IN DEGREE PROJECT MATHEMATICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2016

Optimal portfolio allocation by the martingale method in an incomplete and partially observable market

EMIL KARLSSON

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Optimal portfolio allocation by the martingale method in an incomplete and

partially observable market

E M I L K A R L S S O N

Master’s Thesis in Mathematical Statistics (30 ECTS credits) Master Programme in Applied and Computational Mathematics (120 credits Royal Institute of Technology year 2016

Supervisors at Ampfield AB: Torbjörn Hovmark Supervisor at KTH: Henrik Hult Examiner: Henrik Hult

TRITA-MAT-E 2016:51 ISRN-KTH/MAT/E--16/51-SE Royal Institute of Technology SCI School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Optimal portfolio allocation by the martingale

method in an incomplete and partially observable

market

Emil Karlsson

Abstract

In this thesis, we consider an agent who wants to maximize his ex-pected utility of his terminal wealth with respect to the power utility bythe martingale method. The assets that the agent can allocate his capitalto are assumed to follow a stochastic differential equation and exhibitsstochastic volatility. The stochastic volatility assumption will make themarket incomplete and therefore, the martingale method will not have aunique solution. We resolve this issue by including fictitious assets thatcomplete the market and solve the allocation problem in the completedmarket. From the optimal allocation in the completed market, we willadjust the drift parameter for the fictitious assets so that our allocationdon’t include the fictitious assets in the portfolio strategy. We consideralso the case when the assets also has stochastic drift and the agent canonly observe the price process, which makes the information in the mar-ket for the agent partially observable. Explicit results are presented forthe full and partially observable case and a feedback solution is obtainedin the full observable case when the asset and volatility are assumed tofollow the Heston model.

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Contents

1 Introduction 5

2 Model 92.1 Main Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The Martingale Method 113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Example: Power Utility . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Section Summary . . . . . . . . . . . . . . . . . . . . . . . 143.3 Incompleteness issue . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 Incomplete markets . . . . . . . . . . . . . . . . . . . . . 153.4 Fictitious Completion . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Martingale Method for the Fictitious Market 194.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Martingale Method . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Solution to the portfolio problem 235.0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 235.0.2 Deriving the PDE . . . . . . . . . . . . . . . . . . . . . . 23

5.1 Explicit solution for the Heston model . . . . . . . . . . . . . . . 245.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 245.1.2 The Heston Model . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Section summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 The partially observable case 286.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Introduction to filtering . . . . . . . . . . . . . . . . . . . . . . . 306.4 The Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.5 Deriving the PDE . . . . . . . . . . . . . . . . . . . . . . . . . . 346.6 Section Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7 Summary 38

A Appendix 39A.1 The Dual problem . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.2 Connection to Hamilton Jacobi Bellman Equation . . . . . . . . 41

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Sammanfattning

I denna uppsats har vi en individ som vill maximera sin forvantade nyttan medavsende till en potens nyttofunktion med martingal metoden.

De tillgangar som agenten kan fordela sitt kapital till antas folja en stokastiskdifferentialekvation och dessa tillgangar antar stokastisk volatilitet.

Den stokastiska volatilitet gora att marknaden blir ofullstandig och im-plicerar att martingal metoden inte kommer anta en unik losning.

Vi loser det problemet genom att inkludera fiktiva tillgangar som komplet-terar marknaden och loser allokerings problemet med martingal metoden i dennya fiktiva marknaden.

Fran den optimal allokeringen i den fiktiva marknaden, kommer vi att justeradrift parametrarna for de fiktiva tillgangar sa att var allokering inte inkluderardem i portfoljstrategin.

Vi undersoker ocksa fallet nar tillgangarna har stokastisk drift och individenkan bara observera pris processen, vilket gor informationen pa marknaden forindividen delvis observerbar.

Resultat presenteras for de fullt och delvis observerbara fallet och en losningerhalls i det fullt observerbara fallet nar tillgangen och volatiliteten antas foljaen Heston modell.

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Acknowledgements

I would like to express my gratitude to Torbjorn Hovmark and the whole staffof Ampfield AB for granting me the opportunity to study this interesting topic.Without their guidance and patience, this thesis would not have been possible.I would also like to thank my parents for their continuous support.

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1 Introduction

Continuous optimal portfolio problem is the study of how an agent should dis-tribute his wealth over time when he is faced with different investment oppor-tunities that provide either a stochastic value gain or endowment in the futurein order to maximize his future expected utility. The utility the agent obtaincan either come from an increase in his personal wealth/consumption or a com-bination of them both.This area of study started with Merton in his paper [28] and have sprung offto different extensions of his model. In his paper, he solved the optimizationproblem with setting up the Hamilton-Jacobi-Bellman equation (HJB). Themethod provides a way to transform the stochastic optimization problem intoa PDE for which the optimal allocation can be obtain after it has been solved.The formulation of the PDE from the optimization problem is simple and thishas made the method popular for solving allocation problem of the type wemention before. Its drawback with it is that the resulting PDE usually becomesa non-linear one for which analytical solution often can’t be obtained and hasto be solved numerically. It is also limited to Markovian systems, which meanthat the future value of the different investment opportunities only depends onits current value and not their historical values.

Due to these limitations of the HJB, another method called the Martingalemethod has been developed to handle problems that the HJB approach can’thandle, such as non-Markovian system.

With the Martingale method, one is focusing on how the portfolio valueevolves over time and it is treated as a variable rather than as a consequencefrom the allocation in it as one do in the HJB approach.

When one has obtained what the optimal portfolio process is, one use theidea of replication in order to derive what the allocation in it has to be. If themarket is complete, meaning that there exists a unique risk-neutral measurethat makes all discounted assets to martingales, the replication of the portfolioprocess has to be unique in order to prevent arbitrage and that assures that onecan obtain a unique representation of the allocation.

Some early papers of when it has been applied to consumption/investmentproblems we mentioned in the beginning are [22], [12] and [33]. They have asimilar setup in their paper, they assume that the agent can invest in a risk-freeasset and other risky assets that follow a geometric Brownian motion over a finitetime horizon. They also establish under what conditions the problem obtainsa solution. In [12], they also provided a verification theorem for the allocation.The verification theorem involves finding solutions to a linear partial differentialequations, which compare to the HJB would be a non-linear one. The methodhas also been used to price options ( [17] and [18] )

Equilibrium models has been treated with this method in [11], where theagent only gains utility from consumption, where he can gain additional meanfor consuming by holding physical goods, production operations or derivatives.These models try to capture the value of these assets from a supply-demandperspective rather than assuming that the agent is a price taker. The returnfrom these different goods and enterprises is set by the agent’s demand for themin equilibrium when the number of derivative contracts for them exist in zeronet supply.

Since this method is based on the ability of replicating the optimal portfolio,

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this method is only applicable in a complete market setting. If that is not thecase, then it can exist a contingent claim that is not fully replicable by theassets in the market and can therefore not be marked by trading the existingassets. This can give rise to arbitrage opportunities which is equivalent to thatthe market is not complete. Due to this, the allocation from the derived optimalportfolio does not have a unique representation and can therefore not be solved.

Some cases when an incomplete market setting can arise are when the agentis facing transaction cost from trading; trading constraints imposed upon themfrom governmental institutions; volatility that can increase from unexpectednews and fluctuate over time depending on the market sentiment.

The issue with incompleteness is something that can be handled by the HJBapproach. Some papers that deals with the different cases we mention can befound in [27], [32] and [42].

One way to resolve this issue with incompleteness while still using the moreprobabilistic approach that one does with the martingale method is to formulatethe dual of the maximization problem. It provides a upper bound for the primalproblem and if one can minimize it to obtain a zero duality gap between thedual and primal objective function, one have found a solution for the primalproblem from the dual solution.

As we have comment before, if the risk from a claim can not be hedgedaway completely by the assets in the market, the price of risk the agent is facingwill depend on his own utility preference and it has to be incorporated in thevaluation.

In paper [19], the author studies the incomplete market problem with shortsales restriction and introduces the notation of min-max measure when theassets diffusion part is driven by more Brownian motions than that are assets.

In paper [29], the author study the case when asset’s exhibits stochasticvolatility by the dual formulation. In the paper from [25], the author studiesfor what conditions have to be satisfied for an optimal solution to exists for thedual.

In [23], they showed that there is an equivalency between finding the optimaldual measure and finding a feasible solution in a completed market, where theunhedgable risk is represented by fictitious assets. The idea they have, is thatthe agent should create fictitious assets that contain the risk that he can nothedge by the assets already in the market and add them to the market. In thatway, the agent has fictitiously completed the market and the martingale methodcan be applied. Since the fictitious assets are artificially created, the agent canthen adjust their properties as he desires and by so doing, making it inferior tohold any of them in his portfolio and obtain a feasible allocation.

Another aspect that limits the earlier models is that one assumes that themodel’s parameters are constant. As we said before about the volatility, thereis an abounding evidence that volatility is not constant over time when oneconsolidates empirical data for different assets classes. Other factors such asshort-term growth are also varying over time due to business cycles in the econ-omy.

These factors and effects should be included in the model to increase itsexternal validity. However, even after one has taken these effects into account,one might have to make the assumption that these effects are directly measurableand observable in order to use them.

This becomes a problem when one is trying to empirically implement those

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models since there are factors that cannot be observed or at least be partiallyobserved. For instance, economic factors that affect the long term growth ofasset’s prices are not directly visible in the short run due to short term fluctu-ations.

These variables should be taken into account in the optimization problemto enhance performance but since they are not directly observable, one has toestimate them, which result in an estimation error that has to be taken intoconsideration when doing the allocation.

