Financial Modeling Under Incomplete Information

U.U.D.M. Project Report 2015:3

Examensarbete i matematik, 30 hp

Handledare och examinator: Erik Ekström Maj 2015

Department of Mathematics

Financial Modeling Under Incomplete Information

George Bliatsios

I would like to dedicate this thesis to my parents Athanasios and Paraskevi

and my brother Stilianos

for their endless support during my master studies.

”There is nothing impossible to him who will try.”

Alexander the Great

Acknowledgements

I would like to thank my supervisor Erik Ekstr¨om for his priceless support, guidance and knowledge that offered me throughout this project. Each of our meetings was a valuable lesson to me.

Moreover, I would like to thank my professors Johan Tysk, Maciej Klimek and G¨oran ¨Osterholm. Their lecture notes along with their comments and fruitful discussions during the classes were a catalyst for this project.

Finally, I would like to thank Laertis Vaso for viewing my project and for his recommendations during the writing.

Abstract

In this project our main assumption is that the price of a stock is modeled as a Geometric Brownian Motion with unknown drift and known constant volatility.

In particular, we examine two cases for the drift. In the first one, we assume that the drift is modeled as a random variable with known distribution which is not directly observable by an agent in the market and it can take two values. In the second case, we assume that the drift is modeled as an Ornstein-Uhlenbeck process with known initial distribution which again is not directly observable.

In view of the above assumptions, we address the following two problems. In the first one, we assume that an agent has a short position on a stock and we identify when is the optimal time to buy it back. In the second one, we assume that an agent has a self-financing portfolio consisting of one stock and a bank account and we present the optimal wealth allocation between the risky asset and the risk-free one.

Contents

Chapter 1 - Introduction

Introduction 7

Chapter 2 - Modeling the Market

2.1 - Introduction 9

2.2 - The Digital Drift Case 9

2.2.1 - Market Model 9

2.2.2 - The Filtering Problem 9

2.3 - The Ornstein - Uhlenbeck Drift Case 13

2.3.1 - Market Model 13

2.3.2 - The Filtering Problem 13

Chapter 3 - Optimal Closing of a Short Position

3.1 - Introduction 17

3.2 - The Digital Drift Case 17

3.3 - The Ornstein - Uhlenbeck Drift Case 21

Chapter 4 - The Portfolio Problem

4.1 - Introduction 22

4.2 - Portfolio Rebalance 24

4.2.1 - The Market Under Complete Information 25

4.2.2 - The Digital Drift Case 27

4.2.3 - The Ornstein - Uhlenbeck Drift Case 34

Chapter 5 - Conclusions

Conclusions 35

Chapter 1

Introduction

In this project we address the problems of Optimal Closing of a Short-Position as well as The Portfolio Problem (Chapters 3 and 4 respectively) for 2 different stock price models of the form:

dX_t= θ(t, X_t)X_tdt + σX_tdW_t (1)

where the drift θ(t, Xt) is an unobserved quantity while the volatility σ is considered as a known positive constant.

In Chapter 2, we introduce the stock price models and the main assumptions under which we work through out this project. Specifically, in paragraph 2.2 we assume that the drift is modeled as a random variable with known distribution and 2 possible outcomes, the Digital Drift case. Moreover, we assume that in the beginning of time (t = 0) a hypothetical coin toss occurred but an agent in the market is not able to observe the outcome and the only available informa- tion to her is given by the price process of the stock Xt. In view of the general theory of the Filtering Problem (see [8]), the Girsanov theorem (see [7], [8]) and the Bayes formula (see [8], [10]), we find the best estimate for the drift.

Then, we transform (1) in such a way where all the parameters are observable, that is, equations 2.2.2.13 and 2.2.2.14. In paragraph 2.3, we assume that the drift is modeled as an Ornstein-Uhlenbeck process and same as above, it cannot be observed. In view of the general theory of the Filtering problem and the Kalman-Bucy theorem (see [8]), we transform (1) in a Stochastic Dif- ferential Equation where all the parameters are observable quantities, that is, equations 2.3.2.12 and 2.3.2.13 (similar work can be found in [1], [2], [3], [11] ).

In Chapter 3, we address the optimal stopping problem (see [7], [8]) of clos- ing a short position for both cases described above. The main mathematical machinery that we use is the Girsanov theorem as well as the Feynmann-Kac theorem (see [7], [8]). The optimal execution boundary for both models is given by theorems 3.2.1 and 3.3.1 respectively (similar work can be found in [1], [2]).

In Chapter 4, we address the portfolio optimization problem (see [7]) un- der the assumption that an agent in the market possesses a portfolio consisting of a stock and a bank account growing with interest rate r > 0. Since this is a stochastic optimal control problem, the main mathematical machinery that we use is the Hamilton-Jacobi-Bellman equation (see [7], [8]) as well as the Feynmann-Kac theorem. We derive the portfolio’s optimal wealth allocation as

well as the agent’s optimal value function for both market models described in Chapter 2. In addition, we solve the problem for the standard case where the drift is a known constant and we introduce theorem 4.2.2.2 which compares the value functions between the standard case and the Digital Drift case. The main results of this chapter are given by theorems 4.2.1, 4.2.2.1, 4.2.2.2 and 4.2.3.1 (similar work can be found in [4], [5], [6]).

Chapter 2

Modeling the Market

2.1 Introduction

In this chapter we formulate the main models that we will use through out this thesis. Our main assumption is that an agent in the market can only observe the price process of an asset without having any information about its drift or its driving process. Moreover, we assume that the asset’s volatility is an estimated known positive constant (for example historical volatility).

2.2 The Digital Drift Case

2.2.1 Market Model

Suppose that the price of a stock is modeled by the following stochastic differ- ential equation:

dX_t= θX_tdt + σX_tdW_t, X₀= x > 0

where, σ > 0 is constant and Wt is a Wiener process. Let Mt be the informa- tion generated by Xs, s ≤ t. Moreover, we assume that θ is a random variable which can only take the values θ1 and θ2 ( θ1< 0 < θ2 ) and it is not directly observable. Furthermore, we assume that at time t = 0 an agent in the market has information about the probabilities of the events {θ = θ1} and {θ = θ2} , that is, P [{θ = θ2}] = ρ and P [{θ = θ1}] = 1 − ρ. Notice that only Xt is observable.

