optimal liquidation of stock

U.U.D.M. Project Report 2016:34

Examensarbete i matematik, 30 hpHandledare: Erik EkströmExaminator: Kaj NyströmJuli 2016

Department of MathematicsUppsala University

Optimal Liquidation of Stock

Di Liu

MASTERS THESIS


Author:Di LIU

Supervisor:Erik EKSTRÖM

A thesis submitted in fulfillment of the requirementsfor a Master of Science Degree

at the

Department of Mathematics

July 13, 2016

iii

Declaration of AuthorshipI, Di LIU, declare that this thesis titled, “Optimal Liquidation of Stock”and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature fora research degree at this University.

• Where any part of this thesis has previously been submittedfor a degree or any other qualification at this University or anyother institution, this has been clearly stated.

• Where I have consulted the published work of others, this isalways clearly attributed.

• Where I have quoted from the work of others, the source is al-ways given. With the exception of such quotations, this thesisis entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly withothers, I have made clear exactly what was done by others andwhat I have contributed myself.

Signed:

Date:

v

“No man has lived in the past, and none will ever live in the future; thepresent alone is the form of life, but it is also life’s sure possession which cannever be torn from it. The present always exists together with its content;both stand firm without wavering, like the rainbow over the waterfall.”

Arthur Schopenhauer

vii

UPPSALA UNIVERSITY

AbstractDepartment of Mathematics

Master of Science


by Di LIU

The problem studied in this thesis is how to liquidate stock hold-ings under risk-aversion. The thesis studied two cases: the indivis-ible stock and the infinitely divisible stock . In the indivisible stockwith an infinite horizon a theoretical analysis was performed and aconstant boundary was characterized at which it is optimal to liq-uidate. If, on the other hand, the horizon is finite, then numericalcalculations were carried out to find a time-dependent liquidationboundary. In the infinitely divisible case the optimal strategy wasdescribed. I also intuitively clear property that the liquidation of adivisible stock begins earlier than the liquidation of an indivisibleone.

ix

AcknowledgementsAfter two years, my studies with the financial mathematics Mas-

ter Program at Uppsala University has come to an end. Writing thisthesis is a mile stone for this period of my life. It has been an pro-cess that has impacted not only on scientific aspects, but also on apersonal level. After an intensive period of six months, I am nowending the thesis with the writing of this acknowledgment.

My deepest appreciation goes to Prof. Erik Ekström, who hasbeen very patient and supportive throughout the course of this the-sis. Discussions with Prof. Ekström were illuminating; the inspira-tion that he gave me was invaluable. This thesis would not havebeen possible Without his guidance and tireless help. The tuition hegave me in scientific writing was systematic and this will now staywith me in the future.

I would like to express my gratitude to Prof. Maciej Klimek andSenior Lecturer Elisabeth Larsson for the solid Matlab skills and knowl-edge that I have learned from their course.

I would like to offer my special thanks to my Study CounselorAlma Kirlic for all her insightful comments and suggestions. Shemanaged to disperse the mists every time that I felt doubtful aboutmy studies.

xi

Contents

Declaration of Authorship iii

Abstract vii

Acknowledgements ix

1 Introduction 1

2 The Time-independent Model 32.1 Indivisible Stock Case . . . . . . . . . . . . . . . . . . . 3

2.1.1 The objective . . . . . . . . . . . . . . . . . . . . 32.1.2 The Optimal Stopping Problem . . . . . . . . . . 42.1.3 The Verification . . . . . . . . . . . . . . . . . . . 5

2.2 Divisible Stock Case . . . . . . . . . . . . . . . . . . . . 72.2.1 The objective . . . . . . . . . . . . . . . . . . . . 72.2.2 The General strategy . . . . . . . . . . . . . . . . 72.2.3 The Explicit Solution . . . . . . . . . . . . . . . . 92.2.4 The Verification . . . . . . . . . . . . . . . . . . . 11

2.3 Comparison of results . . . . . . . . . . . . . . . . . . . 122.3.1 Numerical examples . . . . . . . . . . . . . . . . 13

3 The Time-dependent Model 153.1 Indivisible Stock Case . . . . . . . . . . . . . . . . . . . 15

3.1.1 The objective . . . . . . . . . . . . . . . . . . . . 153.1.2 The Grid . . . . . . . . . . . . . . . . . . . . . . . 163.1.3 The Crank–Nicolson Discretization . . . . . . . 173.1.4 Finite Difference Method - PSOR . . . . . . . . . 20

3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 213.3 The Divisible Case . . . . . . . . . . . . . . . . . . . . . 22

4 Conclusion 23

A Figures from The Time-independent Model 25

B Figures from The Time-dependent Model 27

Bibliography 29

xiii

List of Figures

2.1 Function of b with fixed parameters. . . . . . . . . . . . 122.2 The thresholds of the time-independent model . . . . . 13

3.1 The value V evaluation in the mesh . . . . . . . . . . . 213.2 The threshold S∗t . . . . . . . . . . . . . . . . . . . . . . . 22

A.1 The function of b with µ varying . . . . . . . . . . . . . 25A.2 The function of b with ρ varying . . . . . . . . . . . . . . 25A.3 The function of b with σ varying . . . . . . . . . . . . . 26A.4 The function of b with w varying . . . . . . . . . . . . . 26A.5 The thresholds with 10000 trajectories . . . . . . . . . . 26

B.1 The value V evaluation in the 50 years mesh . . . . . . 27B.2 The threshold S∗t of 50 years period . . . . . . . . . . . . 27

xv

To Liu Yu - my mother and my friend,for your love and for educating me, a

seemingly complicated combination, yet youmade it seem effortless.

