Fictitious Completion as a Tool to Solve Utility Maximisation Problems

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Fictitious Completion as a Tool to

Solve Utility Maximisation

Problems

Candidate Number: 244754

Mathematical Institute

University of Oxford

A thesis submitted in partial fulfillment of the MSc in

Mathematical and Computational Finance

June 26, 2015

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Abstract

In this paper, we apply the technique of fictitious completion (Karatzas et al. (1991)) and the martingale approach to study the problem of utility maximisation in a Markovian stochastic volatility model. We show that this method gives a solution to our incomplete market problem which is consistent with the distortion power solution (Zariphopoulou (2001)) obtained from the dynamic programming approach.

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Acknowledgements

I would like to express my gratitude to my supervisor, Professor Michael Monoyios, for giving me the opportunity to study this interesting topic. Without his guidance, support and patience, this thesis would not have been possible.

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Chapter 1 Introduction

Optimal investment and portfolio selection is one of the classical problems in financial economics. In his seminal paper, Merton (1971) considered the problem of maximising the expected utility of terminal wealth of an agent investing in a risky asset and a risk-free asset. Since then, many variants of the original problem have been studied.

There are generally two main approaches to the Merton problem. The traditional approach uses dynamic programming. This method transforms the original stochastic optimal control problem into the problem of solving a non-linear deterministic partial differential equation (PDE), namely the Hamilton–Jacobi–Bellman (HJB) equation. The probabilistic nature of the problem disappears once the HJB equation has been formulated. This method assumes a Markovian structure but allows us to study the problem using PDE techniques.

However, in general, the associated HJB equations are difficult to solve analytically and explicit solutions are rare. Merton (1971) obtained explicit solutions for the complete market case with constant parameters by applying the dynamic programming principles. In the context of utility maximisation in a stochastic volatility model, Zariphopoulou (2001) derived an explicit solution, known as a distortion power solution, by using a power transformation to show that the value function of the optimisation problem is a solution to a linear PDE.

An alternative approach to the Merton problem is based on martingale theory. This approach is more general and does not assume a Markovian structure, thus allowing weaker assumptions on the dynamics of the underlying. In this method, we first look for an optimal terminal wealth by solving a static optimisation problem. The optimal portfolio is then recovered by replicating the optimal terminal wealth. This method was developed by Harrison and Kreps (1979), and Harrison and Pliska (1981) in the context of option pricing. The martingale approach was first explicitly applied to the problem

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of utility maximisation by Pliska (1986), Karatzas et al. (1987) and Cox and Huang (1989). Cvitanić et al. (2001) solved a more general problem where an incomplete market with an unhedgable endowment is only assumed to follow a semi-martingale model.

A financial market is complete if any contingent claim on an underlying asset can be perfectly replicated by a hedging portfolio in the underlying asset. It is relatively easy to solve the complete market problem. In many cases, the conditions for the existence of an optimal portfolio can be found and a representation of the optimal policy can be derived. The incomplete market case is much more complex. A market can be incomplete for many reasons. It may be the case that an underlying factor is not a tradeable asset, there may be a short-selling prohibition on an asset or other constraints may be imposed on an agent’s portfolio.

The idea of a fictitious completion was first introduced by Karatzas et al. (1991) to study the problem of utility maximisation in an incomplete market. This is a set of artificial stocks whose role, when added to the original stocks, is to serve to complete the market. The resulting complete market optimisation problem can then be solved. By adjusting the drift of the fictitious stocks, so that the optimal position in these stocks is null, one in principle obtains the solution to the original incomplete market utility maximisation problem.

This procedure is intimately related to the selection of an optimal dual measure that achieves the infimum in the dual to the primal utility maximisation problem. Delbaen et al. (2002) studied the dual problem, establishing a relationship between utility maximisation and minimising entropy measure. Hobson (2004) and Monoyios (2006) studied the optimal dual measure in a stochastic volatility model. Monoyios (2006) derived representations for the optimal dual measure by using the distortion

power solution.

There are few, if any examples of using the method of fictitious completion in an explicit example of an incomplete market. In this paper, we apply the technique of fictitious completion to solve the utility maximisation problem in a stochastic volatility model.

There is strong empirical evidence that the volatility in financial markets is not constant over time. Econometric research on stock return volatility (survey articles by Hobson (1998) and others) finds that changes in stock return volatility are persis-tent, and that stock price movements tend to be negatively correlated with volatility. Research also suggests that volatility exhibits mean reversion and that stock returns follow a distribution with a heavier tail than a normal distribution.

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In this paper, we assume a two-factor Markovian model adopted from Monoyios (2006) containing a stock S and a stochastic volatility Y, driven by two correlated Brownian motions. We introduce a fictitious stock P to create an artificially complete market. By using the martingale approach in the extended market, we derive the market price of risk such that the optimal position in the fictitious stock is zero. The resulting optimal portfolio in this choice of market completion should give us the solution to our original problem since it can be realised in the incomplete market.

We then approach the problem directly using dynamic programming to obtain the distortion solution. We study the relation between the function H appearing in the distortion solution and the function Hffrom the solution given by fictitious completion,

and show that the two approaches give us the same solution. We also study the dual to the primal utility maximisation problem. Using the representation of the dual optimal measure in terms of H derived by Monoyios (2006), and the relation between H and

f

H, we verify that the dual optimal measure corresponds with the martingale measure characterising the least favourable market completion.

Finally, we look at the optimal consumption problem briefly. We apply the fictitious completion method to derive a condition on the market price of risk in the least favourable completion where the optimal position in the fictitious stock is zero.

The rest of this paper is organised as follows. In Chapter 2, we briefly review the idea of a fictitious completion and the results from Karatzas et al. (1991). Chapter 3 presents our stochastic volatility model. In Chapter 4, we present the primal utility maximisation problem. We derive both the solution by the fictitious completion method and the distortion solution. We then show that they are consistent. Chapter 5 is concerned with the dual to our primal utility maximisation problem. Chapter 6 briefly extends the fictitious completion method to the problem of optimal consumption. Finally, Chapter 7 concludes.

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Chapter 2 Fictitious Completion

In this chapter, we present the idea of fictitious completion developed by Karatzas et al. (1991) and briefly review their main results.

In their paper, they considered a model, not necessarily Markov, which consists of a bond and m stocks, with the stocks being driven by a d-dimensional Brownian motion. It is assumed throughout that m≤d. The interest rate, the stock appreciation rates and the stock volatility coefficients are taken to be random processes adapted to the

d-dimensional Brownian motion.

The problem studied is that of an agent, endowed with a fixed initial capital, seeking to maximise the expected utility of the terminal wealth by investing in the bond and the stocks.

The market is incomplete whenm is strictly smaller than d. It is then typically not possible to construct a portfolio consisting of the bond and the stocks which completely hedges the risk associated with this market.