2.2.2 The Filtering Problem

As we can only observe X_t, we would like to find an estimate ˆθ_tof θ given Mt. Let ˆθ_t= E[θ|Mt]. Now, since our observations satisfy

dX_t= θX_tdt + σX_tdW_t (2.2.2.1) we have that,

1 Xt

dX_t= θdt + σdW_t.

If we set Zt=

t

R

0

1 Xs

dXs, we have that our observations get the form:

dZ_t= θdt + σdW_t (2.2.2.2).

According to the general theory of the filtering problem we can define our in- novation process as: Nt= Zt−

t

R

0

θˆsds. Thus,

Nt=

t

Z

0

θds + σ

t

Z

0

dWs−

t

Z

0

θˆsds ⇔

Nt= σWt+

t

Z

0

θ − ˆθsds ⇔

1

σNt= Wt+

t

Z

0

1

σ(θ − ˆθs) ds (2.2.2.3).

Let ˆWt= Wt+

t

R

0

1

σ(θ − ˆθs)ds, which is a Wiener process. Then we have that, d ˆWt= dWt+ 1

σ(θ − ˆθt)dt ⇔ d ˆWt= 1

σ(θdt + σdWt− ˆθtdt) ⇔ d ˆWt= 1

σ( 1

X_tdXt− ˆθtdt) ⇔ dXt= ˆθtXtdt + σXtd ˆWt (2.2.2.4).

Therefore, our observations will be of the form:

dZ_t= ˆθ_tdt + σd ˆW_t (2.2.2.5).

Now, we would like to find the Stochastic Differential Equation that ˆθtsatisfies.

Let Zt be the information generated by the process (Z_s)_s≤t as appearing in (2.2.2.2). In view of the Girsanov theorem if we let:

dQ

dP = MT = exp



−1 2

T

Z

0

θ² σ²ds −

T

Z

0

θ σdWs



=

exp θ² 2σ²T −θ

σZT



, on Zt (2.2.2.6)

we have that Ztis a Q- Wiener process independent of θ. Moreover, notice that Mt=Ztso ˆθt= E[θ|Mt] = E[θ|Zt]. Let (Ω, M, Q) be a probability space and φ be the distribution of θ, i.e., φ(B) = Q(θ⁻¹(B)), B ⊂ R Borel. By the Bayes formula we have that:

θˆt= E[θ|Mt] = E^Q[θ ^dP_dQ|Mt]

E^Q[^dP_dQ|Mt] = f(t, Zt) (2.2.2.7) where,

E^Q[θ dP dQ|Mt] =

Z

R

x exp(− x² 2σ²t + x

σ²Zt)dφ(x) (2.2.2.8) and

E^Q[dP dQ|Mt] =

Z

R

exp(− x² 2σ²t + x

σ²Zt)dφ(x) (2.2.2.9)

Now, by applying Ito’s formula (with differentiation under the integral sign) on ˆθt = f(t, Zt) and using (2.2.2.5) as the dynamics of Zt as well as Fubini’s theorem combined with the independence of θ and Z we end up to:

dˆθt= σ fz(t, Zt)d ˆWt (2.2.2.10) By the Bayes formula we have that:

fz(t, Zt) = 1 σ²







 R

R

x²exp(−_2σ^x22t +_σ^x2Zt)dφ(x) R

R

exp(−_2σ^x22t +_σ^x2Zt)dφ(x) −





 R

R

x exp(−_2σ^x22t + _σ^x2Zt)dφ(x) R

R

exp(−_2σ^x22t +_σ^x2Zt)dφ(x)







2







⇔ fz(t, Zt) = 1

σ² (E[θ²|Mt] − E[θ|Mt]²) = 1

σ²var(θ|Mt) (2.2.2.11) Thus, (2.2.2.10) becomes

dˆθt= 1

σvar(θ|Mt)d ˆWt (2.2.2.12)

In addition, if we let ρ_t= P [{θ = θ₂}|M_t] for t > 0, that is that the estimate for the probability of the drift changes over time, then (2.2.2.4) becomes:

dX_t= [ρ_t(θ₂− θ1) + θ₁]X_tdt + σX_td ˆW_t (2.2.2.13) where,

dρ_t= (θ₂− θ₁)

σ ρ_t(1 − ρ_t)d ˆW_t (2.2.2.14)

Regarding equation (2.2.2.14) see [2], equation (2.5). Finally, notice that under this representation of Xtall the coefficients are observable quantities.

2.3 The Ornstein-Uhlenbeck Drift Case

2.3.1 Market Model

Suppose that the price of a stock is modeled by the following stochastic differ- ential equation:

dXt= θtXtdt + σXtdWt, X0= x > 0

where, σ > 0 is constant and Wtis a Wiener process. Let Mtbe the information generated by Xs, s ≤ t. Moreover, we assume that the process θt satisfies the Ornstein-Uhlenbeck stochastic differential equation:

dθt= (k − θt)dt + udVt (2.3.1.1)

where, Vtis a Wiener process, u > 0, k < 0 constants. Furthermore, we assume that the initial value of the drift process has normal distribution, θ0∼ N[µ0, σ²₀] which is independent of W_tand V_t and that only X_tis observable.

2.3.2 The Filtering Problem

Since the drift process θ_tsatisfies the Ornstein-Uhlenbeck stochastic differential equation we have that:

dθt= (k − θt)dt + udVt⇔ dθt− θtdt = kdt + udVt⇔ e^tdθt− e^tθtdt = ke^tdt + ue^tdVt⇔

d(e^tθt) = ke^tdt + ue^tdVt⇔

θt= θ0e^−t+ k(1 − e^−t) + u

t

Z

0

e^s−tdVs

with E[θt|Nt] = θ0e^−t+ k(1 − e^−t), where Nt is the information generated by θs, s ≤ t. As we can only observe Xt, we would like to find an estimate ˆθtof θt

given Mt. Let ˆθt= E[θt|Mt]. Now, since our observations satisfy dX_t= θ_tX_tdt + σX_tdW_t (2.3.2.1)

we have that,

1 Xt

dX_t= θ_tdt + σdW_t.

