Binomial approximation methods for option pricing303745/FULLTEXT01.pdf · Binomial Approximation...

Binomial approximation methods for optionpricing

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

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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.


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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


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


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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 0S . 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].


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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


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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.


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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.


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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:

+−= )(),( KSTSC TTE

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

Figure 1.2: Negative Payoff function of the call option


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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:

+−= )(),( TTE SKTSP

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.


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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


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The binary put option has a payoff 1 if TS K< ( )B TP S = 0 if TS K≥ and the binary call option has a payoff, 0 if TS K≤ ( )B TC S = 1 if TS K> . The binary options are example of options with a discontinuous payoff.


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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 0A and 0B respectively. We will assume that, 00 BA = if ba > . 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 0C will be: 000 BAC −= 00 =C ,


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while at time T the value of portfolio P(C) will be: TTT BACP −=)( bTaT

T eBeACP 00)( −= , ( )[ ] 01)( 0 >−= − TbabT

T eeACP . As we assumed that 00 BA = and 1)( )( >= − Tba

T eCP , 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 00 BA = 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.


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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 )( tTreX −− . Portfolio B: one European put option plus one share. Both are worth ),max( XST at expiration of the options. Hence the portfolios must have the same value at the present time. Therefore: SPeXC tTr +=+ −− )( , 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 TS 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 rTe− : rT

BB eCP −=+ , where T is the remaining time until expiry.


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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 $ )( tTre −− is invested at time t. 6. Borrowing and lending at the risk-free rate in possible.


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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

tS . The spot price 0S 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

tt 2,( σμ ) respectively. The rate of return over some time interval t is given below:

0

0

SSSt − .

The rate of return can be expressed as:

0

0

SSSt − Ztt σμ += , (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 by tμ and also jumps up or down by a random amount Ztσ . We will denote the random component of equation 3.1 by a single variable. Let us write: ZtWt = . If we make time intervals smaller and smaller then it means the limit 0⎯→⎯t , 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

tt dWdt

SdS

σμ +=0

, (3.2)

where tW is a stochastic variable. Letting

0S

dSdX t

t = ,


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equation 3.2 becomes: tt dWdtdX σμ += , (3.3) where the variable µ is called the drift rate. Using the fact that

)(log0

tt Sd

SdS

= ,

and

0S

dSdX t

t = ,

we can write tS as tX

t eSS 0= . (3.4) This means that the logarithm of tS is normally distributed, hence we say that the distribution of tS is lognormal.


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3.1.3 Brownian Motion The variable tdW 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.

Equation 3.2 tells us that the logarithm of 0S

St is a Brownian motion with drift dtμ and

gives us a way to describe the changes tdS in the asset price tS as time passes.

Figure 3.1: Graphical representation of Brownian motion


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3.1.4 Ito’s formula We now consider the function f depending on the asset price tS and time t. If the asset price S were a deterministic variable we would simply expand ),( 0 tSSf ΔΔ+ at )0,(Sf in Taylor series:

......)21(...)

21( 2

2

22

2

2

++Δ∂∂

+Δ∂∂

++Δ∂∂

+Δ∂∂

=Δ SS

fSSft

tft

tff

The Ito calculus is stochastic process equivalent to Newtonian differential calculus. In the limit tΔ 0→ , terms of tΔ 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 ZSttSS 00 Δ+Δ=Δ σμ , because tSΔ depends on tΔ in case of random process. Then consider 2

002 )()( ZSttSS Δ+Δ=Δ σμ ,

since Z is standard normal, Z 2 is distributed with gamma distribution with mean 1. Therefore [ ]2

02 )( tSZtE Δ−Δ σσ

[ ] )( 23

20

2 tOZtES +Δ= σ ,

)( 23

02 tOtS +Δ= σ .

