CiteSeerX — Continuous time limit of the Binomial Model

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Continuous time limit of the Binomial Model

Timothy Kevin Kuria Kamanu May 26, 2004

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1

ACKNOWLEDGEMENT

I would like to acknowledge my sponsors, The African Institute for Mathematical Sciences, and it donors for giving me an opportunity to undertake the Diploma programme, my supervisor, Dr. Diane Wilcox (UCT), for her advice and kindness with reading materials, my fellow col- leagues; special regards to Ikleel, academic and non-academic staff for creating and enabling environment for study.

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Abstract

The limit of the Cox, Ross and Rubinstein formula as the length of time-steps goes to zero is the Black- Scholes formula. In this paper we use the de Moivre Laplace central limit theorem to demonstrate the convergence of option prices in the binomial model to the price given by the latter.

We also show the convergence of option prices to the geometric Weiner process which is a pre-assumption of the Black-Scholes model by relating the parameters of the binomial model (up and down return) with the parameters of the log-normal distribution of the stock prices (drift and volatility).

In addition, we show that the option price given by the Black-Scholes formula satisfies the partial differential equation of Black-Scholes using the continuous time limit of the binomial model.

In chapter four, we consider an alternative model of stock prices; the Merton model while we prove the Poisson asymptotics.

Prior to undertaking the tasks, we review the Gaussian distribution, Poisson distribution, Weiner process, Poisson process, central limit theorem (de Moivre-Laplace)- with proof, The binomial model, option pricing in the binomial model, Cox-Ross-Rubinstein formula, partial differential equations of second order, classification and the notion of well posed problems.

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Introduction

1.1 Background

In 1973, Fisher Black and Myron Scholes in their landmark paper,‘ The pricing of options & corporate liabilities ’ presented the first satisfactory equilibrium pricing model. Later in the same year, Robert Merton extended their model by incorporating other features in his paper ‘ Theory of rational pricing ’.

The success of the Black-Scholes model was because it’s pricing formulas do not require knowledge of the investors attitudes towards risks and it uses observable variables.

The fundamental assumption in their model was that stock prices follow a continuous-time diffusion process whose sample paths is always continuous with probability 1 so as to avoid risk-free profits. This latter requirement is due to Modigliani and Miller (1958) in ’ The cost of capital, corporation finance and the theory of investment ’ who showed that, in equilibrium, packages of financial claims which are, in essence similar must command the same price.

However in real markets, no empirical time-series has a continuous sample path, therefore the solution based on Black-Scholes model is compromised. Merton and Samuelson in their paper ’ Fallacy of the log-normal approximation of optimal portfolio decision-making over many periods ’ showed that contin- uous trading is solution is a valid asymptotic approximation to that of a discrete-trading model, provided that the price dynamics have a continuous sample path.

As a consequence, the ’risk-less’ portfolio hypothesis in discrete time implies that, the portfolio so considered has an element of risk, but its magnitude is a bounded continuous function of the interval length.

As the length of an interval tends to zero, so does the associated risk.

If the stochastic process implied by the price dynamics cannot be represented as a continuous sample

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1.2 Fundamental concepts 4

path, the Black-Scholes solution is invalid. However, Merton [9] showed that the validity of model depends on whether the price changes satisfy a localized Markov property. That is, in a short time interval, the stock price can only change by a small amount.

In this paper, we demonstrate convergence to the Black-Scholes model. We use the William Sharpes’

model. This approach uses elementary mathematics to derive the Black and Scholes formula and presents the underlying economic principles, while we analyse and review the continuous time limit of the binomial model. In the last chapter, we consider convergence of jump stochastic processes; The Merton model.

1.2 Fundamental concepts

1.2.1 Financial Instruments

An Asset (security) may be defined as an item of economic value owned by an individual or corporation.

If an asset can be traded independently, it is referred to as a primary security, otherwise it is a derivative security. Derivative securities are assets that derive their value from an underlying security to which they relate. They cannot exist in their own right and therefore are contingent to trading and value of the underlying. In particular, Let Stdenote the price of the underlying security say, a stock then,the value of the derivative D can be expressed as D = Φ(St)where Φ is a known function. Derivative are also re- ferred to as contingent claims and are characterized by legal contracts conferring certain financial rights or obligations. Examples include; options and forward contracts.

In general, securities may be categorized as either risky or risk-less. A risky security (stocks or shares,foreign exchange) is one whose future price is unknown today, whereas a risk-less security (bank deposits and government bonds) is one whose future value is deterministic. A combination of both is termed as a portfolio.

1.2.2 Underlying Assumptions

To undertake the task on hand, we proceed to introduce basic terms, fundamental principles, concepts and assumptions. The imposition of assumptions will cater for the limitations and simplifications of our mathematical model, and make it tractable to the complexity of the real-world.

• Randomness and Positivity of prices

The price of the securities is strictly positive at all time. t ≥ 0. At the present time t = 0, all prices are

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1.2 Fundamental concepts 5

known to all investors. The future price of a risky security is unknown (Stochastic); It can be modelled as a positive random variable, S(t) such that for a probability space¹Ωwhich contains all feasible events ω, S(t, ω) : Ω−→ (0, ∞). The value of a portfolio say, Vtat any time t, containing φ risky assets (stocks) and risk-less bonds or cash say ψ can be expressed as:

V_t = φ S_t+ ψ β_t (1.1)

where βt denotes the value of a bond at time t. At time t = 0, β0 = 1

• Divisibility and Short-selling

From (1.1) it is mathematically convenient and practical to note that φ and ψ can take any values;

φ, ψ ∈ R. This implies that, all investor is unrestricted to buy or sell any number of different shares or bonds at any time: φk∈ R or ψk∈ R, k = 1, 2...n

• Rationality and Solvency

We assume that all investors intend to maximise their returns while still remaining risk averse. In particular the portfolio value Vt ≥ 0 for all t ≥ 0. Such a portfolio is referred to as an admissible portfolio.

Again, there is no limit on the price and trading (liquidity). This not practical since in real world markets the number of possible different prices is finite and bounded.

• Complete Market and Efficiency

The share prices reflect all the information currently available and the market is efficient in adjusting this prices to instantaneous information so as to remain or tend to equilibrium. This follows from it being deemed to be liquid at all times. In this regard, Investors’ expectations about future returns are based on the information currently available and the information is uniform to all. For simplicity we assume that there are no transaction costs.

