Optimal exercise of an American Optionunder drift uncertainty

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Examensarbete i matematik, 30 hp Handledare: Erik Ekström

Examinator: Kaj Nyström Juni 2016

Department of Mathematics Uppsala University

Optimal exercise of an American Option under drift uncertainty

Marta Rita Braojos Peláez

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Optimal exercise of an American Option under drift uncertainty

Marta Rita Braojos Pel´ aez

Supervised by Erik Ekstr¨ om

Master Thesis in Financial Mathematics, 30 hp

June 2016

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2

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Abstract

The purpose of this work is to provide an optimal exercise strategy for American option, within a market model with uncertainty about the drift. The criteria will be the maximization of the gain resulting of the exercise at every time.

We will approach the problem first theoretically, using Optimal Stopping theory and studying the dynamics of such a model to detail our context

Next, the implementation of numerical techniques to this modeling will provide some interesting pictures of the Optimal Exercise Boundary.

The results provided will answer the question of how to behave to maximize an investor’s profits from an American option where the underlying asset is driven by a random drift.

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2 Acknowledgements

Foremost, I would like to express my gratitude to my supervisor, Prof. Erik Ekstr¨om, for all his great help, ideas and encouragement that enabled me to write this thesis. I could not have had a better support.

Besides my advisor, I would like to thank Martin Vannest˚al, whose previous work and help with my questions allows me to advance on my results.

My sincere thanks also goes to my friends in Sweden and Spain, who has been a constant source of happiness and motivation during all my studies, specially to ´Alvaro Correales, whose help with LaTex saved me hours of research, and Peter Cassars, for all those long days at the study room.

And last but not least, to my father Ram´on Braojos, whose support and inspiration are in everything I do.

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Introduction

The concern of this work is simple to state. Having a stock X driven by a drift whose value we are not certain about, what would be the optimal time to exercise an American Option we own on this stock?

Our focus of interest will be the behavior of this unknown drift, having as first data which values do we consider it is possible for it to take, and the degree of belief on each of them.

Based on this information, that we will keep actualizing over time, we wil try to draw a boundary that separates where is better, in terms of profits expectation, to hold the option and when it does not worth any more and we should exercise it (Continuation and Exercise regions).

This way we will have a guideline for an optimal behavior in this market model.

The matter we are going to study in this thesis is the behavior of a ”Gain” function, that represents the profits provided by the exercise of an Option on a particular time. The fundamental idea is to maximize this Gain on time, that is, to choose the optimal moment to exercise an American option we own in order to make the most profit from this action.

Our context, tough, is different than the usual Black Scholes market model. We are going to consider a very special version of the market model, and this is what is going to make our problem new and interesting.

Generally, in Pricing Option problems, we are given the equation that drives the dynamics of the associated stock. It would typically have the form of a Geometric Brownian motion we can work with, with a constant drift equal to the interest rate of the market.

However, in our problem, the approach is different. The idea is the investor does not agree with this formulation, and instead he think that the drift of the stock is driven by another constant, µ, with a certain value. He does not know for sure this value, but instead he has several candidates and a certain degree of belief in each of them.

As time passes and he has more information, he keeps actualizing those beliefs and acts consequently.

This behavior may provides him with an advantage over the market, and increase his hypothetical benefits.

As this work advance, we are going to keep introducing concepts and necessary theory for the correct understanding of the subject. We hope this way to make the lecture as clear and nice as possible.

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1.1 The market model

First of all, we will define the context we are working in.

We will need to make some essential assumptions about the market, lets state them:

• Constant volatility σ

• Constant and known interest rate r

• The stock pays no dividends

• There are no transaction cost in the operations (However, our only concerned operation is the liquidation of the option)

• The price of the stock X is driven by a geometric Brownian Motion starting at a certain X₀≥ 0, in the form:

dXt= µXtdt + σXtd ˜Wt, t ≥ 0 (1.1) where the drift µ is a random variable independent of the Brownian motion ˜W , that can take N < ∞ possible values: µ = µ1, ..., µN.

We denote the probability of the event µ = µi at time t as:

π^t_i = P (µ = µi|Ft) WithPN

i=1π_i^t= 1 ∀t

In this framework, we will present our problem.

1.2 The problem

In financial investment, it is not always about portfolio hedging. Taxes in the financial operations, lack of liquidity and other issues we do not include in modeling, could make impossible or inconvenient to proceed according to a pattern that is optimal in a theoretical sense. Other times, simply our objective is something different than minimize the risk, e.g. maximize the profits.

For this purpose, we can appeal to the field of optimal stopping theory, a branch of probability theory that face the problem of decide when to interrupt a stochastic process in order to maximize a function associated to it.

We are going to go deeper in this theory in next chapter, but before we are going to state clearly our purposes and goals.

The concern of this thesis is going to be to maximize the profits derived from the exercise of an American Option whose underlying asset is driven by 1.1.

The question is then the maximize of a certain process ”gain” G = (Gt)t≥0 as a function of time.

This process shows the money we would receive from the exercise of our American option exactly at every time point.

We want, shortly speaking, to find when to keep our option and when to exercise it. This would define a Continuation region and a Exercise region in the space of stock values and time moments. We expect a neat and well define boundary between those regions, which allows us to acts consequently.

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CHAPTER 1. INTRODUCTION 6

1.3 Standard (Black-scholes) case for reference

The equivalent problem for the Standard Black-Scholes market model has been widely studied. That is, in the context of an asset with constant drift driven by

dXt= rXtdt + σXtd ˜Wt, t ≥ 0

where r ∈ R is a constant and coincides with the market rate of interest, what should be the moment of exercise this right of sell/buy the asset that our American option provide us?

The studies results in the establishment of certain boundaries defining a continuation vs an exercise region written on the time point and the price of the stock, given us pictures like:

(a) Put option (b) Call option (No early exercise)

Figure 1.1: Parameters r = 0.1, σ = 0.41, T = 1 (years), K = 150

We can as well have the situation where the drift is a known constant, µ, that does not coincide with the market drifft

dXt= µXtdt + σXtd ˜Wt, t ≥ 0 Values of µ ≤ r or µ ≥ r would let us to different pictures:

(a) Put option (b) Call option (No early exercise)

Figure 1.2: Parameters µ = −0.03, r = 0.01, σ = 0.2, T = 2.5 (years), K = 150

We are including this results on our work in aim to compare and find similarities and differences between the ones we are going to get later in our study. It will happen that, in our perspective, this are going to be particular cases (µ=constant with probability 1) of a more general set of possible problems.

