One hundred and eleven years ago, this guy Louis pictured on the right determined mathematically that asset prices have a random nature to their behavior.  In the interim countless fortunes have been made and lost by people who have convinced themselves that they know something about the future.  This gave rise to the industry of financial intermediation and advice, whereby people get paid to make predictions about the future and relate them to asset prices and deduce recommendations about buying and selling them in order to be as wealthy as the pesky guy down the street with the white teeth and the porsche.  The facts show that in fact it is the other way around; he who entices another into taking their advice gets the white teeth at the expense of the punters.

A Martingale is something where the best estimate of the next value is the current value.  (i.e. without any new information, a stock is likely worth what it last traded for)

The point I am trying to make is that the only way to make money in stocks is to hold them until infinity.  Anything short of that introduces risk, with more and more risk the closer you get to intraday trading.  Trading actively creates fees and taxes.

FEES AND TAXES ARE THE ARCHENEMY OF WEALTH.....FACT.    (See   Logarithms)  

The reason for this is that stock prices are random events with "drift".  Drift is the magic associated with capitalism.  It is caused because joint-stock company, or corporation uses it's specialized knowledge to allocate investor's capital better than the investor could otherwise do by himself.  When this works out, assuming nobody has stolen from you, lied to you, or cheated you (rarely the case) over time the value of the stock will increase, to the point where after infinity time the value of the stock is almost certain to be higher than at day one......this is the DRIFT, and this is where insight is helpful.  However anyone who tells you that a stock is going up tomorrow is either lying, cheating or both.

However, I can tell you with near perfect certainty that in 999 years (or infinity, whichever comes first), we will not be paying $4.99 to watch a pay-per-view movie, we will not be using fossil fuels, and I will not need to reboot my operating system.  Those are my assertions, and they will have enormous impact on the Entertainment, Oil, and Software industries.  However, I can also tell you with near certainty that I know almost zero about the direction of the stock prices in these industries tomorrow.....sorry...

Concept of Brownian Motion and Ito’s Lemma

Written by someone who forgot to sign it, (I found it on www.actuarialoutpost.com forums board, with no attribution of author) so it is used without permission of the original author, but if I knew who they were I would certainly have asked for it......If you wrote this, please accept my compliments on a good presentation of these ideas, and let me know if it's OK to keep this here......Didn't your teachers always tell you to write your name on the top of the paper?

Normal and Lognormal Distribution

Understanding of normal and lognormal distribution.

 Whenever we say that X being a random variable follows lognormal distribution, it means that, when we take the logarithm of X, ie, Y= ln X, it becomes normally distributed, ie, Y follows normal distribution. This random variable Y has 2 parameters, which are μ and σ, each being E(Y) and square root of Var(Y). What about random variable X? What is E(X)? E(X) is just exp(μ+0.5σ²), MOD(X) is exp(μ-σ²) and MEDIAN(X) is exp(μ). (In other word, there is more than probability of 50% that the stock price will underperform the expected price. Think about it)

On the other hand, if we are given that X follows normal distribution. What is the relationship of Y and X, such that Y follows lognormal distribution? If the reader is good in probability, straight away it can be recognized that the answer is Y= e^X. Compare this to the above paragraph. Ln Y indeed follows normal distribution. I purposely exchange the use of X and Y, so that clear picture can be seen.

Brownian Motion

Brownian motion is defined to be a kind of random walk, with probability of 0.5 each of moving with a magnitude of 1 and -1. It is applied for modeling price movement. There are some characteristics about this model: 

1. Z(0) = 0
2. It is normally distributed with mean Z(0) and standard deviation σ.
3. Z(t) is continuous with t.
4. The variance is the sum of the time.

 Now, since at t=0, Z(0) also equals to 0, it has to be modified, so that this model can represent price movement. Generally, this model is not used to find the expected price, but the expected movement of price. (Actually, from the expected movement we can find the expected price) This is illustrated by:

 Example 1

Given the price of stock follows Brownian Motion. The price of stock at t=3 is 52. Determine the probability that the price is more than 60 at t=10.

 Answer

Since the last given t is 3, and the last given price is 52, we have Z(3) = 52. The answer wanted is P{Z(10)> 60 | Z(3) = 52}. From the property of Brownian motion, the expected value is the current value, and the variance is the sum of square of time. So, mean= 52, variance= 10-3 = 7. Hence, the volatility, σ  is 7^0.5. So,

 P{Z(10)> 60 | Z(3) = 52}= 1- N[(60-52)/ 7^0.5] = 0.00125

 You can solve it in other way of thinking, which is, to let the difference of price being normally distributed. So, we have P{Z> [(60-52)-0]/ 7^0.5}, which yields the same answer. ***

 However, the expected value of a stock is not necessarily the current price. Some stock is surely going up in the coming time. Let the expected increase rate be β at t. This is called drift. So, having made a modification of this model, the price X(t) is:

X(t)- X(0) = βt + σ Z(t)

This is reasonable and intuitive. The expected change is β multiplying by the time (so, actually β is a rate, that is, the rate of increase of price). In addition to this, there is a deviation, and this is called the noise, which is implied by the latter part of the right hand side of the equation. So, the price follows normal distribution with mean X(t) + βt and standard deviation of σ*t^0.5.

