(for phyllis young) Per our discussion 2 paper different sources

A Comparison of Option Pricing Models

Ekrem Kilic �

11.01.2005

Abstract

Modeling a nonlinear pay o¤ generating instrument is a challenging work. The mod-

els that are commonly used for pricing derivative might divided into two main classes;

analytical and iterative models. This paper compares the Black-Scholes and binomial

tree models.

Keywords: Derivatives, Option Pricing, Black-Scholes,Binomial Tree

JEL classi�cation:

1. Introduction

Modeling a nonlinear pay o¤ generating instrument is a challenging work to

handle. If we consider a European option on a stock, what we are trying to do is

estimating a conditional expected future value. In other words we need to �nd out

the following question: what would be the expected future value of a stock given

that the price is higher than the option�s strike price? If we �nd that value we can

easily get the expected value of the option. For the case of the American options

the model need to be more complex. For this case, we need to check the path that

we reached some future value of the stock, because the buyer of the option might

exercise the option at any time until the maturity date.

To solve the problem that summarized above, �rst we need to model the move-

ment of the stock during the pricing period. The common model for the change

of the stock prices is Geometric Brownian Motion. Secondly, the future outcomes

of the model might have the same risk. Risk Neutrality assumption provides that.

By constructing a portfolio of derivative and share makes possible to have same

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A Comparison of Option Pricing Models 2

outcome with canceling out the source of the uncertainty.

The models that are commonly used for pricing derivative might divided into

two main classes. The �rst classes is the models that provide analytical formulae to

get the risk neutral price under some reasonable assumptions. The Black-Scholes

formula is in this group. The formulae that we have to price the derivatives

are quite limited. The reason is that we are trying to solve a partial di¤erential

equation at the end of the day. But mathematician could manage to solve just

someof thepartialdi¤erential equations; therefore, weareboundedto some limited

solutions.

The second classes models provide numerical procedures to price the option.

Binomial trees that �rst suggested by Cox, Ross and Rubenstein, is in this group,

because we need to follow an iterative procedure called �backwards induction�to

get option price. Monte Carlo simulations are another type of models that belongs

to this class. Also �nite di¤erencing methods are a type of numerical class.

In this paper, �rst I will introduce Black-Scholes and Binomial Tree models for

option pricing. Second I will introduce the volatility estimation methods I used

and calculate some option prices to compare models. Finally I will conclude.

2. Option Pricing Models

2.1. Black-Scholes Model

Black-Scholes formula suggested by Fischer Black and Myron Scholes at 1973.

The Black-Scholes (1973) option pricing formula prices European put or call op-

tions on a stock that does not pay a dividend or make other distributions. The

formula assumes the underlying stock price follows a geometric Brownian motion

A Comparison of Option Pricing Models 3

with constant volatility. The geometrics Brownian motion can be shown as follows;

�S = �S�t+�S�z

where S is the current price of the stock, dS the change in the stock price, � is

the expected rate of return, � is the volatility and dz is the part follows a Wiener

process.

Then using Ito�s lemma the price of the option might be shown as;

�f = � @f @S �S + @f

@t + 1

2 @2f @S2

�2S2 � �t+ @f

@S �S�z

where df is the change in the option value.

This di¤erential equation formulates the movement of the option price over

time. Now note that the source of the risk in the stock and the option is exactly

same (dz). Then if we construct a risk neutral portfolio consisting a short position

in the option and a long position in the stock as follows

derivative : �1

share : @f @S

The pay o¤ function can be shown as;

� = �f + @f @S S

The change of the pay o¤ function is;

�� = ��f + @f @S �S

�� = � � @f @S �S + @f

@t + 1

2 @2f @S2

�2S2 � �t� @f

@S �S�z + @f

@S (�S�t+�S�z)

�� = � � @f @t + 1

2 @2f @S2

�2S2 � �t

A Comparison of Option Pricing Models 4

This portfolio is riskless because we can exclude the source of uncertainty.

Since it is riskless it should yield risk free rate otherwise there could be arbitrage

opportunities. Then we can write;

�� = r��t� @f @t + 1

2 @2f @S2

�2S2 � �t = r

� f � @f

@S S � �t

And

@f @t + @f

@S rS + 1

2 @2f @S2

�2S2 = rf

This partial di¤erential equation is called Black Scholes di¤erential equation.

The solution for this di¤erential equation is the Black-Scholes formulae;

c = S0N (d1)�Ke�rTN (d2)

p = Ke�rTN (�d2)�S0N (�d1)

where

d1 = ln(S0=K)+(r+�2=2)T

� p T

d2 = ln(S0=K)+(r+�2=2)T

� p T

= d1 �� p T

Then the formula needs the current price of the stock S0 , the stirke price K,

risk free short term interest rate r, time to maturity T and volatility of the stock

�(N (:) is the cumulative normal distribution function).

