First, some - If a stock first increases by 13%, then decreases by 13%, it will not return to the
original stock price,
since (1+0.13)(1-0.13)= 0.9831 so the stock returns to just 98.31% of its original value. In order to return to the original price an increase by a factor (1+0.13) must be followed by a decrease by a factor 1/(1+0.13) ... then, when you multiply, you get "1". - Some mathematical notes:
(x+y)^{2}= x^{2}+ 2 xy + y^{2}with binomial coefficients 1,2,1 (x+y)^{3}= x^{3}+ 3 x^{2}y + 3 xy^{2}+ y^{3}with binomial coefficients 1,3,3,1 (x+y)^{4}= x^{4}+ 4 x^{3}y + 6 x^{2}y^{2}+ 4 xy^{3}+ y^{4}with binomial coefficients 1,4,6,4,1 (x+y)^{5}= x^{5}+ 5 x^{4}y + 10 x^{3}y^{2}+ 10 x^{2}y^{3}+ 5 xy^{4}+ y^{5}with binomial coefficients 1,5,10,10,5,1
*etc. etc.* hence the**magic formula** (x+y)^{N}= N_{0}x^{N}+ N_{1}x^{N-1}y + N_{2}x^{N-2}y^{2}+ ... + N_{N}y^{N}where the*binomial coefficients*are or, equivalently: (x+y)^{N}= N_{N}x^{N}+ N_{N-1}x^{N-1}y + N_{N-2}x^{N-2}y^{2}+ ... + N_{0}y^{N} or, equivalently: (1+x)^{N}= N_{0}+ N_{1}x + N_{2}x^{2}+ ... + N_{N}x^{N}
>Those binomial numbers ... aren't they usually called
Very good. How about the number of paths that end up at, say, 1 Up and 4 Down? >I count 5. Good. In fact, the number of ways to arrive at the stock price after 5 time steps: - 5 Up and 0 Down is 5
_{0}or 5_{5}, namely 1 - 4 Up and 1 Down is 5
_{1}or 5_{4}, namely 5 - 3 Up and 2 Down is 5
_{2}or 5_{3}, namely 10 - 2 Up and 3 Down is 5
_{3}or 5_{2}, namely 10 - 1 Up and 4 Down is 5
_{4}or 5_{1}, namely 5 - 0 Up and 5 Down is 5
_{5}or 5_{0}, namely 1
Yes. >But some stock prices are more likely than others, right? Right. Let g be the gain associated with each time step.We let p be the probability that the stock will go Up by the factor U = 1+g.Then 1-p is the probability that the stock will go Down by the factor D = 1/(1+g).
>You're assuming only these two possibilities, eh? Yes, and we've assumed that, after an Up and a Down, our stock returns to the same price ... as per item 1.
So what's the probability
>The risk-free gain factor? Yes. If, for each time step, the risk-free return is r (where r = 0.012
means 1.2%), then R = 1 + r is the risk-free gain factor.Okay, we now attach a probability to each final state of the stock price, after n Ups
and m Downs.Since each of the n Ups has a probability of p and each of the m Downs
has a probability of 1-p, I think it's clear that
the probability of n Ups and m Downs is p
^{n}(1-p)^{m}
>Clear? To who? To whom. Just be quiet and pay attention.
Note something interesting. >Are you saying that's equal to 100%? No, I'm saying it's equal to 1 because, according to the
magic formula in item 2 above (with x replaced by p
and y replaced by 1-p):{ p + (1-p)}^{N}
= N_{0}p^{N} + N_{1}p^{N-1}(1-p) + N_{2} p^{N-2}(1-p)^{2}+ ... + N_{N}(1-p)^{N}
and, of course, { p + (1-p)}^{N} = 1^{N} = 1.
We can write this like so:
p
^{m} (1-p)^{N-m} = 1If the stock price after N steps (involving m Ups and N-m Downs) is
U^{m}D^{N-m}S then ...
>So there are lots of possible stock prices, right? where we now know the associated probabilities and we've again used the magic formula in item 2, above.
We'll rewrite this formula using a more familiar notation for the binomial coefficients:
m Ups (and N-m Downs)
and we've used the formula (*).
Okay, if we have the expected price after N time steps, what's it worth now, today?
We assume it's worth $C today and $E after N steps. If we invested our $C at the
risk-free rate (meaning a gain factor of
Okay, we'll write as
and divide by R and rearrange/combine stuff to get:
^{m}R^{N-m}
Do you recognize this expression?
Notice that there are two curious factors in this summation, namely P = {pU/R}
and {(1-p)D/R}. Hence, we can rewrite the formula above as:
Okay, now for a Look again at formula (1), above, namely: Look carefully at p U + (1-p)D = p (U - D) + D = {(R - D)/(U - D)} (U - D) + D = {R - D} + D = R In other words, if we assume that the probability of an Okay, we now look at formula (2b), above, namely: But, according to our formula >Isn't that obvious? I mean, we already assumed we started with a stock whose price is S, so ...
>So all the stuff above is garbage, eh?
>Wrong! Do you realize that you hardly have any pictures? Okay, here's a picture. It illustrates the terms in the above sum, for small a larger values of N: Figure 3
They are Binomial Probability functions: f(m) = density and F(m) = cumulative.
The binomial distribution, defined by
Actually, binomial probabilities describe a process where there are just two mutually exclusive outcomes, each having an associated probability (namely p and 1-p). Tossing a coin would have p and 1-p both equal to 0.5 and, if we had a pair of dice, each having ten sides, and six of the sides said UP and four said DOWN then, throwing a single die would have ... >Don't tell me! 60% chance of UP and 40% chance of DOWN. Yes, so p = 0.6 and q = 0.4 and ... >So, what's the Mean and Standard Deviation? For a Normal distribution they define the distribution, right?Yes, they do ... so we'll find them for the Binomial distribution: If we let q = 1-p, then the density function for the Binomial Distribution is:
B(N,m,p) = p.^{m}q^{N-m}
Note that this is the probability of getting a value however, the first Now we go through the following ritual:
[
and now we let
and now we stare at
so, finally (!) we conclude that:
>In Figure 3 the means are 0.6 x 15 = 9 and 0.6 x 100 = 60, right?
namely: >Let's just skip that one ...
Now we go through a second ritual (using the first ritual, above):
[ which enables us to rewrite the above expression as:
and each sum equals "
Hence
>So, in Figure 3, the standard deviations are 15 x 0.6 x 0.4 = 3.6 and 100 x 0.6 x 0.4 = 24, right?
>But if you let N get larger and larger then the Mean and SD also become ...
Let's do the following: We'll assume an >This is fiction, right? Up, 70% of the time? And what's with that "equivalent" monthly gain ...?
Continuing:
>A picture would be good.
>What's that horizontal axis?
>I still find it confusing. I mean ...
Note: Using a formula we obtained above, we have (for our example):
^{36} = $1.11
Here's a calculator to play with. Note that it assumes we start with a stock worth $10, not $1.
>Calculators are nice, but pictures are better!
Just Save Target.
See also |