In the limit tΔ 0→ dtSdSt

20

22 σ= . Therefore, if f is a function dependent on tS , it is also a stochastic process tf such that

dtS

fSdSSfdt

tfdf tt 2

220

2

21

∂∂

+∂∂

+∂∂

= σ , (3.5)


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or, writing out tdS , we obtain Ito’s formula for the option, given the underlying asset is a stochastic process with 0 0tdS S dt S dwμ σ= + :

dtS

fSSfS

tfdw

Sfdf tt )

21( 2

220

20 ∂

∂+

∂∂

+∂∂

+∂∂

= σμσ . (3.6)

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

),( tSf 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 is 2σ . We define the volatility of the portfolio byσ . A risk-less portfolio thus has 0=σ , r=μ and no random component. Finding a way to eliminate the random component in the following Ito’s equation:

.)21( 2

220

20 dt

SfS

SfS

tfdw

Sfdf tt ∂

∂+

∂∂

+∂∂

+∂∂

= σμσ

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.


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3.1.6 The Black-Scholes Equation Assume the Ito’s formula

dtS

fSSfS

tfdw

Sfdf tt )

21( 2

220

20 ∂

∂+

∂∂

+∂∂

+∂∂

= σμσ .

We have a stochastic process tf depending on another process tS . Construct a portfolio consisting of one option and a short position G units on the stock. The value of this portfolio is 000 GSf −=π , after one time-step dt, the portfolio will charged by ttt GdSdfd −=π . (3.7) Apply Equation 3.5 to Equation 3.7

.)21( 2

220

20 dt

SfS

SfS

tfdw

Sfdf tt ∂

∂+

∂∂

+∂∂

+∂∂

= σμσ

We will get the following equation

ttt GdSdtS

fSdSSfdt

tfd −

∂∂

+∂∂

+∂∂

= 2

220

2

21σπ ,

dtS

fSdSGSfdt

tfd tt 2

220

2

21)(

∂∂

+−∂∂

+∂∂

= σπ .

To eliminate the random component tdS , we choose SfG∂∂

= at t = 0

dtS

fSdSSf

Sfdt

tfd tt 2

220

2

21)(

∂∂

+∂∂

−∂∂

+∂∂

= σπ

dtS

fSdttfd t 2

220

2

21

∂∂

+∂∂

= σπ ,

dtS

fStfd t )

21( 2

220

2

∂∂

+∂∂

= σπ .


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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 0π at t = 0 inverted in this security yields dtr 0π during time step dt. The replicated option on

deterministic portfolio on the other hand yields tdπ . Their yields must be equal, so tddtr ππ =0 . Put the value of tdπ in the above equation

dtS

fStfd t )

21( 2

220

2

∂∂

+∂∂

= σπ ,

dtr 0π = dtS

fStf )

21( 2

220

2

∂∂

+∂∂ σ . (3.8)

Since we know that 00 GSf −=π ,

00 SSff∂∂

−=π .

Put the values of 0π in equation 3.8

=∂∂

− dtSSffr )( 0 dt

SfS

tf )

21( 2

220

2

∂∂

+∂∂ σ ,

=∂∂

− )( 0SSffr )

21( 2

220

2

SfS

tf

∂∂

+∂∂ 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 0SS = , the equation is usually written in this form

021

2

222 =−

∂∂

+∂∂

+∂∂ rf

SfrS

SfS

tf σ , (3.9)


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or equivalently

rfSfrS

SfS

tf

=∂∂

+∂∂

+∂∂

2

222

21σ .

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 0S . After an infinitesimal timestep dt, the holder of the asset gains dtDS0 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 0S and selling it immediately at dtdSS t+0 after receiving the dividend would yield a risk-free profit of DSdt. Thus we must have this equation, dtDSdtSdWSdS tt 000 −+= μσ . (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:

rfS

fSSfSDr

tf

=∂∂

+∂∂

−+∂∂

2

222

21)( σ . (3.11)


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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 0S is the spot price of a risky asset at time t = 0 after some time period T, it can only assume two distinct values: 0S u and 0S d, 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. Thus, we can say that an investment of 0S dollars at time t = 0 yields rTeS0 dollars at time t = T. The no arbitrage argument is also valid here, we must require that uSeSdS rT

000 << or ued rT << . (3.12)

