URL of this page is: http://business.fortunecity.com//mitchell/257/index.html
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In these webpages I plan to provide information on Probability and Options Investment.
Thank you.
-Bakulesh Thakker
My brief biography: I am born in 1955 unmarried male from India, I graduated in M.Stat from Indian Statistical Institute.
My current interests include investing in options. Since I started with small account, I am especially interested in options investment strategies for small investors.
Currently I am reading Chapter 10 of Bernt Oksendal's Stochastic Differential Equations (5th ed, Springer Verlag). The book is highly mathematical but interesting. Chapter 10 deals with Optimal Stopping Rules. For example, if you owned a stock you might want to find out when is the right time to sell the stock. Right time to sell would maximize your gain discounted at some discounting rate. This might involve maximizing expectation at sell time of following "reward function":
exp(-rho * t) * (X(t) - a)
where rho is discounting rate of a future payment
X(t) is stock price at t
a is cost of sell transaction.
Okesndal actually solves this problem under the assumption that X(t) varies according to stochastic differential equation (SDE):
dX(t) = r * X(t) * dt + alpha * X(t) * dB(t)
where B(t) is 1-dimensional Brownian motion
More on this topic later.
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Probability That Price May Go Up ...(1)
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Hi All,
I have a Masters degree in Statistics from Indian Statistical Institute. In coming days I am going to post a series of articles with title "Probability That Price May Go Up ...". The aim of these articles is to publish ideas of probability and statistics and their relation to investment in options. Computer programs may also be published. Anything useful for options trader.
By way of disclaimer, I want to say following: Mathematics and Statistics are abstract topics and my training in Statistics was long time ago (I graduated in 1981). There may be errors in what I write. Programs may also have errors. No material in these articles comes with any kind of warranties. Copyright is mine. All quotations will be properly attributed to its original authors.
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Although the topic of options pricing is investigated by mathematicians, many books on options trading don't even mention Black-Scholes option pricing model, the most famous of all such models. I have eleven books on options, two of which are by Malkind [1,2]. Malkind's books are targeted to audience averse to mathematics. In fact Malkind writes in [2]:
"Q: Why is the literature on options so complicated? ...
A: ... part of the problem is that option pricing on a theoretical level involves complicated mathematics. ...
An interesting question would be to ask how many of these authors actually trade options and just how successful they are, but I'm going to leave this subject for a later project." [2,p 3]
Of the two books which I have which has a mention of Black-Scholes option pricing model, one is by Trester and the other is by Kolb. Trester doesn't give the Black-Scholes' formula but gives rationale for its derivation:
"The Black & Scholes pricing formula estimates what the market price of an option should be, and it does this by determining the cost of creating a perfect hedge in the market, using options and stock." [3, p 413]
Kolb begins discussion of option pricing with a simplified model of option price which ignores effects of interest rate and volatility. [4, p 82] Although greek symbol sigma for volatility is introduced at the beginning it is not defined and discussed only later. [4, p 98] Here instead of precisely defining sigma an intuitive description precedes a formula where sigma is used:
"... a stock ... that will experience either a 10 percent price rise or a 10 percent price decline over the next year." [4, p 98]
When at last Kolb gives the formula for the price of an European Call by Black-Scholes model, sigma is described as: instantaneous variance rate of the stock's returns. [4, p 103]
What is this sigma? Given prices of a stock in recent past, (say, closing prices of last several days) how do we compute sigma? We need data for how many days? Is it good idea to use all data we have?
On how the formula is derived Kolb says:
"The mathematics of [model assumes] that stock prices follow a certain kind of path through time called a stochastic process. A *stochastic process* is simply a mathematical description of the change in the value of some variable through time. The particular stochastic process used by Black and Scholes is known as a *Wiener process*. The key features of the Wiener process are that the variable changes continuously through time and that the changes that it might make over any given time interval are distributed normally." [4, p 102]
The word "normally" refers to the famous Normal distribution discovered by Gauss. Because word "normally" is also a word of ordinary english to mathematically uninformed words "distributed normally" might be confusing unless this new meaning of word "normally" is also explained.