The author in [26] conducts the portfolio allocation problem when the assetprices the agent can invest in is described by a geometric Brownian motion whosedrift follows a Ornstein-Uhlenbeck process. The problem becomes partiallyobservable since the drift is stochastic and the agent can only observe the priceprocess. The authors in [39] investigates the case when the drift follows acontinuous time Markov chain.

The impact on portfolio performance under partial information compared tothe full information case was conducted in [40]. The authors in [34] managesto represent the allocation using Malliavin calculus for the two cases when thedrift follows a mean-reverting process and a Markov chain. In [7], they solveda portfolio optimization problem under partial information condition when theunderlying drift components followed a general, not necessary a Markovian pro-cess.

All of these papers here have derived a solution for the portfolio problem bythe martingale method, and hence set up their models so that market becomesa complete one.

In this thesis, we will solve the portfolio optimization problem when theassets exhibit stochastic volatility by the martingale method. The informationavailable for the agent will only be the price process generated by the assets inthe market.

The incompleteness issue we face by the stochastic volatility will be resolvedby the fictitious completion idea that was introduced in [23]. We will thenextend our model to include stochastic drift for which the components in themodel become partially observable. That model will be transformed into asystem that is adapted to the information that is available to the agent forwhich he can derive what the allocation should be.

We have structured the thesis following way. Section 2 we will introducethe general framework of our model and introduce the reader to some of thenotations and concepts that we will use throughout this thesis.

Section 3 will cover the basic foundation of the martingale method andthe necessary steps and theory one needs to derive a solution in the completemarket scenario. That section also covers an introduction to different caseswhen incompleteness occurs and how one can resolve it by introducing fictitiousassets.

In Section 4 we will perform the martingale method for this complementedmarket and talk in Section 5 about what criteria need to be fulfilled to obtaina feasible solution. Section 6 will involve implementing those criteria and wewill derive a partial differential equation that need to be solved to obtain anexplicit solution. That section will also contain a case when one can solve theportfolio problem explicit in the Heston model setting and the steps to derivethe solution.

Section 7 contains a section about filtering theory and concepts that has to

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be introduced before we extend our model to the partial observable case. Wewill in that sections reduce our model to a observable one (in that since that itsdynamics can be explained by just observing the asset prices).

Last in that section, we will implement the theory we have introduced aboveto derive the martingale solution in a complemented market and the correspond-ing PDE.

Last, we have included an Appendix with some additional theory and com-ments.

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2 Model

Consider a market under (Ω,F ,F,P), with filtration F = Ft0≤t≤T and prob-ability measure P. The market contains two main sources of randomness, consti-tuting a vector of Brownian motions (W 1

t ,W2t ) = [W 11, . . . ,W 1n,W 21, . . . ,W 2n]T .

In the market, there are n risky assets that we denote as St = [S1t , S

2t , . . . , S

nt ]

T

which will exhibit stochastic volatility. There is also a risk-free bank accountthat provide continuous interest. The model and the dynamics for each processcan be found in the Assumption (1).

The asset’s price process under P is given by

dSt = D(St)µ(t, σt)dt+D(St)D(σt)dW1t (1)

Where the function D(p) denotes the diagonal matrix with p’s entries onits diagonal .The variance process is given by

dσ2t = λ(t, σt)dt+ b(t, σt)

(ρdW 1

t + ξdW 2t

)(2)

where σ2t = [σ1

t2, σ2

t2, . . . , σn

t2]T The bank account that is driven by the

risk-free short term rate r is given by

dBt = rBtdt. (3)

W 1t and W 2

t are two independent Rn dimensional Brownian motionsunder measure P.

Assumption 1

We will assume that the processes (1) and (2) and are such that they admita strong solution. This assumption assures that the solution form of (1) and(2) is obtainable by integration. We also let K0,t denote the discounting factorbetween time zero and t. It will be assumed here that the agent can observe allthe components that affect the different market dynamics in Assumption (1).We will relax this assumption later in this thesis.

2.1 Main Problem

We introduce here how the agent constructs his wealth portfolio and whats hewant to achieve.

At the beginning of the investment horizon, the agent has an initial wealthof x ≥ 0. His wealth process is a portfolio that consists of St and Bt that he cancontinuously trade in. We denote the wealth portfolio as Xt and the allocationthe agent makes in the bank account and stocks as (π0

t , πt) = (π0t , π

1t , . . . , π

nt ).

These π’s are the normalized portfolio weights that represent the relative port-folio values allocated to the bank accounts (π0

t ) and to the stocks (πt).The dynamics of Xt can be written in terms of the diffusion dBt and dSt

since those are the components constituting the portfolio.Using the fact that the portfolio weights have to sum up to 1, we can reduce

the dependency of π0t by replacing it with 1−πt1 in order to obtain an expression

that only depends on the stock weights.

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For our problem, the dynamics of the wealth process will be given by

dXt = Xtπtµtdt+Xt (1− πt1) rdt+XtπtD(σt)dW1t (4)

X0 = x.

The goal of the agent is to obtain the most capital gain during the investmentperiod [0, T ] according to a utility preference he has. The objective criteria hehas is then

maxπt

EP [U(XT )] .

We will impose also that the agent can’t have infinite short and long positionsin any assets. The utility function U(x) the agent has is assumed to fulfill thefollowing properties

The utility function U : (0,∞) → R is strictly increasing, strictly concave,continuous and continuously differentiable, and satisfies

∂U

∂x|x=0 = lim

x↓0

∂U

∂x(x) = ∞,

∂U

∂x|x=∞ = lim

x→∞

∂U

∂x(x) = 0

Assumption 2 (The utility function)

These conditions are widely known as the Inada conditions. They ensure thatthe agent has a diminishing utility gain as his wealth increases as well as theavoidance of the situation in which he ends up with a zero wealth in the end.

There are several versions of the utility functions which satisfy the propertiesin Assumption 2, among the most popular ones are, for instance, the Powerutility, the Log utility and the Exponential utility. In this thesis, we focus onthe Power utility function, which can be found in Definition 1.

A utility function U(x) is called power utility if U : (0,∞) → R and

U(x) =xγ

γ

for a non-zero γ < 1.

Definition 1 (Power Utility)

In the limit case of power utility function, when risk-aversion parameter γ → 0,we obtain the log(x) function, which is the Log utility function.

In the next section, we describe the martingale method, which is going tobe used to obtain the our solution. To simplify the description of the generalframework of the method, we begin by a less intricate model and then extendit after the general methodology and notations are introduced.

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3 The Martingale Method

Since the focus of this thesis is largely concentrated on the martingale method,this section consists of an introduction to the methodology surrounding it andillustrates its shortcomings regarding the model we proposed in the previoussection. The notions and structure of this section has been taken from [7], [6]and [5].

Firstly, we are going to look into the case of a complete market, so theprocess σt in Assumption 1 is equal to a constant, invertable matrix σ and µt isset to a constant µ. Since the volatility in no longer stochastic, the only sourceof risk comes from W 1

t which we will simply write as Wt.After comprehensive description of the given case, we extend the method to

the case of incomplete markets.

3.1 Introduction

The main idea of the martingale method is concentrated on portfolio replication.Consider an agent, who intends to maximize his expected wealth XT .

maxEP [U(XT )] (5)

Where U is the utility function, and x is the starting capital for that agent.This agent has access to different assets which he can invest in from today toT .

If one wants to solve this problem by the means of dynamic programming,one has to introduce the dynamics of Xt to incorporate how his allocation affectthe wealth process and from there maximize his utility.

That procedure reduces the problem into solving the Hamilton-Jacobi-Bellmanequation. However, one can make a more probabilistic approach that does notdepend on the dynamics of Xt.

Consider the space of all the portfolios that can be replicated with an initialcapital x. Our final wealth XT belongs to that space by construction and thatspace can be used as a search region when we optimize with respect to XT .

Now, if the market is complete, there can not exist arbitrage and the assetsin the market can replicate all contingent claims.

The previous statement is equivalent to the existence of a unique probabilitymeasure Q for which all discounted price processes are martingales. That mea-sure is referred to as the risk-neutral measure. Since there exists no arbitrage,all assets have a unique price at all time.

In this complete market setting, our wealth process XT has to have a uniqueprice as well under this measure Q.

Since one can construct it with a starting capital of x, its discounted expectedvalue today has to be equal to x under Q.

We can then now reformulate our objective function (5) as a static optimiza-tion problem, with a constraint that the discounted portfolio has to be equal tox to make it feasible.

maxXT

EP [U(XT )]

subject to EQ [K0,TXT ] = x (6)

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The constraint (6) is widely referred to as the budget constraint in academicliterature.

The focus will now be on finding out what the optimal final wealth XT willbe and then extract the dynamics of it.

The corresponding wealth dynamics of the process (4) has to be equal toXT when one assigns the optimal allocation πt to it. One can then comparethese two processes by the martingale representation theorem and obtain fromit what the explicit allocation will be. We return to this procedure later, afterintroducing more notations and the procedure how one obtains XT from (6).

In case of the complete markets, we can define the unique market price ofrisk and the associated likelihood function from the measure P to Q and this isstated in the Assumption 3.

We will denote the market price of risk as

σ−1 (µ− r) = −φ.

Where r denotes a Rn dimensional vector with r at all positions. Thelikelihood function Lt is given by

Lt = exp

∫ t

0

φds−∫ t

0

||φ||2ds

and its dynamics is given by

dLt = LtφdWt

L0 = 1.

Further, we assume that φ satisfies the Novikov condition.

Assumption 3

The Novikov condition in Assumption 3 acts as a sufficient condition so thatthe transition of a process from P to Q becomes a martingale.

The procedure now one does to solve (6) is to set up the Lagrange relaxationof it. We also have to change the measure in the expectation for the budgetconstraint so it matches with the measure in the objective function. This changeof measure can be done by introducing the likelihood function into the budgetconstraint.

