Binomial Approximation Methods for Option Pricing

Binomial approximation methods for option pricing

Yasir Sherwani U.U.D.M. Project Report 2007:22

Examensarbete i matematik, 20 poäng Handledare och examinator: Johan Tysk Juni 2007

Department of Mathematics Uppsala University

Binomial Approximation Methods for Option Pricing

Acknowledgment

“In the name of Allah, the most beneficent, the most merciful.”

The study was conducted at Center of Mathematics and Information Technology (MIC), Department of Mathematics, Uppsala University, Uppsala, Sweden.

I am pleased to tender my humble gratitude for all the people who were with me from the commencement of study till its termination. I would like to pay my respect to all those teachers and colloquies who helped me to complete this program.

First of all I would like to thanks the al-mighty Allah, who gave me courage and ability to complete this Master program in Mathematical Finance.

I would like to express my thanks for the Prof. Johan Tysk, who accepted me as a Master student and kindly supervised me during the whole study period.

I express my gratitude and high regards for Prof. Leif Abrahamsson, Director of International Master of Science Programme, for giving me an opportunity to participate in the programme.

I give my sincere thanks to Prof. Johan Tysk, my supervisor, for helping and guiding me not only for the thesis work, but also for teaching me the Financial Mathematics course, which boost my interest in this field. I would also like to give my sincere thanks to Prof. Maciej Klimek for teaching me the course Financial Derivatives.

I would like to thanks all the Pakistani students living in Uppsala, I really enjoy your company during my stay in Uppsala, I have spent a lot of wonder moments during my stay in Uppsala with you guys, I would love to recall these moments. I am really blessed and fortunate to have friends like you in my life. I wish you all the success in your lives.

At the end my highest regards for my family, particularly my father for his financial support and encouragement. I am in dearth of vocabulary to express my feeling towards you all.

Binomial Approximation Methods for Option Pricing

Contents

1. Introduction 1

1.1. Introduction 1

2. Option Pricing Theory 2

2.1. Option 2 2.1.1. European Options 2 2.1.2. American Options 2 2.1.3. Bermudan Options 3 2.1.4. Asian Options 3 2.1.5. Underlying Asset 3 2.1.5.1. Stock Options 3 2.1.5.2. Foreign Exchange Options 4 2.1.5.3. Index Options 4 2.1.5.4. Future Options 4 2.1.6. Call Options 5 2.1.7. Put Options 6 2.1.8. Binary Options 7

2.2. Arbitrage Free Pricing 9 2.2.1. Implementation of Arbitrage Free Pricing 9

2.3. Put Call Parity 11

2.4. Binary Put Call Parity 11

3. Option Pricing Models 12

3.1. The Black-Scholes Option Pricing Model 12 3.1.1. The Assumption behind Black-Scholes Equation 12 3.1.2. Lognormal Distribution 13 3.1.3. Brownian Motion 15 3.1.4. Ito’s formula 16 3.1.5. Replication Portfolio 17 3.1.6. The Black-Scholes Equation 18 3.1.7. Constant Dividend Yield 20

3.2. The Binomial Method 21 3.2.1. Binomial Asset Price Process 21 3.2.2. Multi-period Binomial Method 25 3.2.2.1. Example 27

iii Binomial Approximation Methods for Option Pricing

3.2.3. Approximating Continuous Time Prices… 30 3.2.4. The Binomial Parameters 35 3.2.5. Deriving Black-Scholes Equation using Binomial Method 39 3.2.6. Constant Dividend Yield 41 3.2.7. The Black-Scholes formula for European Options 43 3.2.7.1. Example 45 3.2.7.2. BS Pricing Formula for Binary Puts and Calls 46 3.2.7.3. BS Pricing Formula for Constand Dividend Yield 47

3.3. Comparison and Results 47

4. The Binomial Method for European Options 52

4.1. The Recursive Algorithm 52 4.1.1. Assumptions 52

4.2. Comparison and Results 54

5. The Binomial Method for American Options 57

5.1. American Options 57 5.1.1. American call and Put 57

5.2. Comparison and Results 58

6. Summary 60

7. References 61

Binomial Approximation Methods for Option Pricing

Chapter 1

Introduction

1.1 Introduction

Modern option pricing techniques are often considered among the most mathematically complex of all applied areas of finance. Financial analyst has reached a point where they are able to calculate with alarming accuracy, the value of an option.

Because of its simplicity and convergence the binomial method has attracted the most attention and has been modified into a number of variants which resulted in improved accuracy sometimes at the expense of speed. Computational methods have always exchange speed and accuracy, i.e. maximizing speed and minimizing errors.

The thesis deals with binomial approximation methods for option pricing to price European and American options.

We consider the lognormal model of asset price dynamics and the arbitrage free pricing concept through these we can uniquely determined the price of an option, given the risk- free yield r, the volatility σ, and the spot price of the asset S0 .

In this thesis we will discuss two equivalent ways of determine the arbitrage free value of the option. i.e. Risk-neutral expectation formula and the Black-Scholes partial differential equation.

We will consider binomial model to derive the risk-neutral expectation formula, we will also discuss multi-period binomial model and approximates continuous time prices with discrete time models, we will derive the Black-Scholes equation using the binomial model given the risk-neutral expectation formula, further, we will derive Black-Scholes pricing formula for European options and see results for different expiries.

Finally we will price European and American options using binomial models. We will price the options by using recursive algorithm, compare their accuracy and stability.

The main reference in this thesis is [1], in the chapter four the main references is [6] and [1].

Binomial Approximation Methods for Option Pricing

Chapter 2

Option Pricing Theory

In this chapter we will discuss some basic concepts about option theory and study the Principal of No-Arbitrage.

2.1 Option

An option gives the holder the right to trade (buy or sell) a specified quantity of an underlying asset at a fixed price (also called the strike price) at any time on or before a given date or expiration date. Since it is a right and not an obligation the owner of the option can choose not to exercise the option and allow the option to expire.

In any option contract, there are two parties involved. An investor that buys an option (that is, an option holder) and an investor that sells an option (that is, an option writer). The option’s holder is said to take a long position while the option’s writer is said to take a short position.

2.1.1 European Options

The European option are options which are only exercisable at the expiry date of the option and can be valued using Black-Scholes Option Pricing formula. There are only five inputs to the classic Black-Scholed Model: spot price, strike price, time until expiry, interest rate and volatility. As such European options are typically the simples option to value. The dividend or yield of the underlying asset can also be an input to the model.

The term European is confined to describing the exercise feature of the option (i.e. exercisable only on the expiry date) and does not describe the geographic region of the underlying asset. For example, a European Option can be issued on a stock of a company listed on an Asian exchange.

2.1.2 American Options

An American option is an option which can be exercised at any time up to and including the expiry date of the option. This added flexibility over European options results in American options having a value of at least equal to that of an identical European option, although in many cases the values are very similar as the optimal exercise date is often the expiry date.

The early exercise feature of these options complicates the valuation process as the standard Black-Scholes continuous time model cannot be used. The most common model

Binomial Approximation Methods for Option Pricing

for valuating American Options is the binomial model. The binomial model is simple to implement but is slower and less accurate than 'closed-form' models such as Black Scholes.

2.1.3 Bermudan Options

Bermudan options are similar in style to American style options in that there is a possibility of early exercise, but instead of a single exercise date there are predetermined discrete exercise dates. They are commonly used in interest rate and FX markets but we generalise them in this case for any type of options.

The main difficulty in determining suitable valuation for Bermudan options comes in the form of the boundary problem. Because of multiple exercise dates, determining the boundary condition in order to solve the pricing problem can be difficult. By determining the optimal exercise strategy and the respective boundary condition, one can generally use simulation methods to determine suitable prices for Bermudans.

2.1.4 Asian Options

An Asian option is based on the average price of the underlying asset over the life of the option and not set a strike price. Asian options are often used as they more closely replicate the requirements of firms exposed to price movements on the underlying asset.

For example, an airline might purchase a one year Asian call option on fuel to hedge its fuel costs. The airline will be charged the market rate for fuel throughout the life of the option and so an Asian option based on the average rate during the period would be preferable to an option based on a single strike price.

2.1.5 Underlying Assets

Options can be traded on a wide range of commodities and financial assets. The commodities include wool, corn, wheat, sugar, tin, petroleum, gold etc. and the financial assets include stocks, currencies, and treasury bonds. Exchange traded options are currently actively traded on stocks, stock indices, foreign currencies and future contracts.

2.1.5.1 Stock Options

Some exchanges trading stock options include OMX (Swedish-Finnish Financial Company, it operates six stock exchanges in the Nordic and Baltic countries), FTSE (London stock exchange), CBOE (Chicago board option exchange) among others. Options are traded on numerous stocks, more than 500 different stocks in fact. The options are standardized in such a way that an option contract consists of 100 shares. Example is IBM July 125 call. This is an option contract that gives a right to buy 100 IBM’s shares at a strike price of $125 each in three months time.

