporarily) denote V (t, Z(t)) by V (t, z) or just V . In this section we will use a simplified, general form of the HJB (again for notational convenience), given by:

≠ˆV ˆt = –(t, z) ˆV ˆz + Â(t, z) ˆ2V ˆz2 + f(t, z), V(T, z) = ›(T, z), V(t, 0) = 0. (3.8)

Here we simplified –(t, z) := –(t, Z(t), ‹ú(t)),Â(t, z) := Â(t, Z(t), ‹(_t)) and f(t, z) :=

f(t, Z(t), ‹ú(t)) (see previous section). In order to solve this PDE numerically, we

start by discretizing the time-state space into Z = (z0, z0 + z, · · · , z0 + k z)T

and t = (t0, t0 + t, · · · , t0 + N t)T. Here we let z0 = 0, z0 + k z = zf = 1

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that during the discussion of our model the threshold Z(t) = 1 has a significant meaning since it separates the Z(t) domain into two region’s Z(t) Æ 1 and Z(t) > 1 which are associated to the authority as a costly - and a costless region. The above normalization of the Z(t)-domain has no impact on this threshold, and we simply will consider the threshold value of Z(t) to be at 1

10. Using the above discretization,

we can write the derivatives of V w.r.t. z (approximately) as a discrete central

difference, given by:

ˆV ˆz ¥ Vj+1≠ Vj≠1 zj+1≠ zj≠1 = Vj+1≠ Vj≠1 2 z (3.9) and ˆ2V ˆz2 ¥ Vj+1≠ 2Vj+ Vj≠1 z2 (3.10)

Here zj+1 = zj + z. We chosen the first order, central (also known as centered)

difference since it is a second order accurate approximation - the error is proportional to z2 _{and hence is much smaller than the error in a forward/backward first order}

approximation when z is small. We can now rewrite Eq 3.10 as:

≠ˆV ˆt = –j(t) 2 z (Vj+1≠ Vj≠1) + Âj(t) z2 (Vj+1≠ 2Vj + Vj≠1) + fj(t) (3.11)

Finally we discretize V w.r.t. t and find:

≠V n+1 j ≠ Vjn t = –n_j+1 2 z 1 V_jn₊₁+1_{≠ V}_jn_≠1+12+Â n+1 j z2 1 V_jn₊₁+1_{≠ 2V}_jn+1+ V_jn_≠1+12+ f_jn+1 (3.12) Letting Vn j = V (n t, j z), ⁄nj+1 = –n_j+1 t 2 z ,fjn+1 = f((n + 1) t, j z) and ÷jn+1 = Â_jn+1 t z2 , we finally find: V_jn= V_jn+1+ ⁄_jn₊₁+11V_jn₊₁+1≠ Vn+1 j≠1 2 + ÷n+1 j 1 V_jn₊₁+1≠ 2Vn+1 j + Vjn+1 2 + tfn+1 j = (÷n+1 j ≠ ⁄nj+1)Vjn≠1+1+ (1 ≠ 2÷nj+1)Vjn+1+ (÷nj+1+ ⁄jn+1)Vjn+1+1+ tfjn, = pn+1 j≠1Vjn≠1+1+ pnj+1Vjn+1+ pnj≠1+1Vjn+1+1+ tfjn. (3.13) We note that qk=≠1,0,1pnj+k+1 = 1 for n + 1 = 1, · · · , N ≠ 1 which look somewhat

like probabilities. As will be discussed later in this section will we apply an upwind method which will ensure that pn+1

j+k Ø 0 for k = ≠1, 0, 1 and again for n + 1 =

1, · · · , N ≠ 1. This is also known as the Markov chain form or also referred to as the positive coefficients method. The letter has an interesting feature, namely that it can be shown to be numerically, unconditionally stable (P.A.Forsyth,Numerical

Methods for Controlled Hamilton-Jacobi-Bellman PDEs in Finance p.30). Below

the computational stencil is shown which depicts the iterative nature of the used, implicit method. The backward iterative method is particularly useful since the type of problem faced by the firm has a terminal boundary condition which is given by V (T, z) = ›(T, z) = e≠—T_z“. In matrix notation, we can rewrite Eq. 3.13 into:

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V_jn+1

j, n

V_jn_≠1+1 V_jn₊₁+1

Figure 3.1: Computational stencil of the upwind implicit numerical method.