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(a) Put option (b) Call option (No early exercise) Figure 1.3: Parameters µ = 0.02, r = 0.01, σ = 0.2, T = 2.5 (years), K = 150

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Chapter 2

Theoretical approach

In this chapter we are going to proceed to a first theoretical approach of solving the optimal stopping problem as a free boundary problem.

We are going to try to reduce the information as much as possible, following the steps of Martin Vannestal [2] in a more general case with N possible drifts.

From now on, we will work with an underlying probability space (Ω, F , P ) and a standard Brownian motion defined on it, ( ˜W , F^W^˜).

In this context we will assume as usual that the asset price process X follows a geometric Brownian motion starting at a certain X0≥ 0. This gives us the dynamics:

dX_t= µX_tdt + σX_td ˜W_t, t ≥ 0 (2.1)

where the volatility σ is a positive constant.

The interesting part is that the drift µ is a random variable independent of the Brownian motion W , that can take N < ∞ possible values: µ = µ˜ ₁, ..., µ_N. In principle, there is no restriction for the possibles values of µ, and they are not necessary proportional or related in any sense with the market interest rate, r.

We will denote the conditional probability of the event µ = µ_i at time t in the following way:

π_i^t= P (µ = µi|F_t^X) For coherence, we will obviously havePN

i=1π_i^t= 1 ∀t

Before going further in the study of the model, we are going to introduce some necessary theory and concepts.

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2.1 Filtration Theory, foundations and our case

We are going to start defining a mathematical concept that appears in a system where information keeps being revealing as time pass: A filtration.

Definition 2.1 (Filtration)

Given a Probability space (Ω, F , P ), a filtration is a sequence of σ-algebras F_t,t≥0 where F_t⊆ F ∀t and F_t₁ ⊆ F_t₂ ∀t₁≤ t₂

In a Probability space, the sets of measurable things are the elements of the σ-algebra associated.

Intuitively, a filtration comes to model how, as time pass, the amount of measurable sets increases, coherently preserving a σ-algebra structure. We incorporate new information and, furthermore, we do not forget the one we had before.

Together with the structure of a probability space, (Ω, F , P ), we obtain a filtered probability space, (Ω, F , Ft, P ).

In our model the investor has some initial beliefs for the probabilities of each of the possible values of the drift, and they keep actualizing this beliefs after every time step, having observed how the asset price X has actually behaved.

In other words, we will have a filtration of information F_t^X, where given π⁰_i = P (µ = µi|F₀^X), that we know as an assumption, we can calculate at every time step π_i^t= P (µ = µi|F_t^X) as a function of t, π⁰_i and Xt, the actual observable value of the stock at time t.

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CHAPTER 2. THEORETICAL APPROACH 10

2.2 Optimal Stopping problems, foundations and our case

Our objective is to maximize the gain a market asset can provide us. This gain depends on what time do we choose to exercise it.

The theory of Optimal Stopping comes to adress this question; to find the best time to stop a process in order to maximize the profits resulting of this.

It results necessary to introduce some important concepts before. (See [9] for more details) Definition 2.2 (Stopping time)

A non negative random variable τ is called a stopping time with respect to the filtration F if it satisfies the condition

{τ ≤ t} ∈ Ft ∀t

Intuitively, this means that the random time τ is non anticipative, in the sense that at any time t we can actually determine whether the instant τ in time has occurred or not.

We now present what is typically the structure of an Optimal Stopping problem.

Lets present first our ”Gain” process, G.

Let (Ω, F , Ft,t≥0, P ) a filtered probability space, and G = (Gt)t≥0a stochastic process defined on it, adapted to the filtration.

Intuitively, this process represent our gain if we stop the contract now.

The general structure of an optimal stopping problem will be on the following form:

V_t= sup_{t≤τ ≤T}E[G_τ] (2.2)

where τ is a stopping time and T is either finite or infinite. In our problem, T is going to be the expiring date of the option, the maturity.

We need some technical assumptions over the process G; We will assume it to be right-continuous, and left-continuous over stopping times. We will assume as well that E[sup_0≤t≤T|Gt|] < ∞

That will allow us to go to the main theorem about the solution of this problem.

Before, we need to define S = (St)t≥0as follows:

St= esssupτ ≥tE(Gτ|Fτ) (2.3)

where τ is a stopping time. The process S is called the Snell envelope of G. (For the concept of essenctial supremum, see [2]).

Now let

τ_t= inf {s ≥ t : S_s= G_s} (2.4)

Theorem 2.1 (For proofs and reference, see [11], pag.3) Consider the Optimal Stopping problem 2.2 and assume E[sup_n≤K≤N|G_k|] < ∞. Assume furthermore that P (τ < ∞) = 1 where τ ≥ 0 we have:

St≥ E[Gt|Ft] for each τ ∈ Mt St= E[Gτ_t|Ft] (2.5) where Mtdenotes the family of all stopping times satisfying t ≥ 0. Moreover, if t ≥ 0 is given and fixed, then we have:

The stopping time is optimal in 2.2

If τ^∗ is an optimal stopping time in 2.2 then P [τt≤ τ^∗] = 1

The process (Ss)s≥t is the smallest right-continuous supermartingale which dominates (Gs)s≥t. The stopped process (Ss∧ τt)s≥t is a right continuous martingale.

If P (τt= ∞) > 0 then, with probability q, there is no optimal stopping time in 2.2.

The concern of this thesis is to find the optimal stopping time for an American Put option in a market model where the drift is a discrete random variable.

The objective of our investor is to exercise the Option in the moment when it reports more benefit, i.e., when the Gain function reaches a maximum.

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2.3 Arbitrage free pricing of American options, foundations and our case

Definition 2.3 (American Option)

An American Put Option with strike K and maturity T is a market contract that gives the owner the right (not the obligation!) to sell an asset at price K at any time point before time T (respectively, an American Call gives the right of buying the asset).