A random variable X is distributed according to the density function f(x) if probability that X will take a value between x and (x + dx) is: f(x).dx. By notation "dx" is meant: infinitesimally small neighborhood. If h > 0 then dx is limit of h as h is reduced to 0(zero) and f(x).dx is limit of f(x).h as h is reduced to to 0(zero). Both these quantities are numerically same: 0(zero) but mathematically denote separate things: dx/dx is 1(one) and f(x).dx/dx is f(x). This is because dx/dx is limit of h/h as h is reduced to 0(zero) and f(x).h/h is f(x) as h is reduced to 0(zero). What all these really means is: f(x) is defined as the limit of probability that X takes value between x and (x + h) as h reduced to 0(zero). We are forced to define our density function like this becasue X can be a continuous variable. For a continuous variable Probabilty(X=a particular x) = 0 for all x, but Probability(X is between x and x + h) is nearly proportionate to h. And constant of proportionality is f(x).
Probability that X will take a value less than or equal to x is:
F(x) = Integral(xi = -infinity to xi = x) of f(xi).d(xi)
By integral of f(x).dx is meant the area under the curve f(x) between the two boundary values. Thus Integral(xi = a to xi = b) of f(xi).d(xi) is area under curve f(x) between a and b. F(x) is called cumulative function. Since X will take a value between -infinity and +infinity, F(+infinty) = 1. Probability of 1 means 100 percent certainty.
[1] Malkind, Samuel N. (1995) Options Are Easy to Understand, New York: Vantage Press.
[2] Malkind, Samuel N. (1998) Option Strategies for Beginners, New York: Dorrance Publishing Co.
[3] Trester, Kenneth R. (1998) The Complete Option Player, 3rd ed., Lake Tahoe: Institute for Options Research, Inc.
[4] Kolb, Robert W. (1993) Financial Derivatives, Paramus: New York Institute of Finance.
Copyright 1999, Bakulesh Thakker (Email: [email protected])
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Probability That Price May Go Up ...(2)
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The density function of Gauss's Normal distribution is:
f(x) = (1 / sqrt(2.pi.sigma).exp^( -(x - myu)^2/(2.sigma^2))
where pi stands for ratio of circumference of a circle to its diameter and exp stands for exponential e which is limit of (1 + n)^n, as n is increased to +infinity. sqrt stands for positive square root of. Thus sqrt(x)^2 = x. myu and sigma are two parameters.
One use of exponential e is to model an investment which earns interest which is continuously compounded. Assume that you loan a person a principal sum of P at an annual interest of i percent. If interst is compounded every year, then after 5 years he owes you:
P.(1 + R)^5, where R = i/100.
This is because interest is compounded (in other words added to principal) every year. If interest is compounded monthly (that is interest is added to principal every nth of a year for n times a year, where n=12) then he owes you:
P.((1 + R/12)^12)^5 or: P.((1 + R/n)^n)^5
The quantity: (1 + R/n)^n is e^R in limit as n is increased to +infinity.
Kolb has mentioned the assumption that stock prices follow a stochastic process known as Wiener process a key feature of the which is: that the variable changes continuously through time and that the changes that it might make over any given time interval are distributed normally." [4,p 102]
A college text for Statistics (Hsu [5]) defines Wiener process as:
"A random process {X(t), t>=0} is called a Wiener process if
1. X(t) has stationary independent increments.
2. The increment X(t) - X(s) (t>s) is normally distributed.
3. E[X(t)] = 0.
4. X(0) = 0.
The Wiener process is also known as the Brownian motion process ..."[5,p 172]
If we write S(t) for price of a stock t years from now then clearly S(t) is not a Wiener process as defined by Hsu because S(0) not = 0. But probably S(t) - S(0) is. But even S(t) - S(0) cannot be a Wiener process as defined by Hsu. That is because S(t) - S(0) cannot be less than -S(0), that is: in worst case scenario S(t) falls to zero, but it cannot go below zero. Wiener process permits a negative X(t) unrestrictedly. Almost everyone who has thought about stock prices have come to conclusion that absolute dollar value of change in the price of a stock is not true indicator of how much the price has moved. It is the value of change in price *relative* to current price which is a more valid indicator of change. We suspect Professor Kolb is way off mark in describing stock prices as following a Wiener process. Another book clears up confusion on this matter [6].
The authors Les Clewlow and Chris Strickland mention in their book:
"Most models of asset price behaviour for pricing derivatives are formulated in continuous time framework by assuming a stochastic differential equation (SDE) describing the stochastic process followed by the asset price. For example, the most well-known assumption made about asset price behaviour, which was made by Black and Scholes (1973), is *geometric* Brownian motion."[6,p 3]
The reference to 1973 work of Black and Scholes is perhaps most referred journal article in options pricing literarture: "The Pricing of Options and Corporate Liabilities", Journal of Polical Economy, 81, pp 637-659. The word "geometric" is highlighted to draw attention to the key difference between the description of Kolb and that of Clewlow et al.
