BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS
Connection between discrete time random walks and stochastic
processes by Donsker’s Theorem
by
Zandra Bernergård
Kandidatarbete i matematik / tillämpad matematik
DIVISION OF APPLIED MATHEMATICS
MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN
Bachelor thesis in mathematics / applied mathematics
Date:
2020-06-05
Project name:
Connection between discrete time random walks and stochastic processes by Donsker’s The-orem Author: Zandra Bernergård Supervisor: Linus Carlsson Reviewer: Anatoliy Malyarenko Examiner: Ying Ni Comprising: 15 ECTS credits
Abstract
In this paper we will investigate the connection between a random walk and a continuous time stochastic process. Donsker’s Theorem states that a random walk under certain conditions will converge to a Wiener process. We will provide a detailed proof of this theorem which will be used to prove that a geometric random walk converges to a geometric Brownian motion.
Contents
Definitions . . . 3
1 Introduction 4
1.1 Background and literature review . . . 4 1.2 Problem formulation and outline . . . 5
2 Binomial model 6
2.1 One step binomial model . . . 7 2.2 Two step binomial model . . . 11 2.3 Multiperiod binomial model . . . 14
3 Wiener process 18
3.1 Convergence in probability . . . 22 3.2 Compactness and tightness . . . 23
4 Donsker’s Theorem 30
4.1 Convergence of a geometric random walk . . . 36
Definitions
In this thesis we will use the following notations:
N – 1,2,3... The natural numbers
N(µ, σ ) – Normal distribution with mean µ and standard deviation σ . f(t) = ft, to improve readability we will sometimes denote f (t) with ft.
C– Continuous functions X : [0, 1] → R. C – The Borel σ-field defined on C. S– An arbitrary metric space. S – The Borel σ-field on S. B – The Borel σ-field on R. B– The complement of a set B. A− – The closure of A.
Chapter 1
Introduction
Stochastic differential equations (SDE) have many applications, particularly in finance. SDE’s are often difficult or impossible to solve analytically, most of them does not have a closed form solution. Instead we have to compute the solution numerically for which the equation has to be discretized in time. To determine if a certain discrete time model can be used as an approximation of an SDE we will study the limit. If the limit of the discrete time model goes towards the SDE it means that as the length of the time intervals approaches zero, the discrete time model goes towards the SDE. In this thesis we will analyse a discrete arithmetic random walk as an approximation of a continuous SDE, namely the Wiener process. This is used to prove the convergence of a geometric random walk to a geometric Brownian motion.
1.1
Background and literature review
The Wiener process is also called Brownian motion since the model was first described by the botanist Robert Brown in 1828 [1]. It was not until 1921 that the Brownian motion was studied mathematically by Norbert Wiener [2]. He constructed a mathematical model of the Brownian motion that describes the movements of a random variable in continuous time. The Wiener process has several applications in applied mathematics, especially in the well known Black–Scholes model.
The Black–Scholes model was first derived by F. Black and M. Scholes in 1973 [3]. This model assumes that the stock price follows a geometric Brownian motion. To solve this we have to solve a stochastic differential equation that consequently can be difficult or impossible to solve analytically. In 1979 J.C. Cox, S.A Ross, and M. Rubenstein described a model that discretizes the stochastic process. This model became known as the CRR-model, also called binomial option pricing model [4]. The CRR-model describes the stock movements as a geo-metric random walk.
A. De Moivre was the first one to prove that the binomial distribution under certain conditions can be approximated by a normal distribution. This lead to what today is known as the Central Limit Theorem. This is also a special case of the theorem that was stated by M.D. Donsker
in 1949. He provided a proof of the idea that a random walk converges to a Wiener process when the length of each step in the random walk goes towards zero. This was referred to as Donsker’s invariance principle [5]. Donsker developed this proof and achieved more general-ized result which he published in his article of 1951 [6] and this is what now is referred to as Donsker’s Theorem.
1.2
Problem formulation and outline
To price options we can use stochastic differential equations defined in continuous time which are often difficult or even impossible to solve explicitly. Instead we use numerical methods to approximate the result which means that we have to construct the problem in discrete time. We will describe the discrete time model used for option pricing but the main question of this thesis is to determine if the discrete time random walk converges to the correct solution of the stochastic differential equation. The transition from a discrete time stochastic process to a continuous time model will be the primary study of this thesis.
The next chapter will describe the binomial model used for option pricing. This is by defini-tion a random walk that will be of much interest later in this report. We will derive a general formula for pricing options with the n-step binomial model.
In Chapter three we will introduce the continuous time stochastic process, the Wiener pro-cess that lead us to the geometric Brownian motion, also widely used to price options. We will also cover the most important definitions and theorems needed in Chapter four, which is the main part of this report, where we provide a detailed proof of Donsker’s Theorem. Donsker’s Theorem states that a random walk converges in distribution to a Wiener process when we let the number of time steps go to infinity.
In the original proof the random walk is defined with a continuity correction term. The proof provided in this thesis will be done without the extra term. This is since we can not see how the increments of the random function can be independent if we keep the continuity correction term. We will see that the independence of the increments will be an important part of the proof, and when taking the limit the extra term will not effect the result we will get.
The last part of Chapter four is to prove, by using Donsker’s Theorem, that a geometric ran-dom walk converge to a geometric Brownian motion.
Since the main study of this thesis is the mathematical part describing the convergence of a random walk, the reader is assumed to have basic knowledge in finance to understand Chapter two. For the reader who wants to get more familiar with the basic concepts in finance we refer to [7]. We also assume some familiarity with mathematical conventions, such as supremum and limit superior and so forth, the reader who needs to refresh such concepts is referred to [8].
Chapter 2
Binomial model
In this chapter we will describe how to use the binomial model when pricing options. This chapter should not be seen as the main part of this thesis, it is included to give the reader an idea of how the random walk is constructed for a better understanding of the proof provided in Chapter four.
The binomial model is a discrete time model that can be used when analyzing the financial market. We will construct a portfolio consisting of one bond and one stock. The price at time t is denoted st for the stock and bt for the bond. In the binomial model there are only two
possible outcomes of each time step, the stock value can move either up or down.
Assumption 2.1. The following are assumed to be true when we analyze the binomial model: • Short sells are allowed.
• No dividend is paid.
• Fractional holdings are possible.
• No bid-ask spread exists, the selling and the buying price is the same for all assets. • There are no transaction costs of buying or selling.
• The market is always liquid so that there is always possible to buy or sell an unlimited amount of assets. It is always possible to short sell and borrow unlimited amounts from the bank.
• There exists no arbitrage portfolios:
Definition 2.1. A self-financing portfolio P is said to be an arbitrage portfolio if:
P0= 0
Pt > 0 with probability 1, for t > 0.
If such a portfolio exists it means that we can make a profit without investing any money. We start with no money and with positive probability our investment will increase.
2.1
One step binomial model
The price of the bond in the one step model can be described as:
b0= 1, b1= 1 + r,
where r is the rate of the bond of one time period. The price of the stock:
s0= s, s1= ( s· u, with probability pu, s· d, with probability pd, where d < u and pu+ pd= 1.
Definition 2.2. Let α denote the number of bonds and β the number of stocks which are held in the portfolio at time t. This portfolio is denoted as P(α, β ) and the value of the portfolio at time t is:
Pt= αbt+ β st, t= 0, 1. (2.1)
For the one step binomial model this gives the value of the portfolio:
P0= α1 + β s, and the value at t = 1:
P1= (
α (1 + r) + β su, with probability pu,
α (1 + r) + β sd, with probability pd.
This can be illustrated by the binary tree:
α 1 + β s
α (1 + r) + β su
α (1 + r) + β sd pu
Since we have the assumption that there exists no arbitrage portfolios, we will prove the con-dition for which the market is arbitrage free:
Theorem 2.1. The binomial model is arbitrage free if and only if the inequality d < 1 + r < u holds.