The resulting relaxation will then be

L = EP [U(XT )]− Λ(EP [LTK0,TXT ]− x

), Λ > 0.

Setting the first partial derivative of L to zero, we conclude that an interiorsolution to our problem is

∂U

∂x(XT ) = ΛK0,TLT or XT = I (ΛK0,TLT ) . (7)

where I =(∂U∂x

)−1.

The term Λ is still undetermined but can be easily found by inserting thesolution XT into the budget constraint.

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3.2 Example: Power Utility

We concretize the previous section by looking into the scenario, in which theagent’s utility is represented by the power utility function. He has a startingcapital of x and he can choose to invest in a bank account with instantaneousrisk free rate r and among the stocks St with the dynamics

dSt = D(St)µdt+D(St)σdWt.

In the power utility case, the function I(y) in (7) will be equal to I(y) =

y−1

1−γ .Inserting I(y) and the optimal wealth portfolio (7) into the budget con-

straint, we extract what the value of Λ needs to be so the initial capital require-ment is fulfilled.

In our case this value is equal to

Λ− 11−γ = xEP

[(K0,TLT )

− γ1−γ

]−1

.

Replacing the value of Λ− 11−γ in the optimal wealth process (7) gives us a

feasible optimal final wealth, which is equal to

XT =x

H0(K0,TLT )

− 11−γ (8)

where

H0 = EP[(K0,TLT )

− γ1−γ

].

From now on, we will let β = γ1−γ , which is used several more times in this

thesis.As we noted before, under the measure Q, all discounted assets are martin-

gales. This fact implies that our discounted portfolio at time t has to be equalto

K0,tXt = EQ[K0,T XT |Ft

]. (9)

Hence, by inserting the optimal portfolio expression (8) into (9), we canobtain the portfolio value Xt. This will in our case be given by

Xt =x

K0,tH0EQ[K0,T (K0,TLT )

− 11−γ |Ft

](10)

=x

K0,tH0

EP[(K0,TLT )

−β |Ft

]EP [LT |Ft]

(11)

=Ht

H0(K0,tLt)

−11−γ . (12)

Here, Ht is defined as

Ht = EP[(Kt,TLt,T )

−β |Ft

].

So far, we have been working with finding the optimal wealth process forthe portfolio. In order to be able to replicate it, we would require the allocationfor each asset in it. We can obtain the allocation by using the martingalerepresentation theorem, which we mentioned before. We include it as a theoremand it can be found in Theorem 4.

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Let Wt be a n-dimensional Wiener process, and assume that the filtrationF is defined as

Ft = FWt t ∈ [0, T ]

Let M be any Ft-adapted martingale. Then there exists uniquely de-termined Ft-adapted process h such that M has the representation

Mt = M0 +

∫ t

0

hsdWs

Theorem 4 (Martingale Representation Theorem)

To see how we apply Theorem 4, we first write down the dynamics of thetwo representations of the optimal wealth process that we have for Xt.

Furthermore, we remember that the dynamics of the optimal portfolio withits optimal allocation πt will be given by the combined dynamics of the differentassets in it, as discussed in Section 1.

dXt = Xtπtµdt+ Xt(1− πt1)rdt+ XtπtσdWt

The discounted process K0,tXt will then be equal to

d(K0,tXt

)= K0,tXtπ(µ− r)dt+K0,tXtπσdWt. (13)

On the other hand, we have from Ito’s formula that the dynamics of the righthand side of (10) after discounting it will be

d

(Ht

H0K−β

0,t L−11−γ

t

)= (. . . ) dt− 1

1− γ

(Ht

H0K−β

0,t L−11−γ

t

)φT dWt. (14)

Since these processes are identical, they have to have the same martingalerepresentation. In other words, the function ht in front of the diffusion term inTheorem 4 is the same.

If we represent these two processes under the measure Q, the only thing thathappens is that the dt term vanishes, but the term associated with diffusion staysthe same.

So, in order to reach equality between (13) and (14), we conclude that πt

has to be equal to

πt = − 1

(1− γ)φTσ−1. (15)

and we obtain the optimal allocation for the portfolio.

3.2.1 Section Summary

In this section it was discussed that in order to derive the optimal portfolio allo-cation, the martingale method relies on the existence of the unique measure Qthat allows us to connect the optimal wealth process with the portfolio dynam-ics. However, this reasoning falls apart when the market is incomplete, since in

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that case there is no unique risk-neutral measure. In our original model, thereare in fact infinitely many measures that would allow us to write the discountedportfolio as a martingale.

In the coming section, we discuss some of the cases when an incompletemarket can arise as well as the means of dealing with the ambiguity in thechoice of measure by the introduction of fictions assets.

3.3 Incompleteness issue

Now, having looked into the martingale method for the case of a completemarket, we turn to our original formulation of the problem, when the market isincomplete, using the idea presented in [23]. We give first an introduction onhow incompleteness turns up in the market and from that get an idea of how[23] resolves it by introducing fictitious assets.

3.3.1 Incomplete markets

We present the examples of different cases when markets become incomplete.The three main categories of factors causing the discussed problem are statedin below.

Market Friction: Market frictions such as transaction costs, portfolio con-strains and the lack of liquidity are largely the most discussed factorscausing market incompleteness.

Transaction costs create a cost for the replication procedures. Since thereplication is determined by the future price behavior of the assets the in-vestor chose for his portfolio, the accumulated transaction cost is stochas-tic and cannot be naturally hedged. It can also be the case that thereplication procedure becomes too expensive due to the first variation ofthe Brownian motion one might assume the assets are following ([21] [10][13]).

Portfolio constrains, such as no short selling, create a natural limitation onthe portfolio allocation, which can make the full replication not feasible.

Low levels of liquidity in the market would create the incentives for themarket makers to quote prices with a bid-ask spreads. As in the case withtransaction costs, it would introduce an additional cost for trading thathas to be introduced when valuing the strategy one want to do.

Unhedgeable risk: In some cases, there might be an insufficient number offinancial instruments on the market for obtaining the desired risk profileby the means of hedging. The articles [2] and [9] discuss some of the issuesregarding the given problem.

Jumps in the asset’s prices make hedging difficult and lead to severalcomplications in options pricing. In a Black-Scholes setting, the jumpscannot be hedged away by the usual delta hedging method, because itrelies on a linear approximation of the option price changes.

Therefore, the standard delta hedging method does no longer hold in thesituation of drastic price changes due to the non-linearity in the optionsvalues.

15

Stochastic volatility is another factor preventing the investor from theeffective risk hedging, mostly because volatility is per se not a tradableinstrument. The non-constant volatility causes the distribution of the as-set return to vary over time, which in turn affects the price of a contract,which has that asset as underlying instrument. In [14], they tried to cap-ture the scale of the volatility effect on the option market by introducingthe notion of local volatility in order to obtain a market calibrated volatil-ity surface to price different options. However, the model relies on theassumption that volatility is a function of the asset price itself, which re-duces the whole concept of volatility risk, since it is possible to hedge waythe volatility with the asset.

Ambiguity: Even though it may be possible to hedge away the risk, there isstill an issue of ambiguity to tackle. One instance when this problem arisesis when full information about the effect from different market scenariosis not know in advance. The case of the non-constant volatility illustratesthis issue very well. In a continuous time model, the volatility is perfectlyobservable. But when one is limited to discrete time data, one has tofilter it out which is not a easy task [43]. The issue of ambiguity maycreate a hedging error, which can potentially harm the overall investingperformance [8].

We focus on the issue of unhedgeable risk, because it is the main feature ofour model. In the next section, we introduce the concept of complementing themarket with fictitious assets and discuss how they influence our model.

3.4 Fictitious Completion

As discussed in the previous section, if there is risk in the market that can notbe hedged away, there is an ambiguity about what the market price of risk forit should be. But what would happen if the agent could actually trade it? Thenthe market would be complete and that risk can be dealt with in the replicationprocedure.

That is the main idea of [23], that the agent should introduce a fictitious setof assets into his portfolio that contains the risks that the agent can not hedgeaway by only using the assets that are available in the market place.

After the agent decided on a desired allocation of the market and fictitiousassets in his portfolio, he can adjust his portfolio position by varying the marketprice of risk that is associated with the unhedgable market factors. If he managesto create a portfolio that does not contain the fictitious assets, he has managedto obtain a set of portfolios that are feasible. Meaning that he can obtain anallocation with the assets that are available to him. From this set of feasibleportfolios, he would pick that one that maximize his utility.

We will come back to this reasoning later and justify why it indeed yield aoptimal portfolio for the agent.

Let us introduce the additional fictitious assets to our original problem,which are denoted as Ft with the dynamics illustrated by (16).

dFt = λ(t, σt)vdt+ b(t, σt)

(ρdW 1 + ξdW 2

)(16)

16

With the new assets at hand, the market price of risk for our model inAssumption 1 together with the fictitious asset (16) will be given by Assumption5.

17

We denote the market price of risk for completed market as

σ−1t (µt − r) = −φt (17)

where r denotes a 2n-vector with r at all positions and

σt =

[D(σt) 0btρ btξ

], µt =

[µt

λvt

]. (18)

The inverse of σt is explicitly given by

σ−1t =

[D(σt)

−1 0−ξ−1ρD(σt)

−1 ξ−1b−1

].

Further, we assume that φt satisfies the Novikov condition.