• No-Arbitrage principle

This is the most fundamental assumption about the market on which the main tools in Financial Mathe- matics rely. It is to the effect that, no investor can lock up profit without taking any risk, or without an initial endowment. That is, the market does not allow risk free profits with no initial investment. (“ No free Lunch” )

In practice Arbitrage opportunities are rare, short-lived and therefore immaterial with respect to the volume of transactions. In particular, there is no admissible strategy such that V0 < 0 at t = 0 and P (V_t ≥ 0) > 0 for all t ≥ 0.

This principle allows us to determine a self financing, replicating portfolio. A self financing portfolio is

1This notation is introduced in the following section

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1.3 Mathematical Tools 6

one whose value changes only as a result of a change in value of the underlying asset and not because of change in the portfolio structure (assets held) whereas a replicating portfolio is one whose value tracks the target value exactly overtime, It is constructed to have the same terminal value as the derivative.

1.2.3 Option pricing

The Option pricing problem is based on the fact fore-mentioned that, the future value of assets cannot be determined with certainty. In fact, even from past history as a financial time series, one cannot predict on the future movement in prices.

A financial option (plain vanilla) is a contract which gives the holder the right but not the obligation to buy or sell the underlying asset for a strike or exercise price, K (determined say now, time t = 0) at a future date, T called the expiry date. If the holder has the right to buy the asset only at the expiry date, the option is known as a European call option. If he can do so at any time (0 ≤ t ≤ T ) the option is known as an American call option.

Analogously, if the holder has the right to sell the asset, the option is known as European put and American put option respectively for cases mentioned above.

Since the holder has the right and not the obligation to buy or sell the asset he will only exercise it if it is profitable to him. In the case of a European call; he will exercise the option if the market price Stis greater than K, while in the case of a European put; if the market price is below the strike price. The difference between the two prices at the time of exercise gives the payoff of the option.

Since the market price of the asset is generally unbounded the payoff of the call options is also an unbounded random variable. Suppose Ct and Pt denote the payoff of a European call and put option respectively, then:

C_t = Max(St− K, 0) = (St− K)⁺ (1.2)

P_t = Max(K − S^t, 0) = (K− St)⁺ (1.3)

1.3 Mathematical Tools

As above, the future stock price St for all t > 0 is unknown and can therefore be modelled as a random variable. In this section we derive the required mathematical tools necessary to aid us model it’s dynamics. In particular, we will consider a (Ω, F) where Ω is the sample space (non-empty set) of the random variable, containing all scenarios (events) and F is a σ-field or a collection of subsets of Ω and where

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A∈ F is the event.

Definition 1.3.1 - σ-field

A σ-field F is a family of subsets of Ω such that:

• The empty set is contained in it; ∅ ∈ F

• For all A ∈ F, then A^c∈ F

• F is closed under the operation of countable unions; If A1, A₂...∈ F thenS

n>1A_n∈ F Definition 1.3.2 - Measure

A let Ω be a non-empty set, Let F be a σ-algebra, Then (Ω, F) is measurable space if there exists a positive measure say ρ on (Ω, F), defined as a function such that,

ρ : F7−→ [0, ∞) (1.4)

A7−→ ρ(A) and (1.5)

• ρ(∅) = 0

• For any sequence An∈ F where n ∈ N then ρ [∪_n≥1A_n] =P

n≥1ρ (A_n)The set (event) A ∈ F is called a measurable set and ρ(A) its measure. It is said to occur almost surely whenever ρ[A] = 1. In essence, we have define the domain of the random variable for which to work, this is formally known as a σ-algebra.

Definition 1.3.3 - Probability measure

ρis a probability measure denoted as P if ρ = P : F 7−→ [0, 1] where P satisfies the following axioms due to Kolmogrov,

• For all A ∈ F then 0 ≤ P[A] ≤ 1 notably P[∅] = 0

• P[Ω] = 1

• If A1, A₂...are pairwise disjoint sets in F, that is Ai∩ Aj =∅ for all i 6= j then:

P[A₁∪ A2∪ ...] = P[A1] + P[A₂] + ...

• If An ∈ F for all n ∈ N and A1 ⊆ A2... then: P[An] ↑ P [∪n>1A_n]almost surely as n → ∞. The triple (Ω, F, P) is called a probability space

Definition 1.3.4 - Measurable function

Let Ω, F) and (Ω, G) be two measurable spaces, A function f : F → G is called measurable if for any measurable set A ∈ F the set f⁻¹(A) = {ω ∈ Ω, f(ω) ∈ A} is a measurable subset of Ω. For our purpose, since the price St can take any values in R⁺ ≡ [0, ∞) we restrict this to the σ-algebra generated by all open subsets on the real line R. That is, F = B(R) where B(R) is the smallest σ-field

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on R containing all open intervals in R. We will consider measurable functions f : Ω → R which are of the form. f = Pⁿ_j=1c_j1A_j n ∈ N where Aj are measurable functions, cj ∈ R and 1^Aj is the Dirac measure δxassociated to a point x ∈ A such that,

δ_x(A) =







1 if x ∈ A

0 if x /∈ A (1.6)

Definition 1.3.5 - Random Variable

Given a probability triple (Ω, F, P), A random variable is a real valued function on X : Ω → R which is F-measurable. It can either be discrete (taking on countable distinct values) and therefore

{ω ∈ Ω : X(ω) = x} ∈ F or continuous {ω ∈ Ω : X(ω) ≤ x} ∈ F (1.7) The latter defines the probability distribution of X denoted F (x) = P[X ≤ x] = 1

ω∈ Ω is called a scenario of randomness and X(ω) represents an outcome of the random variable if the scenario ω happens.