They began by solving the problem in the complete market case (m =d) by using the martingale approach. The unique risk neutral measure which exists in an arbitrage-free market is determined. Under this measure, the expectation of the discounted terminal wealth must equal the initial endowment. This condition, called the budget constraint, must hold for any terminal wealth which can be replicated by an admissible portfolio. Hence, the budget constraint characterises the set of attainable terminal wealth. The optimal terminal wealth is then determined from this set. Finally, it is shown that an optimal portfolio which replicates the optimal terminal wealth can be constructed. The martingale representation theorem, which states that any martingale with respect to a Brownian filtration can be represented as a stochastic integral with respect to the Brownian motion, ensures that an optimal portfolio can always be found. However, this line of argument fails in the incomplete market.

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They then approached the problem in an incomplete market by the method of fictitious completion. Fictitious stocks, parametrised by their rates of appreciation, are introduced to create an artificially complete market. Intuitively, under an optimal portfolio which also invests in the fictitious stocks, an agent will hold a long position in a stock with a high appreciation rate. Whereas for a stock with a low appreciation rate, the agent is likely to hold a short position. Therefore, we expect to be able to adjust the appreciation rates of the fictitious stocks so that the agent, by optimal choice, does not invest in them at all, giving us a portfolio which must then be the solution to the original incomplete market problem. The fictitious completion resulting in an optimal portfolio which does not invest in the fictitious stocks is called the least favourable completion. This completion is the least advantageous because the portfolio which is optimal under this completion is available under every other fictitious completion.

So, the incomplete market problem can be solved in two steps. First, for every fictitious completion, the optimal portfolio which maximises the expected utility of terminal wealth is determined. Then, the portfolio from the completion which minimises the maximum expected utility is chosen.

In their paper, they also parametrised the market completions by a set of local martingales before giving various equivalent characterisations of the local martingale corresponding to the least favourable fictitious completion. They then studied the dual to the problem of utility maximisation and its relations to the primal problem. They showed that a solution to the dual problem induces one for the primal problem. So, showing the existence of a solution to the dual problem is sufficient to prove the existence of an optimal portfolio for the primal problem.

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Chapter 3 The Market Model

Let (Ω,F,_F := (Ft)(0≤t≤T),P) be a filtered probability space, where the filtration

F is the P-augmentation of that generated by a two-dimensional Brownian motion

W = (W(1)_{, W}(2)₎∗ _and_T _{is the terminal time. The superscript}∗ _{denotes the transpose}

of a vector or a matrix.

In our model, a traded stock price S := (St)(0≤t≤T) and a stochastic volatility

Y := (Yt)(0≤t≤T) follow the dynamics

dSt=YtSt(λ(Yt)dt+dW

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t ),

dYt=a(Yt)dt+b(Yt)dBt,

(3.1)

where the Brownian motionB is correlated with W(1) _{according to}

d⟨B, W(1)⟩t=ρdt, B =ρW(1)+

q

1−ρ2_W(2)_, _ρ_∈₍₋₁_,₁₎_.

We assume that λ, a andb are such that (3.1) admits a unique strong solution. The interest rate is taken to be zero for simplicity. There is no loss of generality in cases where interest rates are deterministic.

Forq ∈(0,1], consider the setMf of local martingale measures_Q(q,ψ) defined via

the density processes Z(q,ψ)_{:= (}_Z(q,ψ)

t )(0≤t≤T) given by

Z_t(q,ψ) :=E(−qλ·W(1)−qψ·W(2))t, 0≤t≤T, (3.2)

whereE denotes the stochastic exponential,

(λ·W(1))t = Z t 0 λ(Ys)dWs(1), (ψ·W (2)₎ t = Z t 0 ψsdWs(2),

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and ψ := (ψt)(0≤t≤T) ∈ Ψ is a process satisfying

RT

0 ψ2tdt < ∞ almost surely. If,

in addition, Z(q,ψ) _{is a martingale , then we can define probability measures}

Q(q,ψ) equivalent to _P by d_Q(q,ψ) d_P |Ft =Z (q,ψ) t , 0≤t ≤T,

and we assume this is the case here. (A sufficient condition for this is_E[exp{1 2

RT

0 ψt2dt}]<

∞.)

By the Girsanov theorem, the two-dimensional processWQ(q,ψ) _{= (}_WQ(q,ψ),(1)_{, W}Q(q,ψ),(2)₎∗

defined by WtQ(q,ψ),(1) :=W (1) t +q Z t 0 λ(Ys)ds, WQ (q,ψ)_,₍₂₎ t :=W (2) t +q Z t 0 ψsds, 0≤t≤T,

is a two-dimensional _Q(q,ψ)_{-Brownian motion. Lastly, we define the subset} _{M ⊆}_M_f _to be the set containing measures of the form _Q(1,ψ)_.

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Chapter 4 Optimal Investment: The Primal

Problem

We begin this chapter by outlining the problem of utility maximisation. We then combine the technique of fictitious completion (Karatzas et al. (1991)) and the mar-tingale approach to study our problem in Section 4.2. In the next section, we study the problem using the alternative approach of dynamic programming, deriving the distortion power solution (Zariphopoulou (2001)). In Section 4.4, we compare the solutions obtained from the two approaches and show that they are consistent with each other.

4.1 The Primal Problem

Consider a self-financing portfolio investing in the stock S with the corresponding wealth process X := (Xt)(0≤t≤T). Let π(1) := (π

(1)

t )(0≤t≤T) be the proportion of wealth invested in the stock, representing a trading strategy. We assume that π(1) _{is an} adapted process satisfying RT

0 (Ytπt)2dt <∞ almost surely. Then, X has dynamics

dXt=π(1)t XtYt(λ(Yt)dt+dWt(1)). (4.1)

Given a concave utility function U and an initial capital x, our problem is then to maximise the expected utility of terminal wealth at time T:

u(x) := sup

π(1)_∈AE

[U(XT)]. (4.2)

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4.2 The Martingale Approach with Fictitious Completion

4.2 The Martingale Approach with Fictitious

Com-pletion

We introduce a fictitious stock whose price process P := (Pt)(0≤t≤T) satisfies

dPt=Pt(µ(Yt)dt+β(Yt)dBe_t), (4.3)

whereβ(Yt)̸= 0 and the Brownian motion Be is correlated withW(1) according to

d⟨B, We (1)⟩_t=γdt, Be =γW(1)+ q

1−γ2_W(2)_, _γ _∈₍₋₁_,₁₎_,

thus creating a fictitious complete market. γ and β will be held fixed throughout and the drift µof the fictitious stock will be considered a parameter.