1

Chapter 1

Introduction

This thesis is devoted to the problem of stock liquidation for a risk-averse agent. In a complete market, an agent can sell a unit of stockwith a single stopping time. On the other hand, based on Hendersonand Hobson’s (Henderson and Hobson, 2008) incomplete market set-ting, the agent is allowed to liquidate fractions of a unit with a familyof stopping times.

The study begins with an optimal liquidation problem for thetime-independent diffusion price process. First the simplest case wasstudied - the case of indivisible stock, which is an optimal stoppingproblem, where an optimal barrier was assumed to start with. Af-ter having solved the problem for the optimal barrier, it was verifiedthat the assumed barrier was indeed the optimal one. However, thevalue function with an optimal barrier turned out to be non-explicit,but it was possible to make numerical plotting.

With perfectly divisible stock, agents are expected to sell part oftheir stock at first when the price reaches a barrier level and this de-pends on the amount of stock remaining. The amount to be soldshould be just enough to maintain the remaining value below thebarrier.

The key assumptions that make the problem independent of timeand make it possible obtain an explicit solution are:

• The work is with respect to single numeraire and assumes thatan agent’s utility function can be expressed relatively in a timeindependent way;

• The dynamics of the stock are time independent;

• The aim is to maximize the utility of wealth, which refers to therevenue from the sale of stock over an infinite horizon.

In the case in question, these assumptions were without loss of gen-erality with the stock being the numeraire.

However, in the case of divisible stock, there is a problem withthree dimensions,viz, the stock price, the agent’s wealth; and theamount of the remaining stock. The boundary will depend on thewealth and the remaining stock and in consistency of the initial wealthof the agent.

2 Chapter 1. Introduction

In Henderson and Hobson’s (Henderson and Hobson, 2011) , theymanaged to characterize the barrier in an explicit form by solving acalculus of variation problem. In general, their method modified theHamilton-Jacobi-Bellman (HJB) approach by calculating the valuefunction for any strategy of threshold type, and then choosing thebest barrier. Thereby this study made comparison and conclusionconcerning the effects of the coefficients on the boundary conditionsthat were produced during the two progresses.

To be more realistic, the problem was entered into a time depen-dent model. Although it was challenging to include the time depen-dence it was also practical, since it is normal that an agent will expectto liquidate his position within a certain period and his behavior maybe affected by the time to maturity. it was assumed that instead ofa horizontal line the barrier would be a curve that decreased as ma-turity got closer. Agent would behave similarly to what is describedfor a time-independent case.

When it came to indivisible stock, the problem was solvable us-ing a numerical method, e.g. the finite difference method, whosealgorithm is similar to the American option case but with a differ-ent payoff function. This solution encompasses the optimal stoppingtime and the wealth of an agent. The comparison is among the nu-merical results. The divisible case is more complicated, because thebarrier depends on the remaining stocks and the remaining time. Itmay involve a concept such as a high dimensional finite differencemethod, which is out of the scope of this thesis.

The structure of this paper is as follows. The second chapter de-scribes the detail of the time-independent mathematical model, de-rives the general solution for the cases, and then makes a comparisonwhich is a summery with numerical examples. Chapter 3 presentsthe numerical algorithm for the time-dependent case, along with em-pirical results. Chapter 4 contains a conclude to the study and futureproposition.

3

Chapter 2

The Time-independent Model

This chapter considers an agent who is described in the following.A risk-averse agent has a share of stock; his aim is to maximize theexpected utility of total revenue from the liquidation in a perpetualcase.

For a case of stock, Black-Scholes model is very commonly used, as proposed by Fischer Black and Myron Scholes in 1973. A moredetailed description of this model can be found in (Hull, 2011). LetSt be the price of the stock on a filtration (Ω,F , (F)t≥0,P) that sup-porting a Wiener process W . St is assumed to be a time-independentdiffusion process with a log-normal dynamic

dSt = σStdWt + µStdt, S0 = s.

where the drift µ > 0 and the volatility σ > 0 are constant coeffi-cients. Then the infinitesimal operator LS of S is defined as LSf =σ(s)2

2fss + µ(s)fs.

Also, it was assumed that the constant risk aversion utility func-tion U of the agent was: U(x) = x1−ρ/1− ρ with ρ ∈ (0, 1).

2.1 Indivisible Stock Case

2.1.1 The objective

In this section, the market assumed that the stocks could only besold in units, while the agent initially held one share of the stock.Under this circumstance, the aim was to determine the single optimalstopping time τ for the a liquidation strategy.

For the agent with a utility function U , the objective was to maxi-mize the expected utility

V (s) = supτE[U(w + Sτ )] ≥ E[U(w + s)],

with the initial wealth w. This is a typical optimal stopping problem.

4 Chapter 2. The Time-independent Model

2.1.2 The Optimal Stopping Problem

The operator of V reads

LSV = −σ2s2

2ρ(s+ w)−1−ρ + µs(s+ w)−ρ

= s(s+ w)−1−ρ(

(µ− σ2ρ

2)s+ µw

).

This leads to three cases corresponding to the parameters, such that

• µ > σ2ρ/2, it is optimal never to stop;

• µ < σ2ρ/2, it is optimal to stop with an upper boundary;

• µ < 0, it is optimal to stop immediately.

The case of µ < 0 and µ > σ2ρ/2 are trivial, so µ was defined µ <σ2ρ/2 from now on. Following the standard optimal stopping prob-lem solving procedure (Peskir and Shiryaev, 2006, p. 374-378), athreshold b ∈ (0,∞) is assumed, such that

τb = inft ≥ 0 : St ≥ b

was the optimal stopping time in this particular case. Then a free-boundary problem was formulated for the value function V and thethreshold b:

LSV = 0 0 < s < b, (2.1)

V (s) =(s+ w)1−ρ

1− ρs ≥ b, (2.2)

V (s) >(s+ w)1−ρ

1− ρ0 < s < b, (2.3)

V ′(b) = (b+ w)−ρ (smooth fit), (2.4)

For (2.1), it was assumed that the solution was in the form of f(s) =sβ and it was inserted back into (2.1) the result was

β2 − (1− 2µ

σ2)β = 0.