In a classic Black-Scholes model, the price of an American Option has been widely studied. According to Black-Scholes Model, the arbitrage free price of an American put option, with strike price K, volatility σ > 0, interest rate r > 0 and time to maturity T is

V (x) = sup

0≤τ ≤T

Ex(e^−rτ(K − Xτ)⁺) (2.6)

where the stopping times τ are with respect to the filtration F^xand X follows the dynamics

dXt= rXtdt + σXtdBt (2.7)

where B = (B_t)_t≥0 is a standard Brownian motion, and X₀= x > 0 under P_x. In our problem, instead, an investor’s model assumes the dynamics of X given by

dX_t= µX_tdt + σX_tdB_t with µ ∈ {µ₁, µ₂, ..., µ_N} (2.8) that is, the drift is a random variable.

That will obviously change the results on a pricing process, and precisely in this ”disagreement” with the standard market model the investor hopes to have and advantage over the others.

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CHAPTER 2. THEORETICAL APPROACH 12

2.4 Dynamics of the model

Consider the price of an asset X = (Xt)t≥0 as a strong Markov process, which is right-continuous and left-continuous and takes values in R^d. Also, we consider a measurable function G : R^d → R^d, which will be our Gain function. We want to answer the question of when to exercise an American Put option over the asset, in order to maximize G. The optimal stopping problem we thus have is

V (x) = sup

τ

E_x(G(X_τ)) (2.9)

The stopping times are taken with respect to X and P_x(X₀= x) = 1 for x ∈ R.

First step is to state the dynamic of the processes X(t) and π_i(t) ∀i ∈ 1, ..., N .

Through a generalization of the approach in [2], following the theorems 7.12 and 9.1 in [3], we obtain the following results:







dXt/Xt

dπⁱ_t







=





 PN

j=1π_t^jµj

0





 dt +







σ πⁱ_t(µi−PN

j=1π^j_tµj)/σ







d ¯W_t (2.10)

Where

d ¯Wt= d ˜Wt+µ −PN j=1π^j_tµ_j

σ (2.11)

with d ˜W_t a standard Brownian motion in the measure P and µ a real variable that takes values in the finite set µ1, ..., µN

For coherence with our context,

πⁱ_t∈ (0, 1) ∀j, t,

N

X

j=1

π_t^j= 1 ∀t (2.12)

Looking at the dynamics of our processes, we can see it has N underlying spatial dimensions. This would be difficult to handle, but in a closer look we will observe some ligatures between them; all the processes are expressed in terms of the same Brownian motion d ¯W , and the π_tⁱ processes are not depending explicitly of Xt for any i = 1, ..., N . Indeed, our first objective is going to be decrease the dimensions of the problem.

Let be Πt=PN

j=1π_t^jµj. Such a Π should fulfill the equation dX(t) = Π(t)X(t)dt + σX(t)d ¯W

Our aim is to find a function f : Ω×[0, T ] −→ R such that Π(t) = f (X(t), t), so we impose this condition.

Applying Itˆo’s formula,

dΠ(t) = (ft(X(t), t) + f (X(t), t)fX(X(t), t)X(t) +1

2σ²X²(t)fXX(X(t), t))dt + (fXσX(t))d ¯W On the other hand, by definition,

dΠ(t) =

N

X

j=1

µjdπ^j(t) = 1 σ

N

X

j=1

(µ²_jπ^j− µjπ^j

N

X

i=1

µiπⁱ))d ¯W

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From above we derive the following system of equations:











f_t(X(t), t) + f (X(t), t)f_X(X(t), t)X(t) +¹₂σ²X²(t)f_XX(X(t), t) = 0 PN

j=1µ²_jπ^j− f²(X(t), t) = σ²f_X(X(t), t)X(t)

(2.13)

This happens not to have an easy way to be solved or simplified; Through a similar procedure as the one in paper [4], pag. 7, adapting it to our discrete case:

Πt= E[Π|F_t^X] = E[Π|µt] = f (t, µt) And following the steps for formulas (3.1), (3.4), we obtain:

π^t_i = π_i⁰e^µit² ⁻

µ2it 2σ2+^µi

σ2log^Xt

X0

PN j=1π⁰_je

µj t 2 −^µ2^j^t

2σ2+^µj

σ2log^Xt

X0

And, consequently,

Π_t=

N

X

j=1

µ_jπ_j^t= PN

j=1µjπ⁰_je^{µj t}² ⁻

µ2jt 2σ2+^µj

σ2log^Xt_X0

PN

j=1π⁰_je^{µj t}² ⁻

µ2jt 2σ2+^µj

σ2log^Xt

X0

= PN

j=1π_j⁰µje^µj²⁽¹⁻^µj^σ2^)t(^X_X^t

0)^µj^σ2 PN

j=1π_j⁰e^µj² ⁽¹⁻^µj^σ2^)t(^X_X^t

0)^µj^σ2

(2.14)

Plug in this formula on the system 2.13, we check it solves it.

Having done this study on our function Π, we have reduce our problem to a two-dimensional one.

We will now use our results to support a numerical analysis of our problem, which may provide us an answer to the optimal exercise question.

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Chapter 3

Numerical approach

Our objective is to determine the optimal stopping time of a process associated to an American Put option over an asset with unknown drift. what we are going to do at first place is to price the option, to calculate the value of this ”exercising possibility” at every time point.

The dynamic of the risky stock price,

dXt= µXtdt + σXtd ˜Wt, t ≥ 0 (3.1)

is driven by a random variable that can take values in a finite set µ1, ..., µN, and we keep updating the probability distribution of this random variable through time with the help of a certain function Πt

Previous section provided us the understanding of the dynamic of the function

Πt=

N

X

j=1

µjπ^t_j, with π^t_i = P [µ = µi at time t]

Πt= PN

j=1π_j⁰µje^µj²⁽¹⁻^µj^σ2^)t(^X_X^t

0)^µj^σ2 PN

j=1π⁰_je^µj² ⁽¹⁻^σ2^µj^)t(^X_X^t

0)^µj^σ2

This characterized our option. We state as well in previous sections how the market value of such an Option will be described by a system of partial differential equations involving this Πt, 2.13. We have thus the variational inequality:











V_t(X(t), t) + f (X(t), t)V_X(X(t), t)X(t) +¹₂σ²X²(t)V_XX(X(t), t) − rV (X(t), t) = 0

max{V_t(X(t), t) + f (X(t), t)V_X(X(t), t)X(t) +¹₂σ²X²(t)V_XX(X(t), t) − rV (X(t), t), (K − X(t))⁺− V (X(t), t)}

In this section we are going to apply numerical methods to this system 2.13 to solve it, and thus obtain the value of our American Put Option with N possible drifts at every time point from t = 0, the initial, to t = T , the maturity.