0S u

0S d

0S


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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 uderT << , 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)( >− rTedφ at time t = T. If derT = such a market position (long asset, short bond) would yield a positive value. If uerT ≥ then market position will be reversed (long bond, short asset) yield a positive value. Suppose now that the option yields uf and df 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 u

rT feuS =+Δ ψ0 , (3.13 a) d

rT fedS =+Δ ψ0 . (3.13 b) This is system of two equation with two unknowns (Δ ,ψ ) there is a unique solution exist if and only if du ≠

dSuS

ff du

00 −−

=Δ ,

duufuf

e udrT

−−

= −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 0V Ψ+Δ= 00 SV , put the values ofΔ andΨ , we will get,

dudfuf

eSdSuS

ff udrTdu

−−

+−−

= −0

00

,

dudfuf

eSduS

ff udrTdu

−−

+−−

= −0

0 )( ,


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dudfuf

eduff udrTdu

−−

+−−

= − .

Thus

du

dfufeffV ud

rTdu

−−+−

=− )(

0 .

We introduce a new variable

dudeq

rT

−−

= .

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

rT fqqfeV )1(0 −+= − . (3.14) The no arbitrage argument guarantees that 10 << q . Thus the value of the option reduces to a certain kind of expectation formula [ ]fEeV q

rT−=0 , (3.15) where the expectation is taken under the probability measure given by q. This measure has the special property that if TV is the value of option at t = T, [ ] rT

Tq eSVE )( 0 ψ+Δ= , [ ] rT

Tq eVVE 0= . This probability measure is called risk-neutral probability measure.


24

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 380

85

90

95

100

105

110

115

120

125

Figure 3.3: 2-Period Binomial model with constant dividend yield 2.0=σ , 1.0=r , 2=N , 1000 =S , 365/180=T and D = T/N, the graph is generated through matlab.


25

3.2.2 Multi-Period Binomial Model Multi-period binomial models applied to the same total period of time tNT Δ= as the

number of periods increased, time step 0→Δ t the distribution of 0

logSST approaches to

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

002

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

0uSffu = , )( 0udSffm = , )( 2

0dSffd = . Assuming the no-arbitrage and risk-neutral, we can apply the following formula [ ]du

rT fqqfeV )1(0 −+= − , to each of the individual branches in this tree to obtain a value for the option step by step. At time tT Δ= 2 , we will get either of the two values of the option. [ ]mu

rT fqqfeV )1(1 −+= − , or [ ]du

rT fqqfeV )1(1 −+= − .

20S u

0S u d

20S d

0S

30S u

20S u d

20S ud

30S d

0S


26

Applying the formula once again [ ]dutr vqqveV 110 )1( −+= Δ− . (3.16) Inserting the values of du vv 11 , we can write this as ( ))()1()()1()( 2

02

02

02

0 dSfqudSfqquSfqeV rT −+−+= − , or

)()1( 20

2

0

20

jj

j

jjrT duSfqqeV −

=

−− ∑ −= .

For the N-Period model, where tNT Δ= , we obtain

)()1( 00

0jNj

N

j

jNjrT duSfqqJN

eV −

=

−− ∑ −⎟⎟⎠

⎞⎜⎜⎝

⎛= . (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: ( ))()1()()( 1

01

00jNjjNjTrjNj duSfqduSqfeduSf −+−+Δ−− −+= . (3.18)

Replacing 3.18 into 3.17

( ))()1(1

)( 01

)(0

1)(0

jNjjNjN

j

tTrNNTTr duSfqqjN

jN

euSfqeV −−

=

Δ+−+Δ+− −⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑

)()1( 01)( NNtTr dSfqe +Δ+− −+ ,

as we know

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛jN

jN

jN 1

1.

The above equation becomes:

)()1(1 1

0

1

0

1))1((0

jNjN

j

jNjtNr duSfqqjN

eV −++

=

−+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.


27

Figure 3.5: 20-period binomial model with constant dividend yield 2.0=σ , ,20,1.0 == Nr 1000 =S , 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 90 =S K =10 r = 0.06 T = 3 3.0=σ 75.0=Δt , and 297.1075.3.0 === Δ eeu tσ , 771.0075.3.0 === −Δ− eed tσ ,


28

dudeq

tr

−−

=Δ

,

771.0297.1771.075.0*06.0

−−

=eq ,

525.0

771.075.0*06.0 −=

eq ,

523.0=q . 477.01 =− q . Figure below shows the binomial tree for this example. Each node has a formula, like

40uS , 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 )0,max( 0

jNj duSK −− . 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( 00,4jNj duSKf −−=

)0),771.0*297.1*9(10max( 400,4 −=f ,

= max(10 -3.184, 0), = max(6.816, 0), = 6.816.