The roots were β1 = 0 and β2 = 1− 2µ/σ2. Thus the general solutionof (2.1) was

f(s) = Cs1−2µ/σ2

+D,

2.1. Indivisible Stock Case 5

where C,D was an undetermined constant. Then the system of (2.2),(2.4) led to

Cb1−2µ/σ2

+D =(b+ w)1−ρ

1− ρ,

C(1− 2µ

σ2)b−2µ/σ2

= (b+ w)−ρ,

D = f(0) =w1−ρ

1− ρ.

It was deduced to

(1− 2µ

σ2)(b+ w)1−ρ − w1−ρ

1− ρ= b(b+ w)−ρ,

which gave the function of b, as follows

f(b) = b(b+ w)−ρ − (1− 2µ/σ2)(b+ w)1−ρ − w1−ρ

1− ρ, (2.5)

f ′(b) =2µ

σ2(b+ w)−ρ − ρb(b+ w)−1−ρ. (2.6)

The facts were observed

f(0) = 0, f ′(0) > 0; f(∞) < 0;

f ′(b) > 0, b < bmax; f ′(b) < 0, b > bmax,

thus indicating that the function f(b) = 0+ had a unique solution b∗.The constant C and the value function read

C =σ2

σ2 − 2µ

b∗2µ/σ2

(b∗ + w)ρ, (2.7)

V (s) =

Cs1−µ/D + E s ∈ [0, b∗](s+w)1−ρ

1−ρ s ∈ (b∗,∞)(2.8)

Hereby, the stock price threshold was determined by b∗ in this case.

2.1.3 The Verification

As part of standard procedure, the accurateness of this guessed so-lution was verified in this section. To be specific, denote the objec-tive from section (2.1.1) by V ∗(s), we would show that V ∗(s) = V (s)where V (s) came from (2.8) above.

At first, the inequalities between V ∗(s) and V (s) were shown.Since V (s) is C2 on (0, b) ∪ (b,∞), the Ito’s formula was applicable


to V (s),

V (St) = V (s) +

∫ t

0

LSV (Sx)1Sx 6=bdx+

∫ t

0

σSxV′(Sx)dWx. (2.9)

For V (s) with drift µ < ρσ2/2, the operator was deduced to LSV =

s(s + w)−1−ρ(

(µ− σ2ρ2

)s+ µw)< 0 for s > b, so that together with

(2.1) the following inequality held

LSV ≤ 0, s ∈ (0,∞)/b. (2.10)

Therefore considering the fact that Ps(St = b) = 0 for every t and s,the functions (2.2),(2.3),(2.9) and (2.10) together implied that

U(w + St) ≤ V (St) ≤ V (s) +Mt, (2.11)

where (Mt)t≥0 is a continuous local martingale given by

Mt =

∫ t

0

σSxV′(Sx)dWx.

Let (τn)n≥1 be a localization sequence of stopping times for M ,then, with (2.11), inequality is arrived at

U(w + Sτ∧τn) ≤ V (s) +Mτ∧τn , ∀n ≥ 1,∀τ.

Using the optimal sampling theorem (Peskir and Shiryaev, 2006, p.60) and

√τ ∧ τn < ∞, it was concluded that E[Mτ∧τn ] = 0 for all n,

and when n→∞ it was found by Fatou’s lemma that

E[U(w + Sτ )] ≤ lim inf E[U(w + Sτ∧τn ] ≤ V (s),

V ∗(s) ≤ V (s).

For the stopping time τb, the equations (2.1) and (2.9) with theoptimal sampling theorem lead to

E[V (Sτb∧τn)] = V (s), ∀n ≥ 1

Since V (Sτb) = U(w+ Sτb), a limitation was taken for n→∞ and thedominated convergence theorem was applied to the above equation,so that

E[U(w + Sτb)] = V (s).

This indicated that τb was optimal for the objective, and consequentlyV ∗(s) = V (x) for all x > 0 was proved.

2.2. Divisible Stock Case 7

2.2 Divisible Stock Case

The presentation of the current section follows an article by Hender-son and Hobson, 2011. They considered the more general problemof liquidating a portfolio of stocks and options. Based on their theo-retical path, the specific case of liquidating a single stock with time-independent setting is presented here.

2.2.1 The objective

The total number (percentage) of remaining stock (Θt)t≥0 is a de-creasing process with θ0 = 1, 1 ≥ Θt ≥ 0. Hence, with the currentstock price St as the return of selling, the total revenue from the liq-uidation is

R = −∫ ∞

0

StdΘt

Section 2.1 stated that the scale function of St is f(x) = xβ with β =1 − 2µ/σ2 which is the solution to LSV = 0. The objective was tomaximize the value function

V (w, s, θ0) = E[U(w +R)]

with the initial wealth w and the utility function U .

2.2.2 The General strategy

Under the time-independent setting, if it is optimal not to sell stockat a certain price level, then no stock will be sold during this priceperiod, and it will still be optimal not to sell any stock if the price re-turns to that level. Therefore, a set of time-independent thresholds isnecessary for the liquidate strategy herein, such as Θt = H(Xt) whereXt = maxs<t Ss. The H is a non-increasing function and it is assumedto be right-continuous, though jumps or interval of constancy are notprecluded fromH . The left-continuous inverse function ofH was de-fined as Xt = h(Θt) = h(Θt;w), since H may depend on an agent’sinitial wealth w.

