Testing Geometric Brownian Motion

7/27/2019 Testing Geometric Brownian Motion

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Testing Geometric Brownian Motion: An

Analytical Look at the Black Scholes Option

Pricing Model

By Julian LamyAdvisor: John Rust

University of Maryland College Park

Department of EconomicsMay 16, 2008



Section 1: Introduction

Fischer Black and Myron Scholes revolutionized the way investors view financial

markets with their famous paper “The Pricing of Options and Corporate Liabilities” in

1973. The mathematical and economic implications of the paper led the way for new

intuition and creativity in how derivatives are priced. Despite their success however,

many academics have challenged the underlying assumptions behind their options pricing

model and its effective use. This paper adds to this research by testing the underlying

assumption behind the Black Scholes model which is that stock prices follow a

Geometric Brownian motion (GBM) diffusion process and compares this assumption

with alternatives.

Although the Black Scholes Model is quite successful when pricing options, it is

known to have certain biases associated with it. There has been strong evidence that stock

prices experience stochastic volatility over time and that the assumption of constant

volatility creates biases when using the Black Scholes model. Much research has been

dedicated to finding alternative models that account for this stochastic volatility and

prove this result by comparing different models to the Black Scholes model (For

example, see Amin and Ng (1993), Heston (1993), Jiang and Sluis (2000), Louis Scott

(1987)). In all of these studies, different models have been proven to perform better than

geometric Brownian motion as they allow for more movement in returns. This paper adds

to this research by comparing the Black Scholes model (more specifically, the geometric

Brownian motion assumptions underlying it), and compares it to a constant elasticity

model (CEV) which allows the volatility parameter of the diffusion to change over time.

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Lauterbach and Schultz (1990) showed that the CEV model does outperform

GBM over a larger dataset when predicting warrant prices. The goal of the paper is to

show that during specific time periods where prices are in a transition in volatility, the

CEV does indeed outperform GBM in predicting price returns. However, during times of

relatively constant volatility, GBM predicts price returns as well or better than the CEV

model. The particular time intervals of interest are of April 30th, 1998 to February 16,

1999 and March 31st, 1994 to January 17th, 1995. The short intervals of these datasets are

chosen deliberately for this research as it reflects the short intervals that investors on Wall

Street use to predict prices on a regular basis (based on the historical volatility parameters

that they use). This paper aims to give practical advice to investors and therefore the time

intervals are also chosen to reflect a particularly interesting period in the financial

markets.

These particular time intervals are chosen for a specific historical purpose. During

the interval from 1994 to 1995, the US economy experienced economic prosperity and

the returns of S&P Index prices were at a relatively constant volatility. In the time interval

from 1998 to 1999 however, the world was thrown into a credit crisis when Russia

defaulted on its debt and many investors were apprehensive of owning various assets

causing large swings of volatility in asset prices. Contrasting these different periods can

help to explain what different methods are best when pricing options in specific market

conditions.

Each diffusion process is estimated using Maximum Likelihood estimation

methods created by Ait Sahalia (1999, 2002) and is then tested using specification tests

described in this paper (with suggestions from Norman Swanson). The results of this

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paper show that over periods of transition in volatility, the CEV model estimates price

returns more accurately than GBM whereas during times of relatively constant volatility,

the GBM performs equally well or better than the CEV model.

The paper is split up into the following sections: Section 2 gives the reader

common background information about options and about the models that are tested

(those who are familiar with this material may wish to skip this section). Section 3

describes the data used by running certain tests which identify properties of the data.

Section 4 describes the tests used to test the GBM model versus the CEV model. In

Section 5, results are presented. Section 6 analyzes the results and explains their

implications for modern traders and investors. Section 7 concludes.

Section 2: Background

2.1 What is an Option?

An option is an agreement to buy or sell a certain security at a predetermined time

T at a predetermined “Strike price” K. The contracts themselves have a certain value

associated with them which is directly dependent on the value of the underlying security.