Proof. If the inequality does not hold and we have that sd < su ≤ s(1 + r) we will get higher return by invest only in the bond. In this case we will short sell the stock and invest all the money obtain from the short sell in the bond. This gives that α = s and β = −1. From this we obtain:
P0= s · 1 + (−1)s = 0,
P1= (
s(1 + r) − su, with probability pu,
s(1 + r) − sd, with probability pd.
If sd < su ≤ s(1 + r) we can see that the value of p1> 0 and we have an arbitrage portfolio.
The same result will follow if s(1 + r) ≤ sd < su.
Assume instead that the inequality is satisfied and that there is an arbitrage opportunity. Con-sider the portfolio:
P0h= α + β s = 0, ⇒ α = −β s, P1h=
(
β s(u − (1 + r)), with probability pu,
β s(d − (1 + r)), with probability pd.
In order for there to be an arbitrage P1h> 0 must hold which gives the equation:
β s(u − (1 + r)) > 0 ⇒ u > 1 + r, β s(d − (1 + r)) > 0 ⇒ d > 1 + r,
which is a contradiction to the assumption that the inequality holds. This means that the inequality can not be true at the same time as we have an arbitrage opportunity.
Definition 2.3. Let X be the stochastic variable defined as X = g(Y ) where Y is the stochastic variable representing how the stock price is changing. Then X is a contingent claim.
Consider a European call option with strike price k. If s1> k we will exercise the option and
buy the stock for price k, and then sell it for s1, with the profit s1− k. This also makes the
value of the option equal to s1− k. If s1< k, the option will be worth 0, since we in that case
can buy the stock for less than k without exercise the option.
X = (
g(u) = max[su − k, 0] with probability pu,
We observe that the above expression describes a call option. For a put option we have:
X= (
g(u) = max[k − su, 0] with probability pu,
g(d) = max[k − sd, 0] with probability pd.
For simplicity we will consider call options in the rest of this chapter, but the methods provided are easy to apply on put option as well.
Definition 2.4. If there exists a portfolio Phsuch that:
P1h= X ,
then X is said to be attainable and the portfolio P1h is said to be a replicating portfolio. The market is complete if all claims can be replicated.
If X is reachable by some replicating portfolio Ph, then the only possible price for X at time t= 0 is P0h, since all other values of X will provide an opportunity to make an arbitrage profit. From this follows that if we have a complete market, such that every X is reachable, we can price all contingent claims.
Theorem 2.2. If the binomial model is free of arbitrage, then it is also complete.
Proof. If we have an arbitrary claim X , we will try to find a replicating portfolio Phfor which X is reachable. We use Equation (2.1) to get the value of the portfolio at time t = 1, if we can choose α and β so that:
P1h= α(1 + r) + β su = g(u), if Y = u, P1h= α(1 + r) + β sd = g(d), if Y = d,
then we have found a replicating portfolio. We solve the equations above and get:
α = 1 1 + r· ug(d) − dg(u) u− d , (2.2) β = 1 s· g(u) − g(d) u− d . (2.3)
This shows that all claims X are reachable from a replicating portfolio Phconstructed by using α and β as in the equations (2.2) and (2.3). According to Theorem 2.2 the one step binomial model is complete. We can now price every contingent claim under this model.
We denote the price of X at time t by f (t, X ). Since we know that f (0, X ) = P0h we can now use the the equations (2.2) and (2.3) to compute f (0, X ):
f(0, X ) = P0h= α + β s, = 1 1 + r· ug(d) − dg(u) u− d + 1 s· g(u) − g(d) u− d · s, = 1 1 + r ug(d) − dg(u) u− d + (1 + r)(g(u) − g(d)) u− d , = 1 1 + r = (1 + r) − d u− d · g(u) + u− (1 + r) u− d · g(d) . (2.4) By noting that (1 + r) − d u− d + u− (1 + r) u− d = u− d u− d = 1, we can rewrite Equation (2.4) as:
f(0, X ) = 1 1 + r(q · g(u) + (1 − q) · g(d)) , (2.5) where q= (1 + r) − d u− d , and 1 − q = u− (1 + r) u− d , (2.6) where q is called the risk neutral probability. Since q only depends on r, u and d we do not need to know pu or pd, to price the option. We only need to know g(u) and g(d). By
using q as the probability we have a risk neutral valuation of the market, meaning that the discounted average value of the stock at t = 1 will be equal to s0. From Equation (2.5) we see
that f (0, X ) = 1
1 + rE˜(X1) where ˜E(X1) is the expected value of X at t = 1 when using the risk neutral probability q.
Example 2.1. Assume that the initial price of the stock is 50kr, u = 1.1, d = 0.9, r = 0.05, and the strike price k = 52kr. The exercise date is after one year, we let this correspond to t = 1. Then we have the binomial models:
50Kr
55Kr
45Kr Stock price in
the one step binomial model
f(0, X )
g(u) = 3
g(d) = 0 X= max[s − k, 0]
Since d ≤ 1 + r ≤ u is true we know that the market is free of arbitrage and therefore also complete. By using Equation (2.6) we can compute q:
q=1.05 − 0.9
1.1 − 0.9 = 0.75, ⇒ 1 − q = 0.25.
We now use (2.5) to compute f (0, X ):
f(0, X ) = 1
1.05(0.75 · 3 + 0.25 · 0) = 2.14.
We are now able to determine how the replicating portfolio is constructed by using the equa-tions (2.2) and (2.3): α = 1 1.05· 1.1 · 0 − 0.9 · 3 1.1 − 0.9 = −12, 8571, β = 1 50· 3 − 0 1.1 − 0.9 = 0.3.
Meaning that we borrow 12,86 Kr from the bank and invest 0.3 shares in the stock. The value of the portfolio in this one step binomial model is:
P0= −12.8571 + 0.3 · 50 = 2.14,
P1= (
−12.8571 · 1.05 + 0.3 · 55 = 3, if Y = u, −12.8571 · 1.05 + 0.3 · 45 = 0, if Y = d.
We can see that the value of this portfolio at t = 0 is, as expected, the same as the price we derived for the option at this time step.
2.2
Two step binomial model
We will now introduce the two step binomial model to see how to price options with two time steps. In the next section we will generalize the idea and derive a formula for the n-step binomial model. To improve readability when the number of the time steps increases we will now denote g(uu) = guu and g(dd) = gdd.
s su su2 sd sd2 sud= sdu u d u2 d2 ud ud Stock price in the two step binomial model f(0, X ) fu(1, X ) guu= fuu(2, X ) fd(1, X ) gdd = fdd(2, X ) gud = fud(2, X ) Pricing model for the option in the two-step model
In this case with the two step model we have that X is equal to max[s2− k, 0]. Since there is
no arbitrage the option price at the exercise time t = 2: f (2, X ) is equal to X . We name these values guu, gdd and gud respectively. The general pricing model will be discussed in more
detail in the next section where we cover the n-step model. As we can see in the figure the two step model consists of three different one step binary trees:
• f (0, X), fu(1, X ), fd(1, X ), • fu(1, X ), fuu(2, X ), fud(2, X ),
• fd(1, X ), fud(2, X ), fdd(2, X ).
Since fuu(2, X ) = guu, fud(2, X ) = gud and gdd = fdd(2, X ) we have that the last two trees can
be written as:
• fu(1, X ), guu, gud,
• fd(1, X ), gud, gdd.