Vn _{= U}n+1_Vn+1 ≠ tfn+1, V_jT = ›_jT = e≠—Tz_j“, j = 0, · · · , k, V₀n = 0, n = 0, · · · , N (3.14) Here T = N t, Vn _{:= (V (n t, z} 1), · · · , V (n t, zk))T, fn:= (f(n t, z0), · · · , f(n t, zk))T

and Un+1 is given by:

Q c c c c c c a (1 ≠ 2÷n+1 1 ) ÷1n+1+ ⁄n1+1 · · · 0 (÷n+1 k≠1 ≠ ⁄nk≠1+1) (1 ≠ 2÷kn≠1+1) ÷nk≠1+1+ ⁄nk+1+1 · · · · · · · · · · (÷kn≠1+1≠ ⁄nk≠1+1) (1 ≠ 2÷nk≠1+1) ÷kn≠1+1+ ⁄nk+1+1 0 · · · ≠⁄n+1 k 1 R d d d d d d b (3.15)

In the above matrix we note that the 0’th row is "missing", as a result of a boundary condition Vn

0 = 0 , n = 1, · · · , N, and the k’th (final) row, slightly differ from

the others. The k’th row denotes the additional assumption that we have made that ˆ2V

ˆz2

-j=k = 0. This assumption results from the fact that we are dealing with

a bounded state-space and we are forced to make some additional assumption on the behavior of V on the edges of its domain. This assumption implies that ˆV

ˆz is

constant at the edge of z. Having discussed the boundary conditions, discretization scheme and numerical method, which in literature is referred to as backward time

implicit scheme (BTIS) will we now briefly discuss upon the stability, consistency

and convergence of the above described approach.

3.1.2 Stability , consistency and convergence of the

back-ward time iterative, implicit scheme (BTIS)

To analyze the numerical stability conditions, first we begin by noting that Ë(U) < 1 … limkæŒ[Uk]i,j æ 0 Here Ë(U) is the spectral radius of U, given by Ë(U) :=

maxi|⁄i|. Here, ⁄i represent the eigenvalues of U for i = 1, · · · , k. To ensure

numerical stability we will define the (t, z) grid in such a way that |⁄i| < 1 ,i =

1, · · · , k (R.U. Seydel,Tools for computational finance 2006 p244). We ensure that this condition is met by applying the Gershgorin Circle Theorem (GCT). The GCT defines the complex disc Di which contains the i´th eigenvalue of a matrix U in terms

of the ith row elements of U. To find the sufficient condition we need to show that the matrix U is (strictly) diagonal dominant (S.Brakken-Thal ,Gershgorin’s Theorem for

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Ui,i is the i’th diagonal entry of the matrix U. First, consider the first row of the

matrix U, we find: --⁄j ≠ (1 ≠ 2÷jn+1) --Æ |÷jn+1+ ⁄nj+1| ∆ |⁄j| ≠ --1 ≠ 2÷nj+1 --Æ |÷jn+1| + |⁄nj+1| triangle inequality ∆ |⁄j| Æ |÷jn+1| + |⁄nj+1| + --1 ≠ 2÷n0+1 --Æ 1 stability condition ∆|⁄nj+1| Æ |÷jn+1|, ∆|–nj+1| z Æ 2|Âjn+1|. (3.16)

Here we have used the reverse triangle inequality which state that for x, y œ R, |x ≠ y| Ø |x| ≠ |y| holds. Second, let us consider the 2 · · · , k ≠ 1 rows and find:

--⁄j ≠ (1 ≠ 2÷nj+1) --Æ |÷jn+1+ ⁄nj+1| + |÷nj+1≠ ⁄nj+1| ∆ |⁄j| ≠ --1 ≠ 2÷jn+1 --Æ 2|÷n+1 j | triangle inequality ∆ |⁄j| Æ 2|÷jn+1| + --1 ≠ 2÷0n+1 --Æ 1 stability condition ∆0 Æ 0. (3.17)

This condition is always satisfied and hence for these entries the eigenvalues fall within the complex unit-circle. Finally the final row yields:

|⁄j≠ 1| Æ |⁄nk+1| ∆ |⁄j| ≠ |1| Æ |⁄nk+1| triangle inequality ∆ |⁄j| Æ |⁄nk+1| + |1| Æ 1 stability condition ∆|⁄n+1 j | Æ 0. (3.18)

This condition cannot always be satisfied and more importantly are ⁄n+1

j and ÷nj+1

(for all j, n + 1 in our grid) dependent on the optimal control policy ‹ún+1

j and thus

we cannot make any generalized statements on the stability of the implicit method in the current situation. Using the upwind method, will we alternate between central

differencing, forward differencing and backward differencing and this alternating

scheme will ensure a Markovian form as shown in Eq.3.13 with pn+1

j+k Ø 0, k = ≠1, 0, 1

and qk=≠1,0,1pnj+k+1 = 1. This upwind, psitive coefficients scheme can be shown to

to be unconditionally stable ( P.Forsyth, Numerical Methods for Hamilton Jacobi

Bellman Equations in Finance p.25). We note that for the k’th row of the matrix U,

if we switch the central difference to a backward difference, we obtain the following:

--⁄j ≠ (1 + ⁄nk+1 --Æ |⁄nk+1| ∆ |⁄j| ≠ --1 + ⁄nk+1 --Æ |⁄nk+1| triangle inequality ∆ |⁄j| Æ |⁄nj+1| + |1| + |⁄nk+1| Æ 1 stability condition ∆0 Æ 0. (3.19)

Switching from a centered difference to a backward difference scheme again ensures the k’th eigenvalue of our matrix U to be contained within the complex unit-circle and hence complies with the stability condition. Having discussed stability we can now review consistency and convergence of our method.

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We say that a method is consistent with the partial differential equation and boundary conditions if ||‘ z_{|| æ 0 as z æ 0. Here ‘} z is the truncation error of our

method and z is a grid step-size in the z-direction (R.J. Le Vegue, Finite Difference

Methods for Differential Equations p.16). This simply says that we have a sensible

discretization of the problem. We observe that our method has (depending on the "upwind state", centered differences, backward - or forward differences), truncation errors which are O( z)1 _{or O( z}2_{) the method satisfies the consistency condition}

as stated above.

For convergence of our method we note that consistency+stability ∆

conver-gence. We would like to refer the the more interested reader on this topic to more

detailed literature such as R.Forsyth, Numerical Methods for Controlled

Hamilton-Jacobi-Bellman PDEs in Finance.

3.1.3 The procedure for the firm

In the above section we have described the assumptions, boundary conditions and discretization schemes we will assume the firm to use in order to solve its optimiza-tion problem. To summarize the above discussion in terms of the firm, we can now simply substitute back the parameters, functions and boundary conditions of the firm to obtain the following discretized HJB and "solve scheme":