There are numerous types of options with different underlying assets but for brevity, I

will define only the following:

Call Option: The right to buy a security at time T for price K Put Option: The right to sell a security at time T for price K European Option: A call or put option that can only be exercised at time T American Option1: A call of put option that can be exercised at or before time T

1 American and European options can also be thought of as the same asset in this paper. It has beenthoroughly proven that the value of a European and American call option are the same because it isoptimal to exercise a call option at time T regardless of its classification (Ross, 2003).

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At the expiration time T, the payoff of a call option can be defined as

Max { S T – K , 0 }Where: S T = underlying security’s price at time T

K = Strike Price

To see that this is the case, notice that if S T > K, then the option holder could

exercise the option and purchase the security for price K and immediately sell the

security on the open market for S T making a profit of S T – K. If S T <= K, then the option

is not exercised and there is a 0 profit. Notice that the exact opposite relationship exists

for a put option. Mainly, the payoff becomes:

Max { K – S T , 0 }Where: S T = underlying security’s price at time T

K = Strike Price

2.2 Geometric Brownian motion and the Black Scholes Options Pricing Model

Inspired by the research and development of Brownian motion from Robert

Brown and further by Norbert Weiner, Louis Bachilier who applied Brownian motion to

stock prices and other economists who tried using Brownian motion to model market

behavior, the Black Scholes model assumes that security prices follow a geometric

Brownian motion diffusion process. This assumption implies the following:

(2) dS t = uS t d t + σS t dW t

S t is the security’s price at time t d t is the time changedS t is the change in St over time d t u is a drift parameter σ is the volatility parameter dW t is a Weiner Process ~N(0, d t )

Notice that one can then rearrange the equation (2) as follows:

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(3) [dS t ]/ S t = ud t + σ[dW t ]

The left hand of the equation in (3) is simply the stock price return during a given

time t. Geometric Brownian motion therefore assumes that stock price returns are defined

by the constant drift u plus the random element defined by the Weiner process times a

constant volatility parameter. Looking at how [dS t ]/ S t is defined above, it is clear that

stock returns follow a normal distribution with mean dt*u and volatility parameter dt*σ 2

when assuming GBM. (To see this, note that dW t ~ N(0,dt) and that u and σ are defined as

constants).

Some interesting observations can then be made about (3). Notice that returns are

therefore assumed to have a very well-behaved normal distribution with constant drift

and volatility parameters. This assumption may be impractical when looking at actual

market data. Later in this paper, this issue will be analyzed thoroughly with historical

datasets.

Another observation that can be made by assuming geometric Brownian motion

for stock prices is when the equation (3) is rearranged in the following way:

Notice that mathematically, one can also write [dS t ]/ S t as log(S t / S t – h ) where h is thelength of dt. Rearranging,

(4) log(S t ) = log(S t-h ) + uh + σ[dW t ]

OR log(S t ) = log(S t-h ) + uh + W t Where W t ~N(0, hσ 2 )

From (4), letting h=1, one can interpret the equation as being a random walk with

a drift parameter u. Under Geometric Brownian motion, the log of stock prices should

therefore be independent of all lag variables except the first lag. This observation is

important as it implies that past prices (beyond lag 1) should not affect the present price.

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One last important observation to make is when (4) is rearranged as follows:

(5) S t = S t – h *E(W t )Where: W t = uh + √h σZ t and Z t ~N(0,1)

From the above specification of stock prices in (5), one can then find the expected

value of prices in the future; E(St). If one then prices an option based on the expected

value of its future profit, then one can use this specification to compute the expected

value of [St – K, 0] which is the return of a call option with expiration t and strike price

K. This procedure is that which the Black Scholes Options Pricing Model is based. It is

therefore a critical assumption to allow stock prices to follow a geometric Brownian

motion. If this assumption does not hold, than pricing options using the Black Scholes

model will produce inaccurate results.