Definition 1.3.6 - Conditional Expectation

If X is an F-measurable random variable with E[|X|] < ∞, and G ⊆ F is a σ-field. The conditional expectation of X given G is the G-measurable random variable with the property that for any A ∈ G

E[[X/G]; A] = Z

A

E[X/G] dP = Z

A

E[X]dP (1.8)

Definition 1.3.7- Characteristic Function

The characteristic function of a random variable X is the function φX : R→ C defined by for all t ∈ R such that

φ_X(t) = E[e^itX] = Z

R

e^itXdF (x) where i =√

−1 (1.9)

Two random variables with the same characteristic functions are identically distributed. Its properties can be analysed as:-

If X and Y are independent random variables then

φ_X+Y(t) = E[e^itX] E[e^itY] = φ_X(t) φ_Y(t) (1.10) If a, b ∈ R and Y = aX + b then,

φ_Y(t) = E[eît(aX+b)] = E[eîtb] E[eî(at)X] = eîtbφ_X(at) (1.11)

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Moreover, If E |X^k| < ∞ then φX has k continuous derivatives at t = 0. In particular, for any random variable X and k ∈ N

If φ^(k)_X (0)exists then







E|X^k| < ∞ if k is even

E|X^k−1| < ∞if k is odd (1.12) If E |X^k| < ∞ then

φ_X(t) =

k

X

j=0

E[X^j]

j! (it)^j+ o(t^k) (1.13)

and so φ^(k)_X (0) = i^kE[X^k].

Definition 1.3.8 - Almost surely convergence

A sequence {Xn}, n ∈ N is of random variables on (Ω, F, P) is said to converge almost surely to a random variable X if

P h

n→∞lim X_n= Xi

= 1 (1.14)

It is commonly known as pointwise convergence and requires that for each ω ∈ Ω the sequence {Xn(ω)}_n≥1 converge to X(ω). The variables {Xn}n≥1have to be defined in the same probability space (Ω, F, P) Definition 1.3.9- Convergence in probability

A sequence {Xn}n≥1, n ∈ N is of random variables on (Ω, F, P) is said to converge in probability to a random variable X if for each > 0

n→∞lim P[|Xn− X|| > ] = 0 (1.15)

Almost surely convergence implies convergence in probability but the two are not equivalent. The latter only puts a condition on the probability of events when n → ∞. Convergence in probability also requires that the random variables are defined in the same probability space. It is denoted Xn

P

n→∞−→ X.

Definition 1.3.10 - Convergence in distribution

A sequence {Xn}n≥1, n ∈ N is of random variables taking values in F is said to converge in distribution to a random variable X if for every bounded continuous function f : F → R (Ω, F, P) is said to converge almost surely to a random variable X if for each > 0

E[f (X_n)] ^−→

n→ ∞E[f (X)] (1.16)

It is also known as Weak Convergence of measures {ρn}n≥1on F.If ρn→ ρ then Xn

D

−→X

Therefore, if {Xⁿ}n≥1 converges in distribution to X, then for any continuous function f, f(Xn)converges in distribution to f(X).

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For our purpose, this notions will be applicable in studying the numerical approximations to the Bino- mial and Poisson distribution in a continuous time framework.

{Xn}n≥1convergences in distribution to X if and only if, for every t ∈ R, their characteristic functions converge. That is, if φXn(t)−→ φX(t).

Definition 1.3.11 - Filtration

A filtration or information flow on a time interval say [0, T ] denoted {F}_{t∈[0,T ]} on a probability space (Ω, F, P)is an increasing sequence of σ-fields containing information on the evolution of the price process up to time T such that for all 0 ≤ s ≤ t then Fs ⊆ Ft ⊆ F

A probability space (Ω, F, P) equipped with a filtration is called a filtered probability space and is de- noted (Ω, F, {F}_{t∈[0,T ]}, P)

Definition 1.3.12 - Stochastic Process

Given a filtered probability space (Ω, F, {F}_{t∈[0,T ]}, P), where {F}_{t∈[0,T ]} is the natural filtration, A stochastic process is a family of (P, {F}t≥0)-adapted real valued functions indexed by time, X_{t∈[0,T ]} on Ω. {Xt}_{t∈[0,T ]}is adapted to {F}_{t∈[0,T ]} if X_{t∈[0,T ]}is Ft-measurable. For each realisation of randomness ω, the trajectory X(ω) : t → X^t(ω) defines a function of time and is called a sample path. More formally, A stochastic process is a function

X : [0, T ]× Ω 7−→ F (1.17)

It therefore follows that stochastic processes are random functions taking values in function spaces. For this reason, we have to clarify that the stochastic processes considered in chapters (2) to (3) are all real valued functions defined on the space of continuous functions C([0, T ], R), with its usual topology kfk = sup_{t∈[0,T ]}kf(t)k. To accommodate the Poisson process in chapter (4) and allow for discontinuous functions, we adopt the following definition.

Definition 1.3.13 - Cadlag function

A function f : [0, T ] → R is said to be Cadlag if it is right continuous with left limits for each t ∈ [0, T ] that is for all s, t ∈ [0, T ] then

f (t−) = lim

s→t ,s<tf (s) and f(t+) = lim

s→t ,s>tf (s) exist and f(t) = f(t+) (1.18) Definition 1.3.14 - Martingale

Given a filtered probability space (Ω, F, {F}t≥0, P). The sequence of random variables {X^t}t≥0 is a Martingale with respect to P and the filtration {F}t≥0if for all 0 ≤ s ≤ t:

• E[|Xt|] < ∞ for all t

• E^P[X_t/X_s] = X_s

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Otherwise, {Xt}_t≥0is a P-super-martingale if EP[X_t/X_s] ≤ Xs

{Xt}t≥0is a P-sub-martingale if EP[X_t/X_s]≥ Xs

Martingale application in the setting of the binomial model can be observed if a change in probability measure (P → Q) is effected such that if { ¯S_t}t≥0is the discounted stock price: ¯S_t = e^rt (0≤ t ≤ T ) then:

E_Q[ ¯S_t+1/ ¯S_t] = ¯S_t

This probability measure Q is called an ‘ Equivalent Martingale Measure (EMM) ’. Any other probability measure, say P based on the preferences of an investor(s) in a market is called ‘ market measure ’. The equivalent martingale measure is a tool which facilitates pricing and hedging in the binomial model.

Definition 1.3.15 - Markov process

A stochastic process {Xt}t≥0with its natural filtration {F}t≥0is a Markov process if for all 0 < s < t:

P(X_t+s∈ A/Ft) = P(X_t+s∈ A/Xt) for all A ∈ F (1.19)

1.3.1 Gaussian Distribution

This distribution is also known as normal distribution, and is characterised by two parameters, mean µand variance σ². It is denoted as N(µ, σ²). The density function of a normally distributed random variable, sayX, with the above parameters is described by a ‘ bell-shaped ’ curve and is given by:

f (x) = 1

√2πσ e⁻^(x−µ)2^2σ2 (−∞ < x < ∞) (1.20)

It is symmetrical about the mean, and the mean, mode and median are all equal. The domain bounded by its distribution function denoted F (x) is equal to 1 and can be interpreted as probability. A standard normal distribution function has mean 0 and variance 1 and hence,

F (x) = P (X ≤ x) = 1

√2π Z _+∞

−∞

e⁻¹²^x²dx (1.21)

The characteristic function of standard normal random variable X is given by;

φ_X(t) = e⁻¹²^t² (1.22)

Approximately two thirds of the area under the curve lies within one standard deviation about the mean.

