2.2.2.1 Calculus of the Deterministic Model

In deterministic calculus of functions, f(x), of a single variable x, there are some important results, commonly called theorems, available for relating the variability characteristics of such functions. These theorems include the following:

• The Mean Value Theorem:

  If f(x) is continuous in the closed interval [a, b], and is differentiable in the open interval (a, b), then there is a value ξ of x, namely, a < ξ < b, such that

  (2.1)
• The Higher Mean Value Theorems:

  For 0 < θ < 1

  (2.2)

  If f′(x) is continuous for a ≤ x ≤ b, and if f″(x) exists for a < x < b, and if h =b – a, and 0 < θ₂ < 1, then

  (2.3)
• The General Mean Value Theorem (Taylor's Theorem):

  If f⁽ⁿ⁻¹⁾(x) is continuous for the closed interval a ≤ x ≤ b, and f⁽ⁿ⁾(x) exists for a < x < b, then

  (2.4)

  where a < ξ < b; and if b = a + h, then

  (2.5)

  The continuity of f⁽ⁿ⁻¹⁾(x) involves that of f(x), f′(x), f″(x),…, f⁽ⁿ⁻²⁾(x).

• Taylor's Theorem for Functions of Several Variables:

  Taylor's formula for functions of one variable, Equation (2.5), may be generalized to functions of several variables having continuous derivatives of order n in a given region of space. The following procedure is outlined for functions of two variables – the result has all the features of more general cases.

  Let f(x, y) be a function of two variables (x, y) having continuous derivatives of order n in some region containing the point.

  

  and let

  

  be a nearby point in the region. The equations of a straight line segment joining these two points may be expressed in the form

  (2.6)

and, along this segment,

(2.7)

(2.8)

The successive derivatives of F(t) may be computed from Equation (2.7) and by the chain rule for differentiating composite functions:

If u = f(x, y), then

(2.9)

Thus,

(2.10)

where in the last step, one recalls Equation (2.6).

Similarly,

and

where C_n,r ≡ n!/[r! (n – r)!] are the binomial coefficients.

From Equation (2.6), it is seen that t = 0 corresponds to the point: x = a, y = b, so that

and so on.

Substituting these values in Equation (2.8) gives

(2.11)

where R_n = [F⁽ⁿ)(θt)/n!] tⁿ, 0 < θ < 1.

Now, from Equation (2.6), αt = x − a and βt = y − b, so that Equation (2.11) takes the form

(2.12)

which is the Taylor formula (for two variables, in this case).

2.2.2.2 The Geometric Brownian Motion (GBM) Model and the Random Walk Model

Historically, the GBM had been developed in terms of the concept of the random walk, W_n(t), which has the following characteristics:

For a positive integer n, the binomial process W_n(t) may be defined by the following characteristic properties:

1. W_n(t) = 0, at t = 0
2. W_n(t) has layer spacing 1/n
3. There exist up and down jumps, equal and of magnitude 1/√n
4. W_n(t) has measure P, given by up and down probabilities ½ everywhere

Thus, if X₁, X₂, X₃,…, X_i is a sequence of independent binomial random variables taking values +1 or −1 with equal probability, then the value of W_n at the ith step is

2.2.2.3 What Does a “Random Walk” Financial Theory Look Like?

The following chart is a simple example.

Random walk hypothesis test by increasing or decreasing the value of a fictitious stock based on the odd/even value of the decimals of π. The chart resembles a stock chart. (Figure 2.3)

Remarks:

1. A Brownian motion wanders randomly with zero mean, but a company stock usually grows at some rate (even if it just increases with normal inflation, etc.).
2. Thus one may choose to add in a drift factor μ. For example, the random walk process may be described as

(2.13)

in which

1. W_t is the log of the asset price at time t
2. μ is a drift constant
3. ∊_t is a random disturbance term for which the exceptions

It had been demonstrated that the random walk hypothesis is inaccurate, there are trends in the stock market, and the stock market is somewhat predictable.

For progressing from a pure GBM to a modifiable deterministic–probabilistic model for the stock market, the following path may be followed:

• A deterministic model versus a probabilistic model
• A combined deterministic and probabilistic model

2.4.2.1 The Ito Lemma

The Ito Lemma is named after its discoverer, the brilliant Japanese mathematician Prof. Dr. Kiyoshi Ito, who passed away on November 10, 2008, at the age of 93. His work created a field of mathematics that is a calculus of probabilistic, or stochastic, variables (http://www.sjsu.edu/faculty/watkins/ito.htm).

Changes in a variable, such as the price of a stock, involve two components:

1. A deterministic component that is a function of time.
2. A probabilistic, or stochastic, component that depends upon a random variable.

Let S be the stock price at time t, and let dS be the infinitesimal change in S over the infinitesimal interval of time dt. The change in the random variable z over this interval of time is dz. Hence, the change in stock price is given by

(2.23)

where a and b may be functions of S and t as well as other variables; that is,

(2.24)

The expected value of dz is zero, so the expected value of dS is equal to the deterministic component adt. Concomitantly, the random variable dz represents an accumulation of random contributions over the interval dt. Applying the central limit theorem, which implies that dz has a normal distribution and hence is completely characterized by its mean and standard deviation:

• The mean, or expected value, of dz is zero.
• The variance of a random variable that is the accumulation of independent effects over a given interval of time is proportional to the length of the interval: in this case dt. Hence, the standard deviation of dz is proportional to the square root of dt, namely, (dt)^½.

All of this means that the random variable dz is equivalent to a random variable w(dt)^½, where w is a standard normal variable with mean zero and standard deviation equal to unity.