The method we are going to use is a Finite-Differences Method.

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3.1 Finite-Differences Method foundations and our case

Finite differences methods for the solution of a problem of partial differential equations have three main steps: A discretization (grid) of the space and time space, the use of a derivative approximation method and, provided some boundary and final conditions, the application of a recursive algorithm to find the value of the function in all points of this grid.

We can find more details and foundations of this family of numerical methods in [6], pag.158.

As first step, we are going to proceed to a time and space discretization on our environment.

Our time space moves from t = 0, where we situate the present moment, to t = T , maturity time of our option, when it expires and lost its value if it is not exercised. We separate this line in n equidistant time steps, having ∆t = ^T_n.

Our space step will move form x = xmin = 0, the minimum value of the stock assumed to be zero, to a maximum reasonable point, x = xmax. A good idea could be let it be four times the value of the strike K (this is proportional to the drift value though). Now we separate this line in m equidistant space steps, giving us ∆x = ^xmax−xmin_m

Placing this discretization in two axes, we are going to obtain a ”grid” of the space, similarly as in [6] pag.160.

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♠It is needed to mention that through this discretization we are implicitly going to change the view of our American Option to a Bermudean Option, this is, it can not be exercise at any time but just at the time points that appears in the grid. Hopefully, a sufficient big number of steps will make this approximation insignificant.

Note: For stability reasons, it is important to keep a proportion not very far from n ∼ m²

Second step will proceed to use a discretization of the derivatives; based on some of the theoretical

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CHAPTER 3. NUMERICAL APPROACH 16 definition of derivative,

f⁰(t) = lim

→0

f (t) − f (t − )

(3.3)

f⁰(x) = lim

→0

f (x + ) − f (x − )

2 (3.4)

f⁰⁰(x) = lim

→0

f (x + ) − 2f (x) + f (x − )

² (3.5)

we approximate

df (x(t), t)

dt 'f (x, t) − f (x, t − ∆t)

∆t (3.6)

and

df (x(t), t)

dx ' f (x, t) − f (x − ∆x, t)

∆x (3.7)

d²f (x(t), t)

dx² 'f (x + ∆x) − 2f (x) + f (x − ∆x)

(∆x)² (3.8)

and then substitute it in the system 2.13.

Now it is possible to the third and last step of the method: Those results will make possible to relate the value of f (xi, tj−1) with f (xi+1, tj), f (xi, tj) and f (xi−1, tj) in a diagram similar to the one in [6], that we can represent:

(3.9) In the next pages we are going to get deeper in this last step of the method, adapting it to our particular problem.

Our aim is now to calculate the arbitrage free price of the American Put Option at every point of this ”grid” we have constructed.

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3.2 Our algorithm

Now we have all the ingredients the Backward Finite Differences method requires. We are going to plug in the discretization and derivative approximation we have choose in our system 2.13, obtaining (after rearrangement of the terms):

f (xi, tj) =∆txi

2∆x

σ²xi

∆x + Π^j_i

f (xi+1, tj)+

1 −∆tσ²x²_i (∆x)² − ∆tr

f (xi, tj)+∆txi

2∆x

σ²xi

∆x − Π^j_i

f (xi−1, tj)

The value of each Π^j_i, adapting our theoretical analysis in the previous section, is here:

Π^j_i = PN

k=1π_k⁰µ_ke^µk²⁽¹⁻^µk^σ2^)t^j(_x^xⁱ

0)^µk^σ2 PN

k=1π_k⁰e^µk² ⁽¹⁻^µk^σ2^)t^j(^x_xⁱ

0)^µk^σ2 being x0 the initial stock value, directly observable.

We need as well some initial and boundary conditions, in order to be able to calculate the values for all points in the grid. Those are going to be:

INITIAL CONDITIONS:

f (x, T ) ∀x = max(K − x, 0)

The payoff function of an European Put Option (this points correspond to t = T , the time of maturity)

BOUNDARY CONDITIONS:

f (x₀, t) ∀t = K

Where the stock value drops to zero, the exercise of a Put will give us a gain equals to the strike of the option.

f (x_max, t) ∀t = 0

Were the stock value rises very high (we are choosing a xmax > K), it have no sense to exercise a Put option; it loose its value.

? It is important, after every step in our algorithm, remember the nature of our derivative; as it is a right, not an obligation, we always have the possibility of not to exercise if it is inconvenient. This will shows by forcing

f (xi, tj) = max(f (xi, tj) provided by the formula , (K − xi)⁺)

What we have done up to now is to determinate the value of our American Option at every point in time and every price of the stock.

This will result determinant information in the next chapter, where is going to be answer the obvious question that comes: Where is that value, actually, profitable?

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Chapter 4

Drawing a boundary

Next step in our work is going to be, having previously established the price of our American Put Option, to define the regions where we should keep it and where we should exercise it.

The aim is to differentiate when it worth to hold the option and when it start do not. The motivations are easy to state:

We are going to define the Continuation region as the set of time and space points where the value of the option is greater than the actual profit that its exercise would provided us.

Consequently, we should not exercise it but to keep it.

CONTINUATION REGION={(t, x)/V (t, x) > (K − x)⁺}

Similarly, the Exercise region will be then the set of time and space points where the value of the option is the same as the profit we obtain from its exercise.

So, we should exercise it at this points

EXERCISE REGION={(t, x)/V (t, x) = K − x}

Having established that, we will proceed to plot those regions with different parameters, and observe the results.

From now on, we are keeping the initial price of the stock x₀= 45, the range of spatial steps moving from x_min= 0 to x_max= 225, the strike K = 150 and time of maturity T = 1 (year).