We can consider these sub trees as three one-step binomial models and if the market is ar-bitrage free we can price them as we did in the one-step model. To satisfy the assumption of absence of arbitrage we have the same requirement as for the one step model, the condi-tion d < 1 + r < u has to hold. We will prove in the next seccondi-tion that this is a necessary and sufficient condition for every binomial model. By using (2.5) for each of these sub trees we get: f(0, X ) = 1 1 + r(q · fu(1, X ) + (1 − q) · fd(1, X )) , (2.7) fu(1, X ) = 1 1 + r(q · guu+ (1 − q) · gud) , (2.8) fd(1, X ) = 1 1 + r(q · gud+ (1 − q) · gdd) . (2.9)
We substitute fu(1, X ) and fd(1, X ) in Equation (2.7) by the equations (2.8) and (2.9) and
obtain the pricing model for the two step model:
f(0, X ) = 1 1 + r q· 1 1 + r(q · guu+ (1 − q) · gud) + (1 − q) · 1 1 + r(q · gud+ (1 − q) · gdd) , = 1 (1 + r)2 q 2g uu+ 2q(1 − q)gud+ (1 − q)2gdd , (2.10) where: guu= max[su2− k, 0], gud = max[sud − k, 0], gdd= max[sd2− k, 0].
Example 2.2. Let s = 50, u = 1.1, d = 0.9, r = 0.05, the strike price is 52. This give the model: 50 55 60.5, X = 8, 5 45 40, 5, X = 0 49.5, X = 0
Stock price for Example 2.2
The inequality d < 1 + r < u holds which imply that the market is arbitrage free. From Equa-tion (2.6)
q= 1.05 − 0.9
1.1 − 0.9 = 0.75, ⇒ 1 − q = 0.25. From the equations (2.8) and (2.9):
fu(1, X ) = 1
1.05(0.75 · 8.5 + 0.25 · 0) = 6.07, fd(1, X ) = 1
1.05(0.75 · 0 + 0.25 · 0) = 0. We now have the pricing model:
f(0, X ) 6.07 8.5 0 0 0
Pricing model to find f(0, X )
We can now use the given values for fu(1, X ) and fd(1, X ) to calculate f (0, X ), we use
Equa-tion (2.7)
f(0, X ) = 1
1.05(0.75 · 6.07 + 0) = 4.34,
or if we compute f (0, X ) directly from Equation (2.10):
f(0, X ) = 1
1.052 0.75
2· 8.5 + 0 + 0 = 4.34.
2.3
Multiperiod binomial model
In the multiperiod binomial model, also called the n-step binomial model we will generalize the result of the previous sections. In this model time is running between
t0 = 0 and the exercise time T , where each time step is of length h = Tn. We construct the
portfolio as before with one stock and one bond. The price process for the bond is given by:
b0= 1,
bt+1= (1 + r)bt,
and for the stock:
s0= s, st+1= stYt,
where Yt is the movement of the stock for t = 0, 1, ..., T with
P(Yt = u) = pu,
P(Yt= d) = pd.
For this model we will consider the portfolios that are self financing. This means that all trades are financed by selling or buying some assets, no other money are withdrawn or added after constructing the initial portfolio.
Definition 2.5. A portfolio P(α, β ) is said to be self-financing if:
αt(1 + r) + β st = αt+1+ βt+1st, t= 0, 1, ..., T − 1.
This expression means that the portfolio constructed at time t − 1, that is held until time t, has at that point the same market value as the portfolio that is created at time t and will be held until time t + 1.
Definition 2.6. If there exists a self-financing portfolio Phsuch that:
PTh= X ,
then X is said to be reachable. The portfolio Phis said to be a replicating portfolio. The market is complete if all claims can be replicated.
Assume X is reachable by any self-financing replicating portfolio Ph then the only possible price for X at t is Pth. If it is possible to buy X to a price lower than Pth at some time t, there exists an arbitrage opportunity. Same result follows if X is priced higher than Pth at time t, since we then can sell X and invest in Pthwith a profit made. We define the price process for X under the n-step binomial model:
Definition 2.7. If a claim X is reachable with the replicating portfolio Ph, the price process of X is given by:
f(t, X ) = Pth, t= 0, 1, ..., T.
Theorem 2.3. The n-step binomial model is complete. For every contingent claim there exists a replicating portfolio.
Proof. The n-step binomial model is complete which can be shown by using induction back-wards. Start at time T and let PTh = X , we then apply the one-period method to find the replicating portfolio PT−1. This is done by considering each node at time step T − 1 as the
initial time step of a one-period model. We can then calculate PT−1for every node at that time
step. We will then use this result to find PT−2 for every node at this time step, and continue
this way until we have found P0.
As mentioned when discussing the one step model, using q as the probability measure give us a risk free valuation of the stock. This means that the stock value is, on average, the same at each time step t in the future as it was at the initial time step, which is equivalent to say that the discounted expected value of the stock at every time step is equal to s0.
Theorem 2.4. If the market is free of arbitrage an European option that has expiration date t= T , has the price model:
f(0, X ) = 1
Proof. We know that f (T, X ) = XT since we have no arbitrage. This implies that ˜E( f (X , T )) =
˜
E(XT). Consider each sub tree as one separate one step tree. We then know that every node at time step T can be replicated from their connecting nodes at time step T − 1. We can calculate the value of the respectively replicating portfolios at T − 1 according to the model derived in the previous section for the one step model: f (T − 1, X ) =1+r1 E(X˜ T). Now every node at time step t = T − 1 has a value corresponding to their replicating portfolio. We will now continue this way to price every node at step t = T − 2 using the values we just calculated for time step T− 1. This give us:
f(T − 2, X ) = 1 1 + rE( f (T − 1, X )) =˜ 1 1 + r 1 1 + rE(X )˜ = 1 (1 + r)2E(X˜ T).
If we do this T times until we reach the first node give us:
f(0, X ) = 1
(1 + r)TE(X˜ T).
The different values s can take are described as s0ukdt−k, at time t, with k up-movements, this
is known as a geometric random walk:
Definition 2.8. Let s0 be the value of the stock at the initial time step, and Yk is i.i.d random
variables such that:
sn= s0 n
∏
k=1
Yk,
for n ∈ N, then the sequence {sn} is a geometric random walk.
There are only two outcomes of each time step: u with the risk neutral probability q, or d with probability 1 − q. This describes the binomial distribution:
P(st= ukdt−ks0) =
t k
qk(1 − q)t−k,
The contingent claim X in the n-step model is described as max[sT − k, 0] and it follows that:
P(X = max[ukdT−ks0− k, 0]) =T k
qk(1 − q)T−k.
The expected value of X is then:
˜ E(XT) = T
∑
k=0 T k qk(1 − q)T−k· max[ukdT−ks0− k, 0]. (2.12) From Theorem 2.4 we know that the n-step binomial model with exercise time T has the pricing model given by:f(0, X ) = 1
which from Equation (2.12) gives: f(0, X ) = 1 (1 + r)T · T
∑
k=0 T k qk(1 − q)T−k· max[ukdT−ks0− k, 0].By using the pricing model defined in Equation (2.11) we can now prove the requirement needed for the absence of arbitrage in the n-step model:
Theorem 2.5. The n-step binomial model is arbitrage free if and only if the condition d < 1 + r < u is true.
Proof. Assume that the inequality is satisfied, then the pricing model derived above hold. We want to show that this implies absence of arbitrage for the n-step model. Let h be an arbitrary portfolio with the following condition:
P(PTh≥ 0) = 1, P(PTh> 0) > 0.
This implies that:
E(PTh) = ˜E(X ) > 0,
and since:
P0h= f (0, X ) = 1
(1 + r)tE(X ),˜
it follows that in every case where PTh> 0 it implies that ph0> 0 and Ph is not an arbitrage portfolio.
If the inequality does not hold we proved for the one step case that we will have an arbitrage opportunity at t = 1. This proof is therefore obviously also applicable for the n-step model, since, if we have arbitrage opportunity at t = 1, we do not have an arbitrage free market.