Figure 3.6: Tree used to value a stock option


29

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: [ ]jiji

trji fqqfef ,11,1, )1( +++

Δ− −+= . 10 −≤≤ Ni ij ≤≤0 For example in the case of node B (i = 3, j = 1) in figure, we have that the value of the option is [ ]1.1311,131,3 )1( +++

Δ− −+= fqqfef tr

[ ]1.42,475.0*06.0

1,3 477.0523.0 ffef += − , [ ])647.4*477.0()1*523.0(956.01,3 +=f , [ ]217.2523.0956.01,3 +=f , 740.2*956.01,3 =f , 619.21,3 =f . Similarly, for the case of node C(i = j = 0), we have that the value of the option is [ ]0.1010,100,0 )1( +++

Δ− −+= fqqfef tr

[ ]0.11,175.0*06.0

0,0 477.0523.0 ffef += − , [ ])438.2*477.0()753.0*5230.0(956.00,0 +=f , [ ]163.1394.0956.00,0 +=f , 557.1*956.00,0 =f , 488.10,0 =f . This is a numerical estimate for the option’s current value of 1.473 SEK.


30

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 ?0→Δ→∝ torN Consider the value of the underlying asset (the stock) after n periods has passed. There has been a random number say nX , ‘up-jumps’ and nXn − ’down jumps’ the value of asset is then ( )nn XnX

n duSS −= 0 . The lognormal process introduced in the previous section involves a stock price tX

t eSS 0= , where tX is given by its differential equation, it is a stochastic process with drift tμ and variance t2σ . Let the value of u, d will be following tteu Δ+Δ= σμ , tted Δ−Δ= σμ . It is clear that d < u, if we assume that arbitrage requirement ued rT << satisfies then the asset price after n periods will be: ))(()(

0nn XNttXtt

n eeSS −Δ−ΔΔ+Δ= σμσμ

tnXtn

neSS Δ−+= σμ )2(0 .

If the limit →∝n then (2 )nt X n tμ σ+ − Δ ?nX→ 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

dudeq

tr

−−

=Δ

,

tttt

tttr

eeeeq

Δ+Δ+

Δ−Δ

−

−=

σμσμ

σμ

.


31

Expanding the denominator and numerator in Taylor Series

)()

21()

21(

)()21(

23

22

23

2

tOtttttt

tOttttrq

Δ+Δ+Δ−Δ−Δ+Δ+Δ

Δ+Δ+Δ−Δ−Δ=

σσμσσμ

σσμ,

)(2

)()21(

23

23

2

tOt

tOturtq

Δ+Δ

Δ+Δ−−+Δ=

σ

σσ,

)(21

21

2

tOtr

q Δ+Δ−−

+=σ

σμ.

The random component of nS is tnX n Δ−= σ)2(

ntnX n σ)2( −= ,

tn

nX n σ)2( −

= .

Let

n

nXY n

n)2( −

= ,

now we will find the mean and variance of nY .

nX 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 nX is nq and the variance of nX is nq(1-q).


32

The mean of nY is then

[ ] [ ]n

nXEYE n

n)2( −

=

n

nnq −=

2 ,

n

qn )12( −= ,

nq )12( −= . Put the value of q in the above equation and we will get the mean of nY :

[ ] )()

21( 2

tOr

tYE n Δ+−−

=σ

σμ.