2.3 Constant Elasticity of Variance

A Constant Elasticity of Variance (CEV) diffusion assumes that stock prices are

governed by the following diffusion process:

(6) dS t = ud t S t + σS t b/2dW t

Notice that the process above is a generalization of geometric Brownian motion

specified in (1) when the constant b = 2. Notice that when b does not equal 2, the

volatility parameter will vary with the stock price. Rearranging (6) to reflect stock

returns:

(7) [dS t ]/S t = ud t + σS t b/2 - 1dW t

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The volatility parameter then becomes σS t b/2 – 1which may be a more realistic

model for stock returns. Assume for example the economy is experiencing a recession;

one would expect the volatility to change over this period because investors are more

apprehensive. Since the volatility parameter depends on the stock price in the CEV

model, the model should therefore be able to capture some of these changes in volatility.

This opposes the GBM model which would assume the same volatility no matter what

external events may be happening in the market. For this reason, this paper uses this

diffusion process to model market prices.

Section 3: Defining and testing the dataset

3.1 Defining the dataset

The dataset that is analyzed in this paper is S&P 500 Index adjusted closing prices

from April 1998 to February 1999 and from March 1994 to January 1995. This data

reflects the general performance of equity prices over these time interval and therefore

provides a generalization of this analysis to various equities. Instead of just choosing the

500 largest companies, the S&P Index selects those that are among the leaders in the

major industries in the economy. Since the S&P Index is also one of the most widely used

measures of stock market performance, using these datasets is consistent with previous

research in modeling stock returns (see Lo and Mackinlay, Ait-Sahalia, etc.).

These particular time periods were chosen in order to reflect certain situations in

market behavior as they specifically reflect different trends in price volatility over time.

In the period from 1994 to 1995, the volatility is relatively constant whereas the period

from 1998 to 1999 experiences volatility that moves from low to high in transition. The

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claim of this paper is that if one were to predict returns during a period of transitions in

volatility, he will benefit from using the CEV model as opposed to the GBM diffusion

which assumes a constant volatility parameter. However, during periods of relatively

constant volatility when an investor is modeling returns, the GBM diffusion will be

sufficient and accurate. It is only during transitions between these time intervals that

merit the use of the CEV model over GBM.

Plotted below are the price returns from June 4 th, 1990 to June 8th, 2007 in order to

demonstrate the trend of S&P Index returns over the past two decades and give some

intuition as to why these particular intervals were chosen.

Graph 1: S&P Index Prices June 1990 to June 2007

0 500 1000 1500 2000 2500 3000 3500 4000 4500-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

(Plotted on the y-axis are daily S&P returns. Plotted on the x-axis is the number of business days starting from the date June 4th , 1990.)

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Judging by the graph of returns, it is clear that volatility is not constant throughout

this extended time period. There are several positions in the data where the volatility is

transitioning from low to high and vise versa. The shifts in volatility can be described by

the following table:

Table 1

Time period (t) Corresponding dates High/low volatility Interval

(0, 500) June 90 – June 92 High 1

(500, 1900) June 92 – Jan 1998 Low 2

(1900, 3300) Jan 98 – July 03 High 3

(3300, 4290) July 03 – June 07 Low 4

Each of these time periods correspond to periods where the US economy is

experiencing economic downturn or prosperity. Interval 1 corresponds to the recession

1990 – 1991, Interval 2 corresponds to the economic boom in the mid 1990’s, Interval 3

corresponds to the credit crisis in the late 1990’s and the technology bubble and Interval 4

corresponds to the recent economic boom that the United States has experienced. In each

of these time periods, the price returns transition from different levels of volatility. Based

on these observations, if one were to assume constant volatility over the time period

when modeling returns, he could experience significant errors.

One can notice then that this paper chooses to compare time periods from interval

2 and interval 3. The overall results of this paper indicate that in interval 2, GBM is

sufficient for predicting returns whereas the CEV model is a better model for interval 3.

The rest of this section is dedicated to testing the datasets for statistical properties of

normality and heteroskedasticity.

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3.2 Testing the datasets

This section performs tests for normality and homoskedasticity of the time

periods of March 1994 to January 1995 and of April 1998 to February 1999. The analysis

below provides evidence that each dataset has different properties that will ultimately

affect the performance of the GBM model.

