### 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

*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

*Binomial Approximation Methods for Option Pricing *

iv

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 assetS . *_{0}

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 *

2

**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

*Binomial Approximation Methods for Option Pricing *

4
**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:

− +

=( )

) ,

(*S* *T* *S* *K*

*C*_{E}_{T}_{T}

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.

*K *
*K *

S_{T}*Payoff *

Figure 1.1: Payoff function of the call option

Strike Price If the asset value is less than the strike price, you lose what you paid for the call.

Price of underlying asset

Net payoff on call option

*Binomial Approximation Methods for Option Pricing *

6

**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:

− +

=( )

) ,

( _{T}_{T}

*E* *S* *T* *K* *S*

*P*

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.

*K *
*K *

S_{T}*Payoff *

Figure 1.3: Payoff function of the put option

Strike Price

Price of underlying asset Net payoff on

call option

Figure 1.4: Negative Payoff function of the put option

If the asset value is greater than the strike price, you lose what you paid for put.

*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.

*K *
*K *

S* _{T}*
Figure 1.6: Payoff function of the binary put option

*K *
*K *

S_{T}*Payoff *

Figure 1.5: Payoff function of the binary call option

*Payoff *

*Binomial Approximation Methods for Option Pricing *

8 The binary put option has a payoff

1 if *S** _{T}* <

*K*

*P S*

*( )*

_{B}*=*

_{T} 0 if *S** _{T}* ≥

*K*and the binary call option has a payoff,

0 if *S** _{T}* ≤

*K*

*C S*

*( )*

_{B}*=*

_{T} 1 if *S** _{T}* >

*K*.

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 areA and*_{0} *B respectively. *_{0}
We will assume that,

*A*_{0} =*B*_{0} 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 *C will be: *_{0}

*C*_{0} = *A*_{0} −*B*_{0}
*C*_{0} =0,

*Binomial Approximation Methods for Option Pricing *

10
*while at time T the value of portfolio P(C) will be: *

*P*(*C*)* _{T}* =

*A*

*−*

_{T}*B*

_{T} *P*(*C*)* _{T}* =

*A*

_{0}

*e*

*−*

^{aT}*B*

_{0}

*e*

*,*

^{bT}*P*

^{(}

*C*

^{)}

*=*

_{T}*A*0

*e*

^{bT}### [

*e*

^{(}

^{a}

^{− T}

^{b}^{)}−

^{1}

### ]

>^{0}.

As we assumed that*A*_{0} =*B*_{0}and 1*P*(*C*)* _{T}* =

*e*

^{(}

^{a}

^{− T}

^{b}^{)}>

*, 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

*A*_{0} =*B*_{0} if *a b*< .

*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(*S** _{T}*,

*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*S** _{T}*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} *P** _{B}* +

*C*

*=*

_{B}*e*

^{−}

*, where T is the remaining time until expiry.*

^{rT}*Binomial Approximation Methods for Option Pricing *

12

**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

*S . The spot price **t* *S is the current price of the asset. *_{0}

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

*t*
*t*, ^{2}

(μ σ ) respectively. The rate of return over some time interval t is given below:

0 0

*S*
*S*
*S** _{t}* −

. The rate of return can be expressed as:

0 0

*S*
*S*
*S** _{t}* −

*Z*
*t*
*t* σ
μ +

= , (3.1)

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 byμ and also jumps up or down by a random amount*t* σ *tZ*.

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

*W** _{t}* =

*tZ*.

If we make time intervals smaller and smaller then it means the limit*t*⎯⎯→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

^{t}*dt* *dW*_{t}*S*

*dS* =μ +σ

0

, (3.2)

where *W is a stochastic variable. ** _{t}*
Letting

*S*0

*dX** _{t}* =

*dS*

*,*

^{t}*Binomial Approximation Methods for Option Pricing *

14 equation 3.2 becomes:

*dX** _{t}* =μ

*dt*+σ

*dW*

*, (3.3)*

_{t}*where the variable µ is called the drift rate.*

Using the fact that

(log )

