the rate of return is viewed as a constant
value subjected to perturbations
http://www.fintools.com/wp-content/uploads/2012/02/StochasticStockPriceModeling.pdf
Stochastic Modeling of Stock Prices
© Montgomery Investment Technology, Inc. / Sorin R. Straja, Ph.D., FRM
May 1997
Sorin R. Straja, Ph.D., FRM
Montgomery Investment Technology, Inc.
200 Federal Street
Camden, NJ 08103
Phone: (610) 688-8111
sorin.straja@fintools.com
www.fintools.com
ABSTRACT
The geometric Brownian motion model is widely used to explain the stock price time series. The
following sections summarize its main features. The stochastic model may be viewed as an
extension of the usual deterministic model for which the rate of return is viewed as a constant
value subjected to perturbations. We present both the Ito and Stratonovich interpretations of the
resulting stochastic differential equation. The parameters estimation and model predictions
could be done using either interpretation; however, the same interpretation must be used for both
steps (i.e., parameters estimation and model predictions).
INTRODUCTION
Bachelier (1900) seems to be the first to have provided an analytical valuation for stock options.
His work is rather remarkable because by addressing the problem of option pricing, Bachelier
(1900) derived most of the theory of diffusion processes. The mathematical theory of Brownian
motion has been formulated by Bachelier (1900) five years before Einstein’s classic paper
(Einstein 1905). Bachelier (1900) has formulated avant la lettre the Chapman-Kolmogorov
equation (von Smoluchowski 1906; Chapman 1916; Chapman 1917; Kolmogorov 1931), called
today the Chapman-Kolmogorov-Smoluchowski-Bachelier equation (Brown et al. 1995), and the
Fokker-Plank or Kolmogorov equation (von Smoluchowski 1906; Fokker 1914; Fokker 1917;
Plank 1917; Kolmogorov 1931). Moreover, the first-passage distribution function for the driftfree
case was provided by Bachelier (1900) before Schrödinger (1915), and the effect of an
absorbing barrier on Brownian motion was addressed by Bachelier (1900; 1901) prior to von
Smoluchowski (1915; 1916). For a detailed summary of these early results the reader is referred
to von Smoluchowski (1912;1916).
Jevons (1878) pointed out that the chaotic movement of microscopic particles suspended in
liquids had been noted long before Brown (1827) published his careful observations; however, it
should be noted that Brown (1827) was the first to emphasize its ubiquity and to exclude its
explanation as a biological phenomenon. A precise definition of the Brownian motion involves a
measure on the path space that was first provided by Borel (1909) and constituted the basis of the
formal theory of Wiener (1921a; 1921b; 1923).
Bachelier assumed stock price dynamics with a Brownian motion without drift (resulting in a
normal distribution for the stock prices), and no time-value of money. The formula provided
may be used to valuate a European style call option. Later on, Kruizenga (1956) obtained the
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same results as Bachelier (1900). As pointed out by Merton (1973) and Smith (1976), this
approach allows negative realizations for both stock and option prices. Moreover, the option
price may exceed the price of its underlying asset.
Kendall (1953), Roberts (1959), Osborne (1959; 1964) and Samuelson (1965) modified the
Bachelier model (also known as the "arithmetic Brownian motion" model) assuming that the
return rates, instead of the stock prices, follow a Brownian motion (also known as the "geometric
Brownian motion" model or the "economic Brownian motion" model). As a result of the
geometric Brownian motion the stock prices follow a log-normal distribution, instead of a
normal distribution as assumed by Bachelier (1900). Sprenkle (1961; 1964) took into account
risk aversion and the drift of the Brownian motion, and based upon the log-normal distribution of
the stock prices, provided a new formula for the valuation of a European style call option that
rules out negative option prices. Boness (1964) improved the model of Sprenkle by considering
the time value of money: the present value of a call option is the discounted value provided by
Sprenkle (1961) using the expected rate of return of the stock as the discount rate. Samuelson
(1965) provided a rigorous review of the option valuation theory and pointed out that an option
may have a different level of risk when compared with a stock, and therefore the discount rate
used by Boness (1964) is incorrect. Samuelson and Merton (1969) provided a general
equilibrium formula that depends on the utility function assumed for a typical investor.
The Black and Scholes (1973) model is often regarded as either the end or the beginning of the
option valuation history. Using two different approaches for the valuation of European style
options, they present a general equilibrium solution that is a function of "observable" variables
only, making therefore the model subject to direct empirical tests. Based on the formula of
Thorp and Kassouf (1967) that determines the ratio of shares of stock to options needed to
construct a hedged position, and recognizing that shares and options can be combined to
construct a riskless portfolio, Black and Scholes (1973) developed an analytical model that
provides a no-arbitrage value for options. An alternative derivation is based upon the capital
asset pricing model that provides a general method for discounting under uncertainty. Merton
(1973) performed a rigorous analysis of the Black and Scholes (1973) model analyzing its
assumptions. The stock price dynamics is described by a Brownian motion with drift. The
manifest characteristic of the final valuation formula is the parameters it does not depend on.
The option price does not depend on the expected return rate of the stock or the risk preferences
of the investors. It is not assumed that the investors agree on the expected return rate of the
stock. It is expected that investors may have quite different estimates for current and future
returns. However, the option price depends on the risk-free interest rate and on the variance of
the return rate of the stock. A detailed analysis of the post Black-Scholes models is presented by
Smithson (1992).
Galai (1978) provided the correct discount rate for options, reconciling the Boness (1964) -
Samuelson (1965) approach with the Black and Scholes (1973) formula. The Black and Scholes
(1973) formula is identical with the Boness (1964) formula if instead of the return rate of the
stock we use the risk-free interest rate. However, this risk-neutral approach may lead to
confusion because it may be inferred that it can be proved that the return rate of the stock equals
the risk-free interest rate (Wilmott et al. 1997).
Stochastic Modeling of Stock Prices
© Montgomery Investment Technology, Inc. / Sorin R. Straja, Ph.D., FRM
May 1997
Page 2 of 19
Barrier options have become increasingly popular over the last years (Johnson and Stulz 1987;
Boyle and Turnbull 1989; Benson and Daniel 1991; Hudson 1991; Hudson 1995; Derman and
Kani 1993; Ravindran 1993; Bowie and Carr 1994; Hart and Ross 1994; Heynen and Kat 1994a;
Heynen and Kat 1994b; Rich 1994; Schnabel and Wei 1994; Broadie and Detemple 1995; Hull
and White 1995; Jarrow and Turnbull 1995; Zhang 1995; Gerber and Shiu 1996). Practitioners
who trade these instruments rely heavily on the numerical solutions provided by McConnell and
Schwartz (1986), Trippi and Chance (1993), Boyle and Lau (1994), Derman et al. (1994), Kat
and Verdonk (1995), Ritchken (1995), Geman and Yor (1996), Dewynne and Willmott (1997).
For European style options with a single barrier, analytical solutions were provided by Merton
(1973), Cox and Rubinstein (1985) and Rubinstein and Reiner (1991). Kunitomo and Ikeda
(1992) provided an analytical solution for options with a double barrier but without taking into
account the stock yield and the rebates corresponding to the barriers.
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