3.2.1 Time period April 1998 to February 1999

Normality

Recall (3) from the previous section, the GBM specification assumes the following:

[dS t ]/ St= ud t + σ[dW t ] ~ N(dt*u, dt* σ 2 )Where u and σ are assumed to be constant

It is therefore necessary to test whether the dataset follows a normal distribution

given constant parameters. Below is a histogram plot of the data estimated with the

sample mean and variance of the data:

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Graph 2: Normal Plot

- 0 . 0 6 6 - 0 . 0 5 4 - 0 . 0 4 2 - 0 . 0 3 - 0 . 0 1 8 - 0 . 0 0 6 0 . 0 0 6 0 . 0 1 8 0 . 0 3 0 . 0 4 2 0 . 0 5 4

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

P

e

r

c

e

n

t

r e t u r n s

The data does appear to stray from the estimated normal distribution which is

given by the blue line. To get a better visualization of how normal the data is, below is a

probability plot of the data. This plot is a graphical technique for assessing whether or not

a dataset is approximately normal. The data are plotted against an estimated normal

distribution where the blue line represents how close the actual data fit this estimated

distribution.

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Graph 3: Probability Plot

0 . 1 1 5 1 0 2 5 5 0 7 5 9 0 9 5 9 9 9 9 . 9

- 0 . 0 7 5

- 0 . 0 5 0

- 0 . 0 2 5

0

0 . 0 2 5

0 . 0 5 0

0 . 0 7 5

r

e

t

u

r

n

s

No r ma l P e r c e n t i l e s

Notice that the data is close to a normal distribution for values that are in the

middle of the distribution. For percentiles beyond 5% and above 95% however, the data

strays from normal. This phenomenon is common when looking at financial data as it

suggests that the distribution has “fat tails”. The significance of this observation is that

the normal distribution assumption does not capture the extreme values of returns when

there is a large shock to prices as is the case during this period. When thinking of the

volatility transitions mentioned earlier in this section, it is clear that during this period,

the market experienced large shocks to price returns at a changing rate. Assuming a

normal distribution with constant parameters for returns would then largely underestimate

these extreme shocks to the market and would assume much lower volatility than

necessary to model returns.

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Another method for testing the normal assumption is by using statistical normality

tests. Provided below are the p-values for the Kolmogorov-Smirnov and Shapiro-Wilk

tests which test the null hypothesis of normality of the data given the sample mean and

sample variance. Notice that one can fail to reject the null hypothesis of normality for

both tests with p-values less than 1% which indicates a strong rejection of the null. These

results further support the fact that this particular time interval is not a well behaved

geometric Brownian motion based on its underlying assumptions.

Table 2

Test p-valueKolmogorov-Smirnov <0.0100

Shapiro-Wilk <0.0001

Homoskedasticity Tests

Another observation worth making about the data is to test whether the

distribution of the data experiences constant volatility. As assumed by GBM, the

volatility parameter σ is assumed to be constant and therefore, returns [dS t ]/ S t experience

constant volatility. Levene’s test for heteroskedasticity is used in this case because it is

less sensitive to the departures from normality (which, from the previous analysis, is an

issue with this data).

The test requires that one splits the data into subgroups and then test the null

Hypothesis that all the groups have the same variance. This particular dataset is broken

into 6 states that are defined by the transition intervals described in the previous section.

If the geometric Brownian motion holds, then the variance from all these different time

intervals should be the same. From speculation after viewing the data, it seems clear that

this assumption will not hold. Below are the results from running Levene’s test in SAS:

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Table 3

Test P-Value

Levene’s Test <.0001

Welch’s Test 0.7177

Notice that Welch’s test is also provided in the above results. This test checks if

the means across the states are significantly different given that the data does experience

heteroskedasticity.

The p-value for Levene’s test and Welch’s test are equal to <0.001 and 0.7177

respectively. This suggests that one can reject the null hypothesis of homoskedasticity as

expected yet one fails to reject the null hypothesis of equality of means across the

distribution. This result is consistent with previous research that identifies the constant

volatility assumption is the main inconsistency with GBM as opposed to the drift

parameter (Heston, Scott, etc.).