0

*t*

*t* *d* *S*

*dS =S* ,

and

*S*0

*dX** _{t}* =

*dS*

*,*

^{t}we can write*S as *_{t}

*S** _{t}* =

*S*

_{0}

*e*

^{X}*. (3.4) This means that the logarithm of*

^{t}*S is normally distributed, hence we say that the*

*distribution of*

_{t}*S is lognormal.*

_{t}*Binomial Approximation Methods for Option Pricing *

**3.1.3 Brownian Motion **

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

Equation 3.2 tells us that the logarithm of
*S*0

*S*_{t}

is a Brownian motion with drift μ*dt*and
gives us a way to describe the changes *dS in the asset price*_{t}*S as time passes. *_{t}

Figure 3.1: Graphical representation of Brownian motion

*Binomial Approximation Methods for Option Pricing *

16

**3.1.4 Ito’s formula **

We now consider the function f depending on the asset price *S and time t. If the asset ** _{t}*
price S were a deterministic variable we would simply expand

*f*(

*S*

_{0}+Δ

*S*,Δ

*t*)at

*f(S*,0)in Taylor series:

...) ...

2 ( 1

2 ...)

( 1 ^{2}_{2} ^{2} ^{2} _{2} Δ ^{2} + +

∂ + ∂

∂ Δ + ∂ +

∂ Δ + ∂

∂ Δ

= ∂

Δ *S*

*S*
*S* *f*

*S*
*t* *f*

*t*
*t* *f*

*t*
*f* *f*

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

In case of lognormal random walk, we can write equation 3.1 as
Δ*S* =*S*_{0}μΔ*t*+σ Δ*tS*_{0}*Z*,
because *Δ depends on tS** _{t}* Δ in case of random process.

Then consider

(Δ*S*)^{2} =(*S*_{0}μΔ*t*+σ Δ*tS*_{0}*Z*)^{2},

since Z is standard normal, Z^{2} is distributed with gamma distribution with mean 1.

Therefore

*E*

### [

(σ Δ*tZ*−σ2

*S*0Δ

*t*)

^{2}

### ]

=σ2*S*0Δ*tE*

### [ ]

*Z*

^{2}+

*O*

^{(}

*t*

^{3}

^{2}

^{)}, (

^{2})

3 0

2*S* Δ*t*+*O* *t*

=σ .

In the limit Δ*t*→0

*dS*_{t}^{2} =σ^{2}*S*_{0}^{2}*dt*.

*Therefore, if f is a function dependent onS , it is also a stochastic process *_{t}*f such that *_{t}

*dt*
*S*
*S* *f*
*SdS*

*dt* *f*
*t*

*df*_{t}*f* _{t}_{2}

2 2 0 2

2 1

∂ + ∂

∂ + ∂

∂

= ∂ σ , (3.5)

*Binomial Approximation Methods for Option Pricing *

or, writing out *dS** _{t}* , we obtain Ito’s formula for the option, given the underlying asset is
a stochastic process with

*dS*

*=μ*

_{t}*S dt*

_{0}+σ

*S dw*

_{0}:

*dt*

*S*
*S* *f*
*S*

*S* *f*
*t*
*dw* *f*
*S*

*df*_{t}*f* * _{t}* )

2

( 1 _{2}

2 2 0 2

0 ∂

+ ∂

∂ + ∂

∂ + ∂

∂

= ∂ σ μ σ . (3.6)

The stochastic variable *dW present in the formula; this means that the option price ** _{t}*
)

,
(*S* *t*

*f* * _{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 by*σ . 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:

) .

2

( 1 _{2}

2 2 0 2

0 *dt*

*S*
*S* *f*
*S*

*S* *f*
*t*
*dw* *f*
*S*
*df*_{t}*f* _{t}

∂ + ∂

∂ + ∂

∂ + ∂

∂

= ∂ σ μ σ

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 *

18

**3.1.6 The Black-Scholes Equation **

Assume the Ito’s formula

*dt*

*S*
*S* *f*
*S*

*S* *f*
*t*
*dw* *f*
*S*

*df*_{t}*f* * _{t}* )

2

( _{0} 1 ^{2} _{0}^{2} ^{2}_{2}

∂ + ∂

∂ + ∂

∂ + ∂

∂

= ∂ σ μ σ .

