Testing Geometric Brownian Motion

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Transcript of 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

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    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 KPut Option: The right to sell a security at time T for price KEuropean Option: A call or put option that can only be exercised at time TAmerican Option1: A call of put option that can be exercised at or before time T

    1American 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 { ST K , 0 }Where: ST = underlying securitys price at time T

    K = Strike Price

    To see that this is the case, notice that ifST> 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 forST making a profit ofST K. IfST

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    (3) [dSt]/ St= udt + [dWt]

    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 [dSt]/ Stis defined above, it is clear that

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

    when assuming GBM. (To see this, note that dWt~N(0,dt)and that u and 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 [dSt]/ Stas log(St/ St h) where h is thelength of dt. Rearranging,

    (4) log(St) = log(St-h) + uh + [dWt]

    OR log(S t) = log(St-h) + uh + WtWhere Wt~N(0, h2)

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

    a drift parameteru. 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) St= St h *E(Wt)Where: Wt= uh + h Zt and Zt ~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) dSt = udtSt + Stb/2dWt

    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