The variance of nY is

[ ] ⎥⎦

⎤⎢⎣

⎡=

nX

Y nn

2varvar

n

X nvar4= ,

n

qnq )1(4 −= ,

)1(4 qq −= . Put the value of q in the above equation and we will get the variance of nY [ ] )(1var tOYn Δ+= . In the limit 0, →Δ→∝ tN then tYnσ tends to a stochastic process tY distributed

normally with mean tr )21( 2σμ −− and the variance t2σ . We can write this as

tt WtrY σσμ +−−= )21( 2 ,


33

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

tnXtn

neSS Δ−+= σμ )2(0 converted to continuous stochastic process

tYtn eSS += μ

0 . Let tt YtX += μ , then we obtain

tt WtrX σσ +−= )21( 2 . (3.19)

We can thus approximate lognormal asset price process with binomial model by setting

ttr

euΔ+Δ−

=σσ )

21( 2

, (3.20 a)

ttr

edΔ−Δ−

=σσ )

21( 2

, (3.20 b)

dudeq

tr

−−

=Δ

. (3.20 c)

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

tt dWdtdtrdX σσσ ++−= 22

21)

21(

tt dWdtdtrdtdX σσσ ++−= 22

21

21 ,

tt dWrdtdX σ+= . This implies that in the risk neutral world the stochastic differential equation is

tt dWdtdX σμ += has drift r=μ . In other words [ ] [ ]tt dWdtEdXE σμ +=

rdtSdS

E t =⎥⎦

⎤⎢⎣

⎡

0

,

or [ ] [ ]tX

t eESSE 0= [ ] .0

trt eSSE = (3.21)

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


34

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.


35

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,

ttr

euΔ+Δ−

=σσ )

21( 2

,

ttr

edΔ−Δ−

=σσ )

21( 2

, and

du

deqtr

−−

=Δ

,

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, TX

T eSS 0= , or

0S

Se TXT = ,

where tX is tt WtrX σσ +−= )21( 2 .

The variance of TXe is: [ ] ( )[ ] [ ]22

var TTT XXX eEeEe −=

( )[ ] [ ]22 TT XX eEeE −= , where TXe has mean rTe as shown in equation 3.4. To find the mean of TXe2 , we will apply Ito calculus:

tt WtrX σσ +−= )21( 2

tt WtrX σσ 2)21(22 2 +−= .


36

To find tdX

tt dWdtdtrdX σσσ 2)2(21)

21(2 22 ++−= ,

tt dWdtdtrdtdX σσσ 2)4(212 22 ++−= ,

tt dWdtdtrdtdX σσσ 222 22 ++−= , tt dWdtrdtdX σσ 22 2 ++= , tt dWdtrdX σσ 2)2( 2 ++= . It means TXe2 has mean Tre )2( 2σ+ and the variance of TXe is given below: [ ] [ ]22 2

var TT XTrTX eEee −= +σ , as we know

0S

Se TXT = ,

thus we can write mean and variance as,

rTT eSS

E =⎥⎦

⎤⎢⎣

⎡

0

, (3.22 a)

2

0

)2(

0

2

⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡ +

SSEe

SSVar TTrT σ . (3.22 b)

We will apply mean and variance to one-period binomial model with tT Δ= and limit

0. →Δ t . In this model the mean and variance are given below:

dqquSS

E T )1(0

−+=⎥⎦

⎤⎢⎣

⎡, (3.23 a)

2

0

22

0

)1( ⎥⎦

⎤⎢⎣

⎡−−+=⎥

⎦

⎤⎢⎣

⎡SSEdqqu

SSVar TT . (3.23 b)

Comparing equation (3.22 a) and (3.23 a)

⎥⎦

⎤⎢⎣

⎡

0SS

E T = ⎥⎦

⎤⎢⎣

⎡

0SS

E T


37

rTe = dqqu )1( −+ , here tT Δ= dqqu )1( −+ = tre Δ . Again comparing equation (3.22 b) and (3.23 b)

⎥⎦

⎤⎢⎣

⎡

0SS

Var T = ⎥⎦

⎤⎢⎣

⎡

0SS

Var T ,

2

0

22 )1( ⎥⎦

⎤⎢⎣

⎡−−+

SSEdqqu T =

2

0

)2( 2

⎥⎦

⎤⎢⎣

⎡−+

SSEe TTr σ ,

22 )1( dqqu −+ = Tre )2( 2σ+ , here again tT Δ= 22 )1( dqqu −+ = tre Δ+ )2( 2σ

22 )1( dqqu −+ = ttre Δ+Δ 22 σ . dqqu )1( −+ = tre Δ (3.24 a)

22 )1( dqqu −+ = ttre Δ+Δ 22 σ (3.24 b) Now we have two equations with three unknowns variables one variable can be chosen

for example 21

=q .