We define a vector-valued process, α := (αt)(0≤t≤T) and a matrix-valued process

σ := (σt)(0≤t≤T) as follows: αt :=   Ytλ(Yt) µ(Yt)  , σt :=   Yt 0 γβ(Yt) √ 1−γ2_β₍_Y t)  . (4.4)

Since γ ̸=±1, σ is invertible. We also define the vector-valued market price of risk process θ := (θt)(0≤t≤T), where θt= (θ (1) t , θ (2) t )∗ :=σt−1αt. Thus, θ_t(1) =λ(Yt), θ (2) t = 1 √ 1−γ2[ µ(Yt) β(Yt) −γλ(Yt)]. (4.5)

We impose the condition thatθ(2)_{must be such that}RT

0 ∥ θt∥2 dt <∞almost surely,

where ∥ θt∥2 = (θ

(1)

t )2 + (θ

(2)

t )2. Then, θ(2) ∈ Ψ. On the other hand, given θ(2) ∈Ψ,

we can obtain the driftµfrom the equation µ_θ(2)(Y_t) :=β(Y_t)(

√

1−γ2_θ(2)

t +γλ(Yt)).

Hence, the fictitious completions of our market can be parametrised by the set Ψ. Again, for a given utility function U, consider an agent with an initial wealthx, trading a self-financing portfolio which now involves both S and the fictitious stock

P. Define a vector process π := (πt)(0≤t≤T), where πt = (π

(1)

t , π

(2)

t )∗ and π(1), π(2)

are the proportions of wealth in S and P respectively. Let X := (Xt)(0≤t≤T) be the corresponding wealth process. Then, X has dynamics

dXt=Xt{[π (1) t Ytλ(Yt) +π (2) t µ(Yt)]dt+ [π (1) t Yt+π (2) t γβ(Yt)]dW (1) t + [π(2)_t q1−γ2_β₍_Y t)]dW (2) t } =Xtπt∗αtdt+Xtπt∗σtdWt. (4.6)

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We now study the optimal investment problem in this artificially complete market. The problem is to maximise_E[U(XT)], with the initial conditionX(0) =x, over KT(x).

Here, KT(x) denotes the set of attainable terminal wealth with an initial capital x. So

our problem is to solve

u(x) = sup

XT∈KT(x)

E[U(XT)].

However, for any FT-measurable random variableXT, the condition

XT ∈ KT(x) (4.7)

is equivalent to the budget constraint

EQ (1,θ(2))

[XT] =x, or E[Z(1,θ (2)₎

T XT] =x. (4.8)

Maximising the Langrangian

L :=_E[U(X)]−w(_E[Z_T(1,θ(2))X]−x)

over XT gives the first order condition for the optimal terminal wealth:

ˆ

XT =I(wZ

(1,θ(2)₎

T ), (4.9)

where I is the inverse ofU′, the gradient of the utility function U, and the Lagrange multiplier w is determined by the budget constraint (4.8).

To recover the optimal portfolio ˆπt = (ˆπ

(1)

t ,πˆ

(2)

t )∗, we can try using the

martin-gale representation theorem. Since for any completion θ(2)_{, we must have ˆ}_X

t =

EQ(1,θ(2))[ ˆXT|Ft], that is, ˆX is a Q(1,θ

(2)₎

-martingale, the martingale representation theorem tells us that

dXˆt=G∗tdWQ

(1,θ(2))

t , (4.10)

for some adapted process Gt = (G

(1) t , G (2) t )∗. But, (4.6) implies dXˆt = ˆXtπˆt∗σt(θtdt+dWt) = ˆXtπˆt∗σtdWQ (1,θ(2)) t . (4.11)

Comparing (4.10) and (4.11), we get ˆ Xtπˆ∗tσt=G∗t, or πˆ ∗ t = 1 ˆ Xt G∗_tσ−_t1. (4.12)

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In principle, we can now try to determine the driftµ so thatπ(2) _≡_{0. However, we} do not know what the process G looks like as the martingale representation theorem merely tells us that such a process exists.

To obtain a more explicit expression for ˆπ, we need to assume a particular form of the utility function U. We will look at the power, exponential and logarithmic utility functions in turn. For brevity, we write L≡Z(1,θ(2)₎

, _QL_≡ Q(1,θ(2)), and Le ≡Z(q,θ (2)₎ , QLe ≡_Q(q,θ(2)).

4.2.1 Power Utility

Here, we assume that U(x) = x_pp, forx >0 with 0< p <1. We let q:= −₁₋p_p. Hence,

I(w) = w−1−1p. The optimal terminal wealth is given by ˆX

T =w − 1 1−p(L T) − 1 1−p while

the budget constraint implies w−1−1p = x e H0 , where f H0 :=E[L q T]. (4.13) So, ˆ XT = x f H0 (LT) − 1 1−p. (4.14)

The optimal expected utility is then given by

u(x) = _E[U( ˆXT)] = x p p Hf 1−p 0 . (4.15) We note that Lq_T = exp{−q Z T 0 θ(1)_s dW_s(1)−q Z T 0 θ(2)_s dW_s(2)− 1 2q Z T 0 ∥ θs∥2 ds} =Le_Texp{ 1 2q(q−1) Z T 0 ∥ θs∥2 ds}. (4.16)

So, we can write

f H0 =E[Le_Texp{ 1 2q(q−1) Z T 0 ∥ θs∥ 2 ds}] =_EQeL [exp{1 2q(q−1) Z T 0 ∥θs∥2 ds}]. (4.17)

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The optimal wealth process can now be derived as follows. Since ˆX is a _QL

-martingale, ˆ Xt=EQ L [ ˆXT|Ft] =_EQL_[ x f H0 (LT)− 1 1−p|F t] = x f H0 1 LtE [Lq_T|Ft] = x f H0 1 LtE [Le_Texp{ 1 2q(q−1) Z T 0 ∥ θs∥ 2 ds}|Ft] = x f H0 e Lt LtE QeL [exp{1 2q(q−1) Z T 0 ∥ θs∥2 ds}|Ft] = x f H0 e Lt Lt exp{1 2q(q−1) Z t 0 ∥ θs∥2 ds}EQe L [exp{1 2q(q−1) Z T t ∥ θs∥2 ds}|Ft] =xHft f H0 (Lt)− 1 1−p_, (4.18) where f Ht :=EQe L [exp{1 2q(q−1) Z T t ∥θs∥2 ds}|Ft]. (4.19) Under _QeL, dYt = (a(Yt)−qb(Yt)(ρθ (1) t + q 1−ρ2_θ(2) t ))dt+b(Yt)dBQe L t , (4.20) where dBQeL t =ρdWQ eL,(1) t + √ 1−ρ2_dWQeL,(2)

t . By Feynman-Kac,Hfsatisfies the PDE f Ht+αeHfy+ 1 2b 2₍_y₎ f Hyy+ 1 2q(q−1)((θ (1) t )2+ (θ (2) t )2)Hf= 0, (4.21) with Hf(T, y) = 1.