Because of its symmetry property N(−x) = 1 − N(x). This distribution occurs in many ways, for example, it can be obtained as a continuous time limit of the binomial distribution as n → ∞.

The formal concept to this effect is presented by the central limit theorem that; The sum of a large number of independent (or at-least not too dependent) identically distributed random variables is approximately normally distributed.

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1.3.2 Log-normal distribution

A random variable X has a log-normal distribution with parameters µ and σ if Z = log(X) has a normal distribution. In particular for all x > 0,

FX(x) = P [X ≤ x] = P [log(X) ≤ log(x)] = Φ (σZ + µ ≤ log(x)) (1.23)

= P

Z ≤ log(x)− µ σ

= Φ log(x) − µ σ

(1.24) It’s density function is given by:

f (x) = d

dxF_X(x) = Φ

log(x)−µ σ

σx = 1

√2πσ² e⁻(log(x)−µ)2

2σ2 (1.25)

In relation to finance when n is large, the random variable Y = log(S_t/S)− µt

σ√

n (1.26)

is normally distributed with mean µt and variance σ²t. This result is clarified in the next section on the central limit theorem (CLT).

1.3.3 Poisson Distribution

An integer valued random variable N is said to follow a Poisson distribution with parameter λ if for all n∈ N, P [N = n] = e^{−λ λ}_n!ⁿ. It is obtained as an approximation to the binomial distribution Ψ(a; n, p) when, the probability of success p of the constituent Bernoulli trials is small and n is large but the product λ = npis moderate. This is an illustration of the concept of convergence in probability. The parameter λ > 0is a physical constant which determines the density of points on the time-axis.

Let Pn(t)denote the probability that exactly n changes occur in a time interval of length t, then nPn→ λ for n subintervals in a unit interval. Therefore for a fixed time t, then nPn→ λt and:

Pn(λt) = e^−λt(λt)ⁿ

n! (1.27)

To accommodate the distribution for our purpose, a sequence of random events is represented as points on the time axis.

1.3.4 Wiener process

It is a basic building block for modelling in continuous time and is also called Brownian motion. It is a real valued stochastic process {Wt}t≥0for t ∈ [0, ∞) and real constant σ such that, • W0 = 0almost

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surely.

• Wtis continuous for t ≥ 0

• For each n ≥ 1 and any times 0 < t1 < t2 < ... < tn the random variables {Wⁱ − Wi−1} are independent.

• For each 0 ≤ s < t the random variable {Wt+s− Ws} has normal distribution with mean 0 and variance σ²t

Its transition density function is defined for all x, y ∈ R and t > 0 as

p(t, x, y) = 1

√2πte

„

−^(x−y)2_2t

«

(1.28)

"SUM3"

Figure 1.1: The generalised Wiener process.

1.3.5 Poisson Process

A Poisson process with parameter λ is a stochastic process which discontinuous trajectories. Given a filtered probability space (Ω, F, {Ft}t>0, P), an adapted sequence {Nt}t≥0is a Poisson process if:

• For any t > 0 then {N}t>0is almost surely finite.

• For any ω ∈ Ω the sample path t 7→ Ntis piecewise constant and increases by unit jumps

• The sample path t 7→ Nt are right continuous and with left limit

• For any t > 0, Nt− = N_twith probability 1

• For any t > 0, Ntfollows a Poisson distribution in (1.27)

• {N}t≥0 has independent increments for any t1 < t₂ < ... < t_n that is, Ntn − Ntn−1, N_t_n−1 − N_t_n−2, .., N_t₂ − Nt1 are independent random variables.

• The increments of Nt are homogeneous in time, For any 0 < s < t then Nt − Ns has the same distribution as N_t−s

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1.4 Central Limit Theorem (de Moivre 1706, Laplace 1812, Lindeberg 1922,..) 14

• {Nt}t>0has the Markov property that is for any 0 < u ≤ s < t then,

E[f (N_t)/N_u] = E[f (N_t)/N_s] (1.29)

1.4 Central Limit Theorem

(de Moivre 1706, Laplace 1812, Lindeberg 1922,..)

Theorem

Let X1, X₂, ...be independently and identically distributed random variables in L², with any common distribution of finite mean µ and variance σ² > 0. Also let: Sn= X₁+ X₂+ ... + X_n, then E[Sn] = n µ, Var(Sn) = n σ²and the standard deviation of Sn= σ√

nand;

Y_n= S_n− n µ σ√

n ∼ N(0, 1) as n → ∞ (1.30)

Proof

E[S_n] = E[X₁+ X₂+ ... + X_n] = E[X₁] + E[X₂] + ... + E[X_n]

=

n

X

i=1

Xi = nµ (1.31)

Var(Sn) = Var(X1+ X₂+ ... + X_n)

=

n

X

i=1

Var(Xi) = nσ² (1.32)

Standard deviation(Sn) = pVar(S_n) = σ√

n (1.33)

Our proof of the theorem is motivated by the concepts or properties of characteristic functions issection 1.3

Proof of Theorem

Let Yi = ^(Xⁱ_σ^−µ) be a standardized random variable for each i = 1, 2, ..n then,

Y_n= S_n− n µ σ√

n = 1

√n

n

X

i=1

Y_i (1.34)

From (1.11) and (1.12) since each E[|Xⁱ|] < ∞ then E[|Yi|] < ∞, i = 1, 2, ..n with

E[X^k] = φ^(k)_X (0)

i^k (k = 1, 2) E[X¹] = µ, and E[X²] = σ²+ µ² (1.35)

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1.4 Central Limit Theorem (de Moivre 1706, Laplace 1812, Lindeberg 1922,..) 15

φ_Y(t) = e^itY

= e⁻ⁱ(σ^t)^µφX(t)

= e⁻ⁱ(σ^t)^µEh

eⁱ(σ^t)^Xi

=

1− itµ

σ − t²µ²

2σ² + o(t²)

E

1 + itX

σ − t²X²

2σ² + o(t²)

= 1−t²

2 + o(t²) (1.36)

and therefore,

φ_Y(t/√

n) = 1− t²

2n+ o t² 2

(1.37) From (1.11), Yiare independent,

Ψⁿ_Y_i(t) = φ_Y₁(t)φ_Y₂(t)...φ_Y_n(t)

=

1− t²

2n+ o t² n

n

−→ e⁻¹²^t² as n → ∞ (1.38) Which is the characteristic function of N(0, 1) distribution in (1.22).