Binomial Approximation Methods for Option Pricing

2.1.5.2 Foreign Exchange Options

This contract gives the holder the right to buy or sell a specific foreign currency at a fixed future time at a fixed price. The size of this contract generally depends on the foreign currency in question. Foreign currency options are important tools in hedging risk, and they can be traded as either an American or a European option.

2.1.5.3 Index Options

This has the same standardized size as a stock option. One contract is to buy or sell 100 times the index at specified strike price. An American of this type of option contract is a call contract on the S&P 100 with a strike price of 980. If the option is exercised, the sum of money equivalent to the payoff of the option is given to the holder of the option.

2.1.5.4 Futures Option

In a future option, the underlying asset is a future contract. The future contract normally matures shortly after the expiration of the option. When a put (call) future option is exercised, the holder acquires a short (long) position in the underlying future contract is addition to the difference between the strike price and the future price.

Binomial Approximation Methods for Option Pricing

2.1.6 Call option

A call option gives the holder of the option the right to buy the underlying asset by a certain price on a certain date, below is our discussion for European options.

The payoff function for a European call option depends on the price of the underlying asset (e.g. a stock) at expiry T and the strike price K. The European call option payoff can be expressed as:

CE (ST ,T) = (ST − K) +

Payoff K

S T K Figure 1.1: Payoff function of the call option

If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. On the other hand if the value of the asset is greater than the strike price the option is exercised, the buyer of the option buys the asset at the exercise price. And the difference between the asset value and the exercise price comprises the gross profit of the option investment.

Net payoff on call option If the asset value is less than the strike price, you lose what you paid for the call.

Strike Price

Price of underlying asset

Figure 1.2: Negative Payoff function of the call option

Binomial Approximation Methods for Option Pricing

2.1.7 Put Option

A put option gives the holder of the option the right to sell the underlying asset by a certain price on a certain date, below is our discussion for European options.

The payoff function for a European put option depends on the price of the underlying

asset (e.g. a stock) at expiry T and the strike price K. The European put payoff P(S T ) can be expressed as:

PE (ST ,T ) = (K − ST ) + Payoff K

S T K Figure 1.3: Payoff function of the put option

If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. On the other hand if the price of the underlying asset is less than the strike price the holder of the put option will exercise the option and sell the stock at strike price. The difference between the strike price and the market value of the asset as gross profit.

Net payoff on If the asset value is greater than call option the strike price, you lose what you paid for put.

Strike Price

Price of underlying asset

Figure 1.4: Negative Payoff function of the put option

Binomial Approximation Methods for Option Pricing

2.1.8 Binary Option

Binary options behave similarly to standard options, but the payout is based on whether the option is on the money, not by how much it is in the money. For this reason they are also called cash-or-nothing options.

As with a standard European style option, the payoff is based on the price of the underlying asset on the expiration date. Unlike with standard options, the payoff is fixed at the writing of the contract.

Payoff K

S T K Figure 1.5: Payoff function of the binary call option

Payoff

S T K Figure 1.6: Payoff function of the binary put option

Binomial Approximation Methods for Option Pricing

The binary put option has a payoff

1 if SKT <

PSBT ( ) =

0 if SKT ≥ and the binary call option has a payoff,

0 if SKT ≤

CSBT ( ) =

1 if SKT > .

The binary options are example of options with a discontinuous payoff.

Binomial Approximation Methods for Option Pricing

2.2 Arbitrage-free Pricing Model

The absence of arbitrage means that no investor in the market is able to make riskless profit by selling and buying securities.

2.2.1 Implementation of Arbitrage-Free Pricing model

The basic assumption in applying the principle of no arbitrage is existence of risk-free security, a default free asset. In this thesis we will assume that there is a risk-free security that has a constant continuously compounded yield r. The traders and investors can borrow and lend at this rate.

Suppose we have a portfolio, which is collection of securities. A portfolio may contain long and short positions of various assets. A position is an item in the portfolio such as a stock or cash, in a money market account. Positions are further classified as long position and short position which are defined below:

Definition: A long position when liquidated creates a positive cash flow, and is thus an asset to us, treasury bond, stock or an option that we purchased are examples of long position.

Definition: A short position creates a potential negative cash flow when liquidated, so it is a liability, borrow stock or a written option are examples of short position.

Suppose we have two risk-less portfolios P(A) and P(B) with respective deterministic

yields a and b over some period of time T. Their current prices are A0 and B0 respectively.

We will assume that,

A0 = B0 if a > b .

Now consider a portfolio P(C) with a long position on portfolio P(A) and a short position on portfolio P(B).

The present value C0 will be:

C0 = A0 − B0

C0 = 0 ,

Binomial Approximation Methods for Option Pricing while at time T the value of portfolio P(C) will be:

P(C)T = AT − BT aT bT P(C)T = A0e − B0e , bT (a−b)T P(C)T = A0e [e −1]> 0.

(a−b)T As we assumed that A0 = B0 and P(C)T = e > 1, it shows that portfolio P(C) does not cost anything and hence involves no risk. We have thus an arbitrage opportunity.

In the above example the excess demand of the high yielding portfolio P(A) and excess supply of the low yielding portfolio P(B) would drive the price of portfolio A up and drag the price of portfolio B down.

Again assume that

A0 = B0 if ab< .

Portfolio P(B) with a long position on portfolio P(C) and portfolio P(A) with a short position on portfolio P(C). In our this case the price of portfolio P(B) would go up and the price of portfolio P(A) would go down. As the price of the portfolios changes, the equilibrium price is obtained when respective yields a = b, it means that the yields of any two risk-less portfolios must be equivalent,

r = a = b.

Binomial Approximation Methods for Option Pricing

2.3 Put Call Parity

By using the principle of no arbitrage we can derive an important relationship between European put and call prices with the same strike price, consider the following portfolios:

Portfolio A: one European call option plus an amount of cash equal to Xe −r(T −t) .

Portfolio B: one European put option plus one share. Both are worth max(ST , X ) at expiration of the options. Hence the portfolios must have the same value at the present time. Therefore:

C + Xe−r(T −t) = P + S , where C is the value of the European call option and P is the value of the European put option.

Hence, calculating the value of the call option also gives us the value of the put option with the same strike price.

2.4 Binary Put Call Parity

By using the principle of no arbitrage we can derive the important relationship between binary put and call prices. Binary put and call have a simple relation. A portfolio consisting of one binary put can one binary call yields 1 regardless of the price of the underlying asset ST at expiry. The portfolio is risk-less. By the no arbitrage argument, the portfolio must grow at the risk free rate r. The present value of the combination portfolio is e −rT :

−rT PB + CB = e , where T is the remaining time until expiry.

Binomial Approximation Methods for Option Pricing

Chapter 3

The Option Pricing Models

In this chapter we will discuss two famous Option Pricing Models: The Black-Scholes model and the Binomial model. We also show that how one can obtain the Black-Scholes equation by approximation with binomial model.

3.1 The Black-Scholes Option Pricing Model

3.1.1 Assumptions behind the Black-Scholes Equation.

There are several assumptions involved in the derivation of the Black-Scholes equation.

1. The stock price S, follows a log-normal random walk.

2. The market is assumed to be liquid, have price continuity, be fair and provide all the investors with equal access to available information. This implies that zero transaction costs are assumed in the Black-Scholes analysis.

3. It is assumed that the underlying security is perfectly divisible and that short selling with full use of proceeds is possible.

4. The principle of no-arbitrage is assumed to be satisfied.

5. We assume that there exists a risk-free security which returns $1 at time T when $ e −r(T −t) is invested at time t.

6. Borrowing and lending at the risk-free rate in possible.

Binomial Approximation Methods for Option Pricing

3.1.2 Lognormal Distribution

Consider a European option with expiry at time T. We will assume the present time is zero. The value of the option depends both on the current value of the underlying asset and the time to expiry T. Let suppose that the price of the asset at some point of time t by

St . The spot price S0 is the current price of the asset.

In the Black-Scholes option pricing model the main assumption is that the relative change of the asset over a period of time is normally distributed, the mean and variance are (μt,σ 2t ) respectively. The rate of return over some time interval t is given below:

S − S t 0 . S0 The rate of return can be expressed as:

S − S t 0 = μt + σ tZ , (3.1) S0 where Z is standard normal random variable with mean and variance (0,1) respectively. The above equation (3.1) tells us that time passes by an amount of t, the asset price changes by μt and also jumps up or down by a random amountσ tZ .

We will denote the random component of equation 3.1 by a single variable. Let us write:

Wt = tZ .