Here, the subscript denotes the partial derivative with respect to the corresponding argument, and e α:=a(y)−qb(y)(ρθ(1)+q1−ρ2_θ(2)₎_, θ(1) =λ(y), θ(2) = √ 1 1−γ2[ µ(y) β(y) −γλ(y)].

We now compute the optimal portfolio. We assume the process Hfhas dynamics

dHf_t=Hf_tµ e

H(t)dt+HftσHe(t)

∗

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4.2 The Martingale Approach with Fictitious Completion whereσ e H(t) = (σ (1) e H (t), σ (2) e H (t))

∗ _{is a two-dimensional adapted process. We know that}

dXˆt = ˆXtπˆt∗αtdt+ ˆXtˆπt∗σtdWt. (4.23) But, (4.18) implies dXˆt = x f H0 d(Hf_t(L_t)− 1 1−p₎ = (. . .)dt+ ˆXt( 1 1−pθ ∗ t +σH_e(t) ∗ )dWt, (4.24)

where we do not care about the exact form of the drift. Comparing (4.23) with (4.24), we have ˆ π_t∗σt= 1 1−pθ ∗ t +σH_e(t) ∗ , or πˆ_t∗ = ( 1 1−pθ ∗ t +σH_e(t) ∗ )σ−_t1. (4.25) The process Hf has the form Hf_t =Hf(t, Y_t). Applying Itô’s Lemma, we get

dHf(t, Y_t) = ∂Hf ∂t (t, Yt)dt+ ∂Hf ∂y(t, Yt)dYt+ 1 2 ∂2Hf ∂y2 (t, Yt)d⟨Y⟩t = (. . .)dt+Hf_yb(Y_t)   ρ √ 1−ρ2   ∗ dWt. (4.26) Therefore, σ e H(t) ∗ = 1 f H f Hyb(Yt)   ρ √ 1−ρ2   ∗ . (4.27)

Our optimal portfolio is then given by

ˆ π(1)_t = 1 Yt { 1 1−pλ(Yt) +ρb(Yt) f Hy f H} − √ γ 1−γ2_Y t { 1 (1−p)√1−γ2( µ(Yt) β(Yt) −γλ(Yt)) + q 1−ρ2_b₍_Y t) f Hy f H }, (4.28) ˆ πt(2) = 1 √ 1−γ2_β₍_Y t) { 1 (1−p)√1−γ2( µ(Yt) β(Yt) −γλ(Yt)) + q 1−ρ2_b₍_Y t) f Hy f H }. (4.29)

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4.2 The Martingale Approach with Fictitious Completion For ˆπ_t(2) = 0, we need 1 (1−p)√1−γ2(γλ(Yt)− µ(Yt) β(Yt) ) =q1−ρ2_b₍_Y t) f Hy f H , i.e. µ(Yt) =β(Yt)[γλ(Yt)−(1−p) q (1−γ2₎₍₁₋_ρ2₎_b₍_Y t) f Hy f H]. (4.30)

Then, we do not invest in the fictitious stock and

ˆ π(1)_t = 1 Yt { 1 1−pλ(Yt) +ρb(Yt) f Hy f H }. (4.31)

In this completion, the market price of risk is given by

θ(1)_t =λ(Yt), θ(2)_t =−(1−p)q1−ρ2_b₍_Y t) f Hy f H , (4.32) where f H(t, y) = _EQ(q,θ(2))_[exp_{1 2q(q−1) Z T t ∥ θs∥ 2 ds}|Yt=y]. (4.33)

Plugging (4.32) into (4.21), we obtain the following non-linear PDE for Hf f Ht+(a(y)−qρλ(y)b(y))Hf_y+ 1 2b 2₍_y₎ f Hyy− 1 2q(1−q)λ 2₍_y₎ f H+1 2 q 1−q(1−ρ 2₎_b2₍_y₎(Hfy)2 f H = 0, (4.34) with Hf(T, y) = 1.

Thus, the optimal portfolio for the original incomplete market is given by the portfolio (4.31) where θ(2) _{satisfies (4.32) and} _H_f_{satisfies (4.34). The optimal expected} utility is given by (4.15).

4.2.2 Exponential Utility

The exponential utility function is given by U(x) =−e−δx, for x∈_R and δ >0. So,

I(w) =−1

δlog( w

δ). The optimal terminal wealth is then ˆXT = −

1

δ(logw+ logLT −logδ),

while the budget constraint is given by _E[LT(−1_δ)(logw+ logLT −logδ)] =x. So,

ˆ XT =x+ 1 δHf0− 1 δlogLT, (4.35)

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4.2 The Martingale Approach with Fictitious Completion where f H0 : =E[LTlogLT] =_EQL_[log_L T]. (4.36)

The optimal expected utility is

u(x) = _E[U( ˆXT)]

=−e−δxe−He0_.

(4.37)

We then compute the optimal wealth process. ˆ Xt=EQ L [ ˆXT|Ft] =x+1 δ(Hf0−E QL_[log_L T|Ft]). (4.38) But, logLt=− Z t 0 θ_s(1)dW_s(1)− Z t 0 θ_s(2)dW_s(2)− 1 2 Z t 0 ∥ θs∥2 ds =− Z t 0 θ_s(1)dWQL,(1) s − Z t 0 θ(2)_s dWQL,(2) s + 1 2 Z t 0 ∥θs∥ 2 ds. (4.39) Thus, f H0 =EQ L [logLT] = 1 2E QL_[ Z T 0 ∥ θs∥2 ds], (4.40) and EQ L [logLT|Ft] = logLt+ 1 2E QL_[ Z T t ∥ θs∥2 ds|Ft]. (4.41) Hence, ˆ Xt=x+ 1 δ(Hf0−Hft−logLt), (4.42) where f Ht:= 1 2E QL_[ Z T t ∥ θs∥2 ds|Ft]. (4.43)

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LetMt:=Hf_t+1

2

Rt

0 ∥θs∥2 ds. This is a QL-martingale. Indeed,

EQ L [Mt] =EQ L [Hf_t+ 1 2 Z t 0 ∥θs∥ 2 ds] =_EQL_[1 2E QL_[ Z T t ∥ θs∥2 ds|Ft] + 1 2 Z t 0 ∥ θs∥2 ds] =_EQL_[1 2 Z T t ∥θs∥ 2 ds+1 2 Z t 0 ∥ θs∥ 2 ds] = 1 2E QL_[ Z T 0 ∥θs∥2 ds] =M0. So, dMt =dHf_t+ 1 2((θ (1) t )2+ (θ (2) t )2)dt = [Hf_t+ (a(Y_t)−b(Y_t)(ρθ(1)_t + q 1−ρ2_θ(2) t ))Hf_y + 1 2b 2₍_Y t)Hf_yy+ 1 2((θ (1) t )2+ (θ (2) t )2)]dt +Hf_yb(Y_t)dBQ L t , where dBQL t = ρdWQ L_,₍₁₎ t + √ 1−ρ2_dWQL,(2)

t . Since the drift term must be zero, we

obtain the following PDE for Hf: f Ht+ (a(y)−b(y)(ρθ(1)+ q 1−ρ2_θ(2)₎₎_H_f y+ 1 2b 2 (y)Hf_yy+ 1 2((θ (1) )2+ (θ(2))2) = 0, (4.44) with Hf(T, y) = 0.