Chapter 3
Wiener process
A stochastic process describes a variable that take on certain values and it can be a discrete- or continuous process. A random walk is a discrete-time stochastic process and when we prove Donsker’s Theorem we will see the connection between a random walk and a continuous-time stochastic process, namely the Wiener process. We therefore start this chapter by introducing the Wiener process by its definition:
Definition 3.1. A Wiener process W : R+→ R is a stochastic process where:
1. W(0)=0.
2. W has independent increments, for r < s ≤ t < u,W (u) − W (t) and W (s) − W (r) are independent stochastic variables.
3. W has Gaussian increments, meaning that W (t + u) −W (t) ∼ N(0,√u). 4. W has continuous paths.
A Wiener process with this properties has drift rate 0 which means that the expected value at any future time is equal to the current value. Variance rate is equal to 1, meaning that the change in variance over a time period of length t is equal to t. Then the generalized Wiener process is defined by the stochastic differential equation:
dX = a dt + b dW. (3.1) In the discrete time model we write this as:
∆X = a ∆t + b ∆W. (3.2)
Since the Wiener process has a variance rate of 1, we get the variance of ∆X :
V[∆X ] = V [a ∆t + b ∆W ], = b2V[∆W ]. = b2∆t.
This implies that the standard deviation σ is equal to b√∆t. This means that the uncertainty about the value of X in the future is increasing in proportion to the square root of time. And from the definition of a Wiener process the expected value of its increments is 0 at each time period:
E[∆X ] = E[a ∆t + b ∆W ], = a · E[∆t] + b · E[∆W ], = a · E[ ∆t],
= a · ∆t.
The drift rate of the generalized Wiener process is then a, and the variance rate is b2. This is how much the mean and variance respectively changes per unit of time. The model just described has a constant drift rate which means that if we have an initial value of x = 40, a drift rate a of 10 Kr per year, we will at the end of year one have E(X ) = 50. This corresponds to a increase in x by 25%. Since the drift rate is constant the expected value after year two will be: E(X ) = 60 which is an increment from the end of year 1 by 20%. This is the reason why this is not directly applicable to stock prices. The expected return we require of a stock is not dependent on the stock price. If we require the return to be 25% when the stock price is 40Kr, we also expect the return to be 25% if the stock price is 50Kr.
When modelling the price of a stock we can no longer assume that the drift rate is constant. We will instead assume that the expected return is constant, which is the drift rate divided by the stock price S, let us denote this µ. Then it follows that the drift rate in this model is µS, and for a short period of time ∆t, we have that the expected change in S equals µS∆t. Let us consider the generalized Wiener process with drift rate µS∆t, and the ∆W term is zero, mean-ing that the variance, also referred to as the volatility of the stock, is zero. From Equation (3.2):
∆S = µ S∆t,
we divide by S:
1
S∆S = µ ∆t,
when taking the limit as ∆t → 0 we get:
1 S(t)· dS = µdt, (3.3) ⇒ 1 S(t)· dS dt = µ.
We integrate both sides with respect to time: Z T 0 1 S(t)· dS dtdt= Z T 0 µ dt, ⇒ ln(S(t)) T 0 = µt T 0 ⇒ ln(S(T )) − ln(S(0)) = µT, ⇒ S(T ) = S(0)eµ T.
If the stock has no volatility, meaning that the variance is zero, the stock price increases ex-ponentially. In real life there is volatility that we have to consider when model the stock movements. The volatility of the stock is proportional to the stock price during a time period ∆t. This is since we will have as much uncertainty when the stock price is 40Kr as when it is 50Kr. For the time period ∆t we get that the volatility of S is σ S. We let ∆t → 0 and combine this with Equation (3.3) which gives:
dS= µSdt + σ SdW, ⇒dS
S = µdt + σ dW. (3.4) The solution of Equation (3.4) is called and known as the geometric Brownian motion. For the discrete time model we have for a small time interval ∆t:
∆S
S = µ∆t + σ ∆W. (3.5)
If we have a normal variable Y with mean µ and standard deviation σ then Z = Y−µ
σ is
nor-mally distributed with mean 0 and variance 1. This also implies that if Z is a standard normal variable then Y = Zσ + µ is normally distributed with mean µ and standard deviation σ [9]. We know from the definition of a Wiener process that it is normally distributed with mean 0 and standard deviation√∆t. Normalizing the Wiener process gives:
Z= (∆W − 0)√ ∆t
,
where Z has a standard normal distribution. Equation (3.5) can be rewritten as:
∆W = ∆S S − µ∆t σ , ⇒ Z = ∆S S−µ∆t σ√ − 0 ∆t , ⇔ Z = ∆S S√− µ∆t ∆tσ , ⇔ ∆S S = Z √ ∆tσ + µ ∆t.
It is now easy to see that ∆S
S ∼ N(µ∆t,
√
∆tσ ). With this model we can estimate the stock price which we can use to calculate the option price.
The next step is to introduce the probability measure used for the Wiener process which will require the definition of the characteristic function:
Definition 3.2. The characteristic function for a random variable Y with probability density function f is:
ϕY(t) = E(eitY) =
Z ∞
−∞
eityf(y)dy.
Theorem 3.1. If Y is a random variable with normal distribution with mean µ and variance σ2the characteristic function becomes:
ϕY(t) = eiµt−
σ2t2 2 .
Definition 3.3. Wiener measure is the probability measureW on (C,C) that satisfies the con-ditions:
1. For t ∈ [0, 1], the random variable Zt on (C,C,W) is defined as Zt(X ) = X (t), for every
X ∈ C and is normally distributed with mean 0 and variance t.
2. The stochastic process Zt has independent increments. For every partition
0 ≤ t1≤ ... ≤ tn= 1, Zti− Zti−1 are independent random variables.
Zt is also called a coordinate-variable process on (C,C,W).
Theorem 3.2. For every 0 ≤ s < t ≤ 1, Zt− Zs has a normal distribution with mean 0 and
variance t− s.
Proof. The characteristic function of Zt and Zsare:
ϕZs(y) = e −isy2 2 = E[eiyZs], ϕZt(y) = e −ity2 2 = E[eiyZt], = E[eiy(Zt−Zs+Zs)], = E[eiy(Zt−Zs)eZs],
= E[eiy(Zt−Zs)]E[eiyZs],
= ϕZt−s(y)ϕZs(y), ⇒ ϕZt−s(y) = ϕZt(y) ϕZs(y) = e −ity2 2 e−isy22 = e−iy2(t−s)2 ,
which shows that ϕZt−s(y) has the characteristic function for a normally distributed random
The following theorem will be important for us to be able to complete the proof of Donsker’s Theorem:
Theorem 3.3. If there exists a Wiener measureW, then the coordinate-variable process Zt on (C,C,W) is a Wiener process.
Proof. We need to prove that if the condition for a Wiener measure holds true, then the Con-ditions 1-4 stated in the Definition 3.1 are satisfied.
Z0 is normally distributed with mean 0 and variance t which is equal to 0, so we can see
thatW(Z0= 0) = 1 and Condition 1 is thereby satisfied. Condition 2 follows from the
defini-tion of Wiener measure. From Theorem 3.2 it follows that Zt+u− Zt∼ N(0,
√
u) which fulfill Condition 3. The last condition holds since the sample space where Zt is defined is C which
by definition consists only of continuous elements.
3.1
Convergence in probability
What we want to show in the next chapter is how a random walk converges when we let the number of time step go towards infinity. This will require us to first define the convergence of random functions and their probability measures which we will cover in this section.
Let (Ω, F, P) be a probability space, let (Ω0, F0) be a measurable space, and let X : Ω → Ω0be the map satisfying X−1(A) ∈ F if A ∈ F0.
Definition 3.4. The distribution of X is the measure on Ω0given by
P ◦ X−1(A) = P{X ∈ A}, A∈ F0.
In the rest of this paper we will use expression like P(X ∈ C : |X (0)| < α) but we will often suppress the notation of the function space and instead write P(|X (0)| < α).
When we talk about convergence of probability measure we will use the definition of weak convergence:
Definition 3.5. Let P and {Pn} be probability measures on the measure space (S,S). If for
every continuous and bounded function X : S → R: lim n→∞ Z S X dPn= Z S X dP,
then we say that Pnconverges weakly to P which denotes: Pn w
−→ P.