The liquidate strategy with the H was as follows. Firstly, if theinitial holding θ0 is greater than H(s), the agent will sell 1 − H(s)stock immediately. Then the agent shall keep Θt = H(Xt) holds forthe rest of time by selling stocks. The strategy is continuous sinceH is continuous, except for the potential initial transaction at t = 0.Stocks are only sold when St is at its historical maximum, which issingular with respect to time.

A general solution was reached by expending the total revenuefrom sales. According to this strategy, stock is sold if Xt ≥ h(Θt),which leads to a higher h, as a result, the balance Xt = h(Θt) can be


regained,

R = −∫ ∞t=0

StdΘt =

∫ θ0

0

1Xt≥h(θ)h(θ)dθ.

The transience assumption on St provides that X = X∞ = maxt St isfinite almost surely. Thus there is the deterministic R conditional onX

R = R(X) =

∫ θ0

H(X)

h(θ)dθ.

Then the value function was expended with integrating by parts andthe change of variable θ = H(x),

V (w, s, θ0) =

∫ ∞s

P(X ∈ dx)U(w +R(x))

= [−U(w +R(x))P(X ≥ x)]∞s +

∫ ∞s

P(Xt ≥ x)U ′(w +R(x))R′(x)dx

= U(w)−∫ ∞s

dxf(s)dH

f(x)dxU ′(w +R(x))x

= U(w) + f(s)

∫ θ0

0

h(θ)

f(h(θ))U ′(w +R(h(θ)))dθ

Thus, if θ0 ≤ H(s), the value function for the agent is

V = U(w) + f(s)

∫ θ0

0

h(θ)

f(h(θ))U ′(w +

∫ θ0

θ

h(φ)dφ

)dθ. (2.12)

Next, the optimal threshold h∗ was chosen with only the relativepart from (2.12) being considered,∫ θ0

0

h(θ)

f(h(θ))U ′(w +

∫ θ0

θ

h(φ)dφ

)dθ. (2.13)

For fixed s, w, θ0, the aim was to maximize (2.13) with the optimalh∗(ψ;w, θ0).

By l’Hopital rule with a continuous stock process St, this con-verged to

h(0)

f(h(0))U ′(w), (2.14)

which was maximized by the optimal initial point h∗(0).For the general threshold, it was assumed that h(θ) was contin-

uous, twice differentiable and denoted D(φ) = −∫ θ0φh(ψ)dψ, then

h(φ) = D′(φ). The goal now was to maximize∫ θ0

0

dφD′(φ)U ′(w0 −D(φ))D(D′(φ)), (2.15)


with the optimal D, where D(z) = 1/f(z). Therefore, a function ofD was obtained and φ: g(D, φ) = D′(φ)U ′(w0 − D(φ))D(D′(φ)). Bythe Euler equation (Arfken, Weber, and Harris, 2006, eq. 17.15) theoptimal D solves

∂g

∂D− d

dφ

∂g

∂Dφ= 0

D′(φ)U ′(w −D(φ))∂D∂φ

(D′(φ)) +d

dφ

[U ′(w −D(φ))D′(φ)2 ∂D

∂D′(D′(φ))

]= 0

where ∂D∂φ

(D′(φ)) = 0. By integration

∂D∂D′

(D′(φ))D′(φ)2U ′(w −D(φ)) = constant.

For a St that is independent of θ then the optimal h∗ has to satisfy

h(φ)2f ′(h(φ))

h′(φ)f(h(φ))2U ′(w +

∫ θ0

θ

h(φ)dφ

)= constant. (2.16)

2.2.3 The Explicit Solution

The general solution (2.16) of h∗ with the specific utility function andscale function leads to this equation of h∗

h(φ)1−β(w +

∫ θ0

φ

h(ψ)dψ

)−ρ=

(qρ

1− β

)−ρ= constant, (2.17)

where the constant term with some q is subject to the optimal h(0)∗

that maximizes (2.14) from last section. Note if β > 0, h∗(0) = 0, itis optimal to sell all the stock immediately. Hereafter, only the case0 < β < 1 was considered with h∗(0) =∞.

A differentiation of (2.17) for the optimal threshold showed thath′ = −h1+η/q with η = (ρ+β−1)/ρ < 1,which arrived at the solutionh∗(φ) = (ηφ/q)−1/η. Under the condition of h∗(0) =∞, the parameterwas η > 0.

Putting the h∗ back into (2.17), the equation must converge to theinitial as φ→ 0:

limφ↓0

(ηφ

q

)(β−1)/η(w +

(η

q

)−1/η ∫ 1

φ

ψ−1/ηdψ

)−ρ=w−ρ

(ηθ0

q

)(β−1)/η

(1− ηq

)−ρ=w−ρ

(ηθ0

q

)(β−1)/η

,


so the constant q = η(η−1 − 1)ηwηθ1−η0 . Therefore, the threshold h∗ is

h∗(θ) = h∗(θ;w, θ0) =w

θ0

(η−1 − 1)

(θ0

θ

)1/η

. (2.18)

Because of the w in the optimal threshold h∗(θ), a wealthier agentwith a higher w would be welling to wait longer before selling ofstocks. Also, there are special cases observed from h∗, e.g. for β < 1,a risk-neutral agent (ρ = 0) will never sell the stocks, an extremelyrisk averse agent (ρ ↑ ∞, η → 1) would sell the position immediately;but for ρ ≤ 1− β, the agent would never sell.

Another character of h∗(θ) is the consistency of the initial endow-ment. For example, an agent with w0, θ0 who applied our strategy,has already sold θ0 − θ1 stock. The equation

h∗(θ;w0, θ0) = h∗(θ;w1, θ1)

provides that his wealth is w1 = w0(θ0/θ1)1/η−1, thus for θ ≤ θ1

h∗(θ;w0, θ0) =w0

θ0

(η−1 − 1)

(θ0

θ

)1/η

=w1

θ1

(η−1 − 1)

(θ1

θ

)1/η

,

= h∗(θ;w1, θ1),

which shows the consistency between the two initial endowmentsw0, θ0 and w1, θ1.