Put the value of 21

=q in equation (3.24 a)

tredu Δ=−+ )211(

21

tredu Δ=+21

21 ,

tredu Δ=+ )(21 ,

tredu Δ=+ 2 .


38

Again put the value of 21

=q in equation (3.24 b)

ttredu Δ+Δ=−+2222 )

211(

21 σ

ttredu Δ+Δ=+2222

21

21 σ ,

ttredu Δ+Δ=+2222 )(

21 σ ,

ttredu Δ+Δ=+2222 2 σ .

Now again we have two equations tredu Δ=+ 2 (3.25 a) ttredu Δ+Δ=+

2222 2 σ . (3.25 b) From these two equations we can get the value of u, d which are given below:

)11(2

−+= ΔΔ ttr eeu σ ,

)11(2

−−= ΔΔ ttr eed σ ,

21

=q .

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


39

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: [ ]fEeV q

Tr−=0 . Consider the single period binomial model. Let the current spot price be S. The risk neutral expectation is given below: [ ] ),()1(),()0,( tSdVqtSuqVSVEq Δ−+Δ= . (3.26) Expand ),( tSuV Δ in Taylor series: ),( tSuV Δ = )),1(( tuSSV Δ−+

= )()1(),(21)1(),(),( 322 uOuStSVuStSVtSV +−Δ′′+−Δ′+Δ .

Expand ),( tSdV Δ in Taylor Series: ),( tSdV Δ = )),1(( tdSSV Δ−+

= )()1(),(21)1(),(),( 322 dOdStSVdStSVtSV +−Δ′′+−Δ′+Δ .

Put the values of ),( tSuV Δ and ),( tSdV Δ in equation 3.26

[ ] ⎥⎦⎤

⎢⎣⎡ +−Δ′′+−Δ′+Δ= )()1(),(

21)1(),(),()0,( 322 uOuStSVuStSVtSVqSVEq

⎥⎦⎤

⎢⎣⎡ −Δ′′+−Δ′+Δ−+ )()1(),(

21)1(),(),()1( 322 dOdStSVdStSVtSVq ,

⎥⎦⎤

⎢⎣⎡ Δ′′−Δ′′+Δ′′+Δ′−Δ′+Δ= 2222 ),(),(

21),(

21),(),(),( StSVuStSVStSVutSVStSVSutSVq

⎥⎦⎤

⎢⎣⎡ Δ′′−Δ′′+Δ′′+Δ′−Δ′+Δ−+ 2222 ),(),(

21),(

21),(),(),()1( StSVdStSVStSVdtSVSdStSVtSVq


40

2222 ),(),(

21),(

21),(),(),( StSVuStSVStSVuqtSVSqtSVSutSqV Δ′′−Δ′′+Δ′′+Δ′−Δ′+Δ=

),(),(),(

21),(

21),(),(),( 2222 tSqVStSVdStSVStSVdtSVSdStSVtSV Δ−Δ′′−Δ′′+Δ′′+Δ′−Δ′+Δ+

2222 ),(),(21),(

21),(),( StSVqdStSVqStSVqdStSVqStSVqd Δ′′+Δ′′−Δ′′−Δ′+Δ′− ,

[ ] )()1(),(21)1)(1()1(),(),( 322 uOuStSVdquqStSVtSV +−Δ′′+−−−−Δ′+Δ= .

We will use the equality tredqqu Δ=−+ )1( , and by risk neutral argument, this amount be equal to )()1)(0,()0,( 2tOtrSVeSV tr δ+Δ+=Δ

= [ ] )()1(),(211),(),( 322 uOuStSVeStSVtSV tr +−Δ′′+−Δ′+Δ Δ ,

)(),(21),(),( 2

322 tOtStSVtSrtSVtSV Δ+ΔΔ′′+ΔΔ′′+Δ= σ .

By the risk-neutral argument, this must be equal to )()1)(0,()0,( 2tOtrSVeSV tr Δ+Δ+=Δ . Rearranging

0)()0,(),(),(21)0,(),( 2

322 =Δ+Δ−ΔΔ′+Δ′′+−Δ tOtSrVttSVrStSVSSVtSV σ ,

when the limit 0→Δt , we will finally get the Black-Scholes partial differential equation

021

2

222 =−

∂∂

+∂∂

+∂∂ Vr

SVrS

SVS

fV σ .