If we make time intervals smaller and smaller then it means the limitt ⎯⎯→ 0 , the random process becomes a continuous random process; we call this a stochastic process and use differentials to describe infinite charges. We can write the equation as

dSt = μdt + σdWt , (3.2) S0

where Wt is a stochastic variable.

Letting

dSt dX t = , S0

Binomial Approximation Methods for Option Pricing equation 3.2 becomes:

dX t = μdt + σdWt , (3.3) where the variable µ is called the drift rate.

Using the fact that

dSt = d(log St ) , S0 and

dSt dX t = , S0

we can write St as

X t St = S0e . (3.4)

This means that the logarithm of St is normally distributed, hence we say that the distribution of St is lognormal.

Binomial Approximation Methods for Option Pricing

3.1.3 Brownian Motion

The variable dWt appearing in the stochastic differential equation 3.2 defines a special kind of a random walk called Brownian motion or Wiener Process. It is a normally distributed random variable having mean 0 and variance t.

S Equation 3.2 tells us that the logarithm of t is a Brownian motion with drift μdt and S0 gives us a way to describe the changes dSt in the asset price St as time passes.

Figure 3.1: Graphical representation of Brownian motion

Binomial Approximation Methods for Option Pricing

3.1.4 Ito’s formula

We now consider the function f depending on the asset price St and time t. If the asset price S were a deterministic variable we would simply expandf (S0 + ΔS,Δt )at f (S,0) in Taylor series:

∂f 1 ∂ 2 f ∂f 1 ∂ 2 f Δ f = ( Δt + Δt 2 + ...) + ( ΔS + ΔS 2 + ...) + ... ∂t 2 ∂t 2 ∂S 2 ∂S 2

The Ito calculus is stochastic process equivalent to Newtonian differential calculus. In the limit Δt → 0 , terms of Δt of higher order than 1, as in the ordinary differential calculus are consider small and can be omitted.

In case of lognormal random walk, we can write equation 3.1 as

ΔS = S0 μΔt + σ ΔtS0 Z ,

because ΔSt depends on Δt in case of random process.

Then consider

2 2 (ΔS) = (S0 μΔt + σ ΔtS0 Z) , since Z is standard normal, Z 2 is distributed with gamma distribution with mean 1. Therefore

2 2 E[(σ ΔtZ −σ S0 Δt) ] 3 2 2 2 = σ S0 ΔtE[]Z + O(t ) , 3 2 2 = σ S0 Δt + O(t ) .

In the limit Δt → 0

2 2 2 dSt = σ S0 dt .

Therefore, if f is a function dependent on St , it is also a stochastic process f t such that

∂f ∂f 1 ∂ 2 f df = dt + dS + σ 2 S 2 dt , (3.5) t ∂t ∂S t 2 0 ∂S 2

Binomial Approximation Methods for Option Pricing

or, writing out dSt , we obtain Ito’s formula for the option, given the underlying asset is a stochastic process with dSt =+μ S00 dtσ S dw :

∂f ∂f ∂f 1 ∂ 2 f df = σdw + ( + μS + σ 2 S 2 )dt . (3.6) t ∂S t ∂t 0 ∂S 2 0 ∂S 2

The stochastic variable dWt present in the formula; this means that the option price

f (St ,t) also moves randomly.

3.1.5 Replicating Portfolio

A replicating portfolio eliminates the randomness of the option and makes the option equivalent to a riskless portfolio. Once we have such a portfolio or else an arbitrage opportunity will occur. We thus assume the absence of arbitrage to determine the fair price of the option.

A risk of a portfolio is the variance of the return on the investment. In our lognormal model, this isσ 2 . We define the volatility of the portfolio byσ . A risk-less portfolio thus has σ = 0 , μ = r and no random component.

Finding a way to eliminate the random component in the following Ito’s equation:

∂f ∂f ∂f 1 ∂ 2 f df = σdw + ( + μS + σ 2 S 2 )dt. t ∂S t ∂t 0 ∂S 2 0 ∂S 2

To find the price of an option is thus the key to finding the ‘fair price’ for the option. The result will be the Black-Scholes equation, which must hold in the absence of arbitrage. Thus if we can eliminate the randomness by a replicating portfolio, we can find the price of the option.

Binomial Approximation Methods for Option Pricing

3.1.6 The Black-Scholes Equation

Assume the Ito’s formula

∂f ∂f ∂f 1 ∂ 2 f df = σdw + ( + μS + σ 2 S 2 )dt . t ∂S t ∂t 0 ∂S 2 0 ∂S 2

We have a stochastic process f t depending on another process St . Construct a portfolio consisting of one option and a short position G units on the stock.

The value of this portfolio is

π 0 = f 0 − GS0 , after one time-step dt, the portfolio will charged by

dπ t = df t − GdSt . (3.7)

Apply Equation 3.5 to Equation 3.7

∂f ∂f ∂f 1 ∂ 2 f df = σdw + ( + μS + σ 2 S 2 )dt. t ∂S t ∂t 0 ∂S 2 0 ∂S 2

We will get the following equation

∂f ∂f 1 ∂ 2 f dπ = dt + dS + σ 2 S 2 dt − GdS , t ∂t ∂S t 2 0 ∂S 2 t

∂f ∂f 1 ∂ 2 f dπ = dt + ( − G)dS + σ 2 S 2 dt . t ∂t ∂S t 2 0 ∂S 2

∂f To eliminate the random component dS , we choose G = at t = 0 t ∂S

∂f ∂f ∂f 1 ∂ 2 f dπ = dt + ( − )dS + σ 2 S 2 dt t ∂t ∂S ∂S t 2 0 ∂S 2

∂f 1 ∂ 2 f dπ = dt + σ 2 S 2 dt , t ∂t 2 0 ∂S 2 ∂f 1 ∂ 2 f dπ = ( + σ 2 S 2 )dt . t ∂t 2 0 ∂S 2

Binomial Approximation Methods for Option Pricing

In this portfolio, there are no random components in it. So it means that this portfolio is deterministic and thus riskless. By the no-arbitrage argument, the yield on this portfolio is as the same as that of a riskless security.

If we assume that the yield of a riskless security is r a portfolio valued π 0 at t = 0 inverted in this security yields rπ 0 dt during time step dt. The replicated option on deterministic portfolio on the other hand yields dπ t . Their yields must be equal, so

rπ 0 dt = dπ t .

Put the value of dπ t in the above equation

∂f 1 ∂ 2 f dπ = ( + σ 2 S 2 )dt , t ∂t 2 0 ∂S 2 ∂f 1 ∂ 2 f rπ dt = ( + σ 2 S 2 )dt . (3.8) 0 ∂t 2 0 ∂S 2

Since we know that

π 0 = f − GS0 ,

∂f π = f − S . 0 ∂S 0

Put the values of π 0 in equation 3.8

∂f ∂f 1 ∂ 2 f r( f − S )dt = ( + σ 2 S 2 )dt , ∂S 0 ∂t 2 0 ∂S 2

∂f ∂f 1 ∂ 2 f r( f − S ) = ( + σ 2 S 2 ) . ∂S 0 ∂t 2 0 ∂S 2

This is the Black-Scholes partial differential equation which must hold for all European options.

Since the point of time t = 0 was arbitrary and S = S0 , the equation is usually written in this form

∂f 1 ∂ 2 f ∂f + σ 2 S 2 + rS − rf = 0 , (3.9) ∂t 2 ∂S 2 ∂S

Binomial Approximation Methods for Option Pricing or equivalently

∂f 1 ∂ 2 f ∂f + σ 2 S 2 + rS = rf . ∂t 2 ∂S 2 ∂S

The drift rate µ is not present in the equation. The price of the derivative security is governed by equation 3.9 given that underlying asset follows lognormal dynamics in the absence of arbitrage. It holds for more general choices of σ, but in this thesis we will consider constant volatilities only, at the time of expiration f equals the payoff function.

3.1.7 Constant Dividend Yield

We consider the case when the underlying asset pays a continuous dividend at some fixed rate D, without loss of generality. Let t = 0 and the spot price of the asset be S0 . After an infinitesimal timestep dt, the holder of the asset gains DS0 dt in dividends.

However, the asset price must fall by the same amount or else there is an arbitrage opportunity: buying the asset at t = 0 at S0 and selling it immediately at S0 + dSt dt after receiving the dividend would yield a risk-free profit of DSdt. Thus we must have this equation,

dSt = σS0 dWt + μS0 dt − DS0 dt . (3.10)

To eliminate the effect of the dividend in the price of the option, the Black-Scholes equation changes to the equation given below:

∂f ∂f 1 ∂ 2 f + (r − D)S + σ 2 S 2 = rf . (3.11) ∂t ∂S 2 ∂S 2

Binomial Approximation Methods for Option Pricing

3.2 The Binomial Model

The binomial model presents another way to describe the random asset price dynamics. Let us take two possible asset prices per time-step, by increasing the number of time- steps in the limit we will eventually arrive at the correct price of the option and find an alternative way to represent the value of the option namely the risk-neutral expectation formula.