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Chapter 2

Option Pricing in the Binomial Model

In modelling of price dynamics, we will consider a fixed length of calendar time [0, T ], with a probability space (Ω, F, P ) as before, consisting of all feasible price movement scenarios ω such that, for all ω ∈ Ω, The price of the risky asset

S_t(ω) : Ω−→ (0, ∞) (2.1)

with St(ω)≥ 0 for all (0 ≤ t ≤ T ). At time t = 0, S0 = S₀(ω)is known to all investors (deterministic).

It can therefore be considered as a constant random variable. Adopting discrete analysis of the price process; Let t = nh, (0 ≤ n ≤ N), where h is a sub-interval in time (minute, day, month or year).

The price dynamics for any conceivable scenario ω ∈ Ω, can be expressed as a vector of non-constant random variables:

[S₀, S_h(ω), S_2h(ω), ...S_nh(ω), ..., S_{N h}(ω) = S_T(ω)] (2.2) For ease of notation we write:

[S₀, S₁, S₂, ...S_n, ...., S_N = S_T] for any (ω ∈ Ω) (2.3) or St(ω_i)≥ 0, (ωi∈ Ω) with i = 1, 2... and (0 ≤ t ≤ T ).

This means that for each t there are at least two scenarios {ω, ¯ω ∈ Ω} such that S^t(ω)6= St(¯ω).

2.1 Binomial Model

The binomial model is a specific type of the above market price dynamics withonly two possible states or scenarios Ω = {ω1, ω₂} of price movement at each time-step. The prices may either go up or down.

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2.1 Binomial Model 17

In fact, the structure of price movement can be represented in tree. A scenario can be seen as a path from the ‘ root ’ to the right most branch tips.

Each path represents a binomial random walk and the prices follow a multiplicative binomial process with the property that, the one-step return on stock, say Ksfor a single step is:-

K_s=







u if prices increase, with probability p

d if prices decrease, with probability 1 − p (2.4) Let r denote the risk-free interest rate over each time-step. We consider a simple one-step market model to introduce and develop the mathematical tools. These tools are generally valid for the complete valua- tion method.

2.1.1 Option pricing in one-step binomial model

This is a model in which there are only two dates, t = 0 and t = 1. We consider a portfolio Φ = (φ, ψ), where φ and ψ denote the number or units of shares (risky asset) and cash (risk-less bonds) held respectively.

Let S0denote the price of the share at time time t = 0 and Ω = {ω1, ω₂} be the two possible states at t = 1such that

S1(ω) =







S₁(ω₁) = S₀(1 + u) with probability p

S1(ω2) = S0(1 + d) with probability 1 − p (2.5) Theorem 2.1

The market is viable (Arbitrage-free) if: −1 < d < r < u Proof - (By contradiction)

Suppose d < u < r, consider a portfolio Φ = (1, −S0). The value of the portfolio at time t = 0 is V0(Φ) = 0. At time time t = 1, if the price of the stock increases by u

V₁(Φ) = S₁(ω₁)− S0(1 + r)

= S₀(1 + u)− S0(1 + r)

= S₀(u− r) > 0

which implies arbitrage, hence r < u. Similarly if the prices decrease, V₁(Φ) = S₁(ω₂)− S0(1 + r)

= S₀(1 + d)− S0(1 + r)

= S₀(d− r) < 0

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2.1 Binomial Model 18 Which implies arbitrage, hence d < r < u

2.1.2 Replicating portfolio - Law of one price

C

C Cu q

1 q_ 0

d

Figure 2.1: One-step tree z

Let Φ = (φ, ψ) be a self-financing portfolio and r be the risk-less interest rate. Let the stock price dynamics be described as in (2.5), again with a one time-step period to maturity. Further let C0 denote the price of a European call option at t = 0, Cu - denote its payoff if the stock prices goes up and Cd if the prices go down as shown in figure (2.1).

If Su= S₀(1 + u)and Sd = S₀(1 + d), then the payoff of a European call at (t = 1) is given by:

Cu = Max{S^u− K, 0} with probability q and/or C_d = Max{Sd− K, 0} with probability 1 − q.

To guarantee that the payoff of the call tracks the value, V1(Φ)of the portfolio at t = 1 (replication), we choose the values of φ and ψ such that:

V₁(Φ) =







φ Su+ ψ (1 + r) = Cu

φ S_d+ ψ (1 + r) = C_d (2.6)

Provided the matrix of the coefficients of Cu 6= Cdin (2.6) is non-singular then:

φ_∗ = C_u− Cd

S_u− Sd

ψ_∗ = C_dS_u− CuS_d

(1 + r)(S_u− Sd) (2.7)

Where Φ = (φ_∗, ψ_∗)chosen as such, is called the Equivalent portfolio.

Bylaw of one price; “ If two assets have the same terminal value, then they must have the same initial value; otherwise and arbitrage profit is feasible ”:

C₀= V₀(Φ) = φ_∗S₀+ ψ_∗ = C_u− Cd

Su− Sd

S₀+ C_dS_u− CuS_d (1 + r)(Su− Sd)

= 1

1 + r

hr− d u− d

C_u+u− r u− d

C_di

(2.8)

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2.2 Multi-step Binomial model 19

2.1.3 Risk-Neutral Probabilities

From the result (2.8), the coefficients of Cuand Cd add up to 1. They can be interpreted as probability.

In fact, the equation can be simplified by defining a subjective probability measure Q = (q, 1 − q) such that, q = _u−d^r−d and 1 − q = ^u−r_u−d. Therefore:

C₀= 1

1 + r[q C_u+ (1− q) Cd] = 1

1 + rE_Q[C₁] (2.9)

The probability measure Q = (q, 1 − q) is called risk-neutral probability or an equivalent martingale measure (EMM). It is a feature of every complete market.

