Making Money From Pascal’s Triangle John Armstrong King’s College London

Pascal’s Triangle

Add up the two numbers above

The Binomial Theorem

Summing the rows

Rescale the entries

A plot of row 0

A plot of row 1

A plot of row 2

A plot of row 3

Running through all the rows...

Using colour instead of height...

Bagatelle

Example: 6 Possible Paths

Counting Paths

Probability of hitting peg

Diffusion

Improving the Resolution

1x1 squares and 0.25x0.25 squares

1x1 squares and 0.5x0.25 squares

D=Direction, X=Position Let D(n) denote the direction at row n D(n) = -1 if ball goes left at row n D(n) = 1 if ball goes right at row n Expected value of D(n) = 0 Variance of D(n) = Expected value of D(n) 2 = 1 Let X(n) denote the x-coordinate at row n X(n)=D(0)+D(1)+D(2)+D(3)+...+D(n-1)

Addition of expectation and variance If A and B are independent random variables then  E (A+B) = E(A) + E(B)  Var(A + B) = Var(A) + Var(B) Conclusion: E(X(n)) = E(D(0))+E(D(1))+...+E(D(n-1))=0 Var(X(n)) = Var(D(0)+D(1)+...+D(n-1) = Var(D(0))+Var(D(1))+...+Var(D(n-1)) = n

Important Result The “width” of the distribution grows at a rate n 1/2 as the row number n increases

1x1 squares and 0.25x0.25 squares

1x1 squares and (0.25) 1/2 x0.25 squares

Important Result The “width” of the distribution grows at a rate n 1/2 as the row number n increases For the diffusion of ink in water, this means that the ink spreads out at a rate t 1/2 where t is time This is a testable conclusion of the atomic theory!

Some history Jan Ingenhousz (1785): coaldust on alcohol Robert Brown (1827): erratic motion of pollen suspended in water Thorvald Thiele (1880): mathematics of Brownian motion described Albert Einstein (1905), Marian Smoluchowski (1906): realised it could be used to test atomic theory Jean Baptiste Perrin (1908): experimental work to confirm Einstein’s theory and calculate Avogadro’s constant. The atomic theory was finally established!

The Central Limit Theorem If you take a sample of n(>30) measurements from a population with mean m and standard deviation s, then the mean of your sample will be approximately normally distributed with – Mean = m – Standard deviation = sn -1/2 Therefore the sum of the sample is normally distributed with – Mean = nm – And standard deviation = sn -1/2

Consequence for Brownian Motion Recall that: X(n)=D(0)+...+D(n-1) So for n>30, X is approximately normally distributed with mean n and standard deviation n 1/2

Consequence for Brownian Motion Recall that: X(n)=D(0)+...+D(n-1) So for n>30, X is approximately normally distributed with mean n and standard deviation n 1/2 This only depends upon the mean and standard deviation of D! Our simple model of unit jumps to the left or to the right is irrelevant. A more complex model would give the same predictions.

Pascal’s triangle is self-similar

10 Time Steps

20 Time Steps

30 Time Steps

400 Time Steps

Rotated

Stock prices If stock price is $100 then may go up or down $1 each day If stock price is $1000 then may go up or down $10 each day These stocks are equally volatile. If log( stock price ) is 2/3 then log( stock price) may go up or down log(101/100)=log(1010/1000) each day

Stock price model Let X(t) follow Brownian Motion Then we can model stock prices by S(t)=A exp( B X(t) + C t )  A = initial stock price  B = volatility  C = drift

Prediction Our scaling properties make a prediction about stock markets: Take a sample of the log of the FTSE 100 at the end of each day for a year. Compute the standard deviation of the day change. Call it S1 Take a sample of the log of the FTSE 100 at the end of each month for a year. Compute the standard deviation of the monthly change. Call it S2 Prediction: S2/S1 ≈ 30 1/2

Test performed on 10 April 2014 S1 ≈ S2 ≈ S2/S1 ≈ /2 ≈ 5.57

DISCLAIMER This is a basic model! Stock prices only follow this model to a crude approximation. Do not invest all your money on the basis of this lecture and then blame me!

Some more history Louis Bachelier (1900) – PhD thesis proposing modelling stocks as Brownian motion Black-Scholes (1973) – Introduced the model of stocks I’ve just described and started modern mathematical finance June 2013 – $692,908 billion notional value of OTC derivatives ($6.9 x )

Summary The same mathematical structure can occur in many places – The formula for (a+b) n – The atomic theory – The stock market One of the most interesting features of Pascal’s Triangle is its scaling behaviour. It is self-similar. It scales with a factor of n 1/2 This allows us to make testable predictions about atoms and stocks.

A path with 400 steps

Infinity Steps