We have a stochastic process *f** _{t}* depending on another process

*S*

*. Construct a portfolio*

_{t}*consisting of one option and a short position G units on the stock.*

The value of this portfolio is

π_{0} = *f*_{0} −*GS*_{0},
*after one time-step dt, the portfolio will charged by *

*d*π* _{t}* =

*df*

*−*

_{t}*GdS*

*. (3.7) Apply Equation 3.5 to Equation 3.7*

_{t}

) .

2

( 1 _{2}

2 2 0 2

0 *dt*

*S*
*S* *f*
*S*

*S* *f*
*t*
*dw* *f*
*S*
*df*_{t}*f* _{t}

∂ + ∂

∂ + ∂

∂ + ∂

∂

= ∂ σ μ σ

We will get the following equation

_{t}_{t}*dt* *GdS*_{t}*S*

*S* *f*
*SdS*

*dt* *f*
*t*

*d* *f* −

∂ + ∂

∂ + ∂

∂

= ∂ ^{2} _{0}^{2} ^{2} _{2}

2 1σ

π ,

*dt*

*S*
*S* *f*
*dS*

*S* *G*
*dt* *f*
*t*

*d* _{t}*f* _{t}_{2}

2 2 0 2

2 ) 1

( ∂

+ ∂

∂ − + ∂

∂

= ∂ σ

π .

To eliminate the random component*dS** _{t}* , we choose

*S*
*G* *f*

∂

= ∂ * at t = 0 *

*dt*

*S*
*S* *f*
*S* *dS*

*f*
*S*
*dt* *f*
*t*

*d* _{t}*f* _{t}_{2}

2 2 0 2

2 ) 1

( ∂

+ ∂

∂

− ∂

∂ + ∂

∂

=∂ σ

π

*dt*
*S*
*S* *f*
*t* *dt*

*d* _{t}*f* _{2}

2 2 0 2

2 1

∂ + ∂

∂

=∂ σ

π ,

*dt*
*S*
*S* *f*
*t*

*d* _{t}*f* )

2

( 1 ^{2} _{0}^{2} ^{2} _{2}

∂ + ∂

∂

= ∂ σ

π .

*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 *r*π_{0}*dt during time step dt. The replicated option on *
deterministic portfolio on the other hand yields *d*π . Their yields must be equal, so _{t}*r*π_{0}*dt* =*d*π* _{t}*.

Put the value of *d*π in the above equation _{t}

*dt*
*S*
*S* *f*
*t*

*d* _{t}*f* )

2

( 1 _{2}

2 2 0 2

∂ + ∂

∂

= ∂ σ

π ,

*r*π_{0}*dt* = *dt*
*S*
*S* *f*
*t*

*f* )

2

( 1 _{2}

2 2 0 2

∂ + ∂

∂

∂ σ . (3.8)

Since we know that

π_{0} = *f* −*GS*_{0},

_{0} *S*_{0}
*S*
*f* *f*

∂

− ∂

π = .

Put the values of π_{0}in equation 3.8

=

∂

− ∂ *S* *dt*
*S*
*f* *f*

*r*( _{0}) *dt*

*S*
*S* *f*
*t*

*f* )

2

( 1 _{2}

2 2 0 2

∂ + ∂

∂

∂ σ ,

=

∂

− ∂ )

( *S*_{0}

*S*
*f* *f*

*r* )

2

( 1 _{2}

2 2 0 2

*S*
*S* *f*
*t*

*f*

∂ + ∂

∂

∂ σ .

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* =*S*_{0}, the equation is usually written in
this form

1 ∂^{2} ∂

∂*f* *f* *f*

*Binomial Approximation Methods for Option Pricing *

20 or equivalently

*rf*
*S*
*rS* *f*
*S*
*S* *f*
*t*

*f* =

∂ + ∂

∂ + ∂

∂

∂

2 2 2 2

2

1σ .