 Note that the above definition is quite confusing. So, let us stick to the principle: Let the difference of the prices be normally distributed. Hence, we have dX= βt + σ Z(t): the difference of the prices is normally distributed with mean βt and standard deviation of σ*t^0.5.

 The bolded sentences mean the same.

 Example 2

Given the price of a stock follows arithmetic Brownian Motion with drift 1 and volatility 0.2. The current price is 40. What is the probability that the price is less than 43 at t=4.

 Answer

Let X(t) be the price of the stock. We have β=1 and σ=0.2. Applying this model we have mean = 40 + 4(1) = 44, and volatility (standard deviation) = 0.2. So,

P{X(4) < 43 | X(0) = 40} = N[(43-44)/0.2(2)] = 0.0062 ***

 However, there is a shortage for this model, which we call arithmetic Brownian Motion, because it may give negative value, which is impossible for a price. Hence, further modification is needed, which, as we will see later, is made by exponentiating arithmetic Brownian motion.

 Let us discuss some logical and theoretical matters with regard to this. When we model stock price, we always have the current price, and based on that price, we would like to find the expected price of the stock after time t. Hence, we always need to compare the current price, X(0) and the future price X(t), which is unknown. How do we compare? Mathematically, we can: 1. Take the difference; 2. Take the ratio.

 Now, we have 2 models to fit these 2 comparisons, which are normal distribution and lognormal distribution. Let’s list them down: 

1. The difference of stock price follows Normal Distribution
2. The ratio of stock price follows Normal Distribution
3. The difference of stock price follows Lognormal Distribution
4. The ratio of stock price follows Lognormal Distribution

 Note that what we have just done so far is the first method, in which we assume the difference of stock price to follow normal distribution. This is intuitively nice, but there is a drawback: the modeled stock price can go negative, which will not happen in real world. For this reason, we seek a better method.

 If we were to use the second method, we have X(t)/X(0) being normally distributed. Notwithstanding, we know that X(t) and X(0) must be positive, and letting their ratio to follow normal distribution doesn’t make sense, since there is 0.5 probability that the ratio can go negative according to normal distribution. Thus, this method is out.

 If we were to use the third method, we have X(t)- X(0) being lognormally distributed. Again, as some of the reader might guess, this doesn’t make sense because Y= X(t)-X(0) might be negative, and lnY is not defined. Hence, this method is also out.

 Finally, we have the fourth method, which as mentioned, is the best way out of 4 to model stock price. In this method, we assume the ratio of the stock price to be lognormally distributed:

                                    X(t)/X(0)~Lognormal OR ln[X(t)/X(0)]~Normal ***

 Notice that arithmetic Brownian Motion follows normal distribution. As we modify it by exponentiation, it becomes lognormally distributed. We call this modified model: Geometric Brownian Motion. So,

                                                lnX(t)- lnX(0) = μt + σZ(t)

                                                                        OR

                                                X(t) = X(0)e^[( α - 0.5σ²)t + σZ(t)]

 Perhaps you may wonder: Why is the mean μ= α - 0.5σ²?

 Recall that α stands for the expected return rate of the stock. It means, if the continuously compounded rate of return of the stock is normally distributed, then the mean of that normal distribution is μ. This implies, stock will follow lognormal distribution with mean e^α = e^(μ + 0.5σ). By this, you get the above bolded equation.

 For this reason, whenever we are asked to find probability related to Geometric Brownian Motion, we need to be careful of the parameter. This is illustrated by example below:

 Example 3

A stock’s price is modeled as geometric Brownian Motion with continuous return α = 0.15, continuous dividends δ = 0.04 and volatility 0.03.

 If the stock’s price is 45 at time 0 and 47 at time 0.6, calculate the probability that the stock’s price is less than 45 at time 1.

 Solution

Notice that this is Geometric Brownian Motion. This implies that lognormal distribution is applied on the ratio of the stock. In order to evaluate probability, we need the parameter of normal distribution μ. This can be easily obtained by equating

                        μ + 0.5σ²= α – δ

=>                                            μ = α – δ – 0.5σ²= 0.065

=>                                            μ*t = 0.065*(1-0.6) = 0.026

=>                    P{X(1)|X(0.6)=47} = N{ [ln(45/47) – 0.026]/ [0.3*0.4^0.5] }= 0.3557***

 Just remember, Arithmetic Brownian Motion says that the difference of stock prices is normally distributed, whereas Geometric Brownian Motion says that the quotients of the stock prices are lognormally distributed.