The optimal portfolio is determined as follows. We assume the process Hf has

dynamics dHf_t=µ e H(t)dt+σHe(t) ∗ dWt. (4.45) From (4.42), dXˆt =− 1 δdHft− 1 δd(logLt) = (. . .)dt+1 δ(θ ∗ t −σH_e(t) ∗ )dWt. (4.46)

Comparing (4.46) with (4.23), we obtain ˆ Xtπˆt∗σt= 1 δ(θ ∗ t −σH_e(t) ∗ ), i.e. πˆ_t∗ = 1 ˆ Xtδ (θ∗_t −σ e H(t) ∗ )σ_t−1. (4.47)

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4.2 The Martingale Approach with Fictitious Completion But Hf_t=Hf(t, Y_t) implies dHf(t, Y_t) = (. . .)dt+Hf_y(t, Y_t)b(Y_t)   ρ √ 1−ρ2   ∗ dWt. (4.48) So, σ e H(t) ∗ =Hf_yb(Y_t)   ρ √ 1−ρ2   ∗ . (4.49)

Our optimal portfolio is then given by ˆ π(1)t = 1 ˆ XtYtδ {λ(Yt)−ρb(Yt)Hf_y} − γ δ√1−γ2_Xˆ tYt {√ 1 1−γ2( µ(Yt) β(Yt) −γλ(Yt))− q 1−ρ2_b₍_Y t)Hf_y}, (4.50) ˆ π_t(2) = 1 δ√1−γ2_Xˆ tβ(Yt) {√ 1 1−γ2( µ(Yt) β(Yt) −γλ(Yt))− q 1−ρ2_b₍_Y t)Hf_y}. (4.51) To get ˆπ(2)t = 0, we need 1 √ 1−γ2( µ(Yt) β(Yt) −γλ(Yt)) = q 1−ρ2_b₍_Y t)Hf_y, i.e. µ(Yt) =β(Yt)[γλ(Yt) + q (1−γ2₎₍₁₋_ρ2₎_b₍_Y t)Hf_y]. (4.52)

In this choice of market completion, ˆ π(1)_t = 1 ˆ XtYtδ {λ(Yt)−ρb(Yt)Hf_y}, (4.53) and θ(1)_t =λ(Yt), θ(2)_t =q1−ρ2_b₍_Y t)Hf_y, (4.54) where f H(t, y) = 1 2E QL_[ Z T t ∥ θs∥2 ds|Yt =y]. (4.55)

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In this completion, Hf satisfies the PDE: f Ht+ (a(y)−ρb(y)λ(y))Hf_y+ 1 2b 2₍_y₎ f Hyy− 1 2(1−ρ 2₎_b2₍_y₎₍ f Hy)2+ 1 2λ 2₍_y_{) = 0}_, _(4.56) with Hf(T, y) = 0.

We conclude that the portfolio (4.53), withθ(2) _{satisfying (4.54) and}_H_f_{solving the} PDE (4.56), is the optimal portfolio for the original incomplete market, giving the optimal expected utility (4.37).

4.2.3 Logarithmic Utility

Now, the utility function is given by U(x) = log(x), for x >0. So, I(w) = _w1. The optimal terminal wealth is then ˆXT = _wL1_T, while the budget constraint isE[LTXˆT] = x.

So,

ˆ

XT =xL−T1. (4.57)

The optimal expected utility is

u(x) =_E[U( ˆXT)]

= log(x)−_E[logLT].

(4.58)

Next, we compute the optimal wealth process. ˆ Xt =EQ L [ ˆXT|Ft] =_E[LT Lt xL−_T1|Ft] =xL−_t1. (4.59)

We then derive the optimal portfolio.

dXˆt=xd(L−t1) = ˆXt((θ (1) t )2+ (θ (2) t )2)dt+ ˆXtθt∗dWt. (4.60)

Comparing (4.60) with (4.23), we obtain ˆ π∗_tσt =θ∗t, or πˆ ∗ t =θ ∗ tσ −1 t . (4.61)

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4.3 The Dynamic Programming Approach Hence, ˆ π(1)t = λ(Yt) Yt − γ (1−γ2₎_Y t {µ(Yt) β(Yt) −γλ(Yt)}, (4.62) ˆ π_t(2)= 1 (1−γ2₎_β₍_Y t) {µ(Yt) β(Yt) −γλ(Yt)}. (4.63)

For ˆπ_t(2) = 0, we must have

µ(Yt) = γλ(Yt)β(Yt). (4.64) Then, ˆ π(1)_t = λ(Yt) Yt . (4.65)

With this choice of completion µ,

θ_t(1) =λ(Yt),

θt(2) = 0.

(4.66)

So, the optimal portfolio for the original incomplete market is given by the portfolio (4.65), with θ_t(2) = 0. Logarithmic utility can be interpreted as the limiting case of the power utility when the risk aversion parameter p tends to zero. Indeed, letting

p→0 in our solution to the power utility problem, we recover the solution above for logarithmic utility.

4.3 The Dynamic Programming Approach

In this section, we consider a self-financing portfolio involving the traded asset S

only, as described in the beginning of this chapter. So the wealth process X has dynamics given by (4.1). We define πe := (πet)(0≤t≤T) to be the wealth invested in S, i.e. e

πt=π

(1)

t Xt.

Given a starting timet ∈[0, T], the objective to be maximised is

J(t, x, y;πe) :=E[U(XT)|Xt =x, Yt =y]. (4.67)

The value function is

u(t, x, y) := sup

e

π∈A

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4.3 The Dynamic Programming Approach

with

u(T, x, y) = U(x). (4.69) Under global Lipschitz and linear growth conditions on the market parameters, Zariphopoulou (2001) established the following distortion power solution for the value function u. This result was extended by Tehranchi (2004) who proved the distortion representation in a non-Markovian model with boundedness conditions on the market parameters. We assume that u is sufficiently smooth and regular, and argue formally.

The Hamilton-Jacobi-Bellman (HJB) equation for the value functionu(t, x, y) is

ut+ max e π∈A[yπλe (y)ux+ 1 2y 2 e π2uxx+a(y)uy + 1 2b 2 (y)uyy+yb(y)πρue xy] = 0, (4.70) u(T, x, y) = U(x).