Data: Market parameters (Table 4.1). Result: V (t, z), (t, z) œ [0, 1] ◊ [0, 1] initialization; VT j = ›jT = e≠—Tz “ j, j = 0, · · · , k, Vn 0 = 0, n = 0, · · · , N, for n + 1 Ω N to 1 do for j Ω 1 to k do central differencing i) uún+1 j = g1(n + 1, j, Vjn≠1+1, Vjn+1, Vjn+1+1), cún+1_j = g2(n + 1, j, Vjn≠1+1, Vjn+1, Vjn+1+1), –n_j+1 = uún+1_j z_jn+1(µ ≠ r(uún+1_j )) + r(uún+1_j )zn_j+1_{≠ c}ún+1_j , Ânj+1 = 1₂(ujún+1zjn+1‡)2. if pn+1 j≠1, pnj+1, pnj+1+1 Æ 0 or q k=≠1,0,1pnj+k+1 ”= 1 then

forward differencing, recalculate step i)

if pn+1

j≠1, pnj+1, pjn+1+1 Æ 0 orqk=≠1,0,1pnj+k+1 ”= 1 then

backward differencing, recalculate step i)

end end ii) Vn j = pnj≠1+1Vjn≠1+1+ pnj+1Vjn+1+ pnj+1+1Vjn+1+1+ tfjn. end end

Algorithm 1: Procedure used by the firm to solve the HJB equation.

1_{Big O notation: A function f : R æ R is big-O(g(x), if for some n, M œ R}+ _{and g : R æ R,}

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3.2. NEWTON RAPHSON’S GRID SEARCH CHAPTER 3. METHODOLOGY

Here we have used uún+1

j and cún+1j to denote the discretized optimal control policies

(uú(t), cú(t)) œ A. The functions g₁_{, g}₂ will be further discussed in the next chapter

and for now are purely for notational purposes. The firm will backward iteratively solve the above set of equations (starting from n + 1 = N to n + 1 = 1) and by doing so it can determine the value-function and from it, its optimal control policies (u(t), c(t)) œ A in its discretized form.

3.2 Newton Raphson’s Grid Search

The authority has to solve a slightly different problem since its objective is not a functional (at least not of the controls available to the authority). The authority can optimize its objective by determining the optimal values for (ﬂ, ¯u). To solve this optimization problem, we will use the Newton Raphson method. If we denote

’k= (ﬂk,¯uk), we can find the optimal value for ’ by iteratively solving:

Data: Market parameters (Table 4.1).

Result: The critical point the maximized objective functional of the

authority. initialization; while ‘ > 1e ≠ 10 do ’k+1 = ’k≠ D’Ja(’k|uú(t), cú(t))≠1Ja(÷k|uú(t), cú(t)), ‘_{Ω ||’}k+1≠ ’k||2. end

Algorithm 2: Newton-Raphson method of the firm.

Here D’Ja(’k|uú(t), cú(t))≠1 is the inverse of the Jacobian matrix of Ja. We note

that we have conditioned on the application of an optimal control policy by the firm. The authority attempts to find a maximizing set of instruments, given that the firm will maximize its objective, given those instruments. During our analysis we will consider three optimization steps for the authority agent. The authority can apply either one of its two instruments ﬂ or ¯u or it can choose to apply both together. We will apply the same method for all three alternatives and determine the (numerical) derivative (Jacobian) and iterate until we reach a tolerance of 1e ≠ 10.

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Chapter 4 Research

4.1 Introduction

In the previous two chapters we have discussed the model setup and how we assume each agent will solve their respective optimization problem. In the following chapter we will analyze the interacting behavior and review the resulting equilibria in our model. Since the authority first has to solve the problem of the firm in order to determine maximizing instruments for its own optimization problem, will we first focus our attention to the problem faced by the firm. For each problem setup we aim to answer the following two questions:

(ii) Necessary condition:Can conditions be stated distinguishing a maximizer from any other element in A (necessary optimality conditions).

(iii) Sufficiency condition: Do the necessary conditions suffice to identify a op-timal control strategy? (sufficient opop-timality conditions)

During the research, we will assess our model in accordance with the above two questions. We will start our discussion with outlining the base model and from there on we will explore by adjusting the model parameters and introducing the authority agent in order to evaluate the impact the authority has on the optimal control policy of the firm. We fix the market parameters in Table 4:

4.2 The Optimization Problem Of The Firm

First we will outline the base model. This can be viewed as a market in which only the firm is an actor. Later, we will introduce restricting assumptions and introduce the authority which will limit the firms set of admissible controls.