In general, the arbitrage price (fair) price of any derivative security X, at time t = 0 in a one step binomial setup is given by:

X₀ = 1

1 + rE_Q[X₁] = E_QX¯₁

(2.10) where ¯X₁ is the discounted price at time t = 1 and E[..] denotes expectation taken with respect to probabilities (q, 1 − q). That is ‘ the present value of a derivative is equal to its discounted expected value under the risk neutral measure. ’ This principle is known to economists as the rational expectation hypothesis

2.2 Multi-step Binomial model

This is an extension one step binomial model to an N-step binomial tree. As earlier discussed, we suppose that the market is observable at times 0 < t1 < t₂ < .. < t_N = T. We still consider a simple market consisting of two financial instruments; A Stock and a Bond. The model can be analysised as below:

• The Bond

Over the time period 0 ≤ t ≤ T the bond price is predictable and the risk-free interest rate r is known and constant. The bond value then increases by a factor of e^rT (compound interest). However, over each mini-period [tn, tn+ 1], n = 0, 1, .., (N − 1), we require that the interest rate over the period is known at the beginning of the interval, that is, at time tn.

• Stock price dynamics

The possible trajectories of the stock price can be encoded in a tree such that over each mini-period [tn, tn+ 1], (0 ≤ n ≤ N − 1) the stocks follow a simple binomial model. The mini-periods are all

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2.3 Cox Ross Rubinstein (CRR) Model 20

of equal length h = _N^T with time tn = nh. Therefore, the stock prices can be given as a vector;

[S₀, S₁, S₂, .., S_n, .., S_N = S_T], (0 ≤ n ≤ N) over the fixed calendar time interval [0, T ]. For ease of notation we consider U = 1 + u and D = 1 + d to denote the growth factors in prices, with the risk-neutral measure Q = (q, 1 − q) representing the probability of an up or downward shift in prices respectively. At any time-step n, (0 ≤ n ≤ N):

• Each scenario (path) with exactly j upward moves and n − j downward moves gives the same stock price; Sn= S₀U^jD^n−j

• There are

n j

such paths and the probability of each is q^j(1− q)^n−jand therefore S_n= S₀U^jD^n−j with probability

n j

q^j(1− q)^n−j

• The stock price is a discrete random variable with n + 1 different values and at each n-step and the stocks have 2ⁿpossible prices

• The number j of upward moves and n − j downward moves are random variable with binomial distribution

2.3 Cox Ross Rubinstein (CRR) Model

The pricing and hedging in a multi-step binomial model is determined using Backward Induction. In particular, suppose that the price of the stock is known at time n − 1. Using the risk neutral probability, S_n−1= Ψ⁽ⁿ⁾E⁽ⁿ⁻¹⁾_Q [S_n], where Ψ⁽ⁿ⁾= _R¹ is the discount factor over each time step

Similarly the value of a derivative X_n−1at time n − 1 can be hedged with respect to the EMM so as to give a value Xnat time n, X_n−1= Ψ⁽ⁿ⁾E⁽ⁿ⁻¹⁾_Q [Xn]

If we again consider X_n−1as a claim at time n − 1 and suppose that we know the price, Sn−2of a stock, we can construct a portfolio at time n − 2 that will replicate the value of the portfolio at time n − 1 with value X_n−1. The cost of such a portfolio is, X_n−2 = Ψ⁽ⁿ⁻¹⁾E⁽ⁿ⁻²⁾_Q [X_n−1], where the expectation is with respect to S_n−2 = Ψ⁽ⁿ⁻¹⁾E⁽ⁿ⁻²⁾_Q [Sn − 1] and Ψ⁽ⁿ⁻¹⁾ = _R¹ This notion of pricing is used to develop the CRR model price for any derivative whose price dynamics can be encoded in a tree structure as shown below.

Let Xn,jdenote the value of a derivative at the nth time-step (0 ≤ n ≤ N) in state j, as shown in figure 2.2. Where j represents the number of times that the price of the underlying stock has had an upward jump (increases). Then using an EMM Q, for all n, (0 ≤ n ≤ N − 1), Xn,jis equal to the discounted

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2.3 Cox Ross Rubinstein (CRR) Model 21

X X

X_{n−1, j−1}

n, j

n, j−1

n+1, j+1

n+1, j

X_{n+1, j−1} X

Figure 2.2: A section of a multi-step binomial tree

expected payoffs of the immediately succeeding time-step n + 1.

That is:

X_n,j = 1

RE_Q(X_n+1)

= 1

R[q X_n+1,j+1+ (1− q) Xn+1,j] and, X_n,j−1 = 1

R[q X_n+1,j+ (1− q) X_n+1,j−1] Similarly

X_n−1,j−1 = 1

R[q X_n,j+ (1− q) X_n,j−1]

= 1

R²[q (q X_n+1,j+1+ (1− q) Xn+1,j) + (1− q) (q Xn+1,j+ (1− q) Xn+1,j−1)]

= 1

R²[q²X_n+1,j+1+ 2 q (1− q) Xn+1,j+ (1− q)²X_n+1,j−1] By induction:

X₀ = 1 Rⁿ

n

X

j=0

n j

q^j(1− q)^n−jX_n,j

At the terminal nodes of the binomial tree, the value of an option XN,j, determined by the price of the underlying asset SN,j, replicates the value of the portfolio.

Let SN,j = S₀U^jD^{N −j} denote the price of underlying stock at the N th-step (expiry) in state j. Also let the payoff of the European call option at expiry be CN,j in state j (0 ≤ j ≤ N + 1) That is,

C_N,j =Max{SN,j − K, 0} = Max{S0U^jD^{N −j}− K, 0}

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2.4 Modelling in Continuous Time 22

where K is the strike price. Assuming that the model is viable, S D^N < K R^N < S U^N where R = 1 + r, then CN,j = (S₀U^jD^{N −j}− K)⁺so that:

C₀ = 1 R^N

N

X

j=0

N j

q^j(1− q)^{N −j}(S₀U^jD^{N −j}− K)⁺

For the claim to be exercisable, we require S0U^jD^{N −j} > K. Suppose Akdenotes the minimum number of up moves required for the option to end up ‘ in the money ’ (where the subscript k denotes the time-step when this is so). To guarantee this condition,

D^N U D

j

> K

S₀ and thus Ak=

ln_S ^K

0D^N−k

ln^U_D

+ 1 where b· · · c denotes the integer part of the result (floor function). Hence:

C₀ = S₀

N −kX

j=Ak

N j

q U R

j

(1 − q) D R

_{N −k−j}

− K

R^N

N −kX

j=Ak

N j

q^j(1− q)^{N −j} (2.11)

Since Q = (q, 1 − q) is risk-neutral, and the model is viable, R = q U + (1 − q) D and we can rewrite, 1 = ^{q U}_R +^{(1−q) D}_R

Let q⁰= ^{q U}_R and 1 − q⁰ = ^{(1−q) D}_R then Q⁰ = (q⁰, 1− q⁰)is a probability and,

C0 = S0 N −kX

j=Ak

N j

q^0j(1− q⁰)^{N −k−j} − K R^N

N −kX

j=Ak

N j

q^j(1− q)^{N −j}

= S₀Ψ(A_k; N, q⁰)− K R^−NΨ(A_k; N, q) (2.12)

(2.12) is called theCox Ross-Rubinstein Formula where,

Ψ(A; N, q) =

N

X

j=A

N j

q^j(1− q)^{N −j}

is the complementary binomial distribution function. Using the continuously compounded risk-free interest rate we can rewrite (2.12) to,

C₀ = S₀Ψ(A_k; N, q⁰)− K e^−rNΨ(A_k; N, q) (2.13)

2.4 Modelling in Continuous Time

Absolute changes in asset prices is not useful for analysis, modelling rate of return is therefore preferred.

Suppose that S denotes the stock price at any time t then the rate or return on the stock can be expressed

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2.5 Ito’s Lemma 23

as the stochastic differential equation:

dS

S = µ dt + σdW (2.14)

where

• µdt - Is the drift term, which gives the deterministic component in the rate of return. It is a measure of the average rate of growth of asset prices.

• σdW - This component models the random change in asset price due to external factors. σ or volatil- ity is a measure of standard deviation about the mean return, dW represents a random sample from a normal distribution N(0, 1). It contains the randomness, which is a feature of asset prices and is formally presented as the Wiener process

The above equation is an example of a random walk. In particular, it defines a markov stochastic process in continuous-time; Ito process.

2.5 Ito’s Lemma

This relates a small change in a function of a random variable to a small change in the variable itself.

Using the Taylor Series expansion, together with the fact that the random term dW in the stochastic differential equation (2.14) is drawn from a normal distribution with variance dt. (implied by the fact that W follows a Wiener process) and that, the order (size) dW² → dt as dt → 0 with probability one, Suppose that f(S) is a function of a random variable S, Then the Taylor series expansion of f(S) to the order, dS is given by:

df = df

dSdS +1 2

d²f

dS²dS²+ o(dS³) (2.15)

Now with, dS² = µ²S²dt²+2µσS²dt dW+σ²S²dW²while dW = O(√

dt), ignoring terms in o(dt²) and o(dt dW), (2.15) becomes:

df = σS df dSdW +

µS df

dS +1

2σ²S² d²f dS²

dt (2.16)

If f(S, t) is a function of S and time t the resulting equation involves partial derivatives and hence the Taylor series of f(S, t) is given by:

df = σS∂f

∂SdW + µS∂f

∂S +1

2σ²S²∂²f

∂S² +∂f

∂t

dt (2.17)

This relation is made up of a random component proportional to the random variable dW and a deterministic term proportional to dt. An application to this lemma is it’s use if f = ln S as illustrated in the Black-Scholes formula.

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2.6 Black -Scholes Formula 24

2.6 Black -Scholes Formula

It is based on the following fundamental assumptions:-

• The asset prices follow a log-normal random walk defined by dS = µS dt + σS dW

• The risk free interest rate r and the asset volatility σ are known functions of time over the life of the security.

• There are no dividends during the life of the derivative security.

• There are no arbitrage opportunities or possibilities

• There are no transaction costs associated to hedging a portfolio.

• Trading of the asset takes place continuously

• Short-selling is permitted, borrowing and lending at the risk free rate is possible and assets are divis- ible.

• The market is liquid and there is no default risk

Let S denote the price of an asset and V (S, t), a function of both price S and time t denote the value of a call option.

The Black-Scholes partial differential equation under the above stated assumptions is given as:-

∂V

∂t +1

2σ²S²∂²V

∂S² − rS∂V

∂S − rV = 0 (2.18)

It is satisfied byany derivative security which is paid for upfront.

2.7 Partial Differential Equations (PDE)

The Black-Scholes equation is a linear parabolic partial differential equation. In general, a second-order- linear partial differential equation in two independent variables x and y,

au_xx+ 2bu_xy + cu_yy+ du_x+ eu_y+ f u = g (2.19) where, we consider a, b, c, d, e, f, g as constants. These may however be functions of x and y. The spatial variables are restricted to a region Ω, with boundary B and the union of the two is called the closure Ω. The time variable runs over and interval t¯ 1 < t < t₂. If the partial derivatives of u up to order m are continuous on the region then, u is class C^m(or u is C^m in Ω). Usually Ω is a subset of the Euclidean n-space Rⁿ.

Classification

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2.7 Partial Differential Equations (PDE) 25

The classification of PDE’s is based on the mathematical concept of characteristic, In 2-dimensional problems (with two independent variables) characteristics are lines along which along which certain properties remain the same (invariant) or certain derivatives may be discontinuous. The invariance property implies that we can introduce a transform,ξ = ξ(x, y) and η = η(x, y), to obtain the canonical forms: [15] [16].

u_ξξ− uηη+ ... = 0 (2.20)

u_ξξ+ ... = 0 (2.21)

u_ξξ+ u_ηη+ ... = 0 (2.22)

where the dots in this context indicate terms involving u and it’s first derivatives uxand uy. The associated polynomial in constants α and β is given as:-

P (α, β) = aα²+ bαβ + cβ²+ dα + eβ + f = g (2.23) Classification proceed from the form the discriminant b²− ac:-

• If b²− ac > 0 we achieve the first canonical form defining hyperbolic PDE’s

• If b²− ac = 0 we achieve the second canonical form defining parabolic PDE’s

• If b²− ac > 0 we achieve the third canonical form defining elliptic PDE’s

Our interest lies in the second type of equations; which describe the general form of the Black-Scholes equation.

Second order parabolic equations are further classified as Backward, if all the terms to the left of the equality in a homogeneous equation have the same sign or Forward, if all the terms to the left of the equality in a homogeneous equation have different signs.