In the case where a random function converges to another random function we will use the definition of convergence in distribution. The connection between the convergence in distri-bution of a function and the weak convergence of the functions probability measure will be important in some of the proofs that follows.
Definition 3.6. Let X be defined on the probability space: (Ω, F, P), and Xnon the probability
space: (Ω0, F0, P) and (S, ρ) is a metric space. X : Ω → S, Xn: Ω0→ S are random elements
with distribution P ◦ X−1 and P ◦ Xn−1 respectively. Then Xn converges in distribution to X ,
which is denoted as Xn−→ X if P ◦ Xd n−1 w
−→ P ◦ X−1.
For random variables we talk about convergence in probability:
Definition 3.7. A sequence of random variables {Xn} is said to converge in probability to a
random variable X if, for every ε > 0:
lim
n→∞P(|Xn− X| < ε) = 1.
This is denoted Xn p
−→ X.
Theorem 3.4. Let {Xn}, {Yn} be sequences of random variables. If Xn −→ X where X is ad
random variable and Yn p − → a where a ∈ R, then: Xn+Yn d − → X + a, XnYn−→ Xa.d
Definition 3.8. Let (S, p) be a metric space and P is a probability measure on (S,S). Let A ∈ S. If P(∂ A) = 0 then A is called a P−continuity set.
3.2
Compactness and tightness
For the proof that follows we will need to determine if a sequence of probability measures is tight, for which we will need the definitions and theorems in this section.
Definition 3.9. For all X ⊆ C and δ > 0 we define the modulus of continuity for the function X of δ as:
ω (X , δ ) := sup
t,s∈[0,1] |t−s|≤δ
|X(s) − X(t)|.
For the following two definitions we define the set of functions X ⊆ C, defined on a domain A⊆ R.
Definition 3.10. X is said to be uniformly equicontinuous if and only if, for every ε > 0 there exists a δ > 0 such that for every x ∈ X it holds that:
|x(s) − x(t)| < ε when |s − t| < δ , for every t, s ∈ A.
Definition 3.11. A set X ∈ C is said to be uniformly bounded if there exists a M ∈ R where M> 0, such that for every x ∈ X and t ∈ A: |x(t)| < M.
Theorem 3.5. Let K ⊂ C with the following properties: 1. sup X∈K |X(0)| = M < ∞, and: 2. lim δ →0 sup X∈K ω (X , δ ) = 0.
Then K is uniformly bounded and uniformly equicontinuous.
The proof of this theorem is not included in this thesis and can be found in Lemma 2.17 in [10].
Definition 3.12. Let (S, ρ) be a metric space. The set K ⊂ S is compact if every open cover contains a finite sub-cover. In particular, Heine-Borel’s theorem tell us that if S = R then K⊂ R is compact iff K is closed and bounded.
Definition 3.13. A set K defined on a metric space S is relatively compact if its closure K− is compact.
By Definition 3.12 we know that a compact set is closed, bounded and equicontinuous. The subset of a compact set is also bounded and equicontinuous and therefore relatively compact according to Theorem 3.6. It might have sequences that converge outside of the set itself, with its limits existing in the closure.
We know from Theorem 3.5 that if a set K is relatively compact it is bounded and equicontin-ous, then K− is bounded and equicontinous and also closed, and therefore compact.
Theorem 3.6. A set K ⊂ C is relatively compact if and only if is satisfies the conditions of Theorem 3.5.
We refer to Theorem 7.2 in [11] for the proof of this theorem since it is beyond the scope of this thesis.
To prove the next theorem we will need the following definition:
Definition 3.14. A sequence Pn where n ∈ N of probability measures defined on a metric
space S is tight if, for every ε > 0, there exists a compact set K ⊂ S and an n0∈ N such that
Pn(K) > 1 − ε for all n > n0.
Theorem 3.7. A sequence Pn, where n∈ N of probability measures defined on (C, C) is tight
if and only if it satisfies the following conditions:
1. There exists an α > 0 such that lim
n→∞Pn(|X (0)| ≥ α) = 0,
2. For every ε > 0 it holds that: lim
δ →0
(lim sup
Proof. We want to prove that if these two conditions are true it implies that for every ε > 0 there exists a compact set K in C and an integer n0> n, such that Pn(K) > 1 − ε for every
n0> n. Condition 1 says by definition of limits that for every ε > 0 there is an n0 such that
for every n > n0it holds that |Pn(|X (0)| ≥ α) − 0| < ε. The probability is not negative so we
write:
Pn(|X (0)| ≥ α) < ε.
By Chebyshev’s inequality we have that:
Pn(|X (0)| ≥ kσ ) ≤
1
k2, (3.6)
where α = kσ , which implies that k2= α
2
σ2 and we can write Equation (3.6):
Pn(|X (0)| ≥ α) ≤
σ2 α2 <
ε 2 < ε.
For any fixed n, σ is just a constant so with α as any real number greater than zero, we can always choose α to be large enough for this to hold, which means that this works for all n ≥ 1. Now define the set B = {X : |X (0)| ≤ α}, since we have just proved that Pn(B) <ε2 it follows
directly that : Pn(B) > 1 −ε2 for all n ≥ 1.
If the second condition holds we know from the definition of limits that for everyε > 0 and ˜ε > 0 there exists a ∆ > 0 and an n0∈ N such that when n > n0and 0 < δ ≤ ∆ we have that
Pn(ω(x, δ ) > ˜ε) < ε. We can then then choose our ˜ε and a sequence {δk}, for k ≥ 1 so that:
Pn(ω(X , δ ) > ˜ε) < 2k+1ε . We now define the set Bk= {X : ω(X , δk) < ˜ε} and we can see that
Pn(Bk) <2k+1ε . To calculate the total probability for all k in the sequence we get:
Pn(∪kBk) ≤ ∞
∑
k=1 Pn(Bk), < ∞∑
k=1 ε 2k+1, =ε 2 ∞∑
k=1 1 2k, =ε 2.The total probability of B ∪ (∪kBk) :
Pn(B ∪ (∪kBk)) = Pn(B) + Pn(∪kBk) <
ε 2+
ε 2 < ε,
and we get:
Pn(B ∩ (∩kBk)) = Pn(B ∪ (∪kBk)),
= 1 − Pn(B∪ (∪kBk)),
> 1 − ε.
Define the set A = Pn(B ∩ (∩kBk)) and let K be the closure of A, then we have that Pn(K) ≥
Pn(A) > 1 − ε. We can see from the definition of the sets B and Bkthat B satisfies Condition 1
of Theorem 3.5 and Bk satisfies Condition 2 of Theorem 3.5. This implies that since A is the
intersection of the two sets, A satisfies the two conditions of Theorem 3.5 and is therefore rel-atively compact, which from Definition 3.13 implies that K is compact. Since Pn(K) > 1 − ε
we know from Definition 3.14 that Pnis tight.
Conversely, suppose that Pn is tight. Then by Definition 3.14 for every ε > 0 there exists
a compact set K ∈C and n0∈ N such that Pn(K) > 1 − ε, for every n > n0. Since a compact
set also is relatively compact we know that the set K satisfies Condition 1 and 2 of Theorem 3.5. Condition 1 says that there exists an α > 0 such that K ⊆ {X ∈ C : |X (0)| < α} and we get that:
Pn(X ∈ C : |X (0)| ≥ α) = 1 − Pn(X ∈ C : |X (0)| < α),
≤ 1 − Pn(K) < ε.
Since this holds for every n ∈ N the first condition of this theorem is satisfied. From Condition 2 of Theorem 3.5 we know that for every ˜ε > 0 there exists a ∆ > 0 such that when 0 < δ ≤ ∆ then:
sup
X∈K
ω (X , δ ) < ˜ε .
This obviously implies that:
ω (X , δ ) < ˜ε . for every X ∈ K
So we know that for any 0 < δ ≤ ∆:
K⊆ {X ∈ C : ω(X, δ ) < ˜ε}. We already know that Pn(K) > 1 − ε which implies that
Pn(X ∈ C : ω(X , δ ) ≥ ˜ε) = 1 − Pn(X ∈ C : ω(X , δ ) < ˜ε),
≤ 1 − Pn(K) < ε.