Example 1:

Boundary for possible drifts µ = (−0.07, 0.01), initial beliefs π0 = (0.8, 0.2), volatility σ = 0.5 and interest rate r = 0.015:

We can appreciate here a boundary that reminds the standard case of an American Put Option with known drift.

Example 2:

Boundary for µ = (−0.05, 0.02, 0.08), π₀= (0.86, 0.13, 0.01), σ = 0.2 and r = 0.01:

Here, however, we start to have an interesting change in the shape; it appears a small upper barrier in our scheme.

It is understandable why this behavior takes place; it appears in situations where we have a small drift with a great initial probability, but also some possible bigger drifts with a small initial probability.

Consequently, we mainly believe in the lower boundary that correspond to the small drift. But still, if the price of the stock becomes big enough, we start to believe in a bigger drift. Then, it appears an upper boundary corresponding to it. (If the price of the stock increase too much, it is not beneficial to exercise a Put Option)

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Figure 4.1: Boundary for Example 1

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CHAPTER 4. DRAWING A BOUNDARY 20

Example 3:

Boundary for µ = (−0.06, 0.02, 0.03, 0.04), π₀= (0.85, 0.13, 0.013, 0.007), σ = 0.17 and r = 0.01:

With this parameters the upper part of the boundary gains importance and suppose a significant difference.

Example 4:

Boundary for µ = (−0.07, 0.05, 0.09), π₀= (0.9, 0.08, 0.02), σ = 0.2 and r = 0.01:

Here we have get to a point where, to avoid early exercise, our stocks would have to move in a band for all time points.

Example 5:

Boundary for µ = (−0.01, 0.15, −0.2, 0.25, −0.3), π₀= (0.2, 0.2, 0.2, 0.2, 0.2), σ = 0.2 and r = 0.01:

For this values, we start already in the stopping region; we should exercise immediately.

Example 6:

Boundary for µ = (0.01, 0.02), π₀= (0.6, 0.4), σ = 0.4 and r = 0.01:

Another case of immediately exercise, that reminds in shape the standard case.

Example 7:

Boundary for µ = (0.24, 2), π₀= (0.9, 0.1), σ = 0.7 and r = 0.015:

Some more extreme values, and immediately exercise recommended.

We could infer that the boundaries that presents ”peaks” corresponds to those parameters that present a high degree of belief in a certain drift value, but still leave a chance to others possible drifts (that has to be quite far away from the first, though).

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CHAPTER 4. DRAWING A BOUNDARY 22

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Chapter 5

Alternative approaches

The method we have developed has shown remarkable results. However, it has result quite difficult to achieve. It implies Filtration theory, and the analysis of some theoretical issues to the requirement of reducing the dimension of our problem.

In this chapter we want to take a look at other possible more simple forms of having approach our problem. Our aim is to study if our approach, more sophisticated, present a significant advantage over this others ways.

For ease on the reading, in our notation V^µ¹^,...,µ^N will correspond to the pricing matrix with our original method.

We define a norm over the matrices as

||M || := max

t,x (M (t, x)) and the associated distance to V

dis(M, V ) := ||M − V ||

||V ||

This will allow us to measure the difference between the methods in a first global way.

Then, it would be interesting to make some experiments in aim to see how the volatility of the market and the distance between the possible values of the drifts affects this differences.

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CHAPTER 5. ALTERNATIVE APPROACHES 24

5.1 V as an average

We approach the value of the option at every moment as V I = V I^µ¹^,...,µ^N(t, x) =PN k=1πkV_k^µ

This way, we are obviating the computation of the function Π, so we are avoiding to actualize the information. But also we are giving away the uncertainty about the drift. Now, we can look at every realization of the random variable µ independently and treat it as a constant.

As a consequence of reduce the uncertainty, we will have V I(x, t) ≥ V (x, t) ∀x, t

Proceeding to measure the distance between this method and our original method, we keep parameters from example 3 in last chapter:

Parameters

Initial price of the stock x₀ = 45, the range of spatial steps moving from x_min = 0 to x_max = 225, the strike K = 150, time of maturity T = 1 (year), volatility σ = 0.17, interest rate r = 0.01, vector of possible drifts µ = (−0.06, 0.02, 0.03, 0.04) and initial beliefs π₀= (0.85, 0.13, 0.013, 0.007).

Distance between methods,

dis(V I, V ) :=||V I − V ||

||V || = 5.4398

150 = 0.0363 (5.1)

5.1.1 Varying the σ

Our aim in this section is to study how the difference between our original approach and VI evolves in relation with the volatility of the market, σ.

So the experiment we are going to realize is the following: We will study the difference between the results of the methods at the initial point, V I(x₀, t₀) − V (x₀, t₀), with different values of the volatility.

Again, we choose the above parameters for it (except σ), obtaining:

Figure 5.1: V I(x₀, t₀) − V (x₀, t₀) for σ varying in (0, 0.5)

It is noticeable the difference is greater up to σ = 0.1, and then it tends to stabilize above zero.

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5.1.2 (µ − , µ + )

In this section, we assume two initial possible values of the drift, µ − and µ + , both with equal initial belief. We will study the behavior of the difference between our original approach V and VI as varies, in the initial point, V I(x₀, t₀) − V (x₀, t₀).

Figure 5.2: V I(x₀, t₀) − V (x₀, t₀) for varying in (0.1, 10)

Observe how the difference stays positive for all values of , becoming continuously increasing for larger values of it from = 1.

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5.2 Π as an average

We take Π asPN

k=1π_k⁰· µK constant in time, and incorporate is in our algorithm this way.

Let the notation for the resulting matrix be V A = V A^µ¹^,...,µ^N

In this approach, we keep the function Π from our previous method, but it takes the form of a constant value; we avoid the filtration, so we do not actualize our initial beliefs about the drifts, but instead we keep them for all t

This method results closer to our than the previous one, showing (for same parameters, i.e. previous section, example 3) a distance of:

dis(V A, V ) := ||V A − V ||

||V || =5.2462

150 = 0.0350 (5.2)

5.2.1 Varying the σ

Lets follow now the procedure in the previous section to compare V and VA as the volatility changes.

We plot the value of V A(x0, t0) − V (x0, t0) for different values of σ.