Referring from the last section, there were two distinct cases of acandidate value function V (w, α, θ0) depending on the initial endow-ment, such that s < h∗(θ0;w, θ0) and s ≥ h∗(θ0;w, θ0).

For the case of s < h∗(θ0;w, θ0), the value function was directlyderived with (2.12) and (2.18)

V (w, s, θ) =w1−s

1− s+ w1−s

(θs

w

)β (1− ηη

)1−β

. (2.19)

If s > h∗(θ0;w0, θ0), the agent shall sold a positive amount of stockinstantly at t = 0. The amount of this sale was chosen in order tosatisfy the following

s = h∗(θ1;w + (θ0 − θ1)s, θ1) =w + (θ0 − θ1)s

θ1

1− ηη

,

thus θ1 = (w + θ0s)(1− η)/s. Hereby, the value function was

V (w, s, θ) = V (w + (θ − θ1)s, s, θ1) =1− ρ+ ηρ

η(1− ρ)(η(w + θs))1−ρ.

(2.20)

The full explicit solution for the problem has been delivered bynow, including the expression for the optimal boundary of stock and


the value function of utility.

2.2.4 The Verification

To complete the theoretical process, the achievements from previoussection were verified are indeed being the optimum.

With the abbreviation β, the generator LSV = σ(s)2Vss/2 + µ(s)Vsyields to LSV = σ2/2(s2Vss + (1 − β)sVs). Let C be the continuousregion with s < h∗(θ;w, θ0) and E be the exercise region with s ≥h∗(θ;w, θ0), then denote the operator LΘ by LΘV = Vθ − sVw. Obvi-ously the equations LSV = 0 in C and LΘV = 0 in E hold.

When s = h∗(θ;w, θ) = w(η−1 − 1)/θ, the value function for thetwo cases (2.19) and (2.20) yield

V(w,w

θ(η−1 − 1), θ

)= w1−ρ βρ

(1− ρ)(ρ+ β − 1).

In region C, this value function were inset to LΘV and the followingwas obtained

LΘV = sw−ρzβ−1[β + (1− β)z − z1−β],

where z = (θsη)/w(1−η), with the boundary condition at z = 1 suchthat LΘV = 0. Thus, given the fact that s < w(η−1 − 1)/θ in C, theresult is z ≤ 1, LΘV ≥ 0.

Trivially, in E , the operator LSV becomes

LSV = −ρθ2s2 + (1− β)(w + θs)sθ,

which provided z ≥ 1 and LSV ≤ 0.By Ito’s formula,

dVt = LΘV dΘt + LSV dt+ dMt ≤ dMt,

where the local martingale Mt was defined as Mt =∫ t

0σSsVxdBs.

Therefore Mt was a supermartingale, since by integrating the equa-tion above Vt ≤ V0 +Mt was obtained which was equivalent to Mt ≥Vt−V0 for a fixed initial endowment. The inequality of V : dVt ≤ dMt

gave E[U(w + R)] = E[(w −∫∞

0SsdΘs)

1−ρ/(1 − ρ)] ≤ V (w, s, θ). Onthe other hand, the calculation for the threshold strategy h providedthe reverse of the inequality.

At this point, the conclusion was that: V (W −∫ t

0SsdΘs, Xt,Θt)

is a supermartingale in general and a martingale under the strategyΘt = H(St;w −

∫ t0SsdΘs). Hence the strategy described around h∗

was optimal, and the expressions in (2.19) and (2.20) gave the valuefunction.


2.3 Comparison of results

As a summary of this chapter, this section compares the thresholdsfor the two cases. Corresponding with the financial concept, thesetwo cases can be interpreted as alternatives for agents under twospecific circumstances as introduced above, viz, a complete marketor an incomplete market. It could literally be taken as two strategiesfor agents with a large portfolio of stock, selling as one piece or sell-ing in separate sections, which was also an reason for comparing thetwo strategies.

Intuitively, agents expect less from the first sale if they have multi-ple chances of liquidating their current position. Therefore, the start-ing point of a threshold in the divisible case h∗(θ0) from (2.18) willbe below the upper boundary of the indivisible case b∗ from (2.5). Toprove this intuition using (2.5) and (2.19), here in the following

h∗(θ0) = w(1

1− 2µ/ρσ2− 1) =

w

ρσ2/2µ− 1,

f ′(h∗(θ0)) =2µ

ρσ2w−ρ

(1

1− 2µ/ρσ2

)−ρ− ρw−ρ

ρσ2/2µ− 1

(1

1− 2µ/ρσ2

)−1−ρ

= w−ρ(

1

1− 2µ/ρσ2

)−1−ρ(2µ

σ2

1

1− 2µ/ρσ2− ρ

ρσ2/2µ− 1

)= 0,

provided that f(h∗(θ)) was at the peak of f(b) and that h∗(θ0) < b∗,since f(b∗) = 0.

FIGURE 2.1: Function of b with fixed parameters.

Since the strategy of divisible stock was continuous, the thresholdrose each time that the stock price hit the boundary according to themethod introduced in section 2.2. Then the divisible threshold mighteventually be above the indivisible one.

2.3. Comparison of results 13

FIGURE 2.2: The thresholds of the time-independentmodel

2.3.1 Numerical examples

To give a forthright example, the strategies were applied numericallyin order to make a further comparison. The following fixed parame-ters were defined as the default set:

σ = o.5, µ = 0.08, ρ = 0.9, w = 0.5.

With the time independent model, M trajectories of stocks were sim-ulated and then the stocks were liquidated following the two strate-gies, which produced the results shown in the figures.