We have shown that for any ε > 0 there exists a ∆ > 0 such that when 0 < δ ≤ ∆ then it holds that Pn(X ∈ C : ω(X , δ ) ≥ ˜ε) < ε for every n, which by definition of limits equals:
lim
δ →0
(lim sup
Definition 3.15. Let K be a set of probability measures defined on (C,C), then we say that K is relatively compact if for every sequence of probability measures {Pn} ⊆ K there exists a
subsequence of {Pn} that converges weakly to some probability measure P on (C,C).
The following definitions will be needed in the next theorem.
Definition 3.16. Let (S, ρ) be a metric space. If M ⊂ S is countable and dense, then S is said to be separable.
From this definition we can conclude that the real numbers are separable. A subset of the real numbers is the rational numbers, that are dense in R, and also countable.
Definition 3.17. A Cauchy sequence is a sequence {an} of real numbers where, for every
ε > 0 there exists a N such that when s, t ≥ N it holds that |as− at| < ε.
Definition 3.18. A metric space (S, ρ) is said to be complete if every Cauchy sequence con-verges to a limit a ∈ S.
Theorem 3.8. Let K be a set of probability measures on the space (S,S). If (S,ρ) is a separ-able and complete metric space, and K is relatively compact, then K is tight.
This Theorem is proven in [11, Theorem 5.2]. To be able to prove the next theorem we will need the following definition:
Definition 3.19. For any partition1T on the interval [0, 1], the projection mapping πT : C → Rk
is defined as:
πT(X ) = (X (t1), ..., X (tk)),
for every X ∈ C.
The following theorem states two conditions that proves if a sequence of random functions converges in distribution. This theorem will be an important part of the proof of Donsker’s Theorem that we will see in the next chapter.
Theorem 3.9. Suppose that X and Xn are random functions on C, defined on a probability
space: (Ω, F, P) and for every partition T ∈ [0, 1]kthe following holds: πT◦ Xn
d
−
→ πT ◦ X, (3.7)
and for any ε > 0
lim
δ →0
(lim sup
n→∞ P(ω(Xn, δ ) ≥ ε)) = 0, (3.8)
then Xn−→ X.d
1By a partition T we mean a set of numbers t
i∈ [0, 1] such that 0 = t0< t1< ... < tk≤ 1, where k ∈ N and
Proof. The vector πT ◦ Xn has the probability distribution P ◦ (πT ◦ Xn)−1 which is
equival-ent to: P ◦ (Xn−1◦ πT−1). By Definition 3.4 the vector πT ◦ X has probability distribution
P◦ (πT ◦ X)−1 = P ◦ (X−1◦ πT−1). We know that composite functions are associative so we
can rewrite this expression as: (P ◦ Xn−1) ◦ πT−1 and (P ◦ X−1) ◦ πT−1.
From the definition of convergence in distribution (Definition 3.6) we know that since the mapping πT◦ Xn
d
−
→ πT ◦ X the probability measure of πT◦ Xnconverges weakly to the
prob-ability measure of πT◦ x. This implies that (P ◦ Xn−1) ◦ πT−1 w
−→ (P ◦ X−1) ◦ πT−1. We will now define the measures µTn and µT:
µTn:= (P ◦ Xn−1) ◦ π0−1, µT := (P ◦ X−1) ◦ π0−1.
Let T = 0 then we have that µ0n−→ µw 0. If we choose any sequence µ nj
0 of µ0n, where the indices
of µ0nj does not have to appear in any special order we have an infinite sequence. Therefore it will always be possible to find a sub sequence µ0nji of µ0nj with indices nj0 < nj1 < ... < njk.
This sequence will of course be a subsequence of µ0n and since µ0n converges weakly to µ0,
µ0nji will converge weakly to µ0.
From this we can see that there exists a subsequence of every probability measure µ0nj ⊆ µn 0
that converges weakly to the probability measure µ0. This implies by Definition 3.15 that µ0n
is relatively compact. Since µ0n is defined on R that is separable and complete, we get from Theorem 3.8 that µ0nis also tight.
Since µ0n is tight there exists a compact set K ⊆ R such that for every ε > 0 and n ∈ N it holds that µ0n(K) > 1 − ε Since the set K is compact it is also closed and bounded by Defini-tion 3.12, so we know that there exists a α > 0 such that K ⊆ {y ∈ R : |y| ≤ α} which implies that µ0n({y ∈ R : |y| ≤ α}) ≥ µ0n(K) for every n ∈ N.
In this proof we assume that f is a function defined on C. From Definition 3.4 we can see that P ◦ Xn−1(| f (0)| > α) means P(Xn ∈ { f : | f (0)| > α}). Recall that the mapping
πT ◦ X = (X(t1), X (t2), ..., X (tk)) for a partition T on [0, 1] So π0◦ Xn= Xn(0) and we can
write: P◦ Xn−1(| f (0)| > α) = P(Xn∈ { f : | f (0)| > α}), = P(|π0◦ Xn| > α), = P ◦ (π0◦ Xn)−1(y ∈ R : |y| > α), = µ0n(y ∈ R : |y| > α), = 1 − µ0n({y ∈ R : |y| ≤ α}), ≤ 1 − µ0n(K) < ε.
Since µ0n(K) > 1 − ε for every ε > 0 and n ∈ N this inequality holds for every n ∈ N. This implies that limn→∞P◦ Xn−1(| f (0)| > α) < ε for any given ε > 0 which show that P ◦ Xn−1
Condition 2 of Theorem 3.7 holds which follows directly from Equation (3.8).
We have now proved that P ◦ Xn−1 is tight. By Condition 1 of this theorem πT ◦ Xn d
−
→ πT ◦ X
which implies that P ◦ Xn−1−→ P ◦ Xw −1. Corollary 2.11 [10] tell us that if these two conditions hold true then Pn
w
−→ P. Then it follows directly form Definition 3.6 that Xn d
− → X.
Chapter 4
Donsker’s Theorem
In the following chapter we will prove Donsker’s Theorem. This theorem states that an arith-metic random walk converges to a Wiener process when we let the size of each time step go to zero. First we will define a function Xtn of a random walk on the time interval [0, 1]. We will also show that there exists a Wiener process on the same interval. Then the proof will be done by the following steps:
i. We will use Theorem 3.9 to prove that Xtn −→ Zd t, where Zt is the stochastic process
defined in Definition 3.3. We have two conditions to prove. The first part of the proof is to show that Xtn− Xn
s converges in distribution to N(0,
√ t− s).
ii. Now step (i) will be used to show Condition 1 of Theorem 3.9: that the mapping πT◦ Xtn
converge in distribution to πT ◦ Zt.
iii. In step (iii) we will prove that the sequence of probability measures of Xtnis tight. We know from Theorem 3.7 that this implies that the second condition of Theorem 3.9 holds.
iv. When we have proved that the two conditions hold true, we have shown that Xtn−→ Zd t.
The only thing left is to draw the conclusion that Zt is a Wiener process and our proof
will be completed.
The proof follows the steps done in Theorem 2.31 [10] but with a change in how we define the random function Xtn. In that article there is a continuity correction term of Xtnwhich we have omitted in this thesis. The reason is that in step (ii) when we want to show that the mappings πT◦ Xnconverge in distribution to πT◦ Z we need Xtni− X
n
ti−1 to be independent for every i ∈ N.
With the continuity correction term these will not be independent and therefore the steps that follows will not be possible to calculate with complete accuracy.
The extra term will be important if we intend to approximate the result numerically. But in this thesis the intention is to show the convergence of a random walk when we let the time step be infinitely small which mean that this continuity correction term will be negligible. We can therefore omit the extra term and the proof will still work.