3.2.2 Time period March 1994 to January 1995

Normality

The results of testing this dataset indicate a stark contrast to that of the period

analyzed above. Indicated by the normal plot, probability plot and normality tests below,

it is clear that returns during this time period do appear to follow a well-behaved normal

distribution. It therefore seems likely that GBM would work well for this particular

period based on its adherence to the underlying assumptions and properties defined by

this model.

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Graph 4: Normal Plot

- 0 . 0 1 6 - 0 . 0 1 2 - 0 . 0 0 8 - 0 . 0 0 4 0 0 . 0 0 4 0 . 0 0 8 0 . 0 1 2 0 . 0 1 6 0 . 0 2

0

5

1 0

1 5

2 0

2 5

3 0

3 5

P

e

r

c

e

n

t

r e t u r n s

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Graph 5: Probability Plot

0 . 1 1 5 1 0 2 5 5 0 7 5 9 0 9 5 9 9 9 9 . 9

- 0 . 0 2

- 0 . 0 1

0

0 . 0 1

0 . 0 2

0 . 0 3

r

e

t

u

r

n

s

N o r ma l P e r c e n t i l e s

Table 4

Test p-value

Kolmogorov-Smirnov 0.0614

Shapiro-Wilk 0.0861

Homoskedasticity

The same tests are then run for testing for homoskedasticity in the data interval

from April 1994 to February 1995. The results below indicate that one can fail to reject

the null hypothesis of equality of variance and equality of means across the distribution.

These results would indicate that the GBM properties exist in this dataset and therefore,

using GBM to price returns should produce accurate results.

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Table 5

Test P-Value

Levene’s Test 0.6353

Welch’s Test 0.9258

3.3.3 Conclusion

From analyzing these datasets, it is clear that one can reject the null hypothesis of

a GBM for the period from 1998 to 1999 based on the properties that are assumed by this

diffusion. The opposite result takes place for the interval of 1994 to 1995. It therefore

seems likely that GBM would work better for the latter interval and not well for the

former.

Section 4: Testing GBM vs CEV models

In this section, the procedures used to estimate and test the two diffusion models

over each interval are described.

4.1 Method of estimation

A challenge faced when testing diffusion models is finding an accurate way to

estimate the parameters in the model. A number of econometric methods have been

recently developed to estimate these parameters. The main issue faced when estimating is

that typically with financial data, there is no continuous time data available. When

considering market prices such as in this paper for example, most data consist of daily

prices rather than trade-for-trade data. In addition, the pure nature of financial price data

is discrete because the amount of time that trading occurs each day is finite. The CEV and

GBM diffusion models are continuous time differential equations therefore, estimating

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the models can be quite difficult using the given discrete datasets used in this study. This

paper uses the maximum likelihood estimation (MLE) methods derived by Ait-Sahalia

(Ait-Sahalia 1999, 2002) in order to evade this problem.

When estimating a continuous time model, a very robust method for estimating its

parameters is by using maximum likelihood estimation. The main caveat when using this

method in the context of this paper is that the estimated likelihood function for discrete

observations generated by the diffusion cannot be determined explicitly for most models.

What Ait-Sahalia is able to do in his paper is to construct closed-form sequences of

approximations to the transition density and therefore, approximations to loglikelihood

functions. From these estimated approximations of the likelihood function, one can then

find the parameters of the desired model. In addition to the method, Ait-Sahalia also

provides Matlab code which estimates certain diffusion models based on his

methodology. This paper uses manipulations and modifications of his code in order to

estimate the parameters of the CEV and GBM diffusions.

4.2 Testing Procedures

In order to test the CEV and GBM diffusions for certain time intervals, I design

the following structure of estimation, simulation and testing. The general approach is to

predict return prices using both models and to then test which model predicts price

returns more accurately. This process is tested by using a test derived and explained in

this paper and also uses likelihood ratio tests1.

1 The testing method was derived with the suggestions and comments from Norman Swanson who is well-known for research in diffusion specification testing (Swanson, 1993).

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4.3 Estimation Procedure

- Start with the interval of interest and estimate both the CEV and GBM modelsusing the MLE method derived by Ait-Sahalia.

- Increase the original dataset by 1 additional day of price returns data and estimate the model again.