3.2.1 Binomial Asset Price Process

The binomial model starts out with an extremely simple two state market model shown in the figure 2.1(a):

Su0

Sd0

Figure 3.2: One period binomial model

If S0 is the spot price of a risky asset at time t = 0 after some time period T, it can only assume two distinct values: Su0 and Sd,0 where u and d are real numbers such that u > d. Moreover we will assume the existence of a risk-less asset with a constant yield r.

rT Thus, we can say that an investment of S0 dollars at time t = 0 yields S0e dollars at time t = T. The no arbitrage argument is also valid here, we must require that

rT S0 d < S0e < S0u or

d < e rT < u . (3.12)

Binomial Approximation Methods for Option Pricing

If it is true than it means that a risk-less investment can be better, worse or even as well as a risky investment. If this is not true, then the risky asset is not risky at all. Ife rT < d < u , one would never prefer the risk-free asset to the risky asset; borrowing φ units of money at the risk-less rate r and buying the risky asset would yield a profit of at least

φ(d − e rT ) > 0 at time t = T. If e rT = d such a market position (long asset, short bond) would yield a positive value. If e rT ≥ u then market position will be reversed (long bond, short asset) yield a positive value.

Suppose now that the option yields fu and f d if the underlying asset goes up or down respectively.

Consider a portfolio consisting of Δ units of the risky asset (e.g. a stock) and ψ units of risk-less asset (e.g. a money market account) forms a replicating portfolio

rT ΔS0u +ψe = fu , (3.13 a) rT ΔS0 d +ψe = f d . (3.13 b)

This is system of two equation with two unknowns ( Δ ,ψ ) there is a unique solution exist if and only if u ≠ d

f − f Δ = u d , S0u − S0 d

uf − uf ψ = e −rT d u . u − d

Since the option payoff at t = T is equal to that of this portfolio, the value of the portfolio must be equal to that of the option. Let say the present value of the option isV0

V0 = ΔS0 + Ψ , put the values of Δ and Ψ , we will get,

f u − f d −rT uf d − df u = S0 + e , S0u − S0 d u − d

f u − f d −rT uf d − df u = S0 + e , S0 (u − d) u − d

Binomial Approximation Methods for Option Pricing

f − f uf − df = u d + e −rT d u . u − d u − d

Thus

f − f + e −rT (uf − df ) V = u d d u . 0 u − d

We introduce a new variable

e rT − d q = . u − d

The value of the option at t = 0 can be expressed as

−rT V0 = e [qfu + (1− q) f d ]. (3.14)

The no arbitrage argument guarantees that0 < q < 1. Thus the value of the option reduces to a certain kind of expectation formula

−rT V0 = e Eq [ f ], (3.15) where the expectation is taken under the probability measure given by q. This measure has the special property that if VT is the value of option at t = T,

rT Eq [VT ] = (ΔS0 +ψ )e ,

rT Eq [VT ] = V0e .

This probability measure is called risk-neutral probability measure.

Binomial Approximation Methods for Option Pricing

125

120

115

110

105

100

80 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Figure 3.3: 2-Period Binomial model with constant dividend yield

σ = 0.2 , r = 0.1, N = 2, S0 = 100, T = 180 / 365and D = T/N, the graph is generated through matlab.

Binomial Approximation Methods for Option Pricing

3.2.2 Multi-Period Binomial Model

Multi-period binomial models applied to the same total period of time T = NΔt as the S number of periods increased, time step Δt → 0 the distribution of log T approaches to S0 normal distribution. In multi-period model the expiry of the option T is divided into two equal time-stepsT = 2Δt , a risky asset moves upward by a factor of u and downward by a factor of d.

2 Su3 Su0 0

2 Sud0 Sud S S0 0 0 2 Sud0

2 3 Sd0 Sd0

Figure 3.4(a): Two period binomial model Figure 3.4(b): Three period binomial model

2 2 This recombining binomial tree has the end asset values (S0u , S0ud, S0 d ) at time t = T = 2Δ t . Suppose now the option payoff function is f(S). Let the three option possible values at t = T be then

2 f u = f (S0u ) ,

f m = f (S0ud ) , 2 f d = f (S0 d ) .

Assuming the no-arbitrage and risk-neutral, we can apply the following formula

−rT V0 = e [qfu + (1− q) f d ], to each of the individual branches in this tree to obtain a value for the option step by step. At timeT = 2Δt , we will get either of the two values of the option.

−rT V1 = e [qfu + (1− q) f m ], or −rT V1 = e [qfu + (1− q) f d ].

Binomial Approximation Methods for Option Pricing

Applying the formula once again

−r Δt u d V0 = e [qv1 + (1− q)v1 ]. (3.16)

u d Inserting the values of v1 ,v1 we can write this as

−rT 2 2 2 2 V0 = e (q f (S0u ) + q(1− q) f (S0ud) + (1− q) f (S0 d )), or

2 −rT j 2− j j 2− j V0 = e ∑ q (1− q) f (S0u d ) . j=0

For the N-Period model, where T = NΔt , we obtain

N −rT ⎛ N ⎞ j N − j j N − j V0 = e ∑⎜ ⎟q (1− q) f (S0u d ) . (3.17) j=0 ⎝ J ⎠

The payoffs at each node in the N-Period model can be expressed as function of the payoffs in an N + 1 period model:

j N − j −rΔT j+1 N − j j N +1− j f (S0u d ) = e (qf (S0u d ) + (1− q) f (S0u d )). (3.18)

Replacing 3.18 into 3.17

N −r(T +ΔT ) N +1 N −r(T +Δt) ⎡⎛ N ⎞ ⎛ N ⎞⎤ j N − j j N − j V0 = e q f (S0u ) + e ∑ ⎢⎜ ⎟ + ⎜ ⎟⎥()q (1− q) f (S0u d ) j=1 ⎣⎝ j ⎠ ⎝ j −1⎠⎦ −r(T +Δt) N +1 N + e (1− q) f (S0 d ) ,

as we know

⎛ N ⎞ ⎛ N ⎞ ⎛ N +1⎞ ⎜ ⎟ + ⎜ ⎟ = ⎜ ⎟ . ⎝ j ⎠ ⎝ j −1⎠ ⎝ j ⎠

The above equation becomes:

N +1 (−r( N +1)Δt) ⎛ N +1⎞ j N +1− j j N +1− j V0 = e ∑⎜ ⎟q (1− q) f (S0u d ) . j=0 ⎝ j ⎠

Which confirms that formula is also valid for N + 1- period model. By the principle of induction, a formula present in equation 3.17 is true for N-periods nodes.

Binomial Approximation Methods for Option Pricing

Figure 3.5: 20-period binomial model with constant dividend yield

σ = 0.2 , r = 0.1, N = 20, S0 = 100, T = 180/365 and D = T/N, the graph is generated on matlab.

3.2.2.1 Example

Consider a three year European put option on a non-dividend-paying stock when the stock price is 9 SEK, the strike price is 10 SEK, the risk-free interest rate is 6% per annum, and the volatility is 0.3. Suppose that we divide the life of the option into four intervals of length 0.75 years. Thus, we have that

S0 = 9 K =10 r = 0.06 T = 3 σ = 0.3 Δt = 0.75 , and

u = eσ Δt = e0.3 .075 = 1.297 ,

d = e −σ Δt = e −0.3 .075 = 0.771,

Binomial Approximation Methods for Option Pricing

e rΔt − d q = , u − d

e0.06*0.75 − 0.771 q = , 1.297 − 0.771

e0.06*0.75 − 0.771 q = , 0.525

q = 0.523.

1− q = 0.477 .

Figure below shows the binomial tree for this example. Each node has a formula, like 4 S0u , and a number, like 6.816. the formula is the stock price at that node and the number is the value of the option at that node. The value of the option at time T is j N − j calculated by the formula max(K − S0u d ,0) . For example, in the case of node A (i = N = 4, j = 0) in figure 3.5, we have that the value of the option is:

j N − j f 4,0 = max(K − S0u d ,0) 0 4 f 4,0 = max(10 − (9*1.297 *0.771 ),0) , = max(10 -3.184, 0), = max(6.816, 0), = 6.816.