For the same parameters as previously:

Figure 5.3: V A(x₀, t₀) − V (x₀, t₀) for σ varying in (0, 0.5)

The behavior reminds the one from previous method.

5.2.2 (µ − , µ + )

Under the same assumptions of previous section, we compare the corresponding results, V A(x0, t0) − V (x0, t0) with a variable .

The difference between the methods keeps negative for all in our range.

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Figure 5.4: V A(x0, t0) − V (x0, t0) for varying in (0.1, 10)

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5.3 µ reduced to a two dimensional vector

We take the values µi such that µi ≤ r and µj such that µj > r, and construct the following two dimensional vector of possible drift:

µ = (µ₁, µ₂) where µ₁=P

i=1π_i· µ_i, µ₂=P

j=1π_j· µ_j

and we apply our algorithm.

We will use the notation V D = V D^µ¹^,...,µ^N for the resulting matrix.

The aim of this procedure is to focus the attention of the problem in the possibilities of having a drift µ bigger or smaller than the market drift r

It is very noticeable how this method results to be very close to our on the results; keeping the parameters we obtain:

dis(V D, V ) := ||V D − V ||

||V || = 0.0889

150 = 5.9235 · 10⁻⁴ (5.3)

Let observe through a plot how similar results to be the boundary for this two methods:

(a) Boundary for V with our approach (b) Boundary for VD with the dual approach

5.3.1 Varying the σ

Once more, we proceed to vary the market volatility and study the impact in the difference between our methods at the initial point, V D(x0, t0) − V (x0, t0), with parameters from above.

The range of variation here is significantly smaller.

5.3.2 (µ − , µ + )

In this case, this experiment would not result meaningful, if we choose µ = r as our centered drift. It would constantly be the same values.

We can conclude that our approach is very significant in the case of special borders, and it can really provides an advantage over other more ”naive” ways, depending of the parameters.

In first case, we ignore to take into the account the values the market provides and the dimensions of our problem. We reduced it to a weighted average of the standard case.

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Figure 5.6: V D(x₀, t₀) − V (x₀, t₀) for σ varying in (0, 0.5)

In the second approach, we conserve our process Π, but we simplify it to an average of initial beliefs, which again does not let any filtration of information from the market.

Last method does not simplify that much the scenario, rather responds to the particular case of two dimensions. However, the focus of interest is it keeps the attention at the values of the possible drifts in relation with the interest rate r.

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Chapter 6

Conclusions

Over this work, we have provided a method to calculate the early exercise boundary of an American Put Option with uncertainty about the underlying asset drift.

In a situation where we do not agree with the values of the market drift, but instead we believe in different ones with uncertainty, the method provides a guide of when to early exercise according to both our believes and the actual market data.

An analysis of our method, and later comparison with other approaches for the same problem have shown interesting results. It could present significant different values from the ones in the standard market model problem.

This framework could give an investor an advantage over the market, allowing direct application of the techniques to the real world.

6.1 Final commentaries

We do not want to conclude this work without mention some details about it.

? In our comparison algorithms, for stability, remains very important to keep ^m_n² = constant << 1

? Although in our numerical approach we have apply a finite difference method with some particular approximations for the derivatives, other approximations can be chosen, as well as other general method (e.g. Monte Carlo techniques).

? In this work we have keep talking about an American Put Option, but the code and the theory are both easily adaptable to the American Call case. See the code in the Appendix for more details.

? For future studies, the door is open to other kind of options, as well as to options over assets with other distributions of random drifts. We could suppose that the choice of a large N could approximate situations of continuous distributions.

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Chapter 7

Appendix

In this section we are going to provide all the MATAB code programmed for this work, and used to generate the figures included.

The reader could thus introduce their own preferred parameters to apply our study to their own problem.

7.1 Function Π

Code for the calculation of the function Π, given the grid of the time-space, the possible drifts and our initial beliefs

f u n c t i o n PI= f u n c t i o n p i M a r t a (MU, P0 , sigma , t j , x i , x0 )

%%

%MU i s a v e r t i c a l v e c t o r o f l e n g t h N w i t h t h e p o s s i b l e s d r i f t s

%P0 i s a v e r t i c a l v e c t o r o f l e n g t h N w i t h t h e i n i t i a l b e l i e f on t h o s e

%d r i f t s

%t j , x i s c a l a r s , c o r r e s p o n d i n g t o t h e time , s p a c e p o s i t i o n i n t h e g r i d

%%

sum1=0;

sum2=0;

PI =0;

f o r k =1: l e n g t h (MU)

sum1=sum1+MU( k ) ∗ P0 ( k ) ∗ ( x i / x0 ) ˆ (MU( k ) / sigma ˆ 2 ) ∗ exp ( (MU( k )/2)∗(1 −MU( k ) / sigma ˆ 2 ) ∗ t j ) ; end

f o r k =1: l e n g t h (MU)

sum2=sum2+P0 ( k ) ∗ ( x i / x0 ) ˆ (MU( k ) / sigma ˆ 2 ) ∗ exp ( (MU( k )/2)∗(1 −MU( k ) / sigma ˆ 2 ) ∗ t j ) ; end

PI=sum1/sum2 ;

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CHAPTER 7. APPENDIX 32

7.2 Backwards Algorithm

Code that implement the backward algorithm with the derivative approximations of our choice, given the grid of the time-space, the type of the option (Put or Call), the possible drifts and our initial beliefs.

f u n c t i o n V = Ba c k w a r d s a lg o r i t h m M ar t a (MU, P0 , sigma , r , K, T, xmax , x0 , x , t , d e l t a x , d e l t a t , t y p e )

%%

%PARAMETERS

%P0 i s a v e r t i c a l v e c t o r o f l e n g t h N w i t h t h e i n i t i a l b e l i e f on t h o s e d r i f t s

%sigma , r , K, T s c a l a r s , ( v a r i a n c e , i n t e r e s t r a t e , s t r i k e , m a t u r i t y t i m e )

%xmax , xmin s c a l a r s t h a t d e t e r m i n e t h e r a n g e where our s t r i k e p r i c e move