If we set Zt=

t

R

0

1 Xs

dXs, we have that our observations get the form:

dZt= θtdt + σdWt (2.3.2.2).

According to the general theory of the filtering problem we can define our in- novation process as: Nt= Zt−

t

R

0

θˆsds. Thus,

Nt=

t

Z

0

θsds + σ

t

Z

0

dWs−

t

Z

0

θˆsds ⇔

Nt= σWt+

t

Z

0

θs− ˆθsds ⇔

1

σNt= Wt+

t

Z

0

1

σ(θs− ˆθs)ds (2.3.2.3).

Let ˆW_t= W_t+

t

R

0

1

σ(θ_s− ˆθ_s)ds, which is a Wiener process. Then we have that, d ˆWt= dWt+ 1

σ(θt− ˆθt)dt ⇔ d ˆWt= 1

σ(θtdt + σdWt− ˆθtdt) ⇔ d ˆWt= 1

σ( 1 Xt

dXt− ˆθtdt) ⇔ dX_t= ˆθ_tX_tdt + σX_td ˆW_t (2.3.2.4).

Therefore, our observations will be of the form:

dZ_t= ˆθ_tdt + σd ˆW_t (2.3.2.5).

Now, we would like to find the Stochastic Differential Equation that ˆθ_t sat- isfies. The system equation is given by (2.3.1.1), while the observations sat- isfy (2.3.2.2). We note that E[θt− k|Mt] = E[θt|Mt] − k = ˆθt− k and that d(θt− k) = dθt. Hence, since the process θthas normal distribution in view of the Kalman-Bucy filter theorem we have that:

dˆθt= σ²+ S(t)

σ² (k − ˆθt)dt +S(t) σ² dZt⇔ dˆθt= [k(1 + S(t)

σ² ) − ˆθt]dt +S(t)

σ d ˆWt (2.3.2.6)

where, the Mean Square Error S(t) = E[(θt− ˆθt)²] satisfies the deterministic Riccati equation:

dS(t) dt = − 1

σ²S²(t) − 2S(t) + u² (2.3.2.7) where,

S(0) = E[(θ₀− ˆθ₀)²] = Var(θ₀) = σ²₀ (2.3.2.8).

The solution of the above equation is:

S(t) = (p

σ⁴+ σ²u²)tanh(A + Bt) − σ² (2.3.2.9) where,

A = tanh⁻¹( σ²+ σ²₀

√σ⁴+ σ²u²)

B = r

1 + u² σ² If we let t → +∞ we have that S(t) →√

σ⁴+ σ²u²− σ² = s^∗ > 0, the steady state of the Mean Square Error.

According to the above result, it can be seen that in the long run the Mean Square Error is a positive constant. Therefore, more information will not have crucial effect on variance reduction of the estimator but it will only renew ˆθt . In addition, the monotonicity of the solution S(t) depends on the choice of the parameters σ, u and σ0.

Namely, if√

σ⁴+ σ²u²− (σ²+ σ₀²) > 0, we have that:

S(t) % and 0 < σ0≤ S(t) ≤ s^∗ (2.3.2.10)

(The unrealistic scenario, since even if the set of our observations is infinite we will never be able to determine the initial distribution of θt )

while if√

σ⁴+ σ²u²− (σ²+ σ²₀) ≤ 0, we have that:

S(t) & and 0 < s^∗≤ S(t) ≤ σ0 (2.3.2.11)

(The realistic scenario, since we know the initial distribution of θt)

Now, under (2.3.2.11) since the Mean Square Error in the long-run will be a positive constant, our market model it is natural to take the form:

dXt= ˆθtXtdt + σXtd ˆWt (2.3.2.12) dˆθ_t= [k(1 + s^∗

σ²) − ˆθ_t]dt + s^∗

σd ˆW_t (2.3.2.13)

Chapter 3

Optimal Closing of a Short-Position

3.1 Introduction

In this chapter we assume that an agent has a short-position on the stock X_t and that there are no dividends and no transaction costs. The problem that we will try to address is to identify when is the best time for the agent to buy back the stock. Let us suppose that she does not execute the order if the price is too high for profit, that is, the agent believes in a small drift. Therefore, there exists a threshold h(t) > 0 such that when the stock price hits h(t), she immediately closes her position. The mathematical interpretation of the above problem is the solution of:

u(x) = infτ ≥0E0,x[Xτ] = E0,x[Xτ^∗] (3.1.1)

where, u is called the value function and τ^∗ is the first exit time of the process X_t from the ”stop-later” region x ∈ (0, h). In the ”stop-now” region where x ≥ h it is u(x) = x. As a result, the solution of (3.1.1) coincides with the determination of the free boundary h(t) between the two regions.

3.2 The Digital Drift Case

Let us assume that our model is governed by (2.2.2.13) and (2.2.2.14). Let {F}t≥0 be the information generated by the process ρt, (Ω, F, P ) be a filtered probability space and T≤ +∞ be a given constant.

Since Xt = x exp(

t

R

0

ρs(θ2− θ1) + θ1ds − ¹₂σ²t + σ ˆWt), we have that under P equation (3.1.1) becomes:

u(x) = inf_{τ ≥0}E_0,x^P [X_τ] = x inf_{τ ≥0}E_0,x^P [exp(

τ

Z

0

ρ_s(θ₂−θ₁)+θ₁ds) exp(−1

2σ²τ +σ ˆW_τ)]

(3.2.1) In view of the Girsanov theorem, if we put:

Mt= exp(−1

2σ²t + σ ˆWt) = exp(−1 2

t

Z

0

σ²ds +

t

Z

0

σd ˆWt), t ≤ T

and dQ

dP = MT, on FT

then, Q is a probability measure on FT and the process:

Wˆ_t^Q = −

t

Z

0

σds + ˆW_t⇔

d ˆW_t^Q= −σdt + d ˆWt

is a Wiener process with respect to Q. Moreover, the stochastic integral repre- sentation of ρtunder Q is of the form:

dρ_t=(θ₂− θ₁)

σ ρ_t(1 − ρ_t)d ˆW_t⇔ dρ_t= (θ₂− θ₁)

σ ρ_t(1 − ρ_t)(d ˆW_t^Q+ σdt) ⇔ dρ_t= (θ₂− θ1)ρ_t(1 − ρ_t)dt +(θ₂− θ₁)

σ ρ_t(1 − ρ_t)d ˆW_t^Q (3.2.2).