Maximising over πe gives the optimal trading strategy ˆπe, where

ˆ

e

π(t, x, y) =−λ(y)ux+b(y)ρuxy

yuxx

. (4.71)

Plugging this back into the HJB equation gives

ut+a(y)uy + 1 2b 2₍_y₎_u yy− 1 2uxx (λ(y)ux+ρb(y)uxy)2 = 0. (4.72)

We now look for an explicit solution for the particular examples of the power utility function and the exponential utility function. We seek a separable solution of the form

u(t, x, y) =U(x)(H(t, y))j (4.73) for some j which transforms (4.72) to a linear PDE for H.

4.3.1 Power Utility

The power utility function is U(x) = x_pp, for x > 0 and 0 < p < 1. We look for a separable solution of the form u(t, x, y) =U(x)(H(t, y))j.

Then, forj = ₁₋1_qρ2, whereq =−

p

1−p, H satisfies the linear PDE

Ht+ (a(y)−qρλ(y)b(y))Hy+ 1 2b 2₍_y₎_H yy− 1 2q(1−qρ 2₎_λ2₍_y₎_H _{= 0}_, _(4.74) with H(T, y) = 1.

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4.3 The Dynamic Programming Approach By Feynman-Kac, H(t, y) = _EQ(q,0)_[exp_{−1 2q(1−qρ 2₎Z T t λ2(Ys)ds}|Yt=y], (4.75) where under_Q(q,0)_, dYt= (a(Yt)−qρb(Yt)λ(Yt))dt+b(Yt)dBQ (q,0) t , (4.76) with dBQ(q,0) t =ρdWQ (q,0)_,₍₁₎ t + √ 1−ρ2_dWQ(q,0),(2) t . Thus, u(t, x, y) = x p p (E Q(q,0)_[exp_{−1 2q(1−qρ 2₎Z T t λ2(Ys)ds}|Yt=y]) 1 1−qρ2_. _(4.77)

Lastly, we compute the optimal portfolio. From (4.71), ˆ e π(t, x, y) = x y(1−p)(λ(y) + ρ 1−qρ2b(y) Hy H ). (4.78) Hence, ˆ π_t(1) = 1 Yt(1−p) (λ(Yt) + ρ 1−qρ2b(Yt) Hy H). (4.79)

4.3.2 Exponential Utility

The exponential utility function isU(x) =−e−δx_{, for} _x_∈

Rand δ >0. Again, we look

for a separable solution of the form u(t, x, y) =U(x)(H(t, y))j_.

Then, it can be shown that forj = ₁₋1_ρ2,H satisfies the linear PDE

Ht+ (a(y)−ρλ(y)b(y))Hy+ 1 2b 2₍_y₎_H yy− 1 2(1−ρ 2₎_λ2₍_y₎_H _{= 0}_, _(4.80) with H(T, y) = 1. By Feynman-Kac, H(t, y) =_EQ(1,0)_[exp_{−1 2(1−ρ 2₎Z T t λ2(Ys)ds}|Yt=y], (4.81) where under_Q(1,0), dYt = (a(Yt)−ρb(Yt)λ(Yt))dt+b(Yt)dBQ (1,0) t , (4.82)

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4.4 Comparison of Results with dBQ(1,0) t =ρdWQ (1,0)_,₍₁₎ t + √ 1−ρ2_dWQ(1,0),(2) t . Hence, u(t, x, y) = −e−δx(_EQ(1,0)_[exp_{−1 2(1−ρ 2₎Z T t λ2(Ys)ds}|Yt=y]) 1 1−ρ2. (4.83)

Next, we calculate the optimal portfolio. From (4.71), ˆ e π(t, x, y) = 1 yδ(λ(y) + ρ 1−ρ2b(y) Hy H). (4.84) Therefore, ˆ π(1)_t = 1 ˆ XtYtδ (λ(Yt) + ρ 1−ρ2b(Yt) Hy H ). (4.85)

4.4 Comparison of Results

We compare the results obtained from the previous sections. We show that the optimal expected utility obtained from the two different methods are consistent with each other.

4.4.1 Power Utility

Consider the non-linear PDE (4.34) satisfied by Hf. We define a function G as follows

G(t, y) := (Hf(t, y))

1−qρ2

1−q , i.e. Hf(t, y) = (G(t, y))

1−q

1−qρ2. (4.86)

Calculating the partial derivatives and plugging them back into (4.34), we obtain the following linear PDE for G:

Gt+ (a(y)−qρλ(y)b(y))Gy + 1 2b 2 (y)Gyy− 1 2q(1−qρ 2 )λ2(y)G= 0, (4.87) with G(T, y) = 1.

So, Gsatisfies the same linear PDE as H with the same terminal condition. Therefore,

G≡H. It then follows from the definition of G that (H(t, y))1−1qρ2 = (Hf(t, y))

1

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4.4 Comparison of Results

From (4.77), the optimal expected utility obtained from the dynamic programming approach is u(x)≡u(0, x, y) = x p p (H(0, y)) 1 1−qρ2 = x p p (Hf(0, y)) 1 1−q (4.89)

which is precisely the solution (4.15) derived from the fictitious completion method.

4.4.2 Exponential Utility

Again, we start with the non-linear PDE (4.56) satisfied by Hfand define a function G

as follows G(t, y) := exp{−Hf(t, y)}1−ρ 2 , i.e. Hf(t, y) = − 1 1−ρ2logG(t, y). (4.90) Calculating the partial derivatives and plugging them back into (4.56), we obtain the following linear PDE for G:

Gt+ (a(y)−ρλ(y)b(y))Gy+ 1 2b 2₍_y₎_G yy− 1 2(1−ρ 2₎_λ2₍_y₎_G_{= 0}_, _(4.91) with G(T, y) = 1.

Gsatisfies the same linear PDE as H with the same terminal condition. Therefore,

G≡H. It then follows from the definition of G that

H(t, y)1−1ρ2 _{= exp}_{−

f

H(t, y)}. (4.92)

From (4.83), the optimal expected utility obtained from the dynamic programming approach is

u(x)≡u(0, x, y)

=−e−δx(H(0, y))1−1ρ2

=−e−δxexp{−Hf(0, y)}

(4.93)

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Chapter 5 Optimal Investment: The Dual

Problem

The method of fictitious completion is closely related to the selection of an optimal dual martingale measure attaining the infimum in the dual to the utility maximisation problem inChapter 4. The fictitious market completions are parametrised byθ(2) _∈_Ψ, and hence by the family of martingale measures M. With fictitious completion, the primal problem is solved by first finding the optimal portfolio for a given completion. We then pick the optimal portfolio from the least favourable completion, that is, the completion which minimises the optimal expected utility.

Karatzas et al. (1991) established a relation between the primal utility maximisation problem and its dual. They showed that the optimal martingale measure for the dual problem induces an optimal portfolio for the primal problem, and that the optimal dual martingale measure corresponds to the martingale measure characterising the least favourable completion.