- Keep estimating the model n times where n is the number of predictions that aredesired. (In this paper, each interval has n = 200 estimated models and corresponding predictions).

4.4 Simulating

Once the desired number of models is estimated, simulations are then run in order

to predict the returns for the next day using the corresponding model with estimated

parameters. A total of n predictions are calculated using the estimated models.

For example, suppose one was to simulate the CEV diffusion given estimated

parameters for a starting interval. The model can be written as follows:

dX t = b(a – X t )d t + c*X t ̂ d* dW t dW t ~ N(0, d t )Where a, b, c and d are the parameters estimated by the MLE method

One can then rewrite the equation as:dX t /X t = [b(a – X t ) d t + c*X t ̂ d* dW t ]/X t

Notice that this form of the diffusion reflects the returns over time dt. Using this

model, one can then take the last observation in the corresponding interval and set it as

Xt. Letting dt = 1/252, the equation then predicts the returns for the following business

day. This provides one of the n desired predictions needed for testing. In order to get the

other predictions, one can then increment the original interval by one more data point and

re-estimate and simulate the model as done above to get the predicted returns of the next

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business day. This procedure is done n times for a total of n estimated models and

corresponding predictions for both the CEV and GBM diffusions.

4.5 Testing

After getting n predictions, tests can be run in order to see which diffusion works

best. This paper creates a simple test for observing the differences between the models

and also uses loglikelihood ratio tests.

4.5.1 Comparing average squared errors

The following test is created in order to check if either the CEV model weakly

dominates GBM or vise versa. Recall that during the simulation process, n predictions are

estimated corresponding to a certain time interval. For each n predictions, one can then

create an array of squared errors:

ErrSQ = [returns_predicted – returns_actual]2

The test takes the difference between the CEV and GBM squared errors and compares

them for each n simulations.

Err_Diff = CEV_ErrSQ – GBM_ErrSQ

This will create an array of n differences in error. The test then counts to see how

many of the entries in this array are negative and checks to see if the count is greater than

or equal to n/2. If the count is greater than or equal to n/2, then the CEV predictions

contain fewer errors than those of the GBM diffusion.

Notice however, that if one were to simulate the models as described by the

simulation procedure multiple times, each simulation would produce different predicted

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returns. This results from the fact that dW t is by definition a randomly generated number

and will therefore effect the results of Err_Diff.

In order to address this problem, the process above is repeated 100 times and the

average number of times that the CEV errors are less than GBM errors is computed. If the

value is greater than 50, then the CEV model weakly dominates the GBM model for the

respective time interval.

4.5.2 Loglikelihood ratio tests

Since the parameter estimation involves a MLE procedure, the loglikelihood

function is readily available for analysis and can be used to compare the two models of

interest. Notice that the CEV diffusion is a generalization of the GBM diffusion. Defining

the CEV diffusion as follows:

dX t = b(a – X t ) d t + c*X t ̂ d* dW t Where a, b, c and d are parameters of the diffusion.

Letting d = 1 and a =0, the model becomes:

dX t = b X t d t + c*X t * dW t

Notice that the above diffusion is GBM.

Because the CEV model is a generalization of GBM, one can therefore run a

loglikelihood ratio test given that one knows the loglikelihood computation for each

estimated model. This computation is given when estimating the parameters from the

method by Ait-Sahalia.

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The null Hypothesis H0 of the test is that the generalized model (CEV) is not

statistically a better predictor than the condensed model (GBM). The test statistic is

defined as follows:

-2*(loglikelihood_CEV – loglikelihood_GBM) ~ ChiSQ(2)

Where ChiSQ(2) is a chi-squared distribution with degrees of freedom equal to the

number of parameter of CEV minus the number of parameters of GBM which equals 2.

Rejecting the H0 means that one can reject GBM as being equivalent to the CEV model

for the given dataset.

Section 5: Results

This section explains the results of the tests described in the previous section.

What is found is that during the interval from April 1998 to February 1999, the CEV

model significantly outperforms the GBM model when predicting price returns 200 days

out of the sample. During the time period of March 1994 to January 1995, the GBM

model is found to predict returns as well as the CEV model.