Hence, under the probability measure Q we have that (3.2.1) becomes:

u(x) = x infτ ≥0E_0,ρ^Q [exp(

τ

Z

0

ρs(θ2− θ1) + θ1ds)]. (3.2.3)

Now, according to equation (3.2.3) it can be seen that the solution of the agent’s original problem can be reduced to the solution of the problem:

V (ρ) = infτ ≥0E_0,ρ^Q [exp(

τ

Z

0

ρs(θ2− θ1) + θ1ds)]. (3.2.4)

Notice that the smaller the probability for the drift to be θ2, the bigger the agent’s profit. Thus, as above we postulate that there exists a threshold h ∈ (0, 1) with the corresponding ”stop-later” region of the process ρt to be A = (0, h). Hence, the agent closes her position immediately when ρt = h. Under the postulated strategy and in view of the Feynman-Kac theorem we have the following:

Theorem 3.2.1

The value function V satisfies the boundary value problem:

1 2

(θ₂− θ₁)²

σ² ρ²(1 − ρ)²V_ρρ^h+ (θ₂− θ1)ρ(1 − ρ)V_ρ^h+ (ρ(θ₂− θ1) + θ₁)V^h= 0, ρ ∈ A V^h(ρ) = 1, h ≤ ρ < 1

where the free boundary h coincides with the positive, with respect to γ, solution of:

1 2

(θ2− θ1)²

σ² γ(γ − 1) + (θ2− θ1)γ + θ1= 0

Proof If we make the change of variables ρ = φ

1 + φ and set V^h(ρ) = 1

1 + φu(φ) we have that:

V_ρ^h= u^h_φ(1 + φ) − u^h, V_ρρ^h = u^h_φφ(1 + φ)³. Thus, the partial differential equation takes the form:

1 2

(θ2− θ1)²

σ² φ²u^h_φφ+ (θ₂− θ₁)φu^h_φ+ θ₁u^h= 0 (3.2.5).

Now, if we make the ansantz u(φ) = φ^γ we have that (3.2.5) becomes:

1 2

(θ₂− θ1)²

σ² γ(γ − 1) + (θ2− θ1)γ + θ1= 0

which has two distinct solutions of opposite signs which we denote them by {γ−, γ+}. Notice that for γ = 1, it is:

1 2

(θ2− θ1)²

σ² γ(γ − 1) + (θ2− θ1)γ + θ1= θ2> 0 so γ+∈ (0, 1). Therefore, (3.2.5) has the following solution:

u^h(φ) = C1u^h₁+ C2u₂^h= C1φ^γ−+ C2φ^γ+

where, u^h₁ and u^h₂ are independent solutions of (3.2.5). Since we want the so- lution to decline due to the fact that the agent will not have incentives to buy back the stock, it suffices to consider:

u^h(φ) = Cφ^γ+.

As a result, the solution of the agent’s initial problem is:

V^h(ρ) = Cρ^γ+(1 − ρ)^1−γ+, ρ ∈ A

V^h(ρ) = 1, for h ≤ ρ < 1

Since V^h(h) = 1, we have that C = h^−γ+(1 − h)^γ+−1. In addition by the

”Higher-Order-Contact” condition we have that:

d^(right)V^h

dρ |h=d^(left)V^h dρ |h

from which we derive that the optimal level h^∗is given by: h^∗= γ+, constant.

Remarks

1) V^h(ρ) is concave in ρ. For ρ ∈ (0, γ+) we have that V^h(ρ) % while for ρ ∈ [γ+, 1) it is V^h(ρ) = 1.

2) For fixed θ1, θ2we have that the smaller the volatility σ, the smaller the γ+. Thus, the margin for profit for the agent is bigger according to our analysis above. Exactly the same effect occurs when σ is fixed, while the difference between θ2and θ1 increases.

3.3 The Ornstein-Uhlenbeck Drift Case

Let us assume that our model is governed by (2.3.2.12) and (2.3.2.13). Let {N}t≥0 be the information generated by the process ˆθt, (Ω, N, P ) be a filtered probability space and T≤ +∞ be a given constant.

Since X_t= x exp(

t

R

0

θˆ_sds −¹₂σ²t + σ ˆW_t), we have that under P equation (3.1.1) becomes:

u(x) = inf_{τ ≥0}E_0,x^P [X_τ] = x inf_{τ ≥0}E_0,x^P [exp(

τ

Z

0

θˆ_sds) exp(−1

2σ²τ + σ ˆW_τ)] (3.3.1) In view of the Girsanov theorem, if we put:

M_t= exp(−1

2σ²t + σ ˆW_t) = exp(−1 2

t

Z

0

σ²ds +

t

Z

0

σd ˆW_t), t ≤ T

and dQ

dP = MT, on NT

then, Q is a probability measure on NT and the process:

Wˆ_t^Q = −

t

Z

0

σds + ˆW_t⇔

d ˆW_t^Q= −σdt + d ˆWt

is a Wiener process with respect to Q. Moreover, the stochastic integral repre- sentation of ˆθtunder Q is of the form:

dˆθ_t= [k(1 + s^∗

σ²) − ˆθ_t]dt +s^∗ σd ˆW_t⇔ dˆθ_t= [k(1 + s^∗

σ²) − ˆθ_t]dt +s^∗

σ(d ˆW_t^Q+ σdt) ⇔ dˆθ_t= [k(1 + s^∗

σ²) + s^∗− ˆθ_t]dt + s^∗

σd ˆW_t^Q ⇔ dˆθt= (α − ˆθt)dt + βd ˆW_t^Q (3.3.2) where,

α = k(1 + s^∗

σ²) + s^∗and β = s^∗ σ.