*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 beS . After an *_{0}
*infinitesimal timestep dt, the holder of the asset gainsDS*_{0}*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 atS and selling it immediately at *_{0} *S*_{0} +*dS*_{t}*dt* after
*receiving the dividend would yield a risk-free profit of DSdt. Thus we must have this *
equation,

*dS** _{t}* =σ

*S*

_{0}

*dW*

*+μ*

_{t}*S*

_{0}

*dt*−

*DS*

_{0}

*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:

*rf*
*S*
*S* *f*
*S*

*S* *f*
*D*
*t* *r*

*f* =

∂ + ∂

∂

− ∂

∂ +

∂

2 2 2 2

2 ) 1

( σ . (3.11)

*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):

Figure 3.2: One period binomial model

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

Thus, we can say that an investment of *S dollars at time t = 0 yields *_{0} *S*_{0}*e** ^{rT}* dollars at

*time t = T. The no arbitrage argument is also valid here, we must require that*

*S*_{0}*d* <*S*_{0}*e** ^{rT}* <

*S*

_{0}

*u*or

*d* <*e** ^{rT}* <

*u*. (3.12)

*S u*0

*S d *0

*S*0

*Binomial Approximation Methods for Option Pricing *

22 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.

If*e** ^{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

0
)
(*d*−*e** ^{rT}* >

φ * at time t = T. If e** ^{rT}* = such a market position (long asset, short bond)

*d*would yield a positive value. If

*e*

*≥ then market position will be reversed (long bond,*

^{rT}*u*short asset) yield a positive value.

Suppose now that the option yields *f and *_{u}*f if the underlying asset goes up or down ** _{d}*
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

Δ*S*_{0}*u*+ψ*e** ^{rT}* =

*f*

*, (3.13 a) Δ*

_{u}*S*

_{0}

*d*+ψ

*e*

*=*

^{rT}*f*

*. (3.13 b) This is system of two equation with two unknowns (Δ ,ψ ) there is a unique solution exist if and only if*

_{d}*u*≠

*d*

*d*
*S*
*u*
*S*

*f*
*f*_{u}_{d}

0

0 −

= −

Δ ,

*d*
*u*

*uf*
*e* ^{rT}*uf*^{d}^{u}

−

= ^{−} −

ψ .

*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 is*V *_{0}

*V*_{0} =Δ*S*_{0} +Ψ,
put the values of Δ and Ψ , we will get,

*d*
*u*

*df*
*e* *uf*

*d* *S*
*S*
*u*
*S*

*f*

*f*_{u}_{d}_{rT}_{d}_{u}

− + −

−

= − _{0} ^{−}

0 0

,

*d*
*u*

*df*
*e* *uf*

*d* *S*
*u*
*S*

*f*

*f*_{u}_{d}_{rT}_{d}_{u}

− + −

−

= − _{0} ^{−}

0( ) ,

*Binomial Approximation Methods for Option Pricing *

*d*
*u*

*df*
*e* *uf*

*d*
*u*

*f*

*f*_{u}_{d}_{rT}_{d}_{u}

− + −

−

= − ^{−} .

Thus

*d*
*u*

*df*
*uf*
*e*
*f*

*V* *f* ^{d}^{u}

*rT*
*d*
*u*

−

− +

= − ^{−} ( )

0 .

We introduce a new variable

*d*
*u*

*d*
*q* *e*

*rT*

−

= − .

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

*V*_{0} =*e*^{−}^{rT}

### [

*qf*

*+(1−*

_{u}*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

^{V}_{0} =* ^{e}*−

*rT*

^{E}*q*

### [ ]

*, (3.15)*

^{f}*where the expectation is taken under the probability measure given by q. This measure*has the special property that if

*V*

_{T}*is the value of option at t = T,*

*E*_{q}

### [ ]

*V*

*=(Δ*

_{T}*S*

_{0}+ψ)

*e*

*,*

^{rT}*E*

_{q}### [ ]

*V*

*=*

_{T}*V*

_{0}

*e*

*.*

^{rT}This probability measure is called risk-neutral probability measure.