Notion of well-posed

To specify a function that represents the solution and characterizes the problem in a particular domain, auxiliary conditions are imposed PDE’s. These may take the form of:-

• Boundary conditions - These specify the conditions that have to be satisfied on the boundary B of the spatial region Ω

• Initial conditions - These must be satisfied throughout Ω at the instance when consideration of the system begins.

The auxiliary conditions, the coefficient functions; (a, b, c, d, e, f) and any inhomogeneous terms in the PDE, comprise its data. The solution is said to depend continuously on the data, if a small change in the data produces a corresponding small change in the solution.

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2.8 Example: Solution - Black Scholes equation for European Options 26

A PDE is said to be well-posed if and only if its’ solution, exists, is unique and depends continuously on the data. Otherwise it is ill-posed.

This implies that for the Black-Scholes equation to be well posed, there must be sufficient auxiliary conditions to guarantee that the solution exists and is unique, the conditions must also be correct to ensure that the solution depends continuously on the data. The backward parabolic equation is well posed but the forward equation is not.

2.8 Example:

Solution - Black Scholes equation for European Options

Let the payoff of the call at expiry time T be V (S, T ). The present value at any time t, (0 ≤ t ≤ T ) is the discounted value restated by a change of variable V to U where V (S, t) = e^{r(T −t)}U (S, t)Hence from (2.18) we have,

∂U

∂t +1

2σ²S²∂²U

∂S² + rS∂U

∂S = 0 (2.24)

Let τ = T − t then dτ = −dt This leads to a backward parabolic equation of the form;

∂U

∂τ = 1

2σ²S²∂²U

∂S² + rS∂U

∂S with (0 ≤ S < ∞) (2.25)

Since prices in the model are assumed to follow a log-normal process. Let ξ = log S then;

∂

∂S = e^−ξ ∂

∂ξ

∂²

∂S² = e^−2ξ ∂²

∂ξ² − e^−2ξ ∂

∂ξ (2.26)

with the change of variables we achieve,

∂U

∂τ = 1 2σ²∂²U

∂²ξ² +

r−1

2σ² ∂U

∂ξ with (−∞ < ξ < ∞) (2.27) With a translation of the coordinates through x = ξ + r −¹₂σ² τ and y = τ, we have:



 ξ τ



=





1 − r − ¹₂σ²

0 1







 x y



 (2.28)

with

∂U

∂τ = ∂x

∂τ

∂U

∂x +∂y

∂τ

∂U

∂y =

r−1

2σ² ∂U

∂x +∂U

∂y

∂U

∂ξ = ∂x

∂ξ

∂U

∂x +∂y

∂ξ

∂U

∂y = ∂U

∂x hence ∂²U

∂ξ² = ∂²U

∂x²

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2.8 Example: Solution - Black Scholes equation for European Options 27

and therefore (2.27) becomes

r−1

2σ² ∂U

∂x +∂U

∂y = 1 2σ²∂W

∂x² +

r−1

2σ² ∂U

∂x (2.29)

For consistency in definition and physical interpretation [2], we retain the variable τ = y and hence we can write in (2.29) in terms of U = W (x, τ) hence:

∂W

∂τ = 1 2σ²∂W

∂x² (2.30)

In summary, the change of variables to the result is:

V (S, t) = e^{−r(T −t)}U (S, t) = e^−rτU (S, T − t) = e^−rτU (e^ξ, T − t)

= e^−rτU

e^x−(r−¹²^σ²^)τ, T− t

= e^−rτW (x, τ )

(2.30) is abackward parabolic equation. Subject to the auxiliary conditions, it can be solved using various methods [2] [18]: similarity solutions, integral transforms etc.

The solution is a Gaussian form;

W (x, τ ) = 1

√2πτ σe⁻^(x−x0)2^2σ2τ (2.31)

where x⁰ is an arbitrary constant. The solution for a European call option at any time t ∈ [0, T ] is:

V (S, t) = SN(d₁)− Ke^{−r(T −t)}N(d₂) (2.32) Where

d₁ = log(S/K) + r + ¹₂σ² (T − t) σ√

T − t and d2 = log(S/K) + r−¹₂σ² (T − t) σ√

T− t

At time t = 0, and for our notational convinience, we write V (S, 0) = C0and therefore from (2.33) we have:

C₀ = S₀N(X)− K e^−rTN(X− σ√

T ) (2.33)

where: X = log(S₀/K) + r +¹₂σ² T σ√

T and N(.) is the normal distribution function.

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Chapter 3

Convergence

3.1 Binomial model to Black-Scholes

The binomial model has several limitations as a model to option prices;

• In the real-world markets, the prices of assets can take on any positive value, but in the binomial model, prices can assume only two values at a particular time.

• Trading can take place almost continuously on 0 ≤ t ≤ T as opposed to the discrete structure considered in the binomial model.

In this chapter, we demonstrate convergence in distribution. We evaluate the effect of considering shorter time intervals in the binomial model, on the option and stock prices. By so doing, we counter both limitations above simultaneously. We consider the price dynamics of the stock introduced earlier, for the fixed time interval [0, T ] divided into N sub-intervals of equal length h, with:

h = T

N (3.1)

The stock price at expiry is given by ST = S₀U^jD^{N −j} where, U = 1 + u and D = 1 + d are the growth factor on each trading interval, if the share prices increased with probability q, or decreased with probability 1 − q respectively. u and d are the rates of return in each instance. The bond value at time t∈ [0, T ] is dependent on the risk-less interest rate.

As the number of trading intervals increases, N → ∞, then h → 0. We have to make adjustments to respective probabilities of increase or decrease in the stock prices, to match our choice of the interval

CiteSeerX — Continuous time limit of the Binomial Model

Continuous time limit of the Binomial Model

ACKNOWLEDGEMENT

Contents

Chapter 1

Introduction

1.1 Background

1.2 Fundamental concepts

1.3 Mathematical Tools

1.4 Central Limit Theorem

Chapter 2

Option Pricing in the Binomial Model

2.1 Binomial Model

2.2 Multi-step Binomial model

2.3 Cox Ross Rubinstein (CRR) Model

2.4 Modelling in Continuous Time

2.5 Ito’s Lemma

2.6 Black -Scholes Formula

2.7 Partial Differential Equations (PDE)

2.8 Example:

Chapter 3

Convergence

3.1 Binomial model to Black-Scholes