Hence, under the probability measure Q we have that (3.3.1) becomes:

u(x) = x inf_{τ ≥0}E^Q

0, ˆθ[exp(

τ

Z

0

θˆ_sds)]. (3.3.3)

Now, according to equation (3.3.3) it can be seen that the solution of the agent’s original problem can be reduced to the solution of the problem:

V (ˆθ) = infτ ≥0E^Q

0, ˆθ[exp(

τ

Z

0

θˆsds)]. (3.3.4)

Notice that the smaller the drift, the bigger the agent’s profit. Thus, as above we postulate that there exists a threshold λ ∈ R with the corresponding ”stop- later” region of the process ˆθ_t to be A = (−∞, λ). Hence, the agent closes her position immediately when ˆθt= λ. Under the postulated strategy and in view of the Feynman-Kac theorem we have the following:

Theorem 3.3.1

The value function V satisfies the boundary value problem:

1

2β²V_{θ ˆˆ}^λ_θ+ (α − ˆθ)V_θˆ^λ+ ˆθV^λ= 0, for ˆθ ∈ A V^λ(ˆθ) = 1, for ˆθ ≥ λ 

Remark

The general solution of the above problem is of the form:

V^λ(ˆθ) = exp θ − 2ˆ r1

2β²+ α

! G −1

4β²+1 2α, 1

2 ; 2(

r1

2β²+ α)(ˆθ − (β²+ α)²)

!

where,

G −1 4β²+1

2α, 1 2 ; 2(

r1

2β²+ α)(ˆθ − (β²+ α)²)

!

is an arbitrary solution of the degenerate hypergeometric ordinary differential equation:

θVˆ _{θ ˆˆ}^λ_θ+ (1

2 − ˆθ)V_θˆ^λ+ (1 4β²+1

2α)V^λ= 0

Chapter 4

The Portfolio Problem

4.1 Introduction

In this chapter we assume that the agent possesses a self-financing portfolio consisting of a risky asset X_t (no dividends, no transaction costs) and a bank account with interest rate r > 0 such that dBt = rBtdt. Under the market models described in Chapter 2 we address the portfolio’s optimal rebalancing problem. In addition, we introduce a comparison theorem for the portfolio value functions between the complete information model where the drift is known con- stant and equal to the expected value of the random variable θ and the Digital Drift model.

4.2 Portfolio Rebalance

Since the portfolio is self-financing that means that at time t and once the new price of the risky asset X_t has been quoted the agent readjusts her position without bringing or removing any wealth. Finally, we introduce the terms of lending and borrowing at a riskless rate. The former can be interpreted as savings in the bank account (by selling the risky asset Xt), while the latter can be considered as withdrawals from the bank account (for investing in the risky asset Xt). Now let T > 0 be the time horizon and Φ(x) = x^γ−1

γ − 1, γ ∈ (1, 2), be the Constant Relative Risk Aversion utility function of the agent’s final time wealth. Φ expresses the agent’s risk aversion, i.e., the higher the γ, the more risk averse is the agent. Let us define P_t as the value of the portfolio at time t ≥ 0 and u⁰(t), u¹(t) the proportions invested in B_tand X_trespectively. Then, we have that the dynamics of P_tare given by:

dP_t= P_t[u⁰(t, ·)dBt

B_t + u¹(t, ·)dXt

X_t ] (4.1.1) where,

u⁰(t, ·) + u¹(t, ·) = 1, t ≥ 0 (4.1.2).

Notice that equation (4.1.2) means that the agent is fully invested and that unlimited short sales are allowed. Moreover, we demand u⁰ and u¹ to be ad- missible control laws, that is, to be adapted to the filtration generated by the risky asset Xt.

4.2.1 The Market Under Complete Information

Let us assume that we are under the following model:

dXt= E[θ]Xtdt + σXtdWt (4.2.1.1) where,

E[θ] = θ₁+ ρ(θ₂− θ1) (4.2.1.2) Theorem 4.2.1

Let V (t, p) = maxu¹Et,p[Φ(PT)], (t, p) ∈ [0, T ] × R+, be the optimal value function of the agent’s portfolio wealth. Then,

1) V (t, p) = p^γ−1 γ − 1exp



r + (E[θ] − r)² 2σ²(2 − γ)



(γ − 1)(T − t)



2) The optimal portfolio wealth allocation is given by:

u¹_∗= E[θ] − r

σ²(2 − γ), u¹_∗= 1 − u⁰_∗ Proof

Under the model given by (4.2.1.1) we have that (4.1.1) becomes:

dPt= Pt{[r + u¹(t, p)(E[θ] − r)]dt + u¹(t, p)σdW } (4.2.1.3)

Therefore, we have that the optimal value function V satisfies the Hamilton- Jacobi-Bellman equation:

Vt+ max_u1{[r + u¹(E[θ] − r)]pVp+1

2(u¹)²σ²p²Vpp} = 0 (4.2.1.5) V (T, ρ, p) = Φ (4.2.1.6)

Thus, if we let

f (u¹) = [r + u¹(E[θ] − r)]pVp+1

2(u¹)²σ²p²Vpp

and maximize it with respect to u¹we have that the optimal portfolio allocation is given by:

u¹_∗(t) = −(E[θ] − r)V_p σ²pVpp

, u⁰_∗(t) = 1 − u¹_∗(t) (4.2.1.7) By (4.2.1.7) we have that (4.2.1.5) takes the form:

Vt+ [r + u¹_∗(E[θ] − r)]pVp+1

2(u¹_∗)²σ²p²Vpp= 0 (4.2.1.8)

Hence, if we try:

V (t, p) = f (t)p^γ−1

γ − 1 (4.2.1.9) where f (T ) = 1, we have that:

f (t) = exp (A(T − t)) (4.2.1.10) where,

A = [r + u¹_∗(E[θ] − r)](γ − 1) +1

2(u¹_∗)²σ²(γ − 1)(γ − 2).

As a result by (4.2.1.7), (4.2.1.9) and (4.2.1.10) we have that, u¹_∗= E[θ] − r

σ²(2 − γ), u¹_∗= 1 − u⁰_∗ (4.2.1.11) and

V (t, p) = p^γ−1 γ − 1exp



r +(E[θ] − r)² 2σ²(2 − γ)



(γ − 1)(T − t)



(4.2.1.12)  Remark

Notice, that in this case the optimal portfolio allocation does not depend on time but only on the agent’s risk aversion level γ and remains constant in time.

Also, notice that the optimal value function is convex in E[θ].

