PowerPoint Presentation

Making MoneyFrom Pascals TriangleJohn ArmstrongKings College LondonPascals 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=PositionLet D(n) denote the direction at row nD(n) = -1 if ball goes left at row nD(n) = 1 if ball goes right at row nExpected value of D(n) = 0Variance of D(n) = Expected value of D(n)2 = 1Let X(n) denote the x-coordinate at row nX(n)=D(0)+D(1)+D(2)+D(3)+...+D(n-1)Addition of expectation and varianceIf A and B are independent random variables thenE (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))=0Var(X(n)) = Var(D(0)+D(1)+...+D(n-1) = Var(D(0))+Var(D(1))+...+Var(D(n-1)) = nImportant ResultThe width of the distribution grows at a rate n1/2 as the row number n increases

1x1 squares and 0.25x0.25 squares

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

Important ResultThe width of the distribution grows at a rate n1/2 as the row number n increasesFor the diffusion of ink in water, this means that the ink spreads out at a rate t1/2 where t is timeThis is a testable conclusion of the atomic theory!

Some historyJan Ingenhousz (1785): coaldust on alcoholRobert Brown (1827): erratic motion of pollen suspended in waterThorvald Thiele (1880): mathematics of Brownian motion describedAlbert Einstein (1905), Marian Smoluchowski (1906): realised it could be used to test atomic theoryJean Baptiste Perrin (1908): experimental work to confirm Einsteins theory and calculate Avogadros constant.The atomic theory was finally established!

The Central Limit TheoremIf 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-1/2Therefore the sum of the sample is normally distributed withMean = nm And standard deviation = sn-1/2

Consequence for Brownian MotionRecall that:X(n)=D(0)+...+D(n-1)So for n>30, X is approximately normally distributed with mean n and standard deviation n1/2Consequence for Brownian MotionRecall that:X(n)=D(0)+...+D(n-1)So for n>30, X is approximately normally distributed with mean n and standard deviation n1/2This 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.Pascals triangle is self-similar

10 Time Steps

20 Time Steps

30 Time Steps

400 Time Steps

Rotated

Stock pricesIf stock price is $100 then may go up or down $1 each dayIf stock price is $1000 then may go up or down $10 each dayThese 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 dayStock price modelLet X(t) follow Brownian MotionThen we can model stock prices by

S(t)=A exp( B X(t) + C t )

A = initial stock priceB = volatilityC = drift

PredictionOur 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 S1Take 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 S2Prediction: S2/S1 301/2

Test performed on 10 April 2014S1 0.0032S2 0.0159S2/S1 5.0301/2 5.57

DISCLAIMERThis 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 historyLouis Bachelier (1900) PhD thesis proposing modelling stocks as Brownian motionBlack-Scholes (1973) Introduced the model of stocks Ive just described and started modern mathematical financeJune 2013 $692,908 billion notional value of OTC derivatives ($6.9 x 1014)

SummaryThe same mathematical structure can occur in many placesThe formula for (a+b)nThe atomic theoryThe stock marketOne of the most interesting features of Pascals Triangle is its scaling behaviour. It is self-similar. It scales with a factor of n1/2This allows us to make testable predictions about atoms and stocks.A path with 400 steps

Infinity Steps