The complete liquidation might taken quite a long period, so aperiod of one year was set for this experiment. Within a fixed pe-riod, two possible outcomes were presented for the threshold (3.2),with the divisible threshold exceeding the indivisible boundary orremaining below it. The remaining stocks for the trajectories hadthe range [0.8094, 1] with a median 0.9689. The amount of liquidationseems small, but, for a large portfolio, this would imply a sale valuedat millions.

In addition, the complete view of this simulation was presentedalong with the evaluation of f(b) through the varying parameter inAppendix A.

15

Chapter 3

The Time-dependent Model

In this chapter, an agent with one unit of stock was aiming for thesame, but he was then expecting to liquidate his position within acertain period T .

Consistent with the previous Black-Schols model, the same Black-Scholes model was applied for a time-dependent stock price processSt which had to satisfy the following stochastic differential equation

dSt = σStdBt + µStdt, S0 = s.

where the constants µ > 0, σ > 0, are the drift and the volatility,and µ < ρσ2/2. For a value function V , the Black-Scholes operatorLSV was defined as LSV = σ(S)2

2VSS +µ(S)VS . With this notation the

Black–Scholes equation reads

Vt + LSV = 0, (3.1)

which gave the solution value consistent with the Black-Scholes frame-work, therefore there was no arbitrage.

The constant risk aversion utility function U of the agent remains:U(x) = x1−ρ/(1− ρ) with ρ ∈ (0, 1).

3.1 Indivisible Stock Case

3.1.1 The objective

As previously, a single optimal stopping time τ was to be determinedwhich maximized the value function for the agent with initial wealthw

V (St, t;w, T ) = supt≤τ≤T

E[U(w + St)] ≥ U(w + s) = U0.

This holds only under the Black-Scholes assumptions (Hull, 2011,p.286-288). The threshold V (St, t;w, T ) ≥ U0 provided a critical stockprice S∗t so that

St ≥ S∗t , V (St, t;w, T ) = U(w + St), (Sell),

St < S∗t , V (St, t;w, T ) > U(w + St), (Hold).

16 Chapter 3. The Time-dependent Model

Curve S∗t is referred to as the boundary between the continuation re-gion C = (S, t)|St ≤ S∗t and the exercise region E - the complementof C.

The critical stock price S∗Am of an American call option with strikeK is defined in (Seydel, 2012, p.175) as

St ≥ S∗Am, V = St −K, (Sell),

St < S∗Am, V > (St −K)+, (Hold).

Since there was a similarity between the two free boundary prob-lems, the resolution American option was adopted. It was revealedthat this is a time-dependent linear complementarity problem (LCP)(Ikonen and Toivanen, 2008), wherein the value function has to sat-isfy the following:

V ≥ g, Vt + LSV ≥ 0,

(Vt + LSV )(V − g) = 0.(3.2)

where g is the utility at exercise. Hence, the objective was to solvethis LCP numerically. For a more theoretical and detailed knowledgebackground of LCP and its solution, refer to (Cottle, Pang, and Stone,2009).

3.1.2 The Grid

To be able to apply finite difference methods, the domain of V hadto be discretized and truncated, resulting in a discrete domain withcorresponding discrete functions. This included time discretizationand space discretization.

The numerical analyses were independent of the financial frame-work, so the model here was reformulated for the simplicity of op-erations. For S = ex, t = T − 2τ/σ2, the Black-Scholes equation (3.1)proved to be equivalent to the equation (ref)

∂v

∂τ=∂2v

∂x2, (3.3)

where v(x, τ) denoted the value after a change of variables, such that

V (S, t) =: v(x, τ), q =2µ

σ2,

v(x, τ) =: exp

−1

2(q − 1)x− 1

4(q − 1)2τ

y(x, τ),


Since the change of variables, the terminal condition became the ini-tial condition y(x.0) such that

V (S, T ) = U(w + S) =(w + S)1−ρ

1− ρ

= exp

−1

2(q − 1)x

y(x, 0).

and thus

y(x, 0) = exp

1

2(q − 1)x

(w + ex)1−ρ

1− ρ.

The transformation of Black-Scholes equation reduced the terms inthe equation which led to a simpler solving system hereafter, butit was also possible to solve the problem directly with the Black-Scholes equation.

The parameters M,N were defined as the number of space stepsand time steps with i = 0, 1, 2, ...,M, j = 0, 1, 2, ..., N. Then a mash(M + 1)× (N + 1) was defined as follows

∆x =xmax − xmin

M, xi = xmin + i∆x,

∆τ =σ2T

2N, τj = j∆τ,

where [xmin, xmax] is the Dirichlet boundary condition for stock price.The Dirichlet boundary condition is a boundary condition that com-monly applied for discretization; it simply specified the derivativevalue explicitly at the boundary points in that mesh. Note that a suf-ficient boundary of x would be far away the interesting region of S∗tto diminish numerical error, but not too far to safe the computationalcost.

3.1.3 The Crank–Nicolson Discretization

In an overview, the implicit method solves a system of linear equa-tions at each time step to achieve one backwards spatial point. Thismethod guarantees the convergence of numerical solution to exactsolution, but it can be very time consuming as it demands ∆s,∆τtend to zero. The explicit method generates forward value usingpart of previous values. This method is simple to implement witha limited stability.(Seydel, 2012, p.160-170)

Based on the grids, the Crank–Nicolson scheme was implementedfor Black–Scholes PDE discretization, which combined the implicitand explicit methods by taking average. The combination of thesetwo led to a faster convergence rate but also to the probability ofoscillations for the Crank–Nicolson scheme. Another undesirable


property was that the Crank-Nicolson is not satisfactory when damp-ening sudden shocks. For a "sufficiently non-smooth" boundary con-dition, the method would deliver a solution with oscillation.(Gilesand Carter, 2006)

There were the discrete approximations: v(xi, τj) = v(i∆x, j∆τ) ≈vi,j , and the central difference approximations for the space deriva-tives

∂vi,j∂x

=vi,j+1 − vi,j−1

2∆x+O(∆x2)

∂2vi,j∂x2

=vi,j+1 − 2vi,j + vi,j−1

∆x2+O(∆x2)

where O denoted the order of the error remaining. Since, as the ∆xtended to zero, the error term O would also tended to zero with afaster convergence rate than any other approximations, the term Owas dropped hereafter.