Assumption 5

We see that we can write down the Girsanov kernel more explicitly as

φt = σ−1(r− µ)

=

[D(σt)

−1(1r − µt)ξ−1b−1(1r − λv

t )− ξ−1ρD(σt)−1(1r − µt)

]=

[φt

vt

]. (19)

We denote the associated likelihood process for this market as Lvt , which is

given by

Lvt = exp

(∫ t

0

φsdW1s +

∫ t

0

vtdW2s − 1

2

∫ t

0

||φs||2 + ||vs||2ds). (20)

The reason that we named λvt with superscript v, is because we will actually

treat it as a parameter that will control the market price of the risk that isassociated with W 2. The other factors associated with Ft’s diffusion have to bechosen in a way that σt spans the whole risk space, otherwise the market wouldstill be incomplete.

Now, with these fictitious assets Ft added to the market, we can proceedwith the martingale method for our original model in Assumption 1. We willin the next section do similar steps we did in the previous one, except thatwe need to introduce new notations on some of the components and take intoconsideration that the volatility is no longer a constant.

18

4 Martingale Method for the Fictitious Market

4.1 Introduction

In this section, we conduct the martingale approach for the problem after thefictitious assets are added. In the first section, we derive the wealth process andthe allocation choice between the tradable and fictitious assets.

Later on, we discuss the result obtained in [23] and its implications for ourproblem.

In the last section, we focus on deriving a PDE for one of the componentsin the resulting wealth process.

4.2 Martingale Method

With the fictitious assets at hand, we can define the new dynamics for ourwealth process as

dΠvt = Πv

t πtµtdt+Πvt (1− 1πt)rdt+Πv

t πtσtdWt. (21)

Where µt and σt are defined as in (18) and πt is the 2n dimensional weightsvector where the first n entries represent the allocation of the tradable assetsand the last n entries describe the fictitious assets we have added.

Wt is structured as

dWt =[dW 1

t dW 2t

]T.

Since the market is complete for any vt that satisfy the Novikov condition,we can set up the static optimization problem as we did before.

max EPv [U(ΠvT )]

EQv [K0,TΠvT ] = x.

We denote the two different measures as Qv and Pv to indicate that theydepend on vt. Also, since Wt is composed of risk components that were adaptedto the filtration Ft, the problem is still in a complete observable market.

We set up the Lagrangian and see that the first order condition implies that

∂U

∂x(Πv

T ) = ΛK0,TLvT or Πv

T = I (ΛK0,TLvT ) .

Where I is as before the inverse of the partial derivative of the utility functionand Λ is determined by the initial condition.

Λ− 11−γ EPv

[K−β

0,TLvT−β]= x

These results imply that the final wealth is equal to

ΠvT =

x

EPv

[K−β

0,TLvT−β] (K0,TL

vT )

− 11−γ =

x

H0(K0,TL

vT )

− 11−γ .

Where β = γ1−γ . Observe that H0 here is different from the H0 we had in

the previous section.

19

The expectation component in the denominator of the final wealth could besimplified further by incorporation of the β term into the likelihood process aswas done in [7]. We do this by introducing the new likelihood process

Lv0t = exp

−∫ t

0

βφTt dW − 1

2

∫ t

0

β2||φt||2. (22)

Note that the original likelihood function can be written as

LvT−β = Lv0

T exp

1

2

∫ T

0

β

1− γ||φt||2dt

.

so that the normalizing expectation H0 becomes

H0 = EPv

[K−β

0,TLvT−β]

= EPv0

[exp

β

∫ T

0

(rt +

1

2(1− γ)||φt||2

)dt

].

Using the fact that a discounted optimal portfolio under Qv has to be amartingale, we derive (with help from the abstract Bayes formula) that theoptimal wealth process can be written as

Πvt =

1

K0,tEQv

[K0,T (K0,TL

vT )

− 11−γ |Ft

] x

H0

= K− 1

1−γ

0,t EQv

[K−β

t,T LvT− 1

1−γ |Ft

] x

H0

= K− 1

1−γ

0,t

EPv

[K−β

t,T LvT−β |Ft

]Lvt

x

H0

=Ht

H0(K0,tL

vt )

− 11−γ x. (23)

Where Ht and H0 have been written under the probability measure Pv0 .

Ht = EPv0

[exp

β

∫ T

t

(rs +

1

2(1− γ)||φt||2

)ds

|Ft

](24)

Using the same arguments as in [15], we assume that Ht can be written asH(t, σ2

t ).Then, we notice that following Ito’s lemma, the dynamics of Ht is

dH(t, σ2t ) =

∂H

∂t+A(H)

dt+

∂H

∂σ2btρtdW

1t +

∂H

∂σ2btξtdW

2t . (25)

where A it the infinitesimal generator.

A(H(t, x)) =n∑

i=1

µi(t, x)∂H

∂xi(x) +

1

2

n∑i,j=1

Cij(t, x)∂2H

∂xi∂xj(x) (26)

C(t, x) = σ(t, x)σ(t, x)T

In addition, the results of [7] allow us to write Ht down as a geometric Brow-nian motion, which comes handy for obtaining the allocation for the optimalportfolio. This result is stated and proved in Lemma 2 below.

20

The process Ht in (24) stochastic differential can be represented as

dH(t, σ2t ) = HtµHdt+HtσHdWt.

Lemma 2

Proof. For simplicity, we write Ht down in (53) as

Ht = EPv0

[exp

∫ T

t

gtds

|Ft

]

Now we use the Bayes formula to write it as

Ht = EPv

[Lv0T exp

∫ T

0

gsds

|Ft

]︸ ︷︷ ︸

=Mt

exp−∫ t

0gsds

Lv0t︸ ︷︷ ︸

=Et

Lv0t

Here we have that Mt is a martingale, and therefore it can be written as Mt =htdWt for a process that is F-adapted. Lv0

t is given by (22) and Et has thedynamics of dEt = −gtEtdt. These facts combined with Ito’s lemma provide uswith the result.

Now we use Theorem 4 to work out what the asset allocation will be for thisportfolio. As we observed before, while discounting the wealth dynamics in (21),the component in front of the diffusion term stays the same. The discountedprocess (23) will however be different from the previous case when volatility wasa constant because of Lemma 2.

Indeed, since Ht is a stochastic process due to the volatility component,it will also contribute to the dynamics as well. Hence, by Ito the discounteddynamics of (23) will be

d

(K0,t

Ht

H0(K0,tL

vt )

− 11−γ x

)= (....)dt

+

(K0,t

Ht

H0(K0,tL

vt )

− 11−γ x

)(σH − 1

1− γφTt )dWt.

As we comment before, we are only interested in the diffusion part of the process.Using the martingale representation theorem again, we see that in order for thewealth process (21) to be equal to the dynamics derived above, πt has to beequal to

πt = (σH − 1

1− γφTt )σt

−1

= σH σt−1 − 1

1− γφTt σt

−1. (27)

Observe that we exclude the hat notation on the allocation πt here. This is toemphasize that even if this allocation is optimal in this completed market, itdoes not have to be feasible for our original problem.

21

If we combine the result from (25) and the fact that we can write the dy-namics of Ht as a GBM. we can explicitly write σH as

σH =1

Ht

[∂H∂σ2 btρt

∂H∂σ2 btξt

]. (28)

It is clear that if we did not have any stochastic volatility, the σH componentwould be zero and we would arrive to the same position formula as in (15).Therefore, it enables us to refer the σH σt

−1 component as the hedging partthat arise from stochastic volatility.

22

5 Solution to the portfolio problem

5.0.1 Introduction

In this section, we use the result from [23] to obtain a partial differential equationthat solves the function Ht from our previous section. Also, we will look into aspecial case for our model called the Heston model, for which we can solve Ht

explicitly.

5.0.2 Deriving the PDE

We now go to the insights from [23] to derive what the parameter vt has to be.We have under this fictitious market, that the agent can gain a higher utilityfrom trading the fictitious assets. i.e

maxEP[U(XT )

]≤ maxEP [U(Πv

T )] for all feasiable vt. (29)

If we now can find a feasible strategy πt that generates portfolio wealth XT

and a parameter value v so that ΠvT = XT a.s, the agent can replicate one of

the portfolios in the fictitious market.However, since the portfolio Πv

T is feasible, we have from the definition of

the optimal portfolio XT that

EP [U(ΠvT )]≤ maxEP

[U(XT )

]. (30)

Indeed, (29) and (30) tell us that if we can find a v that generate a feasiblestrategy, that strategy will be optimal for the agent. This v is referred in [23]as ”least favorable” and showed its existence from the dual formulation of theportfolio problem. We have included in the appendix a short section of the dualformulation of this problem and how it is relates to v.

Considering our results in (27) and our conclusion about σH in (28), if webegin to work out the allocation in (27) by inserting our parameters from themodel, we get that the part that did not depend on Ht becomes

1

1− γφT σ−1 =

1

1− γ

[φTt vTt

] [ D(σt)−1 0

−ξ−1ρD(σt)−1 ξ−1b−1

]=

1

1− γ

[φTt D(σt)

−1 − vTt ξ−1ρD(σt)

−1

vTt ξ−1b−1

]T.

The part that belongs to the hedging component is

σH σ−1t =

1

Ht

[∂H∂σ2 bρ

∂H∂σ2 bξ

]σ−1t

=1

H

[0 ∂H

∂σ2

].

Hence, we conclude that in order to have a feasible solution, meaning thatπt = (πt,0), we have to choose vt such that they cancel each other out.

∂H∂σ2

H− 1

1− γvTt ξ

−1b−1 = 0 ⇒

vt = (1− γ)ξT bT∂H∂σ2

T

Ht

23

This will subsequently give us the position we have to take in the stocks

πt =1

1− γ(µt − 1r)TD(σ2

t )−1 +

∂H∂σ2

HtbρD(σt)

−1. (31)

Since this solution for vt is unique, we here and onward refer to it as vt.The question is what is the value of Ht and its derivatives. One way we can

solve it is by using the Feynman-Kac equation with the underlying process σ2t

under the measure Pv0 . Observe that if we transform the process σ2t from Pv to

Pv0 , it will have this following dynamics.

dσ2t = (λ− βb(ρφt + ξvt)) dt+ b

(ρdW 1

tv0

+ ξdW 2tv0)

From the expression for Ht we have at (24). We can use Feynman-Kacformula to derive a PDE for which Ht satisfies.