Figure 3.6: Tree used to value a stock option

Binomial Approximation Methods for Option Pricing

For all the other nodes, i.e. all the nodes except the final ones, the value of the option is calculated using the following equation:

−rΔt f i, j = e [qf i+1, j+1 + (1− q) f i+1, j ]. 0 ≤ i ≤ N −1 0 ≤ j ≤ i

For example in the case of node B (i = 3, j = 1) in figure, we have that the value of the option is

−rΔt f 3,1 = e [qf 3+1,1+1 + (1− q) f 3+1.1 ] −0.06*0.75 f 3,1 = e [0.523 f 4,2 + 0.477 f 4.1 ],

f 3,1 = 0.956[(0.523*1) + (0.477 * 4.647)],

f 3,1 = 0.956[0.523 + 2.217],

f 3,1 = 0.956* 2.740 ,

f 3,1 = 2.619 .

Similarly, for the case of node C(i = j = 0), we have that the value of the option is

−rΔt f 0,0 = e [qf 0+1,0+1 + (1− q) f 0+1.0 ] −0.06*0.75 f 0,0 = e [0.523 f1,1 + 0.477 f1.0 ],

f 0,0 = 0.956[(0.5230*0.753) + (0.477 * 2.438)],

f 0,0 = 0.956[0.394 +1.163],

f 0,0 = 0.956*1.557 ,

f 0,0 = 1.488 .

This is a numerical estimate for the option’s current value of 1.473 SEK.

Binomial Approximation Methods for Option Pricing

3.2.3 Approximating Continuous Time Prices with Discrete Time Models

The N-period model derived in the previous section is for discrete model, while the Black-Scholes model derived by Stochastic differential equation was continuous, what would happen if N →∝ orΔt → 0 ?

Consider the value of the underlying asset (the stock) after n periods has passed. There has been a random number say X n , ‘up-jumps’ and n − X n ’down jumps’ the value of asset is then

X n (n−X n ) S n = S0u d .

The lognormal process introduced in the previous section involves a stock

X t price St = S0e , where X t is given by its differential equation, it is a stochastic process 2 with drift μt and varianceσ t .

Let the value of u, d will be following

u = e μ Δt+σ Δt , d = e μ Δt−σ Δt .

It is clear that d < u, if we assume that arbitrage requirement d < e rT < u satisfies then the asset price after n periods will be:

(μ Δt+σ Δt ) X n (μ Δt−σ Δt )(N − X n ) S n = S0e e

μt+(2 X n −n)σ Δt S n = S0e .

If the limit n →∝ then μσtXnt+−Δ(2n ) → X n ? if it does, the binomial model can be used for approximating the lognormal process.

By using the values of u, d we can write q as

e rΔt − d q = , u − d erΔt − e μt−σ Δt q = . e μt+σ Δt − e μt+σ Δt

Binomial Approximation Methods for Option Pricing

Expanding the denominator and numerator in Taylor Series

1 3 rΔt − (μΔt −σ Δt + σ 2 Δt) + O(Δt 2 ) q = 2 , 1 1 3 (μΔt + σ Δt + σ 2 Δt) − (μΔt −σ Δt + σ 2 Δt) + O(Δt 2 ) 2 2 1 3 σ Δt + (r − u − σ 2 )Δt + O(Δt 2 ) 2 q = 3 , 2σ Δt + O(Δt 2 ) 1 r − μ − σ 2 1 q = + 2 Δt + O(Δt) . 2 σ

The random component of S n is

= (2X n − n)σ Δt t = (2X − n)σ , n n (2X − n) = n σ t . n

Let

(2X n − n) Yn = , n

now we will find the mean and variance of Yn .

X n can be described as sum of n independent Bernoulli random variables i.e. random variables denoting the number of heads in n flips of a coin, here the coin is risk neutral coin with probability of heads equal to q.

The mean of X n is nq and the variance of X n is nq(1-q).

Binomial Approximation Methods for Option Pricing

The mean of Yn is then

(2E[X n ]− n) E[]Yn = n 2nq − n = , n n(2q −1) = , n = (2q −1) n .

Put the value of q in the above equation and we will get the mean of Yn :

1 (r − μ − σ 2 ) E[]Y = t 2 + O(Δt) . n σ

The variance of Yn is

⎡2X n ⎤ var[]Yn = var⎢ ⎥ ⎣ n ⎦ 4 var X = n , n 4nq(1− q) = , n = 4q(1− q ) .

Put the value of q in the above equation and we will get the variance of Yn

var[]Yn = 1+ O(Δt) .

In the limit N →∝,Δt → 0then Ynσ t tends to a stochastic process Yt distributed 1 normally with mean (r − μ − σ 2 )t and the variance σ 2t . We can write this as 2 1 Y = (r − μ − σ 2 )t + σW , t 2 t

Binomial Approximation Methods for Option Pricing

where Wt is a Brownian motion we have shown that the discrete random process

μt+(2 X n −n)σ Δt S n = S0e converted to continuous stochastic process

μt+Yt S n = S0e .

Let

X t = μt + Yt , then we obtain 1 X = (r − σ 2 )t + σW . (3.19) t 2 t We can thus approximate lognormal asset price process with binomial model by setting

1 (r− σ 2 )Δt+σ Δt u = e 2 , (3.20 a) 1 (r− σ 2 )Δt−σ Δt d = e 2 , (3.20 b) e rΔt − d q = . (3.20 c) u − d

Let us now find dX t , apply Ito calculus to above equation 3.19

1 1 dX = (r − σ 2 )dt + σ 2 dt + σdW t 2 2 t 1 1 dX = rdt − σ 2 dt + σ 2 dt + σdW , t 2 2 t

dX t = rdt + σdWt .

This implies that in the risk neutral world the stochastic differential equation is dX t = μdt + σ dWt has drift μ = r .

In other words

E[dX t ] = E[μdt + σdWt ]

⎡dSt ⎤ E⎢ ⎥ = rdt , ⎣ S0 ⎦ or

X t E[St ] = S0 E[e ] rt E[St ] = S0e . (3.21)

The price of the stock is expected to grow at the risk-neutral rate r.

Binomial Approximation Methods for Option Pricing

Figure 3.7: N-Period Binomial tree Model

Figure 3.7 estimates the binomial model for a set of N. check accuracy for computation, also investigates how long it takes to evaluate tree.

Binomial Approximation Methods for Option Pricing

3.2.4 The Binomial Parameters

In this section we will show the parameters of binomial model for continuous time prices using the lognormal price process, consider the binomial parameters which are defined in equation 3.30,

1 (r− σ 2 )Δt+σ Δt u = e 2 , 1 (r− σ 2 )Δt−σ Δt d = e 2 , and

e r Δt − d q = , u − d which are not the only possible ways to construct a risk neutral binomial tree.

The lognormal model is fully specified by the mean and variance of the random variable,

XT ST = S0e , or S e XT = T , S0 1 where X is X = (r − σ 2 )t + σW . t t 2 t The variance of e XT is:

2 2 var[e XT ]= E[(e XT ) ]− E[e XT ] 2 = E[(e2 XT )]− E[e XT ] , where e XT has mean e rT as shown in equation 3.4. To find the mean of e 2 XT , we will apply Ito calculus:

1 X = (r − σ 2 )t + σW t 2 t 1 2X = 2(r − σ 2 )t + 2σW . t 2 t

Binomial Approximation Methods for Option Pricing

To find dX t 1 1 dX = 2(r − σ 2 )dt + (2σ ) 2 dt + 2σdW , t 2 2 t 1 dX = 2rdt −σ 2 dt + (4σ 2 )dt + 2σdW , t 2 t 2 2 dX t = 2rdt −σ dt + 2σ dt + 2σdWt , 2 dX t = 2rdt + σ dt + 2σdWt , 2 dX t = (2r + σ )dt + 2σdWt .

2 It means e 2 XT has mean e(2r+σ )T and the variance of e XT is given below:

2 2 var[e XT ]= e 2rT +σ T − E[e XT ] , as we know

S e XT = T , S0 thus we can write mean and variance as,

⎡ST ⎤ rT E⎢ ⎥ = e , (3.22 a) ⎣ S0 ⎦ 2 ⎡ST ⎤ (2r+σ 2 )T ⎡ST ⎤ Var⎢ ⎥ = e − E⎢ ⎥ . (3.22 b) ⎣ S0 ⎦ ⎣ S0 ⎦

We will apply mean and variance to one-period binomial model with T = Δt and limit Δ.t → 0 . In this model the mean and variance are given below:

⎡ST ⎤ E⎢ ⎥ = qu + (1− q)d , (3.23 a) ⎣ S0 ⎦ 2 ⎡ST ⎤ 2 2 ⎡ST ⎤ Var⎢ ⎥ = qu + (1− q)d − E⎢ ⎥ . (3.23 b) ⎣ S0 ⎦ ⎣ S0 ⎦

Comparing equation (3.22 a) and (3.23 a)

⎡ST ⎤ ⎡ST ⎤ E⎢ ⎥ = E⎢ ⎥ ⎣ S0 ⎦ ⎣ S0 ⎦

Binomial Approximation Methods for Option Pricing

e rT = qu + (1− q)d , here T = Δt qu + (1− q)d = e r Δt .