In the following definition we will define an arithmetic random walk:
Definition 4.1. Let Ykbe i.i.d random variables and n ∈ N such that:
Sn=
n
∑
k=1
Yk,
then the sequence {Sn} describes an arithmetic random walk.
In this thesis we want to show that an arithmetic random walk converges to a Wiener process, we will therefore define the random walk as in the following definition:
Definition 4.2. Let {Yn} be a sequence of identically and independent distributed random
variable defined on the probability space (Ω, F, P), with mean 0 and standard deviation σ , where σ ∈ (0, ∞). We now define:
Xtn:= 1
σ√nSbntc(ω),
for every t ∈ [0, 1] and n > 0 where S0= 0 and Sn= Y1+Y2+ ... +Ynfor every n ∈ N.
In the following theorem we can see that for the special case where t = 1 we have that Xtn converges to a normal distribution.
Theorem 4.1. Let Snand{Yn} be defined as in Definition 4.2. Then: Sn
σ√n
d
−
→ N(0, 1).
The next two theorems will be used to complete the proof of Donsker’s Theorem: Theorem 4.2. There exists a Wiener measure on (C,C).
The proof of this theorem is omitted of this report and can be found in the proof of Theorem 2.42 in [10].
Theorem 4.3. There exists a Wiener process on [0, 1].
Proof. From Theorem 4.2 we know that there exists a Wiener measure on (C,C), it then follows by Theorem 3.3 that the coordinate-variable random function Z is a Wiener process.
Theorem 4.4. (Donsker’s theorem) Let {Yn} and {Xn} be defined as in Definition 4.2 then
{Xn} converges in distribution to a Wiener process.
Proof. Let Zt be the coordinate variable process on (C,C,W). We want to prove that Xtnand
Zt satisfies the two condition of Theorem 3.9. We start by Step (i), to prove that Xtn− Xn s
converges in distribution to N(0,√t− s). Let 0 ≤ s < t ≤ 1 and n ∈ N. We have the function: Xnt(ω) =Sbntc(ω)
σ √
We know that the defined function depends on (ω), to improve readability we will omit the term (ω). Xtn− Xsn= 1 σ√nSbntc− 1 σ√nSbnsc ,
we can rewrite this as:
Xtn− Xsn= Sbntc− Sbnsc σ√n ,
=Ybnsc+1+Ybnsc+2+ ... +Ybntc σ√n .
We know that Yiare i.i.d and have the same distribution regardless of time, which implies that
Yi= Yi+∆t and we can rewrite the above expression as:
Xtn− Xsn=Y1+Y2+ ... +Ybntc−bnsc σ √ n , = Sbntc−bnsc σ √ n , = Sbntc−bnsc σpbntc − bnsc pbntc − bnsc √ n .
We will now evaluate the limit of these two factors separately. Starting with √
bntc−bnsc √
n . The
limit of bntcn is t as n → ∞, since for every ε > 0, there exists a n1∈ N such that when n > n1:
bntc n − t = bntc − nt n ≤1 n < 1 n1 < ε.
Choose n1> 1ε and the inequality holds true. Similarly we can show that limn→∞bnscn = s.
Then by the sum rule of limits we have that:
lim n→∞ bntc − bnsc n = limn→∞ bntc n − limn→∞ bnsc n = t − s.
By using the power rule of limits we can write:
lim n→∞ r bntc − bnsc n = r lim n→∞ bntc − bnsc n = √ t− s.
This shows that √ bntc−bnsc √ n p − →√t− s.
By Theorem 4.1 we have that Sbntc−bnsc
σ
√
bntc−bnsc converge in distribution to N(0, 1), where N has
standard normal distribution. Now by Theorem 3.4 we can see that since √ bntc−bnsc √ n p − →√t− s and Sbntc−bnsc σ √ bntc−bnsc d − → N(0, 1), we get that: √ bntc−bnsc √ n Sbntc−bnsc σ √ bntc−bnsc d − → N(0,√t− s) which implies that Xtn− Xn s d − → N(0,√t− s).
Now we continue with Step (ii). We will now define Kn= (Xtn
1, X n t2− X n t1, ..., X n tk− X n tk−1) for a
partition T on the interval [0, 1] where t0= 0 and Kn∈ (A1, ..., Ak), where A1, A2, ..., Ak ∈B.
This means that each element, Knj in the vector Kn is an element of the set Aj. We can see
from the definition of Xnthat
Xtn i − X n ti−1 = 1 σ √ nS bntic − 1 σ √ nS bnti−1c , = 1 σ√nYbntic.
Each element of Kn is a linear combination of the random variables Yn. Since each Yioccurs
only in one of the elements of Kn, it follows that the elements of Kn are independent and for every A1, A2, ..., Ak∈B we get: P(Kn∈ (A1, A2, ..., Ak)) = P k \ i=1 ω ∈ Ω : Xtni(ω) − Xtni−1(ω) ∈ Ai ! . (4.1)
Since Xtni(ω) − Xtni−1(ω) is independent for every k Equation 4.1 becomes:
P(Kn∈ (A1, A2, ..., Ak)) = k
∏
i=1 PXtni(ω) − Xtni−1(ω) ∈ Ai ,which from Definition 3.4 is equivalent to:
P(Kn∈ (A1, A2, ..., Ak)) = k
∏
i=1 P ◦ (Xtni(ω) − Xtni−1(ω))−1(Ai), =P ◦ (Xtn1(ω))−1× ... × P ◦ (Xtn k(ω) − X n tk−1(ω)) −1(A 1× ... × Ak),where the product measure P ◦ (Xtn1)−1× ... × P ◦ (Xn tk− X
n tk−1)
−1 is defined on the probability
space (R, B, P ◦ ((Xtn1) −1× ... × P ◦ (Xn tk− X n tk−1) −1)). We know that Xn ti − X n ti−1 converges in
distribution to N(0,√ti− ti−1), which by Definition 3.6 gives that:
Pn◦ (Xtni− X
n ti−1)
−1 w−→ P ◦ (N(0,√t
i− ti−1))−1.
This means that each element of the sequence {P ◦ (Xtn1)−1× ... × P ◦ (Xn tk− X
n tk−1)
−1} converges
weakly to P ◦ (N(0,√ti− ti−1))−1. Since Rk is separable, we can use Theorem 1.13 in [10]
which implies:
We can see that the term on the left hand side is the probability measure of Kn which con-verges weakly to the probability measure of N(0,√t1) × ... × N(0,√tk− tk−1). We again use Definition 3.6 and get that:
Kn d−→ (N(0,√t1), N(0,
√
t2− t1), ..., N(0,ptk− tk−1)),
= (Zt1, Zt2− Zt1, ..., Ztk− Ztk−1).
Now let the mapping g : Rk→ Rkbe the continuous mapping defined as
g(x1, x2, ..., xk) = g(x1, x1+ x2, ..., x1+ x2+ ... + xk) then: (Xtn 1, X n t2, ..., X n tk) = g(K n)−→ g(Zd t1, Zt2− Zt1, ..., Ztk− Ztk−1) = (Zt1, Zt2, ...Ztk), ⇒ (Xtn1, Xtn2, ..., Xtnk)−→ (Zd t1, Zt2, ...Ztk), ⇔ πT◦ Xn d−→ πT◦ Z.
Since this holds for any k ∈ N and any partition T on the interval [0, 1] we can see that Con-dition 1 of Theorem 3.9 is satisfied.
Step (iii) is to show that the second condition of Theorem 3.9 holds which by Theorem 3.7 holds if the sequence of probability measure of Xnis tight. Let µn denote the distribution of
the variable Sn
σ√n. For every n ∈ N we have that:
E[Sn] = (E[Y1] + ... + E[Yn]) = 0.