∂H

∂t+ µ∇Ht +

1

2Tr[σT∇2Htσ

]+ β(r +

1

2(1− γ)||φt||2)Ht = 0 (32)

H(T, σ2T ) = 1

where

µ = λ− βb (ρφt + ξvt)

σ =[bρ bξ

].

Here, µ and σ are the parameters for the process under Pv0-measure.However, we note that the resulting PDE becomes non-linear due to the

HTσ2tH−1

t component in the kernel vt. Those kind of non-linear PDE are hard to

solve and have to be solved by a numerical method. These kinds of equationsare common for portfolio problems with the HJB methodology. It turns out,that when one is deriving the PDE by the Feynman-Kac equation, one obtainsessentially the same equation. The difference is that when one is solving theHJB equation, one is trying to solve the expected utility the agent would obtainwhile with the martingale method, one optimizes the wealth process.

We show in the Appendix that transforming PDE in (32) will result in thesame PDE obtained for the HJB.

One interesting result we can observe from the allocation (31), is that thehedging component disappears when ρ = 0. It will be however that the agentwill reduce his position in the risky assets when the volatility is increasing butno additional hedge is needed for the uncorrelated case.

In the next section, we look into a case when the PDE is analytically solvableand derive what the optimal allocation would be.

5.1 Explicit solution for the Heston model

5.1.1 Introduction

In this section, we shall solve the PDE in (32) when our model is setup as theHeston model. This model was first proposed by Steven Heston in [20] as away to incorporate volatility for the purpose of pricing options. It is possible toderive an explicit solution for it by using the HJB equations (see [24]), and wewill show here that the same solution is obtainable from our PDE in (32).

24

5.1.2 The Heston Model

The Heston model setup is described in Assumption (6). We also assume thatthe agents utility is characterized by the power utility function and his aim isto maximize his final wealth at T .

The asset’s price process under P is given by

dSt =(r + aσ2

t

)Stdt+ σtStdW

1t . (33)

The volatility process is given by

dσ2t = k

(θ − σ2

t

)dt+ bσt

(ρdW 1

t +√1− ρ2dW 2

t

). (34)

and there is a bank account, which is driven by a constant risk free rate rand W 1

t and W 2t are two independent Brownian motions.

Assumption 6 (Heston Model)

The variance process in the Heston model is a Cox-Ingersoll-Ross model [11]which exhibits mean reversion and if 2kθ ≥ b2 holds, one can guarantee thatthe process always remains positive.

We start now by inserting the parameter from the Heston model into (31).

πt =a

1− γ+

∂H∂σ2

Htbρ

The coefficients µ and σ from (32) are equal to the following in the currentmodel setting:

µ = k(θ − σ2) + βbρaσ2 − β(1− γ)σ2(1− ρ2)∂H∂σ2

H

σt =[bσρ bσ

√1− ρ2

]Regarding the component in the exponential of (24). Given that φt = aσ

and vt = (1− γ)√1− ρ2bσ

Hσ2

Ht, it will be equal to

||φt||2 =

(a2 + (1− γ)2b2(1− ρ2)

(Hσ2

H

)2)σ2.

If we insert this component into (32), the resulting equation becomes highlynon-linear due to the squared ∂H

∂σ2 term.The standard procedure to get around this issue is to assume an ansatz for

how the function Ht looks like and to reduce our problem into a simpler onethat is solvable.

Due to the exponent in the expectation and the boundary condition, wecould make the ansatz that Ht should be on the following form.

H(t, σ2) = exp(A(T, t) +B(T, t)σ2

)with A(T, T ) = B(T, T ) = 0. This looks promising since the ratio

Hσ2

H will nowbe equal to B(T, t) and since Ht > 0, we can reduce it by division to get anequation that only depends on A(T, t) and B(T, t).

25

What is left is the variable σ2, but since the whole left-hand side should beequal to zero regardless of the value of σ2, it is feasible to conclude that theterm in front of σ2 has to be zero as well. Given the former, the remaining partthat does not depend on σ2 has to also be equal to zero in order for the equationto balance out.

With our ansatz, we insert these coefficients and restructure the equationsby our previous arguments that the terms has to net out to zero.

We can therefore reduce (32) into these two equations with respective bound-ary conditions.

At + kθB(t, T ) + βr = 0

Bt + (βbρa− k)B(t, T ) +b2

2

(1− γ(1− ρ2)

)B(t, T )2 +

β

2(1− γ)a2 = 0

A(T, T ) = B(T, T ) = 0

We use one more trick to reduce the squared component in the second equa-tion, which only depends on B(t, T ). We introduce a new ansatz involvingrewriting B(t, T ) in the following form

B(t, T ) =1

b2

2

(1− γ(1− ρ2)

)︸ ︷︷ ︸

α

g′(t, T )

g(t, T ). (35)

We see that α will never be zero unless γ = 11−ρ2 , but since γ is always less

than one in the power utility case, this condition will never be fulfilled. Then,taking the time derivative of (35) makes the squared component disappear andwe are left with

g′′(t, T )− kg′(t, T ) + βa2α

2(1− γ)g(t, T ) = 0.

Where k = k − βbρa.This second order differential equation has the solution of the following form

g(t, T ) = c1er+t + c2e

r−t

limt→T

∂g(t, T )

∂t= 0.

The last boundary condition for g(t, T ) if derived from the boundary conditionof B(t, T ) and the roots for the resulting characteristic equation will be

r+,− =k

2± 1

2

√k2 − 2βa2α

(1− γ).

Here, we limit ourselves to the condition, characterized by the expression underthe square root remaining positive, in order to not to deal with complex values.

From the boundary condition, we have that c1 and c2 have to fulfill

c1 = −c2r−r+

e(r−−r+)T .

26

Since c2 is free for us to choose, except for zero value, we set c2 = 1. Insertingthe solution g(t, T ) into (35), we get that B(t, T ) is equal to

B(t, T ) =1

a

r−er−t − r−e

(r−−r+)T er+t

er−t − r−r+

e(r−−r+)T er+t

=1

ar−r+

(1− e(r−−r+)(T−t)

)r+ − r−e(r−−r+)(T−t)

=1

a

βa2a

2(1− γ)

(1− e(r−−r+)(T−t)

r+ − r−e(r−−r+)(T−t)

)=

βa2

(1− γ)

(1− e(r−−r+)(T−t)

r+ − r−e(r−−r+)(T−t)

)=

γa2

(1− γ)2

(eα(T−t) − 1

eα(T−t)r+ − r−

).

where α =√k2 − 2βa2α

(1−γ) .

Hence, the optimal portfolio in the Heston model will be

πt =a

1− γ+

γa2bρ

(1− γ)2

(eα(T−t) − 1

eα(T−t)r+ − r−

). (36)

under the condition that α is a real value number. This result is in line withthe one derived in [24], where it was obtained using the HJB equation.

5.2 Section summary

We have implemented the conclusions of [23] so we could derive a measure forthe non-hedgeable volatility component in the wealth process. After that wehave been able to derive a partial differential equation that needs so be solvedin order to obtain the optimal allocation for the given portfolio. We have solvedthis equation in a Heston model setting and subsequently confirmed that it ispossible to reach the same solution with our method as the one obtainable byusing of the Hamilton-Jacobi-Bellman equation. We have also briefly discussedthe similarities between solving the problem by the martingale method and HJBand referred the reader to the Appendix where we provided a description of theHJB equations and similarities it has with the martingale method.

27

6 The partially observable case

This section is devoted to the case of the drift component being driven by anadditional stochastic factor and the agent only has access to information thatis generated by the price processes in the market (denoted as FS

t ).By doing so, our system becomes partially observable since we cannot fully

recover the factor in the drift and the Wiener processes.This is a more problematic case compared to the previous section, since our

solution was designed for the situation of having full knowledge of the relevantdynamics, which is incorporated into the portfolio. We would like to take intoconsideration this additional factor in the drift, but since we cannot observe it,we have to derive a proxy for it and reduce the model so it becomes FS

t adapted.

6.1 Introduction

Firstly, we set up a new model that is similar to the one described in theAssumption 1 but the difference is that we extended to include stochastic drift.

Later, we introduce the filtering theory and look into at a special case of ourmodel, which is characterized by the stochastic factor in the drift affecting theassets prices in a linear way.

From that model, we will transform it to a system that is adapted to the FSt

filtration for which the agent only have information about.With that FS

t adapted model, we can implement some of the techniquesused in the previous sections and derive what the optimal allocation should bein these cases of incomplete markets and partially observable market.

6.2 Model

The settings for the model can be found in Assumption 7 and are largely similarto what we have setup before, with the exception of an additional process at,which affects the drift of the assets.

28

The asset’s price process under P is given by

dSt = D(St)µ(t, σt, at)dt+D(St)D(σt)dW1. (37)

The variance process is given by

dσ2t = λ(t, σt)dt+ b(t, σt)

(ρdW 1 + ξdW 2

). (38)

and a process at that affects the drift of the assets.

dat = µ1(t, at)dt+ c(t, at)dW3t

With this, there is bank account that is driven by the risk-free rate r.The processes W 1, W 2 and W 3 are independent Rn dimensional Brow-

nian motions under measure P.Here, the system is Ft-adapted, but we will limit ourselves to being only

able to observe the asset prices St, whose filtration will be denoted as FSt .

Assumption 7

We should also make a remark about the volatility process σ2t . Even if we

can only observe the price process St, we derive from the quadratic variationthat we can observe the volatility process too.