*Binomial Approximation Methods for Option Pricing *

24

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

80 85 90 95 100 105 110 115 120 125

Figure 3.3: 2-Period Binomial model with constant dividend yield
σ =0.2, *r*=0.1,*N* =2, 100*S*_{0} = , *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* = *N*Δ*t* as the
number of periods increased, time step *Δt* →0 the distribution of

0

log*S*
*S*_{T}

approaches to
*normal distribution. In multi-period model the expiry of the option T is divided into two *
equal time-steps*T* *= 2 , a risky asset moves upward by a factor of u and downward by a *Δ*t*
*factor of d. *

Figure 3.4(a): Two period binomial model Figure 3.4(b): Three period binomial model This recombining binomial tree has the end asset values

### (

2 0 0^{2}

### )

0*u* ,*S* *ud*,*S* *d*

*S* at time

*t = T* *= 2 . Suppose now the option payoff function is f(S). Let the three option *Δ*t*
*possible values at t = T be then *

*f** _{u}* =

*f*(

*S*

_{0}

*u*

^{2}),

*f*

*=*

_{m}*f*(

*S*

_{0}

*ud*),

*f** _{d}* =

*f*(

*S*

_{0}

*d*

^{2}).

Assuming the no-arbitrage and risk-neutral, we can apply the following formula
^{V}_{0} ^{=}^{e}^{−}^{rT}

### [

^{qf}

_{u}^{+}

^{(}

^{1}

^{−}

^{q}^{)}

^{f}

_{d}### ]

^{, }

to each of the individual branches in this tree to obtain a value for the option step by step.

At time*T* = 2 , we will get either of the two values of the option. Δ*t*

*V*_{1} =*e*^{−}^{rT}

### [

*qf*

*+(1−*

_{u}*q*)

*f*

_{m}### ]

,2

*S u*0

*S u d*0

2

*S d*0

*S*0

3

*S u*0

2

*S u d*0

2

*S ud*0

3

*S d*0

*S*0

*Binomial Approximation Methods for Option Pricing *

26 Applying the formula once again

* ^{V}*0 =

^{e}^{−}

^{r}^{Δ}

^{t}### [

*1*

^{qv}*+(1−*

^{u}*)*

^{q}*1*

^{v}

^{d}### ]

. (3.16) Inserting the values of*v*

_{1}

*,*

^{u}*v*

_{1}

*we can write this as*

^{d} ^{V}_{0} ^{=}^{e}^{−}^{rT}

### (

^{q}^{2}

^{f}^{(}

^{S}_{0}

^{u}^{2}

^{)}

^{+}

^{q}^{(}

^{1}

^{−}

^{q}^{)}

^{f}^{(}

^{S}_{0}

^{ud}^{)}

^{+}

^{(}

^{1}

^{−}

^{q}^{)}

^{2}

^{f}^{(}

^{S}_{0}

^{d}^{2}

^{)}

### )

^{, }

or

^{2} (1 ) ( _{0} ^{2} )

0

2 0

*j*
*j*
*j*

*j*
*j*

*rT* *q* *q* *f* *S* *u* *d*

*e*

*V* ^{−}

=

−

−

### ∑

^{−}

= .

*For the N-Period model, where T* =*N*Δ*t*, we obtain

(1 ) ( _{0} )

0 0

*j*
*N*
*N* *j*

*j*

*j*
*N*
*j*

*rT* *q* *q* *f* *S* *u* *d*

*J*
*e* *N*

*V* ^{−}

=

−

−

### ∑

⎟⎟^{−}

⎠

⎜⎜ ⎞

⎝

= ⎛ . (3.17)