Clewlow et al derive Black-Scholes equation for a European call using formalism of stochastic differential equations. Before Clewlow et al write the Black-Scholes equation on page 7, authors refer to the book: Options, Futures, and Other Derivative Securities by J. Hull (published: 1996 by Prentice-Hall) for more information on justification of assumption that future evolution of the asset depends only on its present level and not on the path taken to reach that level. Along the way Ito's lemma is also mentioned. When I read these works I will post my thoughts on them. But for the present my interest is only in finding a definition and meaning of sigma. And this I get in their conclusion that [6,p 7]:
... the natural logarithm of S [asset price] at time T is normally distributed with following characteristics:
LOG S(T) ~ N(LOG S(t) + (r - d - sigma ^ 2 / 2) * (T - t), sigma * (T - t) ^ 0.5)
where S(t) is asset price at time t,
r is riskless rate of interest
d is per annum continuous dividend yield rate.
That is: LOG S(T), the natural logarithm of S(T) follows a Normal distribution with
Mean = LOG S(t) + (r - d - sigma ^ 2 / 2) * (T - t)
Standard Deviation = sigma * (T - t) ^ 0.5
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Material beyond this line is under construction
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We denote r - d -sigma^2/2 by nu as Clewlow and Strickland do. So our normal distribution becomes:
N( LOG S(t) + nu * (T - t), sigma * (T - t) ^ 0.5)
Or,
LOG(S(T) / S(t)) ~ N(nu * (T - t), sigma * (T - t) ^ 0.5)
We know that E[a * X] = a * E[X] and
and Var[a * X] = a ^ 2 * Var[X].
Since Var[LOG(S(T) / S(t))] = sigma ^ 2 * (T - t)
Var[LOG(S(T) / S(t)) / (T - t) ^ 0.5] = sigma ^ 2 * (T - t) / ((T - t) ^ 0.5) ^ 2
= sigma ^ 2
From this we can say that:
LOG (S(T)/S(t)) / (T - t) ^ 0.5 ~ N ( nu / (T -t) ^ 0.5, sigma)
From this we can work out an estimation procedure for estimating sigma from historical data. Before we give the procedure for estimating we want to make a little change. In options literature we see time measured in years. This makes a period of 1 day = (1/365) = 0.0027397 approximately. So, to simplify we use 1 day as our unit of time. The sigma value that we will determine will be a "dailiized" value. To find corresponding "annualized" value we just multiply this daily sigma by square root of Y, where Y is number of days in a year = 365 approximately.
So our basic data is a quantity like: LOG (S(T) / S) / (T - t) ^ 0.5 which is distributed as a Normal Distribution:
N (nu / (T - t) ^ 0.5, sigma)
Please note that this Normal distribution has a standard deviation = sigma no matter what (T - t) may be.
Let S(T)/S(t) be called: Ratio.
Let LOG (S(T)/S(t)) be called Log-Ratio (LogR).
Let LOG (S(T)/S(t)) / (T - t) ^ 0.5 be called Corrected-Log-Ratio (CLR).
From our database of historical stock price we access last few of records and assemble the figures: S (stock price) and T (time). From this we build a list of pairs (S1,S2) such that all such pairs have same T2 - T1 (T2 > T1). We compute Corrected-Log-Ratios from these pairs. Each of these CLR is supposed to be from same Normal distribution with standard deviation = sigma.
If we have (say) two Corrected-Log-Ratios from same list: X1 = CLR1 and X2 = CLR2, then usual unbiased estimator of sigma ^ 2 is:
(ss - s-square / n) / (n - 1), (here n = 2)
where
s-square = s * s
s = X1 + X2
ss = X1 * X1 + X2 * X2
Example:
(Visit my page sometime later, you will find a numeric example.)
[5] Hsu, Hwei P. (1997) Schaum's outline of theory and problems of Probability, Random Variables, and Random Processes, New York: McGraw-Hill.
[6] Clewlow, Les, and Chris Strickland (1998) Implementing Derivatives Models, Chichester: John Wiley & Sons Ltd.
Copyright 1999, Bakulesh Thakker (Email: [email protected])