We show this in Theorem 8.

The process σ2t in Assumption 7 is FS-adapted.

Theorem 8

Proof. dSit for 1 ≤ i ≤ n is in our case equal to

dSit = µi

tSitdt+ σi

tSidW 1i.

By the transformation f(St) = ln( St

S0), we have from Ito’s lemma that

df(Sit) = (µi

t −1

2σit)dt+ σi

tdW1i.

We have from the partial integration property that

f(Sit)

2 = 2

∫ t

0

f(Sis)df(S

is) +

∫ t

0

σis

2ds.

We can then extract σit2by

d

dt

(f(Si

t)2 − 2

∫ t

0

f(Sis)df(S

is)

)= σi

t

2.

If we do this for all Sit , we would obtain the whole process σ2

t .

29

This conclusion implies that even though we can only observe the priceprocess, we can obtain the volatility process, too.

Therefore, when we derive the proxy for at, we should obtain a better ap-proximation of it by including the volatility process in our calculations, sincethe asset’s prices are correlated with the variance process.

Remark: We point out that this argument on the volatility process beingobservable through the price process is only applicable in continuous time set-ting. Attempting to conduct the steps of the Theorem 8 with real data arisesdifficulties, since then the researcher is exposed to discrete time points. As aresult, the approximation of the derivative in Theorem 8 would give a very poorresult (see [3] for comments about that).

We now proceed with obtaining the estimated value of at. We begin witha brief introduction to the general framework and notations in filtering theorythat we will use for our filtering problem.

6.3 Introduction to filtering

Here we introduce the reader to filtering theory and the general concepts thatwe use in the sections to follow. Most of this section is a summary of what canbe found in [6], [4],[30],[35] and [36].

The basic setup of a filtering problem is as follows; one wants to observe avariable Yt but one can only observe a distorted version of it that we will denoteas Zt. An example of such a system is the following

dYt = µ(t, Yt)dt+ σ(t, Yt)dVt

dZt = b(Yt)dt+ dWt.

Yt can also be referred to as the hidden variable, and one want now to estimatewhat the true value Yt is by observing the history of Zt. We will here denotethe estimated series of Yt from the past history of Zt as Yt.

Given this, the mathematical formulation of the properties that we want Yt

to possess is being FZt -measurable where FZ

t is the σ-algebra generated fromthe observation up to time t.

There are many configurations of this kind, but we are interested in the bestone according to an error criterion.

The criterion usually used in filtering is the squared error criterion, wherethe optimal candidate for the choice is the one with a minimal expected squarederror.

Formally, our objective is to minimize (39).

EP[|Yt − Yt|2

]= inf

EP [|Yt − F |2

], F ∈ Kt

. (39)

where

Kt =F : Ω → Rn;F ∈ L2(Ω) and F is FZ

t -measurable

Given this criterion, we obtain that the optimal solution for Yt is given bythe orthogonal projection of Yt onto Kt.

Yt = EP [Yt|FZt

]30

This formulation will lead us to the Fujisaki-Kallianpur-Kunita filteringequations (FFK), to which we turn to later in this thesis.

Given that the optimal estimation of Yt is created by its projection onto FZt ,

we would like to connect it to the dynamics of the whole system. That enablesus to replace dYt with the more meaningful dYt representing the dynamics ofour filter.

Having the previous in mind, consider these two equations:

dmt = dYt − µtdt = dYt − EP [dYt|FZt

]dvt = dZt − btdt = dZt − EP [dZt|FZ

t

]dmt and dvt represent the detrended versions of the filter and the observation

dynamics.The process dvt is central to the filtering theory; it goes by the name of the

innovation process.From this setup, it is possible to show that these processes are FZ

t -martingalesand that the innovation process is also a Wiener process. The fact that innova-tion process turned out to be a Wiener process is important, because we use itlater while implementing the martingale representation theorem.

From a conjecture known as the innovation hypothesis, we conclude thatthe process dmt dynamics can be written as process of dvt as well as with again process. These facts can be summarized in the FKK equation (Theorem9), which also illustrates the gain process for the system introduced in thissection.

If the filter system is given by

dYt = µ(t, Yt)dt+ σ(t, Yt)dVt

dZt = b(Yt)dt+ dWt.

The the filter equation for Yt is given by

dYt = µtdt+(Ytbt − Ytbt

)dvt

dvt = dZt − btdt.

Where Ytbt − Ytbt is the gain process.

Theorem 9 (The FKK Filtering Equations)

The main idea we should take with us from this section is that the innovationprocess contains the observable dynamics of the agent. Therefore, we should beable to rewrite our observation processes diffusion parts using the innovationprocesses. This would make the system being determined by the process dZt

and hence adaptable to it. We will use the idea later when deriving a system forthe assets prices and variance process that is adapted to the agents informationset.

In order to derive a concrete result, we limit our model in Assumption 7to a linear version and use the Kalman-Busy filter equation to derive the filterdynamics of the underlying processes.

31

6.4 The Linear model

In this section, the underlying dynamics is in linear form and it affects the assetdrift in a linear way. It is also assumed that the agent has some knowledgeabout what the underlying system starting value a0 is at time zero. He canexpress his degree of certainty by a normal distributed around the value a0 witha covariance of Γ0.

We begin now by setting up our model for the linear case and is stated inthe Assumption 10.

The asset’s price process under P is given by

dSt = D(St) (µ0(t) + µ1(t, σt)at) dt+D(St)D(σt)dW1t . (40)

The volatility process is given by

dσ2t = λ(t, σt)dt+ b(t, σt)

(ρdW 1

t + ξdW 2t

)(41)

The process at that affects the drift is

dat = (b0(t) + b1(t)at)dt+ c(t)dW 3t . (42)

The bank account with short term rate r is given by

dBt = rBtdt. (43)

W 1t , W

2t andW 3

t are all independent Rn dimensional Brownian motionsunder measure P.

Assumption 10

The fact that the volatility process σ2t is observable stills holds in this case

from the quadratic variation. Now we set up the observable system for ourmodel.[

dSt

dσ2t

]=

([D(St)µ0

λt

]+

[D(St)µ1

0

]at

)dt+

[D(St)D(σt) 0

btρt btξt

] [dW 1

t

dW 2t

](44)

We can simplify this system further by making the log transformation of theasset prices.[

dZt

dσt

]=

([µ0 − 1

2σt

λt

]+

[µ1

0

]at

)dt+

[D(σt) 0btρt btξt

] [dW 1

t

dW 2t

](45)

Now, we will use the Kalman-Bucy filter (Theorem 11) [36] for (45) to derivethe dynamics of at. Theorem 11 clarifies also the calculation steps we have totake.

32

Let W1 and W2 be two mutually independent Wiener processes, where

W1 =[W11, . . . ,W1k

], W2 =

[W21, . . . ,W1l

].

With a hidden state process and observable process

Yt = (Y1(t), . . . , Yk(t)), Zt = (Z1(t), . . . , Zl(t))

where they have the differential from of

dYt = [a0(t, Zt) + a1(t, Zt)Yt] dt+2∑

i=1

bi(t, Zt)dWi(t)

dZt = [A0(t, Zt) +A1(t, Zt)Yt] dt+2∑

i=1

Bi(t, Zt)dWi(t).

Let (a b) = a1bT1 + a2b

T2 . Then the optimal filter is given by

dYt =[a0(t, Zt) + a1(t, Zt)Yt

]dt+

[(b B)(t, Zt) + ΓtA

T1 (t, Zt)

]((B B))

−1

×[dZt −

(A0(t, Zt) +A1(t, Zt)Yt

)dt]

and the error covariance matrix Γt = E[(Yt − Yt)(Yt − Yt)

T]is the

solution to the following equation

Γt = a1(t, Zt)Γt + ΓtaT1 (t, Zt) + (b b)(t, Zt)−

[(b B)(t, Zt) + ΓtA

T1 (t, Zt)

]× ((B B))

−1(t, Zt)

[(b B)(t, Zt) + ΓtA

T1 (t, Zt)

]T.

With a starting condition Y0 and Γ0.

Theorem 11 (Kalman-Bucy filter)

So, from our observable systems (45) and (42), we derive that the filterdynamics of at is

dat = (b1 + b2at) dt+ ΓtµT1 D(σt)

−1dW 1 − ΓtµT1 D(σt)

−1ρT ξ−T dW 2

where dW 1 and dW 2 is given by[dW 1

t

dW 2t

]=

[D(σt) 0btρt btξt

]−1([dZdσ2

t

]−([

µ0 − 12σt

λ

]+

[µ1

0

]at

)dt

)(46)

and (46) is the innovation process for our system.The (B B)−1 term that turns up in the error covariance matrix will in our

case is equal to

(B B)−1 =

([D(σt) 0btρt btξt

] [D(σt) ρTt b

Tt

0 ξTt bTt

])−1

=

[D(σ2

t )−1 +D(σt)

−1ρTt ξ−T ξ−1

t ρtD(σt)−1 −D(σt)

−1ρT ξ−T ξ−1b−1

−b−T ξ−T ξ−1ρtD(σt)−1 (btξtξ

Tt b

Tt )

−1

]

33

so the error covariance matrix equation is

Γt = a1Γt + ΓtaT1 + ccT − Γtµ

T1

[D(σ2

t )−1 +D(σt)

−1ρT (ξξT )−1ρD(σt)−1]µ1Γt.

We see that in case we reverse and transform (46) back into the form we hadin (44), the resulting system would be[dSt

dσ2t

]=

([D(St)µ0

λt

]+

[D(St)µ1

0

]at

)dt+

[D(St)D(σt) 0

btρt btξt

] [dW 1

t

dW 2t

]. (47)

Equation (47) is a representation of (44) but with respect to the FS filtration.This way of representing the assets and volatility dynamics in terms of theinnovation process and the at have transformed them into FS adapted systemsthat the agent can use when making his allocation decision.