The purpose of this chapter is to verify that the martingale measure characterising the least favourable market completion (i.e. the measure arising from θ(2)_{) from the} previous chapter matches the optimal martingale measure for the dual problem. To derive a representation for the dual optimiser, we follow the work of Monoyios (2006) who characterised the dual optimal measure via distortion. First, a relation between the dual value function and the function H from the distortion solution is established. The dual problem is then approached using dynamic programming to give a representation of the dual optimal measure in terms of H. We present the results and their derivation. Finally, we use the relation between H andHfestablished inSection 4.4 to show that

θ(2) _{characterising the least favourable completion in} _{Section 4.2} _{gives rise to the} optimal dual measure as expected.

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5.1 The Dual Problem

5.1 The Dual Problem

The dual to the utility maximisation problem (4.2) is given by

v(η) := inf

Q∈ME

[V(ηdQ

d_P)], η >0, (5.1)

whereV is the convex conjugate of the utility function U, i.e.

V(η) := sup x∈dom(U) (U(x)−ηx), η >0. (5.2) Thus, U(x) = inf η>0(V(η) +ηx), x∈dom(U). (5.3) We can show that the value functions uand v are conjugate, i.e.

v(η) = sup x>0 (u(x)−ηx), η >0, u(x) = inf η>0(v(η) +ηx), x >0, (5.4)

inheriting this property from U and V.

It can also be established that the optimal terminal wealth ˆXT from the primal

problem is related to the optimal dual measure ˆ_Q by

U′( ˆXT) =u′(x)

d_Qˆ

d_P. (5.5)

For an initial timet∈[0, T], the dual value function is defined by

v(t, η, y) := inf

ψ∈ΨE[V(η

ZT

Zt

)|Yt=y], (5.6)

whereZ ≡Z(1,ψ)_{. Similarly, the value functions} _u₍_{t, x, y}_{) and} _v₍_{t, η, y}_{) are conjugate.} Since v(η) = sup

x>0

(u(x)−xη) for any η > 0, the optimal x is given by ˆx ≡ xˆ(η) satisfying u′(ˆx) = η. But, from Section 4.3, for power and exponential utility functions, we can write u(t, x, y) =U(x)(H(t, y))j_{, for some} _j_{. Writing} _H _≡_H₍₀_{, Y}

0), and evaluating u at t= 0, the condition for the optimality of x becomesU′(ˆx)Hj ₌_η

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5.1 The Dual Problem Thus, v(η) =u(ˆx)−xηˆ =u(I( η Hj))−ηI( η Hj) =Hj[U(I( η Hj))− η HjI( η Hj)] =HjV( η Hj), (5.7)

where the last line follows from the fact thatV(η) = U(I(η))−ηI(η). Let ˆ_Q be the dual optimiser for the value functionv. Then,

v(η) = _E[V(ηd

ˆ

Q

d_P)]. (5.8)

So, (5.7) implies that

E[V(ηd ˆ Q d_P)] =H j_V₍ η Hj). (5.9)

We will now look at the specific examples of the power utility and the exponential utility.

5.1.1 Power Utility

The power utility function isU(x) = x_pp, for x >0 and 0< p < 1. V is then given by

V(η) = −η_qq, where q=−₁₋p_p.

In the case of the power utility, the dual optimiser ˆ_Q maximises_E[(dQ dP)

q_{] over} _M_.

From (5.9), we deduce that ˆ_Q is related to_Q(q,0) by

E[( d_Qˆ d_P) q_{] = (} EQ (q,0) [exp{−1 2q(1−qρ 2₎Z T 0 λ2(Yt)dt}]) 1−q 1−qρ2 =H(0, Y0) 1−q 1−qρ2 (5.10)

A result similar to (5.9) can be derived for v(t, η, y) and H(t, y). They are related as follows:

v(t, η, y) =V(η)H(t, y)

1−q

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Now, we will try to derive the dual optimiser ˆ_Q by approaching the dual problem using dynamic programming. Since

dYt=a(Yt)dt+b(Yt)dBt, dZt=−λ(Yt)ZtdW (1) t −ψtZtdW (2) t ,

the HJB equation for the value function v (assuming sufficient regularity and smooth-ness) is vt+ inf ψ∈Ψ[a(y)vy+ 1 2b 2₍_y₎_v yy+ 1 2(λ 2₍_y_{) +}_ψ2₎_η2_v ηη −ηb(y)(ρλ(y) + q 1−ρ2_ψ₎_v ηy] = 0. (5.12) Minimising over Ψ gives the optimal value of ψ as ˆψ where

ˆ ψ(t, η, y) = √ 1−ρ2 η b(y) vηy(t, η, y) vηη(t, η, y) . (5.13) But, (5.11) implies ˆ ψ(t, y) = − √ 1−ρ2 1−qρ2 b(y) Hy(t, y) H(t, y). (5.14)

So, it follows from a verification theorem that the optimal control for the dual problem is given by ˆψ above. The dual optimal measure is then

ˆ

Q=Q(1,ψˆ), (5.15)

with ˆψ given in (5.14), while H satisfies the linear PDE (4.74).

From (4.32), (5.14), and using the relation (4.88) betweenH and Hf, we get

ˆ ψt=− 1 1−q q 1−ρ2_b₍_y₎Hfy f H =θt(2), (5.16)

where the last line follows from the definition of q. Therefore, ˆψ ≡ θ(2) and ˆ_Q =

Q(1,ψˆ)=Q(1,θ (2)₎

. The martingale measure arising fromθ(2) _{matches the dual optimal} measure.

5.1.2 Exponential Utility

The exponential utility function is U(x) = −e−δx, for x ∈ _R and δ > 0. Thus, V is given by V(η) = η_δ(log(η_δ)−1).

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For exponential utility, the dual optimiser ˆ_Qis such that it minimises _E[dQ dPlog

dQ dP],

the relative entropy between the measures _Q and _P, over M. (5.9) implies that ˆ_Q and _Q(1,0) _{are related by}

E[d ˆ Q d_Plog d_Qˆ d_P] =− 1 1−ρ2log(E Q(1,0)_[exp_{−1 2(1−ρ 2₎Z T 0 λ2(Yt)dt}]) =− 1 1−ρ2logH(0, Y0). (5.17)

A result analogous to (5.9) can be derived forv(t, η, y) and H(t, y). They are related as follows:

v(t, η, y) = V(η)−η

δlog((H(t, y))

1

1−ρ2). (5.18)

We will now derive the dual optimiser ˆ_Q. By repeating the same argument for the power utility case, we get

ˆ ψ(t, η, y) = √ 1−ρ2 η b(y) vηy(t, η, y) vηη(t, η, y) . (5.19) Using (5.18) gives ˆ ψ(t, y) = −√ 1 1−ρ2b(y) Hy(t, y) H(t, y). (5.20)

So, it follows from a verification theorem that ˆψ given above is the optimal control for the dual problem. Hence, the dual optimal measure is

ˆ

Q=Q(1,ψˆ), (5.21)

where ˆψ is as above, while H satisfies the linear PDE (4.80).