4.2.2 The Digital Drift Case

Suppose that the price of a stock is modeled by (2.2.2.13) and (2.2.2.14). Then, (4.1.1) takes the form:

dPt= Pt[u⁰(t, p, ρ)dBt

B_t + u¹(t, p, ρ)dXt

X_t ] ⇔

dP_t= P_t{u⁰(t, p, ρ)rdt + u¹(t, p, ρ){[ ρ_t(θ₂− θ₁) + θ₁]dt + σd ˆW_t}} ⇔ dPt= Pt{ { r + u¹(t, p, ρ)[ρt(θ2− θ1) + θ1− r] }dt + u¹(t, p, ρ)σd ˆWt} (4.2.2.1) . Moreover, let

V (t, p, ρ) = max_u1 Et,p,ρ[Φ(PT)] (4.2.2.2)

be the optimal value function, where V is sufficiently smooth. In order to find the optimal portfolio allocation u¹_∗(t, p, ρ) we will make use of the dynamic pro- gramming argument. Therefore, in order to derive the Hamilton-Jacobi-Bellman equation we apply Ito’s formula on V (t, p, ρ). Notice that the processes Ptand ρt are driven by the same Wiener process. Thus, we have that:

dV =dV dtdt+dV

dρdρ_t+dV dpdP_t+1

2

 d²V

dρ²(dρ_t)²+ 2d²V

dρdp(dρ_t)(dP_t) +d²V dp²(dP_t)²



⇔

dV = Adt + Bd ˆW_t where,

A = Vt+1 2

(θ2− θ1)²

σ² ρ²_t(1 − ρt)²Vρρ+ (θ2− θ1)ρt(1 − ρt)u¹(t, p, ρ)PtVρp+ 1

2u¹(t, p, ρ)²σ²P_t²V_pp+ P_t{ r + u¹(t, p, ρ)[ρ_t(θ₂− θ₁) + θ₁− r] }V_p (4.2.2.3) and

B = (θ2− θ1)

σ ρ_t(1 − ρ_t)V_ρ+ P_tu¹(t, p, ρ)σV_p (4.2.2.4)

As a result, we have that the Hamilton-Jacobi-Bellman equation is of the form:

Vt+ max_u1[L^u1V ] = 0 ⇔ Vt+1

2

(θ₂− θ1)²

σ² ρ²(1 − ρ)²Vρρ+ rpVp+ maxu¹{C} = 0 (4.2.2.5) where,

C = (θ2− θ1)ρ(1 − ρ)u¹pVρp+ pu¹[ρ(θ2− θ1) + θ1− r]Vp+1

2(u¹)²σ²p²Vpp

Hence, by maximizing C with respect to u¹ we have that the optimal portfolio allocation will be given by:

u¹_∗(t, p, ρ) = −(θ₂− θ₁)ρ(1 − ρ)pV_ρp+ [ρ(θ₂− θ₁) + θ₁− r]pV_p σ²p²Vpp

(4.2.2.6) and

u⁰_∗(t, p, ρ) = 1 − u¹_∗(t, p, ρ) (4.2.2.7)

Thus, by substituting (4.2.2.6) into (4.2.2.5) we have that V (t, p, ρ) satisfies:

Vt+1 2

(θ2− θ1)²

σ² ρ²(1 − ρ)²Vρρ+ rpVp+ C_∗= 0 (4.2.2.8) V (T, ρ, p) = Φ (4.2.2.9)

where,

C_∗= (θ2−θ1)ρ(1−ρ)u¹_∗pVρp+pu¹_∗[ρ(θ2−θ1)+θ1−r]Vp+1

2(u¹_∗)²σ²p²Vpp (4.2.2.10).

Let us try a solution of the form:

V (t, p, ρ) = p^γ−1 γ − 1



H(t, ρ)^2−γ, such that H(T, ρ) = 1, H ≥ 0 where (t, p, ρ) ∈ [0, T ] × R+× (0, 1)

(4.2.2.11) Then, we have that:

Vt= p^γ−1 γ − 1



(2 − γ)H^1−γHt, Vρ= p^γ−1 γ − 1



(2 − γ)H^1−γHρ

Vρρ= p^γ−1 γ − 1



(2 − γ)(1 − γ)H^−γH_ρ²+ H_ρρ^1−γ , Vp= p^γ−2H^2−γ V_pp = (γ − 2)p^γ−3H^2−γ, V_ρp= p^γ−2(2 − γ)H^1−γH_ρ

Thus, we have that (4.2.2.6) becomes:

u¹_∗(t, ρ) = (θ2− θ1)ρ(1 − ρ)Hρ

σ²H +ρ(θ2− θ1) + θ1− r

σ²(2 − γ) (4.2.2.12) u⁰_∗(t, ρ) = 1 − u¹_∗(t, ρ) (4.2.2.13)

Now, by (4.2.2.12) we have that (4.2.2.10) takes the form:

C_∗=1 2

(θ2− θ1)²ρ²(1 − ρ)²(2 − γ)p^γ−1H_ρ² σ²H^γ

+(θ2− θ1)ρ(1 − ρ)[ρ(θ2− θ1) + θ1− r]p^γ−1H^1−γHρ

σ² +1

2

[ρ(θ₂− θ₁) + θ₁− r]²p^γ−1H^2−γ σ²(2 − γ)

(4.2.2.14)

Therefore, by (4.2.2.8) in view of (4.2.2.14) and all the above we derive the following:

Theorem 4.2.2.1

The value function of the agent’s problem is given by (4.2.2.11), where H sat- isfies the following linear partial differential equation:

Ht+ 1

2σ²(θ2− θ1)²ρ²(1 − ρ)²Hρρ

+ γ − 1 2 − γ

  θ2− θ1

σ²



{ρ(1 − ρ)[ρ(θ2− θ1) + θ1− r]} Hρ

+ γ − 1 2 − γ

 

r + 1

2σ²(2 − γ)[ρ(θ₂− θ1) + θ₁− r]²

 H = 0 such that: H(T, ρ) = 1.