We restate our conclusion we have drawn so far in this section in Theo-rem 12 where all the relevant components for our optimization problem aredescribed.

The Ft adapted system in Assumption (10) can be represented as a FS

adapted system where the assets and variance process will be given by

dSt = D(St) (µ0(t) + µ1(t, σt)at) dt+D(St)D(σt)dW1t (48)

dσ2t = λ(t, σt)dt+ b(t, σt)

(ρtdW

1t + ξtdW

2t

). (49)

Where the filtered drift is given by

dat = (b1 + b2at) dt+ ΓtµT1 D(σt)

−1dW 1t

− ΓtµT1 D(σt)

−1ρT ξ−T dW 2t (50)

with error covariance matrix as

Γt = a1Γt + ΓtaT1 + ccT

− ΓtµT1

[D(σ2

t )−1 +D(σt)

−1ρT (ξξT )−1ρD(σt)−1]µ1Γt. (51)

Where W 1t and W 2

t are given by (46). The bank account is still the sameas in Assumption (10)

Theorem 12

We will now in the next section move over to the optimization of the agent’sallocation.

6.5 Deriving the PDE

We still face the problems that the market is not complete because of thestochastic volatility. To resolve this, we introduce fictitious assets Ft againbut this time, they are driven by the innovation processes.

dFt = λ(t, σt)vdt+ b(t, σt)

(ρdW 1

t + ξdW 2t

)(52)

34

Having this system and fictitious assets at hand, the whole derivation ofthe wealth process will be conducted in the same fashion as done before in theprevious section.

What difference now is that the driving diffusion factor will be an innovationprocess and the process Ht will now depend on at. To make the distinctionbetween them, we will denote this new process as Ht. One other aspect thatdiffer from our previous steps is the martingale representation that is expressedin terms of dWt rather then the innovation dWt we have now. However, [16]provide a martingale representation theorem where the diffusion part is drivenby the innovation process and it is that one we use here.

Repeating the same steps as we did in the completed market section, weconclude that the wealth process for the optimal portfolio is

Xt =Ht

H0(K0,tL

vt )

− 11−γ x.

Where the Ht is given by

Ht = EPv0

[exp

β

∫ T

t

(r +

1

2(1− γ)||φ(at)t||2

)ds

]. (53)

Here, φ(at)t has the same structure as in (19), except that µ has been replacedwith µ0 + µ1(t, σt)at from (48).

Using the same reasoning as in [15] that the function Ht can be written asH(t, at, σ

2t ) and given the dynamics of (50) and (49). Ito’s lemma tells us that

the dynamics of Ht will be

dHt =

∂H

∂t+A(H)

dt+

(∂H

∂aΓtµ

T1 D(σt)

−1 +∂H

∂σ2bρ

)dW 1v0

t

+

(∂H

∂σ2bξ − ∂H

∂aΓtµ

T1 D(σ)−1ρT ξ−T

)dW 2v0

t .

Where A(H) is the infinitesimal generator.The matrix σt from previous sections has not changed and the fact that the

matrix Ht can be written as a geometric Brownian motion still holds as well(can be proved by repeating the same steps in Lemma 2).

Using Ito’s lemma on the discounted wealth processes once again gives usthat the optimal portfolio allocation is

πt = (σH − 1

1− γφ(at)

Tt )σt

−1

= σH σt−1 − 1

1− γφ(at)

Tt σt

−1.

As before, when extracting vt, we adjust the kernel vt to make the weightsrepresenting the investment in fictitious assets become zero. The hedging de-mand position and the corresponding deterministic position component will berespectively be equal to

35

σH σ−1t =

1

Ht

[∂H∂a Γtµ

T1 D(σt)

−1 + ∂H∂a Γtµ

T1 D(σt)

−1ρT ξ−Tt ξ−1

t ρD(σt)−1

∂H∂σ2 − ∂H

∂a ΓtµT1 D(σt)

−1ρT(ξtξ

Tt

)−1b−1

]T

1

1− γφ(at)

Tt σt

−1 =1

1− γ

[φTt D(σt)

−1 − vTt ξ−1ρD(σt)

−1

vT ξ−1b−1

]TSo, vt has to be chosen so that

0 =∂H∂σ2

Ht−

∂H∂a

HtΓtµ

T1 D(σt)

−1ρT(ξtξ

Tt

)−1b−1 − 1

1− γvT ξ−1b−1 ⇒

vTt = (1− γ)∂H∂σ2

Htbξ − (1− γ)

∂H∂a

HtΓtµ

T1 D(σt)

−1ρT ξ−T .

Under this kernel vt, the resulting optimal allocation for our portfolio willbe

πt =1

1− γ(µ0 + µ1(t, σt)at − 1r)TD(σt)

−1 +∂H∂σ2

HtbρD(σ)−1

+∂H∂a

HtΓtµ

T1 D(σt)

−1. (54)

For this vt, the processes σ2t and at will have the following dynamics under

the Pv0 -measure

dσ2t = (λ− βbρφt − βbξvt) dt+ b

(ρdW 1v0

t + ξdW 2v0t

)dat =

(b1 + b2at − βΓtµ

T1 D(σt)

−1(φt − ρT ξ−T vt

))dt

+ ΓtµT1 D(σt)

−1(dW 1v0 − ρT ξ−T dW 2v0

).

which are the relevant ones, since that is the probability measure that Ht

depends on is Pv0 .Ht can now be determined through the Feynman-Kac equation.

∂H

∂t+ µ∇H +

1

2Tr[σT∇2Hσ

]+ β(r +

1

2(1− γ)||φt||2)H = 0

H(T, σ2T , aT ) = 1.

where

µ =

[λ− βbρφ+ βbξv

b1 + b2a− βΓtµT1 D(σ)−1

(φ− ρT ξ−T v

)]σ =

[bρ bξ

ΓtµT1 D(σ)−1 −Γtµ

T1 D(σ)−1ρT ξ−T

].

This equation is hard to solve. Even if we limit ourselves to the case whenthere is one asset, we would still have to solve the error covariance matrixequation which does not have general solutions for a general process at [1].

36

Some interesting conclusion we can draw from (54) is the impact of notknowing the drift component. In our Assumption (10), the diffusion W 3

t is notcorrelated with W 1

t and W 2t , but we still get an additional hedging component

in (54) from at. As we commented before in the complete observable case whenρ = 0, the hedging component from the volatility disappeared. In this casewhen ρ = 0, the volatility hedge disappeared also but we would still hedge theparameter uncertainty from at.

Another interesting fact we can observe from (54) in the case when at is anunobservant constant. Then dat = 0 and we see that (51) will tend to zero overtime. We can then think of the situation when the agent has a long investmenthorizon, at the furthest end of the period he would allocate his capital almostas if he had complete information about a and the allocation would be that onewe derive in (31).

6.6 Section Summary

In case of the stochastic drift and the delimitation of only observing the priceprocess St, we have been able to filter the stochastic factor in the asset drift andobtain a solution for the portfolio allocation in this incomplete market setting.We have also derived a PDE that solves Ht that is needed to be solved in orderto obtain a solution in a feedback form for the weights.

37

7 Summary

In this thesis, we have successfully overcome the shortcomings of applying themartingale method in incomplete market by fictitious completion when the riskyassets exhibit stochastic volatility and stochastic drift.

We have given an introduction to the martingale method in a completemarket setting and discussed some of the cases when it cannot be used whenthe market is incomplete.

We have showed how one can resolve this issue by fictitious completion andthe path how one derives the optimal allocation from it.

The solution is based on assigning the value of the market price of risk for theunhedgeable risk in the fictitious market so the agent hold none of the fictitiousrisky assets.

This has been showed to lead to the optimal portfolio allocation for theagent.

In case of full information, we have been able to derive an allocation policyfor the agent on a feedback form in the one dimensional case when the riskyasset and volatility dynamic are in the Heston framework.

The solution of the Heston model problem was previous know but has beenderived by solving the Hamilton-Jacobi-Bellman equation. We have in the ap-pendix included the derivation of the HJB equation for our model in the ob-servable case and show the similarities the HJB and our PDE

The two main results from this thesis are the portfolio allocation the agenttakes when the risky assets he can invest in exhibit stochastic volatility in thecases when he has full and partial information about the market.

From our derived allocation policies in (31) and (54), we have gained insightson how the different market settings impact the optimal portfolio allocation.

In case of continuous time when the agent can observe the volatility com-ponent, a non-zero correlation between the driving Wiener processes introducesan additional hedging allocation the agent has to incorporate into his portfolio.

In the case with stochastic drift, the agent also hedge the drift uncertaintyfor the risky assets and the magnitude of it depends on the level of precision bywhich it can be estimated.

38

A Appendix

A.1 The Dual problem

As mentioned before, the martingale approach fails in case there is not a uniquemeasure Q.

We extend the theoretical background for the martingale approach by in-troducing the dual problem formulation and looking into its fitting in in theincomplete market framework, where the fictitious assets are introduced. Thissection represents a summary of the material found in [37],[41], [31] and [38].

We start with the primary problem. The agent wants to maximize his wealthXt over the time horizon T with a starting capital of x.

maxπt

EP [U(XT )] (55)

where

dXt = rXt + πt

(σtdW

1t + (µt − r1)dt

)This constraint problem (with respect to the Xt dynamics) can be turned

into an unconstrained one by introducing the Lagrangian process Yt

dYt = αYtdt+ βYtdW1t

and the two different forms of the integral∫ T

0YsdXs.∫ T

0

YtdXt = XTYT − xY0 −∫ T

0

XsdYs − ⟨X,Y ⟩T (56)∫ T

0

YtdXt =

∫ T

0

Ys(rXt + πt(µt − r1))dt+

∫ T

0

YtπtσtdW1t (57)

Here, the first equality comes from partial integration where ⟨·, ·⟩T stands forquadratic variation and the second one follows from the definition of the wealthprocess.