 Ito’s Process

Now, we can go further to see that we can represent the model with differentials:

1. Arithmetic Brownian Motion: dX(t)= α dt + σ dZ
2. Geometric Brownian Motion: d[lnX] = (α - δ - 0.5σ²)t + σZ(t)

=>dX(t) = α X(t) dt + σ X(t) dZ. Any process of these form are called Ito process.

Before we get into the details of Ito process and Ito’s Lemma, let us spend some time in differential equation and some motivations of learning these.

Now we already have model to forecast the future stock price. We can further modify these model, particularly Geometric Brownian Motion, so that this model can be generalized for any function f[ X(t) ]. For example, the payoff of a call option is also a kind of function with variable X(t). To generalize the function is not easy, but later we will see that Ito’s Lemma helps us much in this matter.

 The point of exposing this is because it leads into the derivation of the Black Scholes formula.

 Perhaps the second differential equation of Geometric Brownian Motion looks weird. However, we have Ito’s Lemma that can help us to relate the value of the changes of the function of asset to the value of changes of asset. In other words, if we have X following Brownian motion, and we would like to know how does Y=e^X behaves, Ito’s Lemma helps us to relate them. In fact, the second is derived by Ito’s Lemma.

 Example 4

Given an Ito process

                                    dS= 0.25 S(t) dt + 0.1 S(t) dZ

Calculate the probability that S(t) is at least 5% higher than S(0) at t=1.

 Answer

Since the given information is on geometric Brownian Motion (refer to the second equation), it is lognormally distributed. To calculate the probability, we have to convert it to normal distribution, or, associated arithmetic Brownian Motion. Hence, we have this “converted” geometric Brownian Motion follows normal distribution with mean μ = 0.25- 0.5*(0.01) = 0.245 and volatility = 0.1. Hence, probability is 1- N[(ln1.05 – 0.0245)/ 0.1] = 0.2206 ***

 Ito’s Lemma

 dC= Cs dS + 0.5 Css (dS) ² + Ct dt

 Where we are given the function C[S(t)], and S(t) is the stock price at time t.

 Example 5

You are given Ito processes dX and dY. dY is defined by:
                                                dY(t)/ Y(t) = 0.1 dt + 0.5 dZ(t)

And X=Ye^0.02t.

 dX can be expressed as

                                             dX(t) = α X(t) dt + σ X(t) dZ(t)

 Using Ito’s Lemma, determine α and σ.

 Answer

 Recognize that X in this question is the C in the Ito’s Lemma.

 Xy  = 0.02t e^0.02t  = X/Y

Xyy = 0

Xt = 0.02 Ye^0.02t = 0.02 X

Therefore, dX(t) = X/Y dY + 0.02X dt

= X/Y [0.1 Y(t) dt + 0.5 Y(t) dZ(t)] + 0.02X(t) dt

= 0.1 X dt + 0.5 X dZ + 0.02X dt

= 0.12X dt + 0.5 X dZ

 Hence, α = 0.12, σ = 0.5 ***

 Note that there is possibility that Css is not zero. In this case, (dS)² will be encountered, and resulting in dt², dZ², and dZdt. Of these, all equal zero except dZ²= dt. Reason? Unknown.

 Sharpe Ratio

 

Sharpe ratio is the risk premium of a stock divided by its volatility, or, standard deviation. Recall from the chapter in which we study option Greeks, where it is mentioned that the risk premium of option equals the risk premium of underlying stock multiply by the absolute value of elasticity:

                                                            γ-r = (α-r)|Ω|

In other words, Sharpe ratio is simply:

                                                            φ = (α-r) / σ

 You need to know the following ideas:

1. For 2 ito processes involving the same dZ, the Sharpe Ratios are equal, where Sharpe Ratio, φ = (α-r)/σ. Remember, the Sharpe Ratios are only equal when the Ito process are in geometric motion form.
2. For 2 Ito’s processes involving the same dZ, the Sharpe Ratio are equal. Remember, when you pick the α and σ from the ito’s process, make sure that they are in geometric Brownian Motion form, ie, dX/ X = (α-δ) dt + σ dZ

 Example 6

For two nondividend paying assets X(t) and Y(t), you are given the following stochastic equations:

                        d[ln X(t)] = 0.08 dt + 0.2 dZ

                        d[ln Y(t)] = A dt + 0.4 dZ

 The risk free rate is 0.04, find A.

 Answer

 Note that these 2 stochastic equations are not in geometric Brownian motion form. It is in arithmetic form. It says that the logarithm of X(t)/X(0) and Y(t)/Y(0) follow normal distribution with μx = 0.08, μy = A. So, we have αx = 0.08 + 0.5(0.2)² = 0.1. Since they are sharing the same dZ, the Sharpe ratio of X and Y should be the same, or,

(αx-r) / σx = (αy-r) / σy

=> (0.1-0.04)/0.2 = (αy – 0.04) / 0.4

=> 0.3 = (αy – 0.04) / 0.4

=> αy = 0.16

=> A = 0.16 – (0.5)0.4² = 0.08***