5.1 April 1998 to February 1999

Comparing average squared errors

On the interval of April 1998 to February 1999, the average number of times that

the CEV errors were smaller than the GBM errors is described by the following table:

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Table 6

Number of Predictions out from the data Average

200 57.74

Notice that the results of this testing procedure are as expected. CEV tends to predict

price returns better than GBM during times of transitioning volatility.

Loglikelihood ratio test

The p-value associated with the likelihood ratio test was 0.0797 indicating that

one can reject the null that the CEV model is not significantly better than the GBM

model.

5.2 March 1994 to January 1995

Comparing average squared errors

On the interval of March 1994 to January 1995, the average number of times that

the CEV errors were smaller than the GBM errors is described by the following table:

Table 7

Number of Predictions out from the data Average

200 52.58

Notice that the results of this testing procedure are as expected. GBM tends to predict

price returns almost as well as CEV during times of relatively constant volatility.

Loglikelihood ratio test

The p-value associated with the likelihood ratio test was 0.3657 indicating that

one can fail to reject that the null that the CEV model is not significantly better than the

GBM model.

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5.3 Graphing the results

The general conclusion of the tests run is that the CEV diffusion weakly

dominates the GBM diffusion during times of transition in volatility. During times of

constant volatility, GBM tends to predict more relatively the same as CEV. Below are

graphs of the squared errors when predicting returns for GBM and CEV plotted on top of

each other for each time period. Notice that for the period of 1998 to 1999, GBM errors

are clearly higher than CEV whereas for the period 1994 to 1995, errors are relatively

similar.

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Graph 6: Squared Errors April 1998 to February 1999

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

3.5x 10

-3

n predictions

S q u a r e d

e r r o r

CEV and GBM

CEV

GBM

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Graph 7: Squared Errors March 1994 to January 1995

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6x 10

-4 CEV and GBM

S q u a r e d

E r r o r s

n predictions

CEV

GBM

Section 6: Interpretation of Results

An important aspect of this research is to provide practical interpretations of these

results for investors and traders. This section therefore aims to highlight the main

conclusions of the results found in this paper and relates them to everyday trading and

investment strategy. In particular, this subsection explains what these results imply about

option pricing.

The Black Scholes model for option pricing is by far the most widely used option

pricing tool in the financial sector. It is therefore interesting to note that this model is

based on the geometric Brownian motion diffusion for the underlying asset of the option

of interest. As it was thoroughly analyzed in this paper, GBM models price returns very

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well during times of relatively constant velocity but is weakly dominated by the CEV

model during times of transition in volatility. The main caveat when using GBM is that

one must assume constant volatility in the dataset which can lead to inaccurate

predictions of price returns. Historically, the use of GBM has been very accurate except

during particular times when it leads to extreme losses.

It is important to note that investors and traders do not blindly choose volatility to

be constant and then price assets based on this value. There are many aspects to deriving

option prices on Wall Street and it is rarely based solely on historical volatility as done in

this paper.

Information from traders in investment banks and private equity firms (who prefer

to remain anonymous) adds a more practical element to this paper. Based on their

comments on how their respective companies conduct trades, there are several volatilities

that are calculated. These traders look at the implied volatility for given assets (solving

the Black Scholes model for volatility given the option’s price), a measure of qualitative

information called a put and call skew and the historical volatility. The implied volatility

captures the market’s interpretation of where the volatility is in the market. The put and

call skews are qualitatively determined and affect the volatility parameter by adding or

subtracting value to it based on what information they know about the asset of interest.

For example, if a positive financial report from a company is to be released in the near

future, then the call and put skews associated with the option based on that company’s

stock would positively increase the volatility parameter in order to account for this event.

The historical volatility captures the past volatility of the asset of interest. Combining all

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measures may account for errors in the constant volatility assumption (GBM) when used

correctly, but they are subject to human error. If investors are able to rely more on the

data by using a more generalized model, then less human errors could result. Therefore,

investors could use another model such as the CEV model or other stochastic volatility

models (Heston, Scott, etc) to predict returns allowing them to rely solely on the data

when pricing options. According to this research, during times of transition in volatility, it

would be even more useful to apply an alternative model in order to insure accurate

pricing. Traders could still apply qualitative analysis but would be able to rely more

heavily on the model for choosing the correct volatility.