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

^{f}^{(}^{S}_{0}^{u}^{j}^{d}^{N}^{−}^{j}^{)}^{=}^{e}^{−}^{r}^{Δ}^{T}

### (

^{qf}^{(}

^{S}_{0}

^{u}

^{j}^{+}

^{1}

^{d}

^{N}^{−}

^{j}^{)}

^{+}

^{(}

^{1}

^{−}

^{q}^{)}

^{f}^{(}

^{S}_{0}

^{u}

^{j}

^{d}

^{N}^{+}

^{1}

^{−}

^{j}^{)}

### )

. (3.18) Replacing 3.18 into 3.17

### (

^{(}

^{1}

^{)}

^{(}

^{)}

### )

) 1

( _{0}

1 ) ( 0

1 ) ( 0

*j*
*N*
*j*
*j*

*N*
*N* *j*

*j*
*t*
*T*
*r*
*N*
*N*

*T*
*T*

*r* *q* *q* *f* *S* *u* *d*

*j*
*N*
*j*

*e* *N*
*u*
*S*
*f*
*q*
*e*

*V* ^{−} ^{−}

= Δ +

− +

Δ +

− ⎥ −

⎦

⎢ ⎤

⎣

⎡ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + −

⎟⎟⎠

⎜⎜ ⎞

⎝ + ⎛

=

### ∑

+*e*^{−}^{r}^{(}^{T}^{+}^{Δ}^{t}^{)}(1−*q*)^{N}^{+}^{1}*f*(*S*_{0}*d** ^{N}*),

as we know

⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛ +

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

*j*
*N*
*j*

*N*
*j*

*N* 1

1 .

The above equation becomes:

1 (1 ) ( _{1} )

0 1

0

1 )

) 1 ( (

0 *N* *j* *N* *j*

*j*

*j*
*N*
*j*

*t*
*N*

*r* *q* *q* *f* *S* *u* *d*

*j*
*e* *N*

*V* ^{+} ^{+}^{−}

=

− + Δ

+

−

### ∑

⎟⎟⎠^{−}

⎜⎜ ⎞

⎝

= ⎛ + .

*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, 100*S*_{0} = *, 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 *

** ***S*_{0} =9** K =10 r = 0.06 *** T = 3 *σ =0.3* Δt* =0.75**, **
*and *

** ***u*=*e*^{σ} ^{Δ}* ^{t}* =

*e*

^{0}

^{.}

^{3}

^{.}

^{075}=1.297

**,**

*d*=

*e*

^{−}

^{σ}

^{Δ}

*=*

^{t}*e*

^{−}

^{0}

^{.}

^{3}

^{.}

^{075}=0.771

**,**

*Binomial Approximation Methods for Option Pricing *

28
** **

*d*
*u*

*d*
*q* *e*

*t*
*r*

−

= ^{Δ} − **, **

** **

771 . 0 297 . 1

771 .

75 0

. 0

* 06 . 0

−

= *e* −

*q* **, **

** **

525 . 0

771 .

75 0

. 0

* 06 .

0 −

= *e*

*q* **, **

** ***q*=0.523**. **
** **1*− q*=0.477**. **

*Figure below shows the binomial tree for this example. Each node has a formula, like *

4
0*u*

*S* *, 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 *
*calculated by the formula *max(*K* −*S*_{0}*u*^{j}*d*^{N}^{−}* ^{j}*,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:**) 0 ,

max( _{0}

0 , 4

*j*
*N*
*j**d*
*u*
*S*
*K*

*f* = − ^{−}

) 0 ), 771 . 0

* 297 . 1

* 9 ( 10

max( ^{0} ^{4}

0 ,

4 = −

*f* **, **

* = 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: *

** ** *r* *t*

### [

*i*

*j*

*i*

*j*

### ]

*j*

*i* *e* *qf* *q* *f*

*f*_{,} = ^{−}^{Δ} _{+}_{1}_{,} _{+}_{1}+(1− ) _{+}_{1}_{,} **. **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 *

** ** *f*3,1 =*e*^{−}^{r}^{Δ}^{t}

### [

*qf*3

_{+}1,1

_{+}1 +(1−

*q*)

*f*3

_{+}1.1

### ]

** ** 0.06*0.75

### [

4,2 4.1### ]

1 ,

3 *e* 0.523*f* 0.477*f*

*f* = ^{−} + **, **

** ** *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 *

** ** *f*_{0}_{,}_{0} =*e*−* ^{r}*Δ

^{t}### [

*qf*

_{0}+

_{1}

_{,}

_{0}+

_{1}+

^{(}

^{1}−

*q*

^{)}

*f*

_{0}+

_{1}

_{.}

_{0}

### ]

** ** *f*_{0}_{,}_{0} =*e*^{−}^{0}^{.}^{06}^{*}^{0}^{.}^{75}

### [

0.523*f*

_{1}

_{,}

_{1}+0.477

*f*

_{1}

_{.}

_{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. *