We can further expand the (56) by inserting the dynamics of Yt and thevalues derived from the quadratic variation, which is equal to

⟨X,Y ⟩T =

∫ T

0

πtσtβtYtdt

Furthermore, those integrals that are with respect to W 1t will vanish when

one take the expectation on them.We can now formulate the Lagrangian relaxation of our problem that we

use as a upper bound to our primal problem by adding and subtracting (57)and (56). Later when we minimize it with respect to the process Yt, we shouldobtain equality.

Λ(Yt) = maxX≥0,π

EP

[U(XT )−XTYT + xY0 +

∫ T

0

Ys (rXt + πt(µt − r1) + αtXt + πtσtβt) dt

]

Now, the process Xt is free from its dynamic’s constraints by this relaxationas it became when we did the martingale formulation of the primal problem inthe Martingale method section.

39

However, since it has to remain positive due to the form of the utility func-tion, this constraint appears in the formula above.

Additionally, when maximizing this expression, we have to make sure thatit remains finite. This will therefore add requirements for the parameters in theintegral expression.

We see that the parts depending on the wealth πt have to cancel each other(to prevent infinitely long and short positions) and the part, which only dependson Xt cannot be positive, since it would lead to an infinite objective value.

Hence, αt and βt have to be chosen so that

r + αt ≤ 0

σtβt + µt − r1 = 0

The part which depends on XT will obtain its maximum in

U(YT ) = maxXT>0

[U(XT )−XTYT ] = U(I(YT ))− YT I(YT ), 0 < y < ∞ (58)

where I = U−1x . The expression (58) is usually referred to as the convex

conjugate or the Fenchel-Legendre transformation.Since U(YT ) is decreasing, YT wants to be as large as possible. We conclude

therefore that the slack condition r + αt ≤ 0 will hold with equality, since YT

is determined to be as large as possible, and it does so by choosing αt to thelargest possible value, −r.

Due to this, the integral expression gives a zero contribution, so our problemis reduced to

infY

Λ(Y ) = infY

EP[U(YT ) + xY0

](59)

Since αt = −r and the dual Yt are in geometric from, we can write thefollowing

YT = Y0 exp

−∫ T

0

rdt

exp

∫ T

0

βtdW1t − 1

2

∫ T

0

β2t dt

The dual Yt consists of a discounting factor and the changes of the measurecomponent. Notice that if σt is invertible, this problem would be identical tothe martingale approach.

Now, we modify this example to fit our extended problem formulation, whichincludes the stochastic volatility and the additional process dW 2

t .Firstly, we add the additional Brownian motion to our dual process.

dYt = αYtdt+ βtYtdW1 + vYtdW

2

Repeating the same steps in the sections above, we conclude that the dualis

YT = Y0 exp

−∫ T

0

rdt

exp

∫ T

0

φtdW1t +

∫ T

0

vtdW2t − 1

2

∫ T

0

||φt||2 + ||vt||2dt

= Y0K0,TL

vT (60)

40

where φt = σ−1(1r − µ) = β but vt is free. If we replace Y0 with Λ, weobtain the same expression for the first order condition in our primary problemin the incomplete case.

U ′(XT ) = ΛK0,TLvT

We can simplify the objective function (59) a bit more. From (60), one canobserve that Y0 will be detainment from the budget constraint of our problem.Hence, it is possible to drop it in (59), and since (60) there is only the processvt that is free in the expression, the optimization problem can be reformulatedas

infY

EP[U(YT )

]= inf

vEP[U(ΛK0,TL

vT )]

(61)

We highlighted before that there were no constraints on vt, which is true. Butin order to interpret Lv

T as a likelihood function, we will impose the constraint

that EP[exp(

∫ T

0||vs||2dt)

]< ∞ for it to become a martingale by fulfilling the

Novikov condition.However, the cases when vt fails the Novikov condition, the objective func-

tion would not be finite as well, which we do not allow.Assume for a moment that we have found the optimal vt for (61). The

optimal wealth process would be given by

Xt = EP[Kt,TL

vT

Lvt

I(ΛK0,TLvT )|Ft

]where Λ has been chosen so that it resolves EP

[LvT I(ΛK0,TL

vT )]= x.

The problem we face now is to find vt, which is not an easy task unlessthe utility function of the agent is chosen attentively so that it simplifies theproblem. For instance, if the utility function is represented by the logarithmicfunction i.e U(x) = log(x) , we would find that the objective function (61) with

the convex conjugate U(y) = −1− log(y) becomes

infvt

EP[U(ΛK0,TL

vT )]= −1− EP [log(ΛK0,TL

vT )]

= −1− log(Λ) +

∫ T

0

rdt+1

2infvt

EP

[∫ T

0

||φt||2 + ||vt||2dt

]

for Λ > 0. We see that the optimal vt for it will be vt ≡ 0. This will not bethat simple for the power utility. Even if this method of deriving the kernel vtis correct, one ends up with solving another complex problem and this pointsout the benefits by treating the problem as they did in [23].

A.2 Connection to Hamilton Jacobi Bellman Equation

Introduction

We concluded that our approach to solving the issue with incomplete marketsleads us to a non-linear partial differential equation. Those kinds of equationsusually appear when the optimal portfolio problem is solved by the dynamicprogramming approach.

41

We will therefore in this section see why these two methods coincide witheach other. Firstly, we give a short introduction of the Hamilton-Jacobi-Bellmanequation and its application to our problem in order to compare it to the PDEwe got from the martingale method. Many of the points and comments aretaken from [42] and [6].

HJB equation for optimal portfolio

In the examples to follow, we limit ourselves to the one-dimensional case of theproblem for the sake of simplicity.

As before, an agent has one risky tradable asset, which exhibits stochasticvolatility, and a bank account, where he can store his wealth. The dynamics ofeach instrument are denoted by

dBt = rBtdt

dSt = µStdt+ σtStdW1

dσ2t = λ(t, σt)dt+ b(t, σt)dW

2

where W 1 and W 2 are one-dimensional Wiener processes that are correlatedwith coefficient ρ. Agents wealth Xt is the following, according to the model

dXt = rXtdt+ (µ− r)πtXtdt+ σtXtπtdW1

as in previous cases, agent intends to maximize his utility over a time period[0, T ].

J(x, σ, 0|πt) = EP [U(XT )|X0 = x, σ0 = σ]

From the above, it is possible to define the value function that correspondsto the utility the agent obtains while choosing the optimal allocation.

u(x, σ, t) = supπt

J(x, σ, t|πt)

The resulting value function has to satisfy this following PDE

ut +1

2b2uσ2σ2 + λuσ2 + rxux +max

πt

[1

2π2t σ

2x2uxx + ρbσπtxuσ2x + (µ− r)xπtux

]= 0

This PDE is referred to as the HJB equation.The allocation πt can be obtained from the first order condition of the com-

ponents dependent on πt. In this case, it equals

0 = πtσ2x2uxx + ρbσxuσ2x + (µ− r)xux ⇒

πt = −(ρbσxuσ2x + (µ− r)xux

σ2x2uxx

)Substituting πt expression into the HJB reduces it to.

ut +1

2b2uσ2σ2 + λuσ2 + rxux − 1

2σ2x2uxx((µ− r)xux + ρbσxuσ2x)

2= 0

42

Here, we look into a good ansatz how the value function looks like in order tosimplify it. One that looks promising is to guess that u is of the following form

u(x, σ2, t) =xγ

γV (σ2, t)

Inserting this and its corresponding derivatives reduces the HJB into

Vt +1

2b2Vσ2σ2 + λVσ2 + γrV +

γ

2(1− γ)σ2V[(µ− r)V + ρbσVσ2 ]

2= 0

V (σ2T , T ) = 1

We can make an additional transformation by replacing V with the following

V (σ2t , t) = v(σ2

t , t)δ

Which turns it into

0 = vt +1

2b2vσ2σ2 +

(λ+ ρ

γ(µ− r)b

(1− γ)σ

)vσ2 +

1

δ

(rγ +

γ(µ− r)2

2(1− γ)σ2

)v

+b2

2

v2σ2

v

((δ − 1) + ρ2

γ

1− γδ

)If we choose δ cleverly, we would be able to reduce some of the more com-

plicated terms.For instance, if we choose δ to be

1− γ

1− γ + ρ2γ

the squared component disappears and we end up with

vt +1

2b2vσ2σ2 + (λ+ c) vσ2 + kv = 0 (62)

with boundary condition v(σ2T , T ) = 1 and where

c = βbρ(µ− r)

σ

k =γ

δ

(r +

(µ− r)2

2(1− γ)σ2

)If we set up the equation we had for our problem with the optimal kernel vt

we derived, we would get

0 =∂H

∂t+

1

2b2Hσ2σ2 +

(λ+ βbρ

(µ− r

σ

))Hσ2 + β

(r +

(µ− r)2

2(1− γ)σ2

)H

− γ

2(1− ρ2)b2

H2σ2

H

1 = H(σ2T , T )

Now, it is not easy to see, but if we replace H(σ2t , t) with v(σ2

t , t)a, where.

a =1

1− γ + γρ2

43

We would obtain the same system as in (62). In [6] and [12], they point out thatwhile solving the portfolio problem by the martingale method, one is essentiallysolving it for the dual variable obtainable from the first order condition aswe discussed in the previous appendix section. In order to find the requiredallocation, one has to invert it back into the primary space where the portfolioweights are.

44

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