The implicit, the explicit, and the Crank–Nicolson method couldbe combined into one equation through a parameter θ. The readilyequation based on the equation (3.3) was as follows

vi,j+1 − vi,j∆τ

=θvi+1,j+1 − 2vi,j+1 + vi−1,j+1

∆x2

+ (1− θ)vi+1,j − 2vi,j + vi−1,j

∆x2,

where θ = 1 is the implicit method, θ = 0 is the explicit method, andθ = 1/2 is the Crank-Nicolson method.

Corresponding to (3.3) the LCP (3.2) was transform to the discreteinequality

vi,j+1 − λθ(vi+1,j+1 − 2vi,j+1 + vi−1,j+1)−− vi,j − λ(1− θ)(vi+1,j − 2vi,j + vi−1,j) ≥ 0,

(3.4)

where λ =: ∆τ/∆x2 was the abbreviation.Applied above, a system for the inner points of the grid was for-

mulated with matrices and vectors. The following notations was de-fined: bi,j = vi,j + λ(1 − θ)(vi+1,j − 2vi,j + vi−1,j), for i = 2, ...,M − 2


and b1,j, bM,j and the boundary conditions was incorporated,

b(j) =:

b1,j

...bM−1,j

v(j) =:

v1,j

...vM−1,j

g(j) =:

g1,j

...gM−1,j

A =:

1 + 2λθ −λθ 0

−λθ . . . . . .. . . . . . . . .

0. . . . . .

∈ R(M−1)×(M−1)

Now, inequality (3.4) was rewritten as Av(j+1) ≥ b(j) for all j.Further, the LCP (3.2) became

v ≥ g, Av − b ≥ 0,

(Av − b)′(v − g) = 0.(3.5)

with the initial condition and boundary conditions

vi,0 = gi,0, i = 1, ...,M − 1;

v0,j = g0,j, vM,j = gM,j, j ≥ 1,

and the realization of boundary conditions in vector b(j) as follows:

b1,j = v1,j + λ(1− θ)(v2,j − 2v1,j + g0,j) + λθg0,j+1,

bM,j = vM−1,j + λ(1− θ)(gM,j − 2vM−1,j + vM−2,j) + λθgM,j+1,

This completed the finite-difference discretization.For the problem about "sufficiently non-smooth" boundary con-

dition, a mathematician Rolf Rannacher suggested a way to achievebetter stability properties. Rannacher’s suggestion was to solve thefirst few time steps implicitly only and then switch back to the Crank-Nicolson. In research of European options, additional time stepswere suggested for the best results, so that if unmodified Crank-Nicolson have taken two steps, then four steps of implicit methodwere taken with the Rannacher modification (Giles and Carter, 2006).This is, implement-wise, θ = 1 for first four steps and then back toθ = 1/2 for the rests.

However, as the error from the oscillations was reduced, the ac-curacy might still be reduced by other cause. Because the issue oflacking smoothness not only related to the discretization but also de-pend on the way of the grid making and the method that was chosento solve (3.5).


3.1.4 Finite Difference Method - PSOR

The bulk of this chapter is to approximate the solution of the LCP sys-tem of equations. The iterative method - Projected Successive OverRelaxation (PSOR) was implemented for this purpose. As shown inthe name, the PSOR method is a variant of the SOR method.

In general, step by step, the iterative algorithm refined a solutionfor the time step when the difference between iterations was below apredefined error tolerance, and assigned the current solution to thattime step, then the algorithm pass to the next time step and repeatsthe process.

For numerical analysis, x := v−g, y := Av−b, and b := b−Agwere defined , then the system (3.5) was transformed to the followingproblem:

Ax− y = b, x ≥ 0, y ≥ 0, x′y = 0.

In order to give a better reasoning on the PSOR method, the SORmethod was introduced firstly which solves a square system of linearequations Ax = b. The (n × n) matrix A can be decomposed asA = D + L + U with diagonal component D, and lower and uppertriangular components L and U with n = M − 1, such that

D =:

a11 0 . . . 00 a22 . . . 0...

... . . . ...0 0 . . . ann

,L =:

0 0 . . . 0a21 0 . . . 0

...... . . . ...

an1 an2 . . . 0

, U =:

0 a12 . . . a1n

0 0 . . . a2n...

... . . . ...0 0 . . . 0

.

The system of linear equations Was rewritten as

(D + ωL)x = ωb− (ωU + (ω − 1)D)x

where content ω < 1 was called relaxation parameter. Intuitively, itarrived at

x(k+1) = (D + ωL)−1(ωb− (ωU + (ω − 1)D)x(k)

).

Then the component-wise iterations of x(k+1) can be written as

r(k)i := bi −

i−1∑j=1

aijx(k)j − aiix

(k−1)i −

M−1∑j=i+1

aijx(k−1)j

x(k)i = x

(k−1)i + ω

r(k)i

aii.

(3.6)

where k and aij denoted the number of the iteration and element ofthe matrix A. The convergence of the iteration was related to thechoice of ω. It was shown that the optimal interval is 1 < ω < 2.