The stochastic representation of H is given by:

H(t, ρ) =

exp



r(γ − 1)(T − t) 2 − γ

 Et,ρ



exp





γ − 1 2σ²(2 − γ)²

T

Z

t

[Ys(θ2− θ1) + θ1− r]²ds







 where the dynamics of Y_t are given by:

dYt= µ(t, Yt)dt + σ(t, Yt)dΠt

where,

µ(t, Yt) = (γ − 1)(θ2− θ1) (2 − γ)σ²



{Yt(1 − Yt)[Yt(θ2− θ1) + θ1− r]}

and

σ(t, Yt) =(θ2− θ1)Yt(1 − Yt) σ

Moreover, the optimal portfolio allocation is given by (4.2.2.12) and (4.2.2.13)



Remarks

1) The stochastic representation of H(t, ρ) can be derived by a straightfor- ward application of the Feynman-Kac theorem. Namely, the solution of the general boundary value problem on [0, T ] × R

H_t+ µ(t, ρ)H_ρ+1

2σ(t, ρ)²H_ρρ+ r(t, ρ)H = 0 H(T, ρ) = Φ(ρ)

by considering the process G_s= H(s, ρ_s) exp

s

R

t

r(u, ρ_u)du



is given by:

H(t, ρ) = Et,ρ



Φ(ρT) exp





T

Z

t

r(s, ρs)ds









2) Notice that in paragraph 3.2 we have identified when is optimal the agent to buy back the stock when she is in a short position.

Now, let as assume that in the beginning of time our agent is fully informed about the distribution of the drift as described in paragraph (2.2.1). Then, by (4.2.2.6) and (4.2.2.8) we derive that:

If P [θ = θ2] = 1 = ρ, then:

V (t, p, 1) = U¹(t, p) = p^γ−1 γ − 1exp



r + (θ₂− r)² 2σ²(2 − γ)



(γ − 1)(T − t)



(4.2.2.15) which is the optimal value function when the drift is known, constant and equal to θ₂, with the optimal wealth allocation to be constant and given by:

u^1,1_∗ = θ₂− r

σ²(2 − γ) > 0 (4.2.2.16) If P [θ = θ2] = 0 = ρ, then:

V (t, p, 0) = U⁰(t, p) = p^γ−1 γ − 1exp



r + (θ₁− r)² 2σ²(2 − γ)



(γ − 1)(T − t)



(4.2.2.17) which is the optimal value function when the drift is known, constant and equal to θ₁, with the optimal wealth allocation to be constant and given by:

u^1,0_∗ = θ1− r

σ²(2 − γ) < 0 (4.2.2.18)

Moreover, if we do not adopt the filtering technique as we did in paragraph 2.2.2 and work solely under the model described in paragraph 2.1, we have that our agent’s portfolio process will be given by:

dP_t= P_t{[r + u¹_θ(t, p)(θ − r)]dt + u¹_θ(t, p)σdW_t} (4.2.2.19)

Let { D}t≥0 be the information generated by the process P_t and notice that u_θ is an admissible portfolio strategy if and only if it is Dt adapted. Then the optimal value function is given by:

V^θ(t, p) = max_u1

θE[Φ(PT)] (4.2.2.20) Now, let Z_t= P_t^γ−1. By Ito’s formula we have that:

ZT = p^γ−1exp



(γ − 1)[r + u¹_θ(θ − r)] +1

2(γ − 1)(γ − 2)(u¹_θ)²σ²

 T



×

exp







−1 2

T

Z

0

(γ − 1)²(u¹_θ)²σ²ds +

T

Z

0

(γ − 1)u¹_θσdWs





 Now, if we define

Mt= exp







−1 2

t

Z

0

(γ − 1)²(u¹_θ)²σ²ds +

t

Z

0

(γ − 1)u¹_θσdWs





 , t ≤ T

and a measure L such that:

dL

dP = MT, on DT

then in view of Girsanov’s theorem we have that (4.2.2.20) becomes:

V^θ(t, p) = max_u1

θE^L

 1

γ − 1p^γ−1exp



(γ − 1)[r + u¹_θ(θ − r)] +1

2(γ − 1)(γ − 2)(u¹_θ)²σ²

 (T − t)



(4.2.2.21)

Lemma 4.2.2.1

For the optimal value functions given by (4.2.2.11) and (4.2.2.21) we have that:

V^θ(t, p) = V (t, p, ρ), ∀(t, p) ∈ [0, T ] × R+

Proof

The result comes immediately by (2.2.2.2), (2.2.2.3), (2.2.2.5), (2.2.2.13), (4.2.2.1) and (4.2.2.19) 

Now, let us assume that our agent decides to follow the admissible portfolio strategy:

u¹_θ= E[θ] − r

σ²(γ − 2) = α, constant

Then, for our agent’s value function, according to (4.2.2.21), we have that:

V_α^θ(t, p) ≤ V^θ(t, p) (4.2.2.22) where,

V_α^θ(t, p) = E^L

 1

γ − 1p^γ−1exp



(γ − 1)[r + α(θ − r)] +1

2(γ − 1)(γ − 2)α²σ²

 (T − t)



Furthermore, notice that the function V_α^θ(t, p) is convex in θ. As a result, in view of Jensen’s inequality we have that:

φ(E^L[θ]) ≤ E^L[φ(θ)] ⇔ p^γ−1

γ − 1

 exp



r +(E^L[θ] − r)² 2σ²(2 − γ)



(γ − 1)(T − t)



≤ V_α^θ(t, p) ⇔ K(t, p, ρ) ≤ V_α^θ(t, p) ≤ V (t, p, ρ) (4.2.2.23)

where,

K(t, p, ρ) = p^γ−1 γ − 1

 exp



r +(ρ(θ₂− θ1) + θ₁− r)² 2σ²(2 − γ)



(γ − 1)(T − t)



(4.2.2.24) Remark

Notice that the function K(t, p, ρ) is the optimal value function derived given by Theorem 4.2.1 where the drift is known constant and equal to the expected value of the random variable θ with the optimal portfolio wealth allocation to be given by α.

Now, in view of (4.2.2.21) we have that:

V^θ(t, p) = max_u1

θE^L

 1

γ − 1p^γ−1exp



(γ − 1)[r + u¹_θ(θ − r)] +1

2(γ − 1)(γ − 2)(u¹_θ)²σ²

 (T − t)



≤ ρ

γ − 1p^γ−1max_u1 θ

 exp



(γ − 1)[r + u¹_θ(θ2− r)] +1

2(γ − 1)(γ − 2)(u¹_θ)²σ²

 (T − t)



+ (1 − ρ)

γ − 1 p^γ−1max_u¹

θ

 exp



(γ − 1)[r + u¹_θ(θ1− r)] +1

2(γ − 1)(γ − 2)(u¹_θ)²σ²

 (T − t)



=

ρ U¹(t, p) + (1 − ρ) U⁰(t, p) (4.2.2.25)

where the last equality comes from (4.2.2.15) and (4.2.2.17).