%x0 i s t h e i n i t i a l v a l u e o f t h e s t o c k

%m, n s c a l a r s , number o f s p a c e / t i m e s t e p s

%t y p e : c a l l o r put

%i n i c i a l i z e t h e m a t r i x w i t h t h e f i n a l and boundary c o n d i t i o n s V=z e r o s ( l e n g t h ( x ) , l e n g t h ( t ) ) ; %(s p a c e , t i m e )

s w i t c h t y p e c a s e ’ put ’

f o r i =1: l e n g t h ( x ) %F i n a l c o n d i t i o n , l a s t column V( i , l e n g t h ( t ))=max (K−x ( i ) , 0 ) ;

end

f o r j =1: l e n g t h ( t )−1 %Boundary c o n d i t i o n s , f i r s t and l a s t row ( but l a s t colum ) V( 1 , j )=K;

V( l e n g t h ( x ) , j ) = 0 ; end

%a c t u a l a l g o r i t h m

f o r j=l e n g t h ( t ) : − 1 : 2 f o r i = 2 : 1 : l e n g t h ( x)−1

PI=f u n c t i o n p i M a r t a (MU, P0 , sigma , t ( j ) , x ( i ) , x0 ) ;

a=d e l t a t ∗x ( i ) ∗ 0 . 5 ∗ ( 1 / d e l t a x ) ∗ ( ( sigma ˆ2∗ x ( i ) / d e l t a x )+PI ) ; b=1− d e l t a t ∗ r −( d e l t a t ∗ sigma ˆ2∗ x ( i ) ˆ 2 / d e l t a x ˆ 2 ) ;

c=d e l t a t ∗x ( i ) ∗ 0 . 5 ∗ ( 1 / d e l t a x ) ∗ ( ( sigma ˆ2∗ x ( i ) / d e l t a x )−PI ) ; Vaux=c ∗V( i −1 , j )+b∗V( i , j )+a ∗V( i +1 , j ) ;

d=max (K−x ( i ) , 0 ) ; V( i , j −1)=max ( Vaux , d ) ; end

j end

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c a s e ’ c a l l ’

f o r i =1: l e n g t h ( x ) %F i n a l c o n d i t i o n , l a s t column V( i , l e n g t h ( t ))=max ( x ( i )−K, 0 ) ;

end

f o r j =1: l e n g t h ( t )−1 %Boundary c o n d i t i o n s , f i r s t and l a s t row ( but l a s t colum ) V( 1 , j ) = 0 ;

V( l e n g t h ( x ) , j ) = xmax − K;

end

d=max ( x ( i )−K, 0 ) ; V( i , j −1)=max ( Vaux , d ) ; end

j end

end end

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7.3 Boundary

Code that calculates the points where it worth to hold the option, and the points where it does not

f u n c t i o n B = FreeBoundaryMarta ( x , t , V, K, t y p e )

%x i s my v e c t o r i n s p a c e

% x : row a r r a y w i t h d i s c r e t e s t o c k v a l u e s . The s t r i k e p r i c e i s

% a p p r o x i m a t e l y i n t h e m i d d l e o f t h i s a r r a y . I f m i s even , t h e n

% x (m/2+1) = K, o t h e r w i s e K l i e s between S ( (m+1)/2) and

% x ( (m+1)/2+1).

%t v e c t o r t i m e

%V m a t r i x o f p r i c e s o f t h e o p t i o n , ( s p a c e , t i m e ) , l i k e mine

%K s t r i k e

%t y p e o f t h e o p t i o n B = z e r o s ( s i z e (V ) ) ;

f o r j = 1 : l e n g t h ( t )

B ( : , j )=V ( : , j )+x’−K; %column#j , l e n g h t ( x)= l e n g t h ( x)=m+1 end

f o r j = 1 : l e n g t h ( t ) B ( : , j )=V ( : , j )−x ’+K;

end end

%G i v e s back a v e c t o r o f x f o r t=j w h e r e i t s t o p w o r t h i n g w i t h j =1:n+1

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7.4 Plotting the boundary

Code to be executed, that needs the previous as a support. It includes the parameters, makes the space-time grid, and plot the optimal exercise boundary

c l o s e a l l c l e a r a l l c l c

%P a r a m e t e r s

MU= [ ] ’ ; %v e c t o r w i t h t h e p o s s i b l e d r i f t s

P0 = [ 1 ] ’ ; %v e c t o r w i t h t h e i n i t i a l b e l i e f i n t h e p r o b a b i l i t y o f e a c h d r i f t , sum ( P0)=1 sigma =; %market v o l a t i l i t y

r =; %market i n t e r e s t r a t e K=; %o p t i o n s t r i k e

T=;% e x p i r i n g t i m e

n=;%number o f t i m e s t e p s

xmax=; %T h i s two d e t e r m i n e s t h e r a n g e where our s t o c k p r i c e moves xmin =;

x0 =;

m=;%number o f s p a c e s t e p s

%Keep n i n t h e r a n g e o f mˆ2 f o r s t a b i l i t y r e a s o n s t y p e = ’ ’ ; %t y p e o f t h e American Option , c a l l o r put

%%

%Grid

d e l t a t=T/n ; t=z e r o s ( 1 , n + 1 ) ; f o r j =1:n+1

t ( j )=( j −1)∗ d e l t a t ; end

d e l t a x =(xmax−xmin ) /m;

x=z e r o s ( 1 ,m+ 1 ) ; f o r i =1:m+1

x ( i )=xmin+( i −1)∗ d e l t a x ; end

V = B a c k w a rd s a l g o r i th m M a r t a (MU, P0 , sigma , r , K, T, xmax , x0 , x , t , d e l t a x , d e l t a t , t y p e ) ; B = FreeBoundaryMarta ( x , t , V, K, t y p e ) ;

e x e r c i s e = B==0;

i m a g e s c ( t , x , ˜ e x e r c i s e ) ; c ol or m ap ( g r a y ) ;

a x i s xy ;

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7.5 Backwards Algorithm for V as an average of V

_k

k = 1, ..., N

Code that implement the backward algorithm with the derivative approximations of our choice, given the grid of the time-space, the type of the option (Put or Call), and a fixed drift.

f u n c t i o n V =

B a c k w a r d s a l g o r i t h m I n d i v i d u a l M a r t a (mu, sigma , r , K, T, xmax , x0 , x , t , d e l t a x , d e l t a t , t y p e )