41

3.2.6 Constant Dividend Yield Suppose we have an asset with constant dividend yield D, buying φ units of the asset at the current spot price 0S worth uSe tD

0Δφ if the price goes up and dSe tD

0Δφ if the price

goes down.

Figure 3.8: Binomial model of an asset with constant dividend yield, buyingφ units of the asset at 0S . If we multiply the tree by tDe Δ− then we will get the binomial tree with ),( 00 dSuS at each node. But the initial value changes to tDeS Δ−

0 . This is illustrated below:

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

0D te S uφ Δ

0D te S dφ Δ

0Sφ

0S uφ

0S dφ

0D tS eφ − Δ


42

Put the values of tDeS Δ−

0 in equation 3.13(a, b) we will get

d

trtD

SuSdSeeS

q00

00

−−

=ΔΔ−

q 0

0)(

)()(

SduSde tDr

−−

=Δ−

,

du

deqtDr

−−

=Δ− )(

.

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

ttDr

euΔ+Δ−−

=σσ )

21( 2

, (3.27 a)

ttDr

edΔ−Δ−−

=σσ )

21( 2

, (3.27 b)

du

deqtDr

−−

=Δ− )(

. (3.27 c)

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.

0S u

0S d

0D tS e− Δ


43

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: [ ]),()0,( 00 TeSVEeSV TXrT−= . The formula will be used a reference when testing the computational methods. Take the European put option ),( tSP t . It payoff at expiry t = T is +− )( TSK , so by the value of the option at t = 0 can be expressed as: [ ]+− −= )()0,( 0 T

rT SKEeSP [ ]+− −= )( 0

TXrT eSKEe . Denote the expectation value restricted to this domain by +E so by the linearity of the expectation operator we can write it as: [ ] [ ])1()0,( 00

TXrT eESKEeSP ++− −= . (3.28)

For standard normal random variable Z, the expectation value [ ])(ZfE+ for 0zZ < is given by:

[ ] ∫∝−

+ −=0

.)21exp()(

21)( 2

z

dzzzfZfEπ

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

tt WtrX σσ +−= )21( 2 ,

for t = T one has

TT WTrX σσ +−= )21( 2 .

Then the standard normal random variable can be defined as:

T

TrXZ

T

σ

σ )21( 2−−

= , (3.29)


44

the TX is restricted to )/log( 0SKX T <∝<−

[ ] dxT

Trx

TE

SK

∫∝−

+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −−−=

0log

22 )

21(

21exp

211

σ

σ

πσ.

By the variable change (3.29), we obtain

[ ] )()21exp(

211 1

21

zdzzEz

φπ

≡−= ∫∝−

+ , (3.30)

where .)(φ is the cumulative normal distribution function defined by the improper integral on the left hand side, and 1z is obtained by the variable change in (3.29):

T

TrSK

zσ

σ )21(log 2

01

−−= . (3.31)

Similarly we will compute

[ ] dxT

Trxx

TeE

SK

XT ∫∝−

+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −−−=

0log

22 )

21(

21exp

21

σ

σ

πσ.

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

[ ] )(21exp

2 22

2

zedzzeeE rTzrT

XT φπ

=⎟⎠⎞

⎜⎝⎛−= ∫

∝−+ ,

where Tzz σ−= 12 ,

T

TrSK

zσ

σ )21(log 2

02

+−= . (3.32)


45

Applying the expectation formulas to equation 3.28 yields the Black-Scholes put option formula: )()()0,( 2010 zSzKeSP rT φφ −= − . (3.33) The Black-Scholes call option formula is derived easily by using the put-call parity formula which is given below: SPeKC tTr +=+ −− )( rTKeSPC −−=− . Putting the values of put option in above equation, we will get: rTrT KeSzSzKeSC −− −=−− 02010 ))()(()0,( φφ )()()0,( 12000 zKeKezSSSC rTrT φφ −− +−−= , ))(1())(1()0,( 1200 zKezSSC rT φφ −−−= − , which is equivalent to )()()0,( 1200 zKezSSC rT −−−= − φφ . (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 420 =S 10=K 1.0=r 5.0=T 2.0=σ ,

T

TrSK

zσ

σ )21(log 2

01

−−= .