The model is based on a normal distribution of underlying asset returns which

is the same thing as saying that the underlying asset prices themselves are log-

normally distributed. A lognormal distribution is right skewed distirbution. The

lognormal distribution allows for a stock price distribution of between zero and

A Comparison of Option Pricing Models 5

in�nity (ie no negative prices) and has an upward bias (representing the fact that

a stock price can only drop 100% but can rise by more than 100%).

In practice underlying asset price distributions often depart signi�cantly from

the lognormal. For example historical distributions of underlying asset returns

often have fatter left and right tails than a normal distribution indicating that

dramatic market moves occur with greater frequency than would be predicted by

a normal distribution of returns�ie more very high returns and more very low

returns.

The main advantage of the Black-Scholes model is speed �it lets you calculate

a very large number of option prices in a very short time.

The Black-Scholes model has one major limitation: it cannot be used to ac-

curately price options with an American-style exercise as it only calculates the

option price at one point in time �at expiration. It does not consider the steps

along the way where there could be the possibility of early exercise of an American

option.

As all exchange traded equity options have American-style exercise (ie they

can be exercised at any time as opposed to European options which can only be

exercised at expiration) this is a signi�cant limitation.

The exception to this is an American call on a non-dividend paying asset. In

this case the call is always worth the same as its European equivalent as there is

never any advantage in exercising early.

2.2. Binomial Model

The binomial model breaks down the time to expiration into potentially a very

large number of time intervals, or steps. A tree of stock prices is initially produced

A Comparison of Option Pricing Models 6

working forward from the present to expiration. At each step it is assumed that

the stock price will move up or down by an amount calculated using volatility and

time to expiration. This produces a binomial distribution, or recombining tree, of

underlying stock prices. The tree represents all the possible paths that the stock

price could take during the life of the option.

At the end of the tree �ie at expiration of the option �all the terminal option

prices for each of the �nal possible stock prices are known as they simply equal

their intrinsic values.

Next the option prices at each step of the tree are calculated working back from

expiration to the present. The option prices at each step are used to derive the

option prices at the next step of the tree using risk neutral valuation based on the

probabilities of the stock prices moving up or down, the risk free rate and the time

interval of each step. Any adjustments to stock prices (at an ex-dividend date) or

option prices (as a result of early exercise of American options) are worked into

the calculations at the required point in time. At the top of the tree you are left

with one option price.

For example we can consider a one step binomial tree. The one step ahead

stock price has two di¤erent values in a binomial model; S0u or S0d where u and

d are the upward and downward size respectively. Then if we construct a risk

neutral portfolio by shorting the derivative and buying the stock;

derivative : �1

share : �

Because the portfolio is risk neutral, whether stock price goes up or down the

pay o¤ would be same, so;

A Comparison of Option Pricing Models 7

S0u��fu = S0d��fd

� = fu�fd S0u�S0d

The cost of settin up this portfolio is;

S0 �f

because we get a riskless position it should yield risk free rate. Then;

S0 �f = e�rT (S0u��fu)

Finally option value is;

f = e�rT (pfu +(1�p)fd)

where

p = e rT �d u�d

To match this formulation with stock volatility, it is a popular way to de�ne

u;d and p as follows;

u = e� p �t

d = 1=u

p = e r�t�d u�d

The big advantage the binomial model has over the Black-Scholes model is

that it can be used to accurately price American options. This is because with the

binomial model it�s possible to check at every point in an option�s life (ie at every

step of the binomial tree) for the possibility of early exercise (eg where, due to eg

A Comparison of Option Pricing Models 8

a dividend, or a put being deeply in the money the option price at that point is

less than the its intrinsic value).

Where an early exercise point is found it is assumed that the option holder

would elect to exercise, and the option price can be adjusted to equal the intrinsic

value at that point. This then �ows into the calculations higher up the tree and

so on.

The binomial model basically solves the same equation, using a computational

procedure that the Black-Scholes model solves using an analytic approach and

in doing so provides opportunities along the way to check for early exercise for

American options.

The main limitation of the binomial model is its relatively slow speed. It�s

great for half a dozen calculations at a time but even with today�s fastest PCs it�s

not a practical solution for the calculation of thousands of prices in a few seconds.

3. Comparison of the Models

3.1. Volatility Estimation

For both models the volatility of the stock is the key factor. Estimation of the

volatilty is another important topic. In this study I used three di¤erent models for

Historical Average(HA), Exponentially Weighted Moving Average (EWMA) and

Generalized Autoregressive Conditional Hetereoscedasitic (GARCH) model.