From (4.54), (5.20), and using the relation (4.92) betweenH and Hf, we get

ˆ ψt = q 1−ρ2_b₍_y₎_H_f y =θt(2). (5.22) Thus, ˆψ ≡ θ(2) _{and ˆ}

Q= Q(1,ψˆ) =Q(1,θ(2)). The martingale measure arising from θ(2)

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Chapter 6 Optimal Consumption: The

Primal Problem

In this chapter, we extend the results from Chapter 4to the problem involving intermediate consumption. As in Section 4.2, we use the technique of fictitious completion to complete the market before approaching the complete market problem by appealing to martingale theory. Finally, we derive a necessary condition on the market price of risk for there to be a solution to our original incomplete market problem.

6.1 The Primal Problem

We assume the same stochastic volatility model from Chapter 3 and introduce the fictitious stock P from Section 4.2having the same dynamics (4.3). But now the wealth processX corresponding to a self-financing portfolio investing in both

S and P satisfies

dXt=−ctdt+Xtπ∗tαtdt+Xtπ∗tσtdWt, (6.1)

where cis a consumption process satisfying RT

0 ctdt <∞ andct≥0 almost surely,

while π,α, σ and θ are as before. Also defineL, _QL_, _L_e _and

QeL as before.

Given utility functions U1 and U2, and an initial capital x, in this artificially complete market, the investor’s objective is to maximise_E[U1(XT) +R0T U2(ct)dt]

over all (π, c)∈ A, where A denotes the set of admissible trading strategies (π, c). The pair (π, c) is an admissible trading strategy if X ≥ 0 almost surely. So, the problem is given by u(x) := sup (π,c)∈AE [U1(XT) + Z T 0 U2(ct)dt]. (6.2)

For our original incomplete market problem, we need the additional constraint

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6.2 The Martingale Approach with Fictitious

Com-pletion

Let I1 and I2 denote the inverses of U1′ and U

′

2 respectively. Given any FT-measurable

random variable XT, the condition that XT can be replicated by an admissible trading

strategy (π, c) is equivalent to the budget constraint

E[LTXT +

Z T

0

Ltctdt] =x. (6.3)

Maximising the Langrangian

L:=_E[U1(X) + Z T 0 U2(ct)dt−wLTX−w Z T 0 Ltctdt+wx]

over XT and ct gives the first order condition for the optimal terminal wealth and the

optimal consumption plan:

ˆ

XT =I1(wLT),

ˆ

ct=I2(wLt).

(6.4)

where the Lagrange multiplierw is determined by the budget constraint (6.3).

As inSection 4.2, we fixγ andβ, and consider the drift µof the fictitious stock as a parameter. As before, the fictitious completions of our market can be parametrised by the set Ψ. We solve the optimal consumption problem above in the fictitiously completed market before adjusting µ so that the optimal position in P is zero. The resulting portfolio should then be the solution to our original incomplete market problem.

6.2.1 Power Utility

Let U = U1 = U2, where U(x) = x

p

p, for x > 0 and 0 < p < 1. Let q = − p

1−p as

before. Then, I = I1 = I2, where I(w) = w

− 1

1−p_{. The optimal terminal wealth and}

the optimal consumption plan become ˆXT =w

− 1 1−pL − 1 1−p T and ˆct=w − 1 1−pL − 1 1−p t . The

budget constraint is given by

E[LTw − 1 1−pL − 1 1−p T + Z T 0 Ltw − 1 1−pL − 1 1−p t dt] =x, i.e. w−1−1p E[LqT + Z T 0 Lq_tdt] =x.

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6.2 The Martingale Approach with Fictitious Completion Let f H0 :=E[ Z T 0 Lq_tdt+Lq_T], (6.5) Then, ˆ XT = x f H0 L− 1 1−p T , ˆ ct= x f H0 L− 1 1−p t . (6.6)

The optimal expected utility is then

u(x) =_E[U( ˆXT) + Z T 0 U(ˆct)dt] =_E[x p p 1 f H₀pL q T + Z T 0 xp p 1 f H₀pL q tdt] = x p p Hf 1−p 0 . (6.7)

The optimal wealth process can now be computed as follows: ˆ Xt= 1 LtE [LTXˆT + Z T t Lsˆcsds|Ft] = 1 LtE [LT x f H0 L− 1 1−p T + Z T t Ls x f H0 L− 1 1−p s ds|Ft] = x f H0 1 LtE [Lq_T + Z T t Lq_sds|Ft]. (6.8) But, Lq_T =Le_Texp{ 1 2q(q−1) Z T 0 ∥ θs∥2 ds}, (6.9)

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6.2 The Martingale Approach with Fictitious Completion which implies E[LqT + Z T t Lq_sds|Ft] =_E[Le_Texp{ 1 2q(q−1) Z T 0 ∥ θs∥2 ds}+ Z T t e Lsexp{ 1 2q(q−1) Z s 0 ∥ θu∥2 du}ds|Ft] =Le_t_EQe L [exp{1 2q(q−1) Z T 0 ∥ θs∥ 2 ds}+ Z T t exp{1 2q(q−1) Z s 0 ∥ θu∥ 2 du}ds|Ft] =Le_texp{ 1 2q(q−1) Z t 0 ∥ θu∥2 du}EQe L [exp{1 2q(q−1) Z T t ∥ θs∥2 ds} + Z T t exp{1 2q(q−1) Z s t ∥ θu∥2 du}ds|Ft] =LqtHf_t, (6.10) where f Ht :=EQe L [exp{1 2q(q−1) Z T t ∥θs∥ 2 ds}+ Z T t exp{1 2q(q−1) Z s t ∥ θu∥ 2 du}ds|Ft]. (6.11) So, ˆ Xt= x f H0 1 Lt LqtHf_t =xHft f H0 L− 1 1−p t . (6.12)

We can compute the optimal portfolio by proceeding exactly as inSection 4.2.1. Assume the processHfhas dynamics

dHf_t=Hf_tµ e H(t)dt+HftσHe (t)∗dWt, (6.13) whereσ e H(t) = (σ (1) e H (t), σ (2) e H (t))

∗ _{is a two-dimensional adapted process. We know that}

dXˆt =−ctdt+ ˆXtπˆt∗αtdt+ ˆXtπˆt∗σtdWt. (6.14) But, (6.12) implies dXˆt = x f H0 d(Hf_t(L_t)− 1 1−p) = (. . .)dt+ ˆXt( 1 1−pθ ∗ t +σH_e(t) ∗ )dWt, (6.15)

Fictitious Completion as a Tool to Solve Utility Maximisation Problems