%%

%PARAMETERS

end

V( l e n g t h ( x ) , j ) = 0 ; end

a=d e l t a t ∗x ( i ) ∗ 0 . 5 ∗ ( 1 / d e l t a x ) ∗ ( ( sigma ˆ2∗ x ( i ) / d e l t a x )+mu ) ; b=1− d e l t a t ∗ r −( d e l t a t ∗ sigma ˆ2∗ x ( i ) ˆ 2 / d e l t a x ˆ 2 ) ;

c=d e l t a t ∗x ( i ) ∗ 0 . 5 ∗ ( 1 / d e l t a x ) ∗ ( ( sigma ˆ2∗ x ( i ) / d e l t a x )−mu ) ; Vaux=c ∗V( i −1 , j )+b∗V( i , j )+a ∗V( i +1 , j ) ;

j end c a s e ’ c a l l ’

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end

V( l e n g t h ( x ) , j ) = xmax − K;

end

a=d e l t a t ∗x ( i ) ∗ 0 . 5 ∗ ( 1 / d e l t a x ) ∗ ( ( sigma ˆ2∗ x ( i ) / d e l t a x )+mu ) ; b=1− d e l t a t ∗ r −( d e l t a t ∗ sigma ˆ2∗ x ( i ) ˆ 2 / d e l t a x ˆ 2 ) ;

c=d e l t a t ∗x ( i ) ∗ 0 . 5 ∗ ( 1 / d e l t a x ) ∗ ( ( sigma ˆ2∗ x ( i ) / d e l t a x )−mu ) ; Vaux=c ∗V( i −1 , j )+b∗V( i , j )+a ∗V( i +1 , j ) ;

j end end

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7.6 Backwards Algorithm for Π as the average of the drifts

Code that implement the backward algorithm with the derivative approximations of our choice, given the grid of the time-space, the type of the option (Put or Call), the possible drifts and our initial beliefs. It uses a constant Π calculated in the beginning as a weighted average of the drifts.

f u n c t i o n V =

B a c k w a r d s a l g o r i t h m A v e r a g e M a r t a (MU, P0 , sigma , r , K, T, xmax , x0 , x , t , d e l t a x , d e l t a t , t y p e )

%%

%PARAMETERS

end

V( l e n g t h ( x ) , j ) = 0 ; end

%a c t u a l a l g o r i t h m PI=P0 ’ ∗MU;

j end

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end

V( l e n g t h ( x ) , j ) = xmax − K;

end

%a c t u a l a l g o r i t h m PI=P0 ’ ∗MU;

j end end

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7.7 Backwards Algorithm for µ as a two dimensional vector

Code that implement the backward algorithm with the derivative approximations of our choice, given the grid of the time-space, the type of the option (Put or Call), the possible drifts and our initial beliefs. It construct first two-dimensional vectors of possible drifts and initial beliefs, concerning about if the values are greater or smaller than the interest rate of the market.

f u n c t i o n V =

BackwardsalgorithmDualMarta (MU, P0 , sigma , r , K, T, xmax , x0 , x , t , d e l t a x , d e l t a t , t y p e )

%%

%PARAMETERS

MU1= [ ] ; MU2= [ ] ; P01 = [ ] ; P02 = [ ] ;

f o r s e p a r a t e c o u n t e r =1: l e n g t h (MU) i f MU( s e p a r a t e c o u n t e r )>= r

MU1( end+1)=MU( s e p a r a t e c o u n t e r ) ; P01 ( end+1)=P0 ( s e p a r a t e c o u n t e r ) ; e l s e

MU2( end+1)=MU( s e p a r a t e c o u n t e r ) ; P02 ( end+1)=P0 ( s e p a r a t e c o u n t e r ) ; end

end

g r e a t e r =(P01/sum ( P01 ) ) ∗MU1’ ; l e s s =(P02/sum ( P02 ) ) ∗MU2’ ; MU=[ g r e a t e r , l e s s ] ;

P0=[sum ( P01 ) , sum ( P02 ) ] ;

V=z e r o s ( l e n g t h ( x ) , l e n g t h ( t ) ) ; %(s p a c e , t i m e ) s w i t c h t y p e

c a s e ’ put ’

end

V( l e n g t h ( x ) , j ) = 0 ; end

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j end c a s e ’ c a l l ’

end

V( l e n g t h ( x ) , j ) = xmax − K;

end

j end end

Optimal exercise of an American Optionunder drift uncertainty

Examensarbete i matematik, 30 hp Handledare: Erik Ekström

Examinator: Kaj Nyström Juni 2016

Department of Mathematics Uppsala University

Optimal exercise of an American Option under drift uncertainty

Marta Rita Braojos Peláez

Optimal exercise of an American Option under drift uncertainty

Marta Rita Braojos Pel´ aez

Supervised by Erik Ekstr¨ om

Master Thesis in Financial Mathematics, 30 hp

June 2016

Contents

Chapter 1

Introduction

1.1 The market model

1.2 The problem

1.3 Standard (Black-scholes) case for reference

Chapter 2

Theoretical approach

2.1 Filtration Theory, foundations and our case

2.2 Optimal Stopping problems, foundations and our case

2.3 Arbitrage free pricing of American options, foundations and our case

2.4 Dynamics of the model

Chapter 3

Numerical approach

3.1 Finite-Differences Method foundations and our case

3.2 Our algorithm

Chapter 4

Drawing a boundary

Chapter 5

Alternative approaches

5.1 V as an average

5.1.1 Varying the σ

5.1.2 (µ − , µ + )

5.2 Π as an average

5.2.1 Varying the σ

5.2.2 (µ − , µ + )

5.3 µ reduced to a two dimensional vector

5.3.1 Varying the σ

5.3.2 (µ − , µ + )

Chapter 6

Conclusions

6.1 Final commentaries

Chapter 7

Appendix

7.1 Function Π

7.2 Backwards Algorithm

7.3 Boundary

7.4 Plotting the boundary

7.5 Backwards Algorithm for V as an average of V

k = 1, ..., N

7.6 Backwards Algorithm for Π as the average of the drifts

7.7 Backwards Algorithm for µ as a two dimensional vector

5.1.2 (µ − , µ + )

5.2.2 (µ − , µ + )

5.3.2 (µ − , µ + )