By Theorem 4.1: Sn σ√n d − → N(0, 1), which by Definition 3.6 implies:
µn w
−→ µN,
where µN is the distribution of a standard normal variable. We define the set Rα = {x ∈ R :
|x| ≥ α}, for every α > 0. Then we can see that the boundary ∂ (Rα) = {x ∈ R : |x| = α}, and
since the boundary of Rα only consists of two points we get:
µN(∂ (Rα)) = Z −α −α e−u22 √ 2πdu+ Z α α e−u22 √ 2πdu= 0,
which by Definition 3.8 shows that Rα is a µN-continuity set. Then we can use Theorem 1.21
of [10]:
lim
From Corollary A.52 in [10] we know that for every α > 0:
µN(Rα) = P(|N| ≥ α) <
3 α4.
By the definition of limits we know that for every ε > 0, there exist a k1such that when k ≥ k1:
|µk(Rα) − µN(Rα)| < ε.
Since this holds for any ε > 0 and since µN(Rα) < α34, we can always find an ε small enough
such that: µk(Rα) < 3 α4. This gives: µk(Rα) = P Sk σ √ k ≥ α = P(|Sk| ≥ σ√kα) < 3 α4. (4.2)
Choose n ∈ N such that k1≤ n and 1 ≤ k ≤ n. If k1≤ k ≤ n we know from Equation (4.2) that:
P(|Sk| ≥ σ √ nα) ≤ P(|Sk| ≥ σ √ kα) < 3 α4.
If k < k1we use Chebyshev’s inequality. The variance of each term in Skis σ , the variance of
Sk= kσ and we get: P(|Sk| ≥ σ √ nα) ≤ kσ 2 σ2nα2 = k nα2 < k1 nα2.
We have concluded that:
max i≤n P(|Si| ≥ σ √ nα) < max 3 α4, k1 nα2 , ⇒ α2max i≤n P(|Si| ≥ σ √ nα) < max 3 α2, k1 n . Since lim sup n→∞ max 3 α2, k1 n = 3 α2, we have that lim sup n→∞ α2max i≤n P(|Si| ≥ σ √ nα) < 3 α2.
If we let α go to infinity we obtain: lim α →∞ lim sup n→∞ α2max i≤n P(|Si| ≥ σ √ nα) = 0,
which by Lemma 2.40 in [10] implies that the set of distributions µXn is tight. If the sequence
is tight the conditions of Theorem 3.7 is satisfied and it follows that the second condition of Theorem 3.9 is true.
This shows that Xtn−→ Zd t and by Theorem 4.2 there exists a Wiener measure on (C,C) which
by Theorem 3.3 implies that Zt is a Wiener process. This completes the final step, Step (iv) of
the proof and shows that Xtn−→ Wd t.
4.1
Convergence of a geometric random walk
We have now proved that the arithmetic random walk, defined in Definition 4.2, converges to the Wiener process, which is a continuous time stochastic process. What we want to show in this thesis is that the random walk described in Chapter 2, used for option pricing in the discrete time case, converges to a geometric Brownian motion, which is found in Equation (3.4). The geometric Brownian motion can be written in the following form:
St= S0e(µ−
σ2
2 )t+σWt. (4.3)
In Chapter two we described a geometric random walk by:
St= s0ukdt−k,
where u and d describe the up and down movement respectively. To allow the rate of return to change in each time step we can rewrite this equation as:
StN= S0 N
∏
k=1 1 +Rk N ,In this model the time is running between 0 and 1, so h = N1. We let Rk = r0+
√ NYk = (µ−σ2 2 )Nt bNtc + √
NYk and Ykis the i.i.d random variables representing the up or down movement
of every time step. r0is equal to µ −σ
2
2 but when N is large we can write r0= (µ−σ2
2 )Nt
bNtc . The
above expression is then equal to:
Stbntc= S0 bntc
∏
k=1 1 +(µ − σ2 2 )t bntc + √ nYk n ! .Taking the limit of this expression: St= S0lim n→∞ bntc
∏
k=1 1 +(µ − σ2 2 )t bntc + √ nYk n ! , = S0lim n→∞ bntc∏
k=1 1 +(µ − σ2 2 )t bntc ! 1 + √ nYk n .We obtain the extra term (µ−σ
2 2 )t(
√ nYk)
nbntc from this rewriting, but we know that Yk is finite for
every k, and in the limit the sum of each term that contains this expression will go to zero faster than the other terms. With the same argument we can add terms that will be arbitrary small as n increases: St= S0lim n→∞ bntc
∏
k=1 1 +(µ − σ2 2 )t bntc ! 1 +√Yk n+ Yk2 (√n)22!+ Yk3 (√n)33!+ ... ! .The right factor is the Taylor expansion of the exponential function e√Ykn, so we have:
St= S0lim n→∞ bntc
∏
k=1 1 +(µ − σ2 2 )t bntc ! e Yk √ n.Since the left factor does not depend on k we can write:
St= S0lim n→∞ 1 + (µ −σ2 2 )t bntc !bntc · bntc
∏
k=1 e√Ykn.We can use the standard limit that a function on the form (1 +Ma)M → ea as M → ∞, from
which we get: St= S0e(µ− σ2 2 )t lim n→∞
∏
k=1 e√Ykn, = S0e(µ− σ2 2 )t lim n→∞e 1 √ n∑ bntc k=1Yk .Since the exponential function is a continuous function we can move the limit inside the func-tion: St= S0e(µ− σ2 2 )telimn→∞ 1 √ n(Y1+Y2+....+Ybntc),
From Definition 4.2 we have that Xtn= 1
σ√n(Y1+ Y2+ ... + Ybntc). We use this to rewrite the
previous expression: St= S0e(µ− σ2 2 )te σ √ nlimn→∞X n t .
We can now use the result of Donsker’s Theorem (Theorem 4.4) from which we know that Xtn converges to the Wiener process Wt:
St= S0e(µ− σ2 2 )te σ √ nWt, = S0e(µ− σ2 2 )t+√σnWt,
which is exactly the solution of a geometric Brownian motion given in Equation (4.3). We have now proved the following theorem:
Theorem 4.5. Let {Yn} and {Xn} be defined as in Definition 4.2 and Stnis defined as:
Snt = S0 N
∏
k=1 (1 +Rk N), where Rk = µ − σ2 2 Nt bNtc + √NYk. Then Snt converges in distribution to a geometric Brownian motion.
Chapter 5
Conclusion
In Chapter two we described a geometric random walk and how it can be used for option pricing in discrete time. In the next two chapters we described a geometric Brownian motion that is used to describe stock movements in continuous time. We then proved that an arith-metic random walk converges to a Wiener process and extended this result and proved that a geometric random walk converges to a geometric Brownian motion. From this result we can conclude that a geometric Brownian motion can be approximated by a discrete time model that in many cases is easier to solve numerically. The same is true for any Wiener process, if we want to calculate it numerically we have proven in this thesis that we can approximate the result by an arithmetic random walk.
Bibliography
[1] Robert Brown. XXVII. a brief account of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants and on the general existence of active molecules in organic and inorganic bodies. The Philo-sophical Magazine, 4(21):161–173, sep 1828.
[2] Norbert Wiener. The average of an analytic functional and the brownian movement. Proceedings of the National Academy of Sciences, 7(10):294–298, oct 1921.
[3] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of political economy, 81(3):637–654, 1973.
[4] John C Cox, Stephen A Ross, and Mark Rubinstein. Option pricing: A simplified ap-proach. Journal of financial Economics, 7(3):229–263, 1979.
[5] Hans Fischer. History of the Central Limit Theorem. Springer-Verlag GmbH, 2010.
[6] Monroe David Donsker. An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc., 6:12, 1951.
[7] Tomas Björk. Arbitrage Theory in Continuous Time (Oxford Finance Series). Oxford University Press, 2009.
[8] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill Education Ltd, 1976.
[9] Masaaki Kijima. Stochastic Processes with Applications to Finance. Taylor & Francis Ltd, 2013.
[10] Pierre Yves Gaudreau Lamarre. Donsker’s theorem. preprint on webpage at http://aix1.uottawa.ca/ rbalan/PierreYves-rapport.pdf, viewed 12 may 2020, 2012.