Let, Z(t, p, ρ) = ρ U¹(t, p) + (1 − ρ) U⁰(t, p). Notice that this represents the optimal value function under the assumption that the agent is not only aware about the distribution of the drift ρ, but also she is fully informed about the outcome of the ”coin toss” occurred in the beginning of time. Therefore, at time t = 0 her decision will naturally be to take position in both possible scenarios with respect to the distribution of the random variable θ and at time t > 0 to follow the strategy indicated by U¹(t, p) or U⁰(t, p).

From all the above we derive the following:

Theorem 4.2.2.2

Let γ ∈ (0, 1) and the agent’s constant relative risk aversion to be given by 0 < γ − 1 < 1. If V (t, p, ρ) is the optimal value function as described in (4.2.2.11) then ∀(t, p, ρ) ∈ [0, T ] × R+× (0, 1) we have that:

K(t, p, ρ) ≤ V (t, p, ρ) ≤ Z(t, p, ρ) 

4.2.3 The Ornstein-Uhlenbeck Drift case

Similarly as in 4.2.2, now we suppose that the market is governed by equations (2.3.2.12) and (2.3.2.13). Therefore, (4.1.1) becomes:

dPt= Pt{[r + u¹(t, p, ˆθ)(ˆθt− r)]dt + u¹(t, p, ˆθ)σd ˆWt} (4.2.3.1).

By defining the optimal value function of the agent’s wealth as:

V (t, p, ˆθ) = maxu¹E_{t,p, ˆθ}[Φ(PT)] (4.2.3.2)

where again V is sufficiently smooth, we have that by Ito’s formula (again ˆθ_t and P_tare driven by the same Wiener process ˆW_t) the Hamilton-Jacobi-Bellman equation is of the form:

Vt+ max_u1[L^u1V ] = 0 ⇔ Vt+ [k(1 + s^∗

σ²) − ˆθ]Vθˆ+ rpVp+1 2(s^∗

σ)²Vθ ˆˆθ+ maxu¹[A] = 0 (4.2.3.3) where,

A = [p(ˆθ − r)Vp+ s^∗pVθpˆ]u¹+1

2σ²(u¹)²p²Vpp

Thus, if we maximize A with respect to u¹ we have that:

u¹_∗(t, p, ˆθ) = −s^∗pV_θpˆ + (ˆθ − r)pV_p σ²p²V_pp (4.2.3.4) and

u⁰_∗(t, p, ˆθ) = 1 − u¹_∗(t, p, ˆθ) (4.2.3.5) As a result, (4.2.3.3) under (4.2.3.4) becomes:

Vt+ [k(1 +s^∗

σ²) − ˆθ]Vθˆ+ rpVp+1 2(s^∗

σ)²Vθ ˆˆθ+ A∗= 0 (4.2.3.6) V (T, p, ˆθ) = Φ (4.2.3.7)

where,

A∗= [p(ˆθ − r)Vp+ s^∗pVθpˆ ]u¹_∗+1

2σ²(u¹_∗)²p²Vpp (4.2.3.8)

Similarly as in 4.2.2, let us apply a solution of the form:

V (t, p, ˆθ) = p^γ−1

γ − 1L^2−γ(t, ˆθ) , such that L(T, ˆθ) = 1, L ≥ 0 where (t, p, ˆθ) ∈ [0, T ] × R+× R

(4.2.3.9)

where L(t, ˆθ) to be determined. Then, (4.2.3.4) becomes:

u¹_∗(t, ˆθ) = s^∗Lθˆ

σ²L +

θ − rˆ

σ²(2 − γ) (4.2.3.10) u⁰_∗(t, ˆθ) = 1 − u¹_∗(t, ˆθ) (4.2.3.11) Now, if we substitute (4.2.3.10) into (4.2.3.8) we derive that:

A_∗=1 2

(s^∗)²(2 − γ)p^γ−1L²_ˆ

θ

σ²L^γ +s^∗(ˆθ − r)p^γ−1L^1−γLθˆ

σ² +1

2

(ˆθ − r)²

σ²(2 − γ)p^γ−1L^2−γ (4.2.3.12)

Thus, by combining all the above we derive the following:

Theorem 4.2.3.1

The value function of the agent’s problem is given by (4.2.3.9), where L satisfies the following linear partial differential equation:

Lt+1 2

(s^∗)² σ² Lθ ˆˆθ+

"

k(1 + s^∗

σ²) − ˆθ + s^∗(γ − 1)(ˆθ − r) σ²(2 − γ)

# Lθˆ

+ γ − 1 2 − γ

"

r +1 2

(ˆθ − r)² σ²(2 − γ)

# L = 0 The stochastic representation of L is given by:

L(t, ˆθ) =

exp



r(γ − 1)(T − t) 2 − γ

 E_{t, ˆθ}



exp





γ − 1 2σ²(2 − γ)²

T

Z

t

(Ys− r)²ds









where the dynamics of Yt are given by:

dYt= µ(t, Yt)dt + σ(t, Yt)dΠt

where Π is a Wiener process and µ(t, Yt) =



k(1 + s^∗

σ²) −s^∗(γ − 1)r σ²(2 − γ)



−



1 − s^∗(γ − 1) σ²(2 − γ)

 Yt



and

σ(t, Y_t) = (s^∗)

σ , constant.

Moreover, the optimal portfolio allocation is given by (4.2.3.10) and (4.2.3.11)  Remarks

1) The stochastic representation of L(t, ˆθ) can be derived exactly as described in the case of theorem (4.2.2.1)

2) If we assume that we are under complete information about the drift, that is Yt = µ0 constant, then by theorem (4.2.3.1) we derive instantly (4.2.1.13) and (4.2.1.14).

3) Notice that in paragraph 3.2 we have identified when is optimal the agent to buy back the stock when she is in a short position.