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4.2. THE OPTIMIZATION PROBLEM OF THE FIRMCHAPTER 4. RESEARCH

Table 4.1: Market parameters as will be used during the research.

Parameter Value

µdrift rate of the book-value proces ₁₀1

r risk-free rate ₁₀₀1

‡ variance rate of the book-value proces 1₂ “ utility parameter/risk-attitude 1₂

” risk premium factor ₁₀₀₀1

— discount rate ₁₀₀1

„ intervention cost factor 1₂

r(u(t)) unequal cost of borrowing-lending ₁₀₀2 + ₁₀₀₀1 (u(t) ≠ 1)I(1,Œ)(u(t))

R required conditions Z(t) œ (₁₀1 ,1)

C Admissible consumption c(t) œ (0, cmax), cmax = 1000

U Admissible leverage-rates u(t) œ (0, umax), umax = 1000

A Admissible leverage-rates (u(t), c(t)) œ U ◊ C

U(c(t)) utility function c(t)14

Ÿ(Z(t)) cost of intervention 1₂(1 ≠ u(t)Z(t))2I_Rc_◊(1,Œ)(Z(t), u(t))

4.2.1 The base SOCP of the firm

Let us review the optimization problem of the firm as we proposed in chapter two.

Jf = E 5 e≠—·U(Z(·)) + ⁄ · t0 e ≠—t_U_(c(t))dt6_, · = inf{t0 Æ t < T, Z(t) = 0} · T, s.t.

dZ(t) = Z(t)[(r(u(t)) + u(t)(µ ≠ r(u(t))))dt + u(t)‡dW(t)] ≠ c(t)dt, Z(t0) = Z0.

(4.1)

We assume that the firm will solve its objective functional by solving the HJB equation as was discussed in the previous chapter. The HJB equation is given by:

≠ˆV ˆt t(t, Z(t)) =(u(t),c(t))œAsup Ë e≠—tc(t)“+ L(u(t),c(t))VÈ, V(T, z) = e≠—Tz“_, V(t, 0) = 0, ˆ2V ˆZ2(t, zf) = 0. (4.2)

Here we require V (t, Z(t)) œ C1,2 _{(regularity condition) i.e. V (t, Z(t)) is (at least)}

once differentiable with respect to t and (at least) twice differentiable with respect to Z(t). The (Dirichlet) boundary conditions are results of the formulation of the objective functional and the von Neuman boundary condition from the finite horizon of the problem, as discussed in the previous chapter. The terminal boundary condi-tion is clear since at the end of the problem period, the firm will have a discounted

value of e≠—·_U(Z(·)). Clearly it can no longer consume since the period has ended

and hence the only remainder utility will be experienced from the remaining book-value Z(·). The boundary condition V (t, 0) = 0 ’t œ T is another result from the

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CHAPTER 4. RESEARCH4.2. THE OPTIMIZATION PROBLEM OF THE FIRM

objective since once the firm reaches a state in which its book-value is equal to zero, the firm is no longer able to engage in business and hence it can no longer experience utility. Before starting our investigation on the behavior of the firm, do we like to underline that the firm is attempting to design an optimal control policy, given the regulatory requirements and conditions imposed by the authority. We thus consider an augmented set of controls for the firm, given by (ur(t), cr(t)) which represent the

restricted controls of the firm. These restricted controls can be rewritten to be: ur(t) = u(t)IR(Z(t)) + ¯uIRc(Z(t)), ,

cr(t) = c(t)IR(Z(t)) + ﬂIRc(Z(t)). (4.3)