3.2. Numerical Results 21

To solve (3.5) using the “projected” SOR method, the SOR methodwas modified to enforce the constraint x(k) ≥ 0, so that for i =1, ...,M − 1

r(k)i := bi −

i−1∑j=1

aijx(k)j − aiix

(k−1)i −

M−1∑j=i+1

aijx(k−1)j

x(k)i = max

0, x

(k−1)i + ω

r(k)i

aii

y

(k)i = −r(k)

i + aii

(x

(k)i − x

(k−1)i

).

Converted x,y back to v,b,g, algorithm under condition ‖v∗−v‖2 >ε was produced as follows

p :=(bi + λθ(v∗i−1 + vi+1))

1 + 2α

v∗i = maxgi,j+1, vi + ω(p− vi)(3.7)

where v∗ denoted an intermediate variable during illustration. Through-out every time step with this algorithm, the solution v := v∗ wasgiven. Therefore, the value function V was determined throughoutthe mesh and the threshold S∗t could be found.

3.2 Numerical Results

For the final step, the algorithm was implemented by Matlab withthe parameters as follow

M = 500, N = 1000, T = 1, σ = 0.5, µ = 0.08, ρ = 0.9, w = 0.5.

The evaluation of value function V was shown in (Figure 3.1).

FIGURE 3.1: The value V evaluation in the mesh

Then the threshold S∗t (Figure 3.2) was found by comparing theexpected value and the instinct exercise utility. As expected, thethreshold is a smooth curve that converges to a certain level at t = 0.


FIGURE 3.2: The threshold S∗t

Figures for the threshold in a longer period and with default pa-rameters were presented in Appendix B.

3.3 The Divisible Case

For the infinitesimal amounts, the agent had to liquidate them con-tinuously and recursively with the maturity took into account. Ide-ally, this is a problem involves time, utility, volume and stock price,which may be solved for a group of formula theoretically.

Due to practical limitations and the scope of the thesis, a numer-ical or theoretical method was not provided for this case. But witha intuition, a possible alternative was proposed based on previoussection.

Let the infinitesimal amounts of stock be discretized intoN pieces,following section 3.1, 1/N of the stock were sold at each time, and theprofit was accumulated to initial wealth for the next sale and thencontinued recursively until the N pieces were all liquidated. How-ever, this method was computationally expensive for a rather accu-rate result.

This proposal was more a temptation to solve the problem than aproper method. This subject will be studied for the future research.

23

Chapter 4

Conclusion

In this thesis, three cases were presented for the problem of optimalexercise boundary of a risk-averse agent seeking to liquidate a posi-tion in stock market. A key assumption in this thesis was that thestock share is infinitely divisible. Of course this is impossible for realpractice, but it is appropriate for a large portfolio, where it may bringsome insight.

The comparison between the divisible and indivisible cases wasnot meant to provide a "better" strategy but rather to highlight theadvantage of each one of them, so the decision for agents with theirown demand would be easier to made . A large portfolio impliesthat it will takes long time to liquidate the whole position, not onlybecause the expectation from a single liquidation may be impracti-cally high but also because the sale may create shock to the marketin financial concept. In this circumstance, the maturity was no longerrelative; the profits became the key factor to consider.

However, liquidation within a certain maturity is a normal andrealistic demand of agents, so the comparison was extended to a timedependent model. Unfortunately, due to the scope of the thesis, awell-established method was not successfully delivered for the liq-uidation of a divisible stock, but an alternative way of liquidationwas proposed based on the indivisible case. This will be left as afuture study direction.

Furthermore, there were a few possible extensions for a futurestudy. As in (Henderson and Hobson, 2011), this study could be ex-tended to option and option with price impact, or a portfolio withdifferent stocks. Part of the general result from this study was appli-cable for the further, and the new factors such as the payoff function,the different strikes and the correlation between stocks were neededto be studied.

25

Appendix A

Figures from TheTime-independent Model

FIGURE A.1: The function of b with µ varying

FIGURE A.2: The function of b with ρ varying

26 Appendix A. Figures from The Time-independent Model

FIGURE A.3: The function of b with σ varying

FIGURE A.4: The function of b with w varying

FIGURE A.5: The thresholds with 10000 trajectories

27

Appendix B

Figures from TheTime-dependent Model

FIGURE B.1: The value V evaluation in the 50 yearsmesh

FIGURE B.2: The threshold S∗t of 50 years period

29

Bibliography

Arfken, George B., Hans J. Weber, and Frank E. Harris (2006). Math-ematical Methods for Physicists. sixth edition. Elsevier AcademicPress.

Cottle, Richard W., Jong-Shi Pang, and Richard E. Stone (2009). TheLinear Complementarity Problem. Classics in Applied Mathematics.Society for Industrial and Applied Mathematics.

Giles, Michael B. and Rebecca Carter (2006). “Convergence analysisof Crank-Nicolson and Rannacher time-marching”. In: Journal ofComputational Finance 9, 89–112.

Henderson, Vicky and David Hobson (2008). “Perpetual AmericanOptions in Incomplete Markets: The Infinitely Divisible Case”. In:Quantitative Finance 8(5), 461–469.

— (2011). “Optimal Liquidation of Derivative Portfolios”. In: Mathe-matical Finance 21, 365–382.

Hull, John C. (2011). Options, Futures and Other Derivatives. seventhedition. Pearson Education Limited.

Ikonen, Samuli and Jari Toivanen (2008). “An operator splitting methodfor pricing American options”. In: Computational Methods in Ap-plied Sciences 16, 279–292.

Peskir, Goran and Albert Shiryaev (2006). Optimal Stopping and Free-Boundary Problems. first edition. Birkhäuser Basel.

Seydel, Rüdiger U. (2012). Tools for Computational Finance. sixth edi-tion. Springer-Verlag London Limited.

optimal liquidation of stock

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