Put the values, we will get

5.02.0

5.0)2.0211.0(

4240log 2

1

−−=z ,


46

=1z -0.6278,

T

TrSK

zσ

σ )21(log 2

02

+−= ,

5.02.0

5.0)2.0211.0(

4240log 2

2

+−=z ,

=2z -0.7693, and 5.0*1.040 −− = eKe rT , 05.040 −− = eKe rT , 049.38=−rTKe . The Black-Scholes put option formula is: )()()0,( 2010 zSzKeSP rT φφ −= − )7693.0(42)6278.0(049.38 −−−= φφ , = 0.81. The Black-Scholes call option formula is: )()()0,( 1200 zKezSSC rT −−−= − φφ )6278.0(049.38)7693.0(42 φφ −= , = 4.76. 3.2.7.2 Black-Scholes Pricing Formula for Binary puts and calls The binary put option value is simply [ ]1)0,( 0 +

−= EeSP rTB .

By equation 3.30 we can write above equation as )()0,( 10 zeSP rT

B φ−= . (3.35) By the binary put call parity formula we obtain the binary call option formula which is given below: )()0,( 10 zeSC rT

B −= − φ . (3.36)


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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 the spot price 0S with DTeS −

0 . 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, the payoff function is shown as a dotted line, the curve touches 0V at TrKeV −=0 .


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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 curve touches 0V at TrKeV −=0 .


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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 →∝TS the value of the call approaches to rTKeS −−0 , in time the value rTKe−− approaches to zero so the value of the call obviously approaches that of the asset itself 0S .


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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, as →∝TS the value of the call approaches to rTDT KeeS −− −0 , in time the value rTKe−− and DTeS −

0 both terms approaches to zero value of the call gradually loses its value.


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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 0=TS the curve touches the 0V at

rTe− .

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 7.0, the payoff function is shown as a dotted line, as KST > , the curve is near rTe− .


52

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 0S is known. Hence at time tt Δ= the possible asset prices are 00

2 ,udSSu and 02Sd .

In general at time titt i Δ−== )1(: there are i possible asset prices which are given below:

0

1SudS nniin

−−= , in ≤≤1 hence, at the expiry time Ttt M == +1 there are M + 1 possible asset prices. The values

inS for in ≤≤1 and 11 +≤≤ Mn form a recombining binary tree. Which is given below:


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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 the actual price of the asset turns out to be less than K. Hence, if the asset has price 1+M

nS at time Ttt M == +1 then then value of the option will be )0,max( 1+− M

nSK . Let i

nV denote the value of the option at time itt = )0,max( 11 ++ −= M

nM

n SKV . We will calculate 1

1V the option value at time t = 0 and 11++i

nV for the later time level. The following recursive formula we will use to calculate the values: ),)1(( 11

1++

+Δ− −+= i

ni

ntri

n VqqVeV in ≤≤1 11 +≤≤ Mn where the constant r is risk-free interest rate. The values of u, d and q are given below:


54

12 −+= AAu , ud /1= ,

du

deqtr

−−

=Δ

.

Where ( )trtr eeA Δ+Δ− += )( 2

21 σ and constant σ represents the volatility of the asset.

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.


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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.


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Figure 4.4: Relative error )/( Sε 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.


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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) )(SPV ≥ If the situation )(SPV < 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: [ ]),)1(max( 1

11 −

+Δ−− −+= n

jnj

nj

trnn fVqqVeV ,

where jf is the payoff of the option. 5.1.1 American Call and Put An American call option’s payoff is +− )( KSτ if it is exercised at time T≤τ , similarly an American put option’s payoff is +− )( τSK if it is exercised at time T≤τ . The value of the American call option at time 0 can be written as: [ ]+− −= )(*max)( 0 KSeESC r

ττ , (5.1)

where the max is over all stopping times .T≤τ For exact justification of equation 5.1 as the appropriate pricing formula, see the reference [23] and [24].


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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.


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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.


60

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.


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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”.


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[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/

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