The above notation nicely depicts the role of the authority as it imposes the re-stricted control policy (instruments) (¯u, ﬂ) whenever the firm does not comply with the requirements imposed by the authority, denoted by R. Given the regulatory restrictions imposed by the authority, can the firm only maximize its objective func-tional funcfunc-tional w.r.t. u(t) and c(t). As we discussed in Chapter 3.1. will the firm first solve the optimality (necessary) condition given by Eq. 3.6. If we, for nota-tional convenience, let ˆV

ˆt = ˆV ˆt(t, Z(t)), ˆV ˆz = ˆV ˆZ(t, Z(t)), ˆ2V ˆz2 = ˆ2V ˆZ2(t, Z(t)) and

use (t, z, u, ur, c, cr) to denote (t, Z(t), u(t), ur(t), c(t), cr(t)) we can write Eq. 3.5 as:

≠ˆV_ˆt = sup (u,c)œA C e≠—tc“_r + zur(µ ≠ r(ur)) ˆV ˆz + (rz ≠ cr) ˆV ˆz + 1 2z2u2r‡2 ˆ2V ˆz2 D . (4.4)

The above maximization can be viewed as a sequential, static optimization problem and given regularity of V (·, ·) and Assumption I, this objective can be evaluated for all (t, z) in the (discretized) domain. In order to do so, we solve the FOC of the RHS of Eq 4.4 w.r.t. c and u, we furthermore note that dur

du = IR(z) and dcr dc = IR(z) which yields: ˆ ˆc C e≠—tc“_r + zur(µ ≠ r(ur)) ˆV ˆz + (rz ≠ cr) ˆV ˆz + 1 2z2u2r‡2 ˆ2V ˆz2 D = 0, ∆e≠—t“c“_r≠1dcr dc ≠ ˆV ˆz dcr dc = 0, ∆cúIR(z) = C 1 “e —tˆV ˆz D 1 “≠1 IR(z). (4.5)

Here we have used that crIR(z) = (cIR(z) + ﬂIRÊ(z))I_R(z) = cI_R(z). Similarly, the

leverage rate control of the firm maximizes Eq. 4.4. with respect to u:

ˆ ˆu C e≠—tc“_r + zur(µ ≠ r(ur)) ˆV ˆz + (rz ≠ cr) ˆV ˆz + 1 2z2u2r‡2 ˆ2V ˆz2 D = 0, ∆z(µ ≠ r(ur)) ˆV ˆz dur du ≠ ru(ur)urz ˆV ˆz dur du + z 2_u r‡2 ˆ2V ˆz2 dur du = 0, ∆z(µ ≠ r(ur)) ˆV ˆz IR(z) ≠ ru(ur)urz ˆV ˆz IR(z) + z 2_u r‡2 ˆ2V ˆz2IR(z) = 0. (4.6)

Here ru(ur) = ”I(1,Œ)(ur) and r(ur) = r + ”(ur≠ 1)I(1,Œ)(ur) and thus find:

uúI_R(z) = ≠(µ ≠ r(ur))

ˆV ˆzIR(z)

z‡2 ˆ_ˆz2V2IR(z) ≠ ”ˆVˆzIR(z)I(1,Œ)(ur)

The stochastic optimal control of market regulation

The Stochastic Optimal Control

Of

Market Regulation

Universiteit van Amsterdam

Master Of Science

Written By

Laurens Voogd

Born Palma de Mallorca, Spain

Abstract

Abreviations

Nomenclature

Contents

Chapter 1

Introduction

1.1 A brief overview

1.2 Research goals

Chapter 2

The mathematical model

2.1 The Market

2.2 The firm

2.3 The Authority Agent

Chapter 3

Methodology

3.1 The Hamilton Jacobi Bellman equation.

3.1.1 Numerical Solution Of The Hamilton Jacobi Bellman

Equation

3.1.2 Stability , consistency and convergence of the

back-ward time iterative, implicit scheme (BTIS)

3.1.3 The procedure for the firm

3.2 Newton Raphson’s Grid Search

Chapter 4

Research

4.1 Introduction

4.2 The Optimization Problem Of The Firm

4.2.1 The base SOCP of the firm