Future research needs to be done in order to verify this claim. A challenge in

answering this question would be the ability to find option price predictions from a

company that uses this approach and compare it to simulated option prices derived from

the CEV model.

Section 7: Conclusion

This paper tests the geometric Brownian motion and constant elasticity of

variance diffusions for S&P 500 Index prices from April 1998 to February 1999 and from

March 1994 to January 1995. The results show that over time periods of strong

transitions of volatility, the CEV model predicts price returns more accurately than the

GBM. During times of relatively constant volatility, both the CEV and GBM models

predict returns equally well. Although the CEV model did not drastically outperform the

GBM model, the results imply that it is necessary to incorporate a stochastic element to

volatility when predicting returns. This could perhaps also be done using other models

30



such as GARCH models or perhaps even Jump Diffusion models which would allow for

the volatility to jump. Further research is necessary to explore these different avenues to

search for an ideal method for modeling price returns and pricing options.

References

Aït-Sahalia, Yacine. “Maximum Likelihood Estimation of Discretely Sampled

Diffusions: A Closed-Form Approximation Approach.” Econometrica. 70 (Jan., 2002):223-262.

Amin, Kaushik I. and Victor K. Ng. “Option Valuation with Systematic Stochastic

Volatility.” The Journal of Finance. 48.3 (Jul., 1993): 881-910.

Black, Fischer and Myron Scholes. “The Pricing of Options and Corporate Liabilities.”The Journal of Political Economy. 81(May - Jun., 1973): 637-654.

Broadie, Mark and Jerome B. Detemple. “Option Pricing: Valuation Models andApplications.” MANAGEMENT SCIENCE. 50 (September 2004): 1145–1177.

Cody, Ronald P. and Jeffrey K. Smith. Applied Statistics and the SAS Programming

Language. Ed. 5. New Jersey: Pearson Education Inc., 2006.

Corradi, Valentina and Norman R. Swanson. “Bootstrap Specification Test for Diffusion

Processes.” Working Paper. (June 2003): JEL No. C12, C22.

Crouch, R. L. “A Nonlinear Test of the Random-Walk Hypothesis.” The American Economic Review. 60 (1970): 199-202.

DePeña , Javier and Luis A. Gil-Alana. “Do Spanish Stock Market Prices Follow a

Random Walk?” Working Paper. No.02/02 (April 2002): JEL No. C22, G14.

Ewald, Christian-Oliver. “Mathematical Finance Introduction to continuous time

Financial Market models.” Continuous Time Finance, School of Economics and Finance

University of St. Andrews. 27 March 2007.

Heston, Steven L. “A Closed-Form Solution for Options with Stochastic Volatility with

Applications to Bond and Currency Options.” The Review of Financial Studie. 6.2(1993): 327 – 343.

Hong,Yongmiao and Halbert White. “Consistent Specification Testing Via Nonparametric

Series Regression.” Econometrica. 63(Sep., 1995): 1133-1159.

31



Jiang, George J. and Pieter J. van der Sluis. “Index Option Pricing Models with

Stochastic Volatility and Stochastic Interest Rates.” Center for Economic Research.(March 2000).

Lo, Andrew W. and A. Craig MacKinlay. “Stock Market Prices do not Follow RandomWalks: Evidence from a Simple Specification Test.” The Review of Financial Studies. 1

(Spring, 1988): 41-66.

Ross, Sheldon M. An Elementary Introduction to Mathematical Finance. Ed. 2. New

York: Cambridge UP, 2003.

Scott, Louis O. “Option Pricing when the Variance Changes Randomly: Theory,Estimation, and an Application.” The Journal of Financial and Quantitative Analysis.

22.4 (Dec., 1987): 419-438.

Swanson, Norman. “Bootstrap Specification Testing for Diffusion Processes.” Journal of Economic Literature. (June, 2003).

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