Again comparing equation (3.22 b) and (3.23 b)

⎡ST ⎤ ⎡ST ⎤ Var⎢ ⎥ = Var⎢ ⎥ , ⎣ S0 ⎦ ⎣ S0 ⎦ 2 2 2 2 ⎡ST ⎤ (2r+σ 2 )T ⎡ST ⎤ qu + (1− q)d − E⎢ ⎥ = e − E⎢ ⎥ , ⎣ S0 ⎦ ⎣ S0 ⎦ 2 qu 2 + (1− q)d 2 = e(2r+σ )T , here again T = Δt

2 qu 2 + (1− q)d 2 = e(2r+σ )Δt 2 qu 2 + (1− q)d 2 = e 2rΔt+σ Δt .

qu + (1− q)d = e rΔt (3.24 a) 2 qu 2 + (1− q)d 2 = e 2rΔt+σ Δt (3.24 b)

Now we have two equations with three unknowns variables one variable can be chosen 1 for example q = . 2 1 Put the value of q = in equation (3.24 a) 2 1 1 u + (1− )d = erΔt 2 2 1 1 u + d = e rΔt , 2 2 1 (u + d) = erΔt , 2 u + d = 2e rΔt .

Binomial Approximation Methods for Option Pricing

1 Again put the value of q = in equation (3.24 b) 2

1 1 2 u 2 + (1− )d 2 = e2rΔt+σ Δt 2 2 1 1 2 u 2 + d 2 = e2rΔt+σ Δt , 2 2 1 2 (u 2 + d 2 ) = e 2rΔt+σ Δt , 2 2 u 2 + d 2 = 2e 2rΔt+σ Δt .

Now again we have two equations

u + d = 2erΔt (3.25 a) 2 u 2 + d 2 = 2e 2rΔt+σ Δt . (3.25 b)

From these two equations we can get the value of u, d which are given below:

2 u = erΔt (1+ eσ Δt −1) ,

2 d = erΔt (1− eσ Δt −1) , 1 q = . 2

This proves that the binomial model approximates the lognormal price process. To conclude the decision about the equivalence of two models.

Binomial Approximation Methods for Option Pricing

3.2.5 Deriving the Black-Scholes Equation using the Binomial Model

In this section we will show that given the risk neutral binomial process, we can derive the Black-Scholes equation from the risk-neutral expectation formula given below:

−rT V0 = e Eq [ f ].

Consider the single period binomial model. Let the current spot price be S. The risk neutral expectation is given below:

Eq []V (S,0) = qV (Su,Δt) + (1− q)V (Sd,Δt) . (3.26)

Expand V (Su,Δt) in Taylor series:

V (Su,Δt ) = V (S + S(u −1),Δt) 1 = V (S,Δt) +V ′(S,Δt)S(u −1) + V ′′(S,Δt)S 2 (u −1) 2 + O(u 3 ) . 2

Expand V (Sd,Δt) in Taylor Series:

V (Sd,Δt) = V (S + S(d −1),Δt) 1 = V (S,Δt) +V ′(S,Δt)S(d −1) + V ′′(S,Δt)S 2 (d −1) 2 + O(d 3 ) . 2

Put the values of V (Su,Δt ) and V (Sd,Δt) in equation 3.26

⎡ 1 2 2 3 ⎤ Eq []V (S,0) = q V (S,Δt) +V ′(S,Δt)S(u −1) + V ′′(S,Δt)S (u −1) + O(u ) ⎣⎢ 2 ⎦⎥ ⎡ 1 ⎤ + (1− q) V (S,Δt) +V ′(S,Δt)S(d −1) + V ′′(S,Δt)S 2 (d −1) 2 O(d 3 ) , ⎣⎢ 2 ⎦⎥

⎡ 1 1 ⎤ = q V (S,Δt) + SuV ′(S,Δt) − SV ′(S,Δt) + u 2V ′′(S,Δt)S 2 + V ′′(S,Δt)S 2 − uV ′′(S,Δt)S 2 ⎣⎢ 2 2 ⎦⎥ ⎡ 1 1 ⎤ + (1− q) V (S,Δt) +V ′(S,Δt)dS − SV ′(S,Δt) + d 2V ′′(S,Δt)S 2 + V ′′(S,Δt)S 2 − dV ′′(S,Δt)S 2 ⎣⎢ 2 2 ⎦⎥

Binomial Approximation Methods for Option Pricing

1 1 = qV (S,Δt) + SuV ′(S,Δt)q − SV ′(S,Δt)q + u 2V ′′(S,Δt)S 2 + V ′′(S,Δt)S 2 − uV ′′(S,Δt)S 2 2 2 1 1 +V (S,Δt) +V ′(S,Δt)dS − SV ′(S,Δt) + d 2V ′′(S,Δt)S 2 + V ′′(S,Δt)S 2 − dV ′′(S,Δt)S 2 − qV (S,Δt) 2 2 1 1 − qdV ′(S,Δt)S + qV ′(S,Δt)S − qd 2V ′′(S,Δt)S 2 − qV ′′(S,Δt)S 2 + qdV ′′(S,Δt)S 2 , 2 2 1 = V (S,Δt) +V ′(S,Δt)S[]q(u −1) − (1− q)(d −1) + V ′′(S,Δt)S 2 (u −1) 2 + O(u 3 ) . 2

We will use the equality

qu + (1− q)d = e rΔt , and by risk neutral argument, this amount be equal to

V (S,0)e rΔt = V (S,0)(1+ rΔt) + O(δt 2 )

1 = V (S,Δt) +V ′(S,Δt)S[erΔt −1]+ V ′′(S,Δt)S 2 (u −1) 2 + O(u 3 ) , 2 1 3 = V (S,Δt) +V ′′(S,Δt)SrΔt + V ′′(S,Δt)S 2σ 2 Δt + O(Δt 2 ) . 2

By the risk-neutral argument, this must be equal to

V (S,0)e rΔt = V (S,0)(1+ rΔt) + O(Δt 2 ) .

Rearranging

1 3 V (S,Δt) −V (S,0) + S 2σ 2V ′′(S,Δt) + rSV ′(S,Δt)Δt − rV (S,0)Δt + O(Δt 2 ) = 0 , 2 when the limit Δt → 0 , we will finally get the Black-Scholes partial differential equation

∂V 1 ∂ 2V ∂V + S 2σ 2 + rS −Vr = 0 . ∂ f 2 ∂S 2 ∂S

Binomial Approximation Methods for Option Pricing

3.2.6 Constant Dividend Yield

Suppose we have an asset with constant dividend yield D, buying φ units of the asset at DΔt DΔt the current spot price S0 worth φe S0u if the price goes up and φe S0 d if the price goes down.

DtΔ φeSu0

φS0

DtΔ φeSd0

Figure 3.8: Binomial model of an asset with constant dividend yield,

buyingφ units of the asset at S0 .

−DΔt If we multiply the tree by e then we will get the binomial tree with (S0u, S0 d ) at each −DΔt node. But the initial value changes to S0e . This is illustrated below:

φSu0

−ΔD t φSe0

φSd0

Figure 3.9: Binomial model with constant dividend yield buyingφ eDΔt units instead makes the end nodes value as in the no- dividend model.

Binomial Approximation Methods for Option Pricing

−DΔt Put the values of S0e in equation 3.13(a, b) we will get

S e−DΔt erΔt − S d q = 0 0 S0u − S0d (e(r−D)Δt − d)S q = 0 , (u − d)S0 e(r−D)Δt − d q = . u − d

This means that in the derivation of u and d, we will use r – D instead of r in the end. We will get following values of u and d

1 (r−D− σ 2 )Δt+σ Δt u = e 2 , (3.27 a) 1 (r−D− σ 2 )Δt−σ Δt d = e 2 , (3.27 b) e(r−D)Δt − d q = . (3.27 c) u − d

Su0

−ΔD t Se0

Sd0

Figure 3.10: If we discount the present value of the asset by its dividend yield D, we can apply the no dividend mode with risk-free rate (r - D) instead of r.

Binomial Approximation Methods for Option Pricing

3.2.7 The Black-Scholes Pricing Formula for European Options

The European option are valued using Black-Scholes Option Pricing formula. The Black- Scholes equation solved European call and put options and European binary option. The most straightforward way to derive the formula is by risk-neutral expectation formula which is given below:

−rT XT V (S0 ,0) = e E[V (S0e ,T )].

The formula will be used a reference when testing the computational methods. Take the

European put option P(St ,t ) . It payoff at expiry t = T is (K − ST ) + , so by the value of the option at t = 0 can be expressed as:

−rT P(S0 ,0) = e E[(K − ST ) + ]

−rT XT = e E[(K − S0e ) + ].