Now consider another variable C, such as the price of a call option: it is a function of S and t, say C = f(S, t): because C is a function of the probabilistic variable S, it will have a probabilistic component S as well as a deterministic component. Thus, C will have a representation of the form

(2.25)

where the coefficients p and q may be functions of S, t, and possibly other variables; that is,

(2.26)

It remains to be determined how the functions p and q are related to the functions a and b in the equation

(2.27)

The answer may be found using the Ito Lemma:

Remark: The Ito Lemma is crucial in deriving differential equations for the value of derivative securities such as stock options.

A Proof of the Ito Lemma:

A derivation of the Ito Lemma is provided herein.

The Taylor series for f(S, t) gives the increment in C as

(2.29)

For example: The increment in stock price dS is given by

(2.30)

where w is a standard normal random variable and v is the scale of the variability of the random element; that is, its standard deviation. Substitution of adt +bvw(dt)^½ for dS in Equation (2.27) yields

(2.31)

With the expansion of the squared term and the product term the result is

(2.32)

Taking into account the infinitesimal nature of dt so that dt to any power higher than unity vanishes, namely, all the terms in red in Equation (2.32) vanish, so that Equation (2.32) may be reduced to

(2.33)

Now, the expected value of w² is unity, hence the expected value of dC is

(2.34)

This, (2.32), is the deterministic component of dC.

The probabilistic, or stochastic, component is the term that depends upon dz, which in (2.31) is represented as vw(dt)^½. Therefore, the probabilistic, or stochastic, component is

(2.35)

In the foregoing derivation, it would seem that there is an additional stochastic term that arises from the random deviations of w² from its expected value of 1; that is, the additional term

(2.36)

However, the variance of this additional term is proportional to (dt)², whereas the variance of the stochastic term given in Equation (2.33) is proportional to (dt). Thus the stochastic term given in Equation (2.34) vanishes in comparison with the stochastic term given in Equation (2.33).

Remarks:

1. The Ito Lemma Is Essential in the Derivation of the Black–Scholes Equation

   An immediate question is whether Black–Scholes equation is an extension of Ito's Lemma for stable distributions of z other than the normal distribution. This question may be investigated in the study of stable distributions.

2. Probabilistic (or Stochastic) Calculus and the Ito Lemma

   Now, a probabilistic process S_t is a GBM if it satisfies the following probabilistic differential equation:

   (2.37)

   where W_t is the Wiener or Brownian motion, and μ the percentage drift and σ the percentage volatility are constants.

   Here, μ is used to model deterministic behavior and σ is used to model a set of unpredictable events occurring during the motion.

3. Solving the Probabilistic Differential Equation

   For an arbitrary initial value S₀ the above probabilistic differential Equation (2.37) has the analytic solution

   (2.38)

This result may be obtained as follows:

• First divide the probabilistic differential (Equation 2.37) by S_t in order to have the choice random variable on only one side.
• Next, write the equation in integral form (known as the Ito form):
  (2.39)
• At this point, it might appear that dS/S_t may be related to ln S_t.

  However, S_t is an Ito process, requiring the use of Ito Calculus.

  Applying the Ito Lemma leads to

  (2.40)

  where dS_t dS_t is the quadratic variation of the SDE, which may be written as d[S]t or <S.>_t. In this case, one has

  (2.41)
• Substituting the value of dS_t in Equation (2.41), and simplifying, one obtains
  (2.42)
• Taking the exponential, and multiplying both sides by S₀ gives the solution (2.40).

2.5.1.1 The Geometric Brownian Motion (GBM) Model and the Random Walk Model

Historically, the GBM had been developed in terms of the concept of the random walk, W_n(t), which has the following characteristics:

For a positive integer n, the binomial process W_n(t) may be defined by the following characteristic properties:

1. W_n(t) = 0, at t = 0
2. W_n(t) has layer spacing 1/n
3. There exist up and down jumps, equal and of magnitude 1/√n
4. W_n(t) has measure P, given by up and down probabilities ½ everywhere.

Thus, if X₁, X₂, X₃,…, X_i is a sequence of independent binomial random variables taking values +1 or −1 with equal probability, then the value of W_n at the ith step is

(2.43)

What does a “Random Walk” look like?

The chart that follows is a simple example.

Random walk hypothesis test by increasing or decreasing the value of a fictitious stock based on the odd/even value of the decimals of π. The chart resembles a stock chart (Figure 2.5a).

Notes:

1. A Brownian motion wanders randomly with zero mean, but a company stock usually grows at some rate (even if it just increases with normal inflation).
2. Thus one may choose to add in a drift factor. For example, the process with a constant noise factor σ may be represented by
   (2.44)

• A Deterministic Model versus a Probabilistic Model
• A Combined Deterministic and Probabilistic Model

2.5.1.2 Other Models for Simulating Random Walk Systems Using R

The following are the two, among many, efficient models for simulating random walk systems that have been found to be useful in financial engineering and econometric data modeling:

1. The Cox–Ingersoll–Ross (CIR) model
2. The Chan–Karolyl– Longstaff–Sanders (CKLS) model

The CIR model is generally used to describe the evolution of interest rates. It is a type of “one-factor model” (aka a short rate model) as it models interest rate movements as driven by only one source of market risk. The model may be used in the valuation of interest rate derivatives. It was introduced in 1985 by J.C. Cox, J.E. Ingfersoll, and S.A. Ross. The CIR model assumes that the instantaneous interest rate follows the stochastic (probabilistic) differential equation. The CIR is an ergodic process, and possesses a stationary distribution.

The CKLS models belong to a class of parametric stochastic differential equations widely used in many finance applications, in particular to model interest rates or asset prices. This model nests a class of asset pricing models such as the CIR. This flexibility of the CKLS model spawns many empirical applications.