3.1.1. Historical Average

If we assume that conditional expectation of the volatility is constant and the

daily returns has zero mean, the proper estimate of the volatility is;

A Comparison of Option Pricing Models 9

� =

vuut 1 n�1

nX i=1

r2

where � is the estimated volatility, n is the sample size and r is the daily return.

The weakest point of this model is of course constant volatility assumption.

This estimate of the volatility could not mimic the big changes in the volatility

and remains nearly constant where the sample size increases.

3.1.2. Exponentially Weighted Moving Average

EWMA past observations with exponentially decreasing weights to esitmate

volatility. Therefore this is a modi�ed version of historical averaging. Instead of

equally weighting, in EWMA weights di¤er. The estimated volatility can shown

as;

�2t = (1��)r2t +��2t�1 �t =

p �2t

By repeated substitutions we can re-write the forecast as;

�2t = (1��) nX i=1

�i�1r2t

equation shows the volatility is equal to a weighting average. The weights de-

crease geometrically. The value of , decay factor, estimated simply by minimizing

the one week forecast errors.

3.1.3. GARCH

The Generalized ARCH model of Bollerslev (1986) de�ned GARCH by;

A Comparison of Option Pricing Models 10

rt = �+�t"t

�2t = �+

qX i=1

�i (rt�i ��) 2 +

pX j=1

�j� 2 t�j

GARCH imposes that the proper volatility estimate is based not only on the

recent volatilities and also previous forecasts which include the previous volatilities.

Then the GARCH model is a long memory model. The parameters of the GARCH

can be estimated by a Maximum Likelihood procedure.

3.2. Some Hypothetic Options

3.2.1. An Stock Index Option on ISE-100

First let us consider a one-month european call option with strike price 27000

(current price is taken as 25300). The price for several combinations is as follows;

Black-Scholes CRR

Historical Average 2207.95 2206.60

EWMA 3068.71 3071.38

GARCH 2562.63 2560.39

The CRR results obtained with 100 steps. As we can see the price of the

option changes according to the volatility estimates. The volatility estimate of

the historical simulation leads us to lowest price (16%). The volatility estimate of

EWMA and GARCH is 30% and 22% respectively.

Now let us consider the case we have a one-month european put option with

strike price 27000. The prices are;

A Comparison of Option Pricing Models 11

Black-Scholes CRR

Historical Average 433.72 432.37

EWMA 1294.48 1297.15

GARCH 788.40 786.16

This type option price change at a higher ratio, then this option is more non-

linear with respective to volatility.

For the american call option (non-divident paying) the price is same as we

calculated before. But for the american put option the value will change and we

can not use the Black-Scholes formula for the american put options. The table

shows the prices by the binomial model;

CRR

Historical Average 856.91

EWMA 1657.57

GARCH 1172.60

As we can see the price of the american option is higher than the european

option.

3.2.2. An FX Option on USD/TRL

First let us start with a one-month european call option again with strike price

1.38 (current price is taken as 1.38). The price for several combinations is as

follows;

A Comparison of Option Pricing Models 12

Black-Scholes CRR

Historical Average 0.1419 0.1418

EWMA 0.1633 0.1631

GARCH 0.1574 0.1572

The binomail model�s results obtained with 100 steps again. The volatility

estimate of HA, EWMA and GARCH is 14% , 22 and 20% respectively.

Another case, we have a one-month european put option with strike price 1.38.

The prices are;

Black-Scholes CRR

Historical Average 0.0106 0.0104

EWMA 0.0320 0.0318

GARCH 0.0261 0.0259

The table shows the prices by the binomial model result for the american

version of the last option;

CRR

Historical Average 0.0220

EWMA 0.0458

GARCH 0.0395

As we can see the price of the american option is higher than the european

option again.

4. Conclusion

Due to its nonlienar pay o¤s pricing an option is quite di¢ cult with respective

to other �nancial instruments like �xed income instruments. One should consider

A Comparison of Option Pricing Models 13

the possible future oturcomes of the underlying asset. If the option is an exotic

that is e¤ected by more than one sources of the uncertainty it becomes more

complicated. In this paper we just introduce an compare two popular models.

The results show that the prices of the two models quite similar where the

number of steps in the binomial. Since Black-Scholes model can not calculate the

price of the american option one can use the binomial model with high number

steps and can similar result as if Black-Scholes.

Another result shows that due to high volatility the option price on the ISE

will be very high. For example a long position on one-month american call with

strike price 27000 might have positive payo¤ if the prices goes somewhere around

30000. On the other hand the price of the FX options is more reasonable.

By the results of the paper we can see that the volatility estimation is also as

important as option pricing. Because the volatilty changes the price might change

exponentially.