Denote the expectation value restricted to this domain by E+ so by the linearity of the expectation operator we can write it as:

−rT XT P(S0 ,0) = e (KE+ [1]− S0 E+ [e ]) . (3.28)

For standard normal random variable Z, the expectation value E+ [ f (Z)] for Z < z0 is given by:

1 z0 1 E f (Z) = f (z)exp(− z 2 )dz. + [] ∫ 2π −∝ 2

If X T is the stochastic process defined in equation 3.19 also given below:

1 X = (r − σ 2 )t + σW , t 2 t for t = T one has 1 X = (r − σ 2 )T + σW . T 2 T

Then the standard normal random variable can be defined as:

1 2 X T − (r − σ )T Z = 2 , (3.29) σ T

Binomial Approximation Methods for Option Pricing

the X T is restricted to − ∝< X T < log(K / S0 )

2 K ⎛ ⎞ log ⎡ 1 2 ⎤ S0 ⎜ x − (r − σ )T ⎟ 1 ⎜ 1 ⎢ 2 ⎥ ⎟ E+ []1 = exp − ⎢ ⎥ dx . σ 2πT ∫ ⎜ 2 σ T ⎟ −∝ ⎜ ⎢ ⎥ ⎟ ⎝ ⎣⎢ ⎦⎥ ⎠

By the variable change (3.29), we obtain

1 z1 1 E 1 = exp(− z 2 )dz ≡ φ(z ) , (3.30) + [] ∫ 1 2π −∝ 2 whereφ(.) is the cumulative normal distribution function defined by the improper integral on the left hand side, and z1 is obtained by the variable change in (3.29):

K 1 log − (r − σ 2 )T S0 2 z1 = . (3.31) σ T

Similarly we will compute

2 K ⎛ ⎞ log ⎡ 1 2 ⎤ S0 ⎜ x − (r − σ )T ⎟ 1 1 ⎢ ⎥ XT ⎜ 2 ⎟ E+ []e = exp x − ⎢ ⎥ dx . σ 2πT ∫ ⎜ 2 σ T ⎟ −∝ ⎜ ⎢ ⎥ ⎟ ⎝ ⎣⎢ ⎦⎥ ⎠

We will obtain the following, by changing the variable and completing the square

z e rT 2 ⎛ 1 ⎞ E e XT = exp − z 2 dz = e rT φ(z ) , + [] ∫ ⎜ ⎟ 2 2π −∝ ⎝ 2 ⎠ where

z2 = z1 −σ T ,

K 1 log − (r + σ 2 )T S0 2 z2 = . (3.32) σ T

Binomial Approximation Methods for Option Pricing

Applying the expectation formulas to equation 3.28 yields the Black-Scholes put option formula:

−rT P(S0 ,0) = Ke φ(z1 ) − S0φ(z2 ) . (3.33)

The Black-Scholes call option formula is derived easily by using the put-call parity formula which is given below:

C + Ke −r(T −t) = P + S C − P = S − Ke −rT .

Putting the values of put option in above equation, we will get:

−rT −rT C(S0 ,0) − (Ke φ(z1 ) − S0φ(z2 )) = S0 − Ke −rT −rT C(S0 ,0) = S0 − S0φ(z2 ) − Ke + Ke φ(z1 ) , −rT C(S0 ,0) = S0 (1−φ(z2 )) − Ke (1−φ(z1 )) , which is equivalent to

−rT C(S0 ,0) = S0φ(−z2 ) − Ke φ(−z1 ) . (3.34)

3.2.7.1 Example

Consider the situation where the stock price six months from the expiration of an option 42 SEK, the exercised price of the option is 40 SEK, the risk free interest rate is 10% per annum, and the volatility is 0.2 per annum. What is the price of European call and put option?

We have that

S0 = 42 K = 10 r = 0.1 T = 0.5 σ = 0.2 ,

K 1 log − (r − σ 2 )T S0 2 z1 = . σ T

Put the values, we will get

40 1 log − (0.1− 0.22 )0.5 42 2 z1 = , 0.2 0.5

Binomial Approximation Methods for Option Pricing

z1 = -0.6278, K 1 log − (r + σ 2 )T S0 2 z2 = , σ T 40 1 log − (0.1+ 0.22 )0.5 42 2 z2 = , 0.2 0.5

z2 = -0.7693, and

Ke −rT = 40e −0.1*0.5 , Ke −rT = 40e−0.05 , Ke −rT = 38. 049.

The Black-Scholes put option formula is:

−rT P(S0 ,0) = Ke φ(z1 ) − S0φ(z2 ) = 38.049φ(−0.6278) − 42φ(−0.7693 ) , = 0.81.

The Black-Scholes call option formula is:

−rT C(S0 ,0) = S0φ(−z2 ) − Ke φ(−z1 ) = 42φ(0.7693) − 38.049φ(0.6278) , = 4.76.

3.2.7.2 Black-Scholes Pricing Formula for Binary puts and calls

The binary put option value is simply

−rT PB (S0 ,0) = e E+ [1].

By equation 3.30 we can write above equation as

−rT PB (S0 ,0) = e φ(z1 ) . (3.35)

By the binary put call parity formula we obtain the binary call option formula which is given below:

−rT CB (S0 ,0) = e φ(−z1 ) . (3.36)

Binomial Approximation Methods for Option Pricing

3.2.7.2 Black-Scholes Pricing Formula for Constant dividend yield

If the underlying asset yields a constant dividend yield D. the formulas we derived above simply modified by replacing the risk-free rate r with r – D. alternatively we can replace −DT the spot price S0 with S0e . This also affects the variables.

3.3 Comparison and Results

Figure 3.11: The Black-Scholes solution for European put option.

Figure 3.11 shows the Black-Scholes solution for European put options using Black- Scholes pricing formula stated in equation 3.33, using these parameters r = 0.05, σ = 0.3, the thick continuous curve has expiry 1.0 and the thin continuous curve has expiry 7.0, −rT the payoff function is shown as a dotted line, the curve touchesV0 at V0 = Ke .

Binomial Approximation Methods for Option Pricing

Figure 3.12: The Black-Scholes solution for European put option with constant dividend yield D.

Figure 3.12 shows the Black-Scholes solution for European put options with constant dividend yield, using Black-Scholes pricing formula stated in equation 3.33, using these parameters r = 0.05, σ = 0.3, D = 0.10, the thick continuous curve has expiry 1.0 and the thin continuous curve has expiry 7.0, the payoff function is shown as a dotted line, the −rT curve touchesV0 at V0 = Ke .

Binomial Approximation Methods for Option Pricing

Figure 3.13: The Black-Scholes solution for European call option.

Figure 3.13 shows the Black-Scholes solution for European call options using Black- Scholes pricing formula stated in equation 3.34, using these parameters r = 0.05, σ = 0.3, the thick continuous curve has expiry 1.0 and the thin continuous curve has expiry 7.0, the payoff function is shown as a dotted line, as ST →∝ the value of the call approaches −rT −rT to S0 − Ke , in time the value − Ke approaches to zero so the value of the call obviously approaches that of the asset itself S0 .

Binomial Approximation Methods for Option Pricing

Figure 3.14: The Black-Scholes solution for European call option with constant dividend yield D.

Figure 3.14 shows the Black-Scholes solution for European call options with constant dividend yield using Black-Scholes pricing formula stated in equation 3.34, using these parameters r = 0.05, σ = 0.3, D = 0.10 the thick continuous curve has expiry 1.0 and the thin continuous curve has expiry 7.0, the payoff function is shown as a dotted line, −DT −rT −rT as ST →∝ the value of the call approaches to S0e − Ke , in time the value − Ke −DT and S0e both terms approaches to zero value of the call gradually loses its value.

Binomial Approximation Methods for Option Pricing

Figure 3.15: The Black-Scholes solution for European binary put option.

Figure 3.15 shows the Black-Scholes solution for European binary put options using Black-Scholes pricing formula stated in equation 3.35, using these parameters r = 0.05, σ = 0.3, the thick continuous curve has expiry 1.0 and the thin continuous curve has expiry

7.0, the payoff function is shown as a dotted line, as ST = 0 the curve touches the V0 at e −rT .

Figure 3.16: The Black-Scholes solution for European binary call option.

Figure 3.16 shows the Black-Scholes solution for European binary call options using Black-Scholes pricing formula stated in equation 3.35, using these parameters r = 0.05, σ = 0.3, the thick continuous curve has expiry 1.0 and the thin continuous curve has expiry −rT 7.0, the payoff function is shown as a dotted line, as ST > K , the curve is near e .

Binomial Approximation Methods for Option Pricing

Chapter 4

The Binomial Model for European Option

A European option gives its holder the right, but not the obligation, to sell or buy a prescribed asset for prescribed price at some prescribed time in the future. The Black- Scholes partial differential equation under some assumptions can be used to determine the value of the option. The binomial method is a simple numerical technique for approximating the price of the European option.

4.1 Recursive Algorithm

We can approximate the value of an European option by using the recursive algorithm. Suppose that the option is taken out at time t = 0 and that the holder may exercise the option. i.e. sell the asset, at a time T called the expiry time.

4.1.1 Assumptions

1. A key assumption is that between successive time levels the asset price either moves up by a factor u > 1 or moves down by a factor d < 1. an upward movement occurs with probability p and a downward movement occurs with probability 1 – p.

2. The initial asset price S0 is known. Hence at time t = Δt the possible asset prices 2 2 are u S0 ,udS0 and d S0 .

In general at time t = ti := (i −1)Δt there are i possible asset prices which are given below:

i i−n n−1 S n = d u S0 , 1 ≤ n ≤ i

hence, at the expiry time t = tM +1 = T there are M + 1 possible asset prices. The values i S n for 1 ≤ n ≤ i and1 ≤ n ≤ M +1 form a recombining binary tree. Which is given below:

Binomial Approximation Methods for Option Pricing

Figure 4.1: Recombining binary tree for asset prices

Let K be the exercise price of the option. This means that holder of the put option may exercise the option by selling the asset price at price K at time T. Clearly the holder of the option will exercise the option whenever there is a profit to be made, that is, whenever M +1 the actual price of the asset turns out to be less than K. Hence, if the asset has price S n M +1 at time t = tM +1 = T then then value of the option will be max(K − S n ,0 ) .

i Let Vn denote the value of the option at time t = ti

M +1 M +1 Vn = max(K − S n ,0) .

1 i+1 We will calculateV1 the option value at time t = 0 andVn+1 for the later time level. The following recursive formula we will use to calculate the values:

i −r Δt i+1 i+1 Vn = e (qVn+1 + (1− q)Vn ), 1 ≤ n ≤ i 1 ≤ n ≤ M +1 where the constant r is risk-free interest rate. The values of u, d and q are given below:

Binomial Approximation Methods for Option Pricing

u = A + A2 −1 , d = 1/ u , e r Δt − d q = . u − d

1 2 Where A = ()e−r Δt + e(r+σ )Δt and constant σ represents the volatility of the asset. 2

4.2 Comparison and Results

Figure 4.2: European put option calculated by 64-Period Binomial method.

Figure 4.2 shows European put option calculated by the Binomial method, using these parameters K = 1.0, T = 1.0, σ = 0.30 and r = 0.05 where N = 64, the binomial method is applied on several spot prices. Each dot shows a spot price, while the continuous thin curve over dots is Black-Scholes solution of European put option.

Binomial Approximation Methods for Option Pricing

Figure 4.3: European call option calculated by 64-Period Binomial method.

Figure 4.3 shows European call option calculated by the Binomial method, using these parameters K = 1.0, T = 1.0, σ = 0.30 and r = 0.05 where N = 64, the binomial method is applied on several spot prices. Each dot shows a spot price, while the continuous thin curve over dots is Black-Scholes solution of European call option.

Binomial Approximation Methods for Option Pricing

Figure 4.4: Relative error(ε / S ) in European call and put option using 64-Period Binomial method.

Figure 4.4 tells us about the relative error occurred while pricing European put option and we have almost the similar results for European call option. We define error as the difference between the Binomial approximation and the value computed by Black Scholes equation.

Binomial Approximation Methods for Option Pricing

Chapter 5

The Binomial Model for American Option

5.1 American Options

Most of the options traded at exchanges are of American type; therefore, an accurate and efficient valuation of American options is important.

Because an American option can be exercised at any time, its value can never be less than the payoff. (Otherwise it would be exercised immediately)

V ≥ P(S )

If the situation V < P(S ) occurred, the option would be exercised immediately, and its payoff (its value) would be P(S).

Furthermore because the early exercise is an additional right to the exercise at expiry, an American option is at least as valuable as a European option.

The recursive formula can be used almost unchanged to value American options. Only one equation has to be modified in the following way:

n−1 −rΔt n n n−1 Vn = e max([qV j+1 + (1− q)V j , f j ]) ,

where f j is the payoff of the option.

5.1.1 American Call and Put

+ An American call option’s payoff is (Sτ − K) if it is exercised at timeτ ≤ T , similarly + an American put option’s payoff is (K − Sτ ) if it is exercised at timeτ ≤ T . The value of the American call option at time 0 can be written as:

−rτ + C(S0 ) = max E *[e (Sτ − K) ], (5.1) where the max is over all stopping times τ ≤ T. For exact justification of equation 5.1 as the appropriate pricing formula, see the reference [23] and [24].

Binomial Approximation Methods for Option Pricing

5.2 Comparison and Results

Figure 5.1: American put option calculated by 64-Period Binomial method

The figure 5.1 shows American put option calculated by the binomial method, using these parameters K = 1.0, T = 1.0, σ = 0.30 and r = 0.05 where N = 64, the value of the corresponding European put option is shown as a smooth curve (partly overlapped by the dots) the payoff function is shown as dotted curve.

Binomial Approximation Methods for Option Pricing

Figure 5.2: American call option calculated by 64-Period Binomial method.

The figure 5.2 shows American call option calculated by the binomial method, using these parameters K = 1.0, T = 1.0, σ = 0.30 and r = 0.05 where N = 64, the value of the corresponding European call option is as same as the American call option. The value of the corresponding European call option is shown as a smooth curve.

Binomial Approximation Methods for Option Pricing

Chapter 6

Summary

In this thesis we study binomial approximation methods for European as well as American options. We study options on stocks, with as well as without dividends. We also include a chapter on how to derive Black-Scholes equation from a binomial model. Our study shows how versatile the binomial method is, both from a theoretical and a practical point of view.

Binomial Approximation Methods for Option Pricing

References

[1] Kerman, Jouni. “Numerical Methods for Option Pricing: Binomial and Finite- difference Approximations”, New York University, 2002.

[2] Björk, Tomas. “Arbitrage Theory in Continuous Time”, Oxford University Press. 1998.

[3] Haugen, A. Robert. “Modern Investment Theory”, 3rd edition, Prentice Hall.

[4] Odegbile, Olufemi Olusola. “Binomial Option Pricing Model”, African Institute of Mathematical Sciences.

[5] Elke Korn and Ralf Korn. “Option Pricing”, Management Mathematics for European Schools MaMaEuSch.

[6] Higham, J. Desmond. “Nine Ways to Implement the Binomial Method for Option Valuation in Matlab”, Society for Industrial and Applied Mathematics 2002.

[7] Khyami, Aidasani. Nitesh. “Option Pricing, Java Programming and Monte Carlo Simulation”.

[8] Chao, Jung Chen. “Pricing European and American Options with Extrapolation”, National Taiwan University.

[9] Widdicks, Andricopoulos. Newton and Duck. “On the enhanced Convergence of Standard Lattice methods for option pricing”, University of Manchestor.

[10] Tian, S. Yisong. “A Flexible Binomial Option Pricing Model”, York University.

[11] Ju, Nengjiu. “An Approximate formula for Pricing American Options”, Journal of Derivatives, Winter, 1999.

[12] Cox, J.C and S.A. Ross. “Option Pricing a Simplified Approach”, Journal of Financial Economics, 1979.

[13] Dai, Min and Y.K. Kwok. “Knock-In American Options”, The journal of Future Markets, 2004.

[14] Venkatramanan, Aanand. “American Spread Option Pricing”, University of Reading, 2005.

[15] AitSahlia, F. and Carr, P, “American Options: A Comparison of Numerical Methods”.

Binomial Approximation Methods for Option Pricing

[16] Barucci, Emilio and Landi, Leonardo, “Computational Methods in Finance: Option Pricing”.

[17] Thompson, Sarah, “Simulation of Option Pricing”.

[18] Benninga, Simon and Wiener, Zvi “The Binomial Option Pricing Model”.

[19] Fischer Black and Myron Scholes. “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, 1973.

[20] Paul Wilmott, Sam Howison, and Jeff Dewyne. “The Mathematics of Financial Derivatives”, Cambridge University Press, 1995.

[21] John C. Hull. “Options, Futures, and Other Derivatives”, Prentice-Hall, 1997.

[22] Paul Wilmott. “Derivatives: The Theory and Practice of Financial Engineering”, John Wiley and Sons Ltd, 1998.

[23] Bensoussan, A. “On the Theory of Option Pricing”, Applied Mathematics 1984.

[24] Karatzas, I. “On the Pricing of American Option”, Applied Mathematics 1998.

[25] http://www.derivativesone.com/

[26] http://www.global-derivatives.com/