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

2. Pascal’s Triangle

3. Add up the two numbers above

4. The Binomial Theorem

5. Summing the rows

6. Rescale the entries

7. A plot of row 0

8. A plot of row 1

9. A plot of row 2

10. A plot of row 3

11. Running through all the rows...

12. Using colour instead of height...

13. Bagatelle

14. Example: 6 Possible Paths

15. Counting Paths

16. Probability of hitting peg

19. Diffusion

20. Improving the Resolution

21. 1x1 squares and 0.25x0.25 squares

22. 1x1 squares and 0.5x0.25 squares

23. 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)

24. 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

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

26. 1x1 squares and 0.25x0.25 squares

27. 1x1 squares and (0.25) 1/2 x0.25 squares

28. 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!

29. 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!

30. 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

31. 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

32. 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.

33. Pascal’s triangle is self-similar

34. 10 Time Steps

35. 20 Time Steps

36. 30 Time Steps

37. 400 Time Steps

38. Rotated

39. 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

40. 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

42. 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

43. Test performed on 10 April 2014 S1 ≈ 0.0032 S2 ≈ 0.0159 S2/S1 ≈ 5.0 30 1/2 ≈ 5.57

44. 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!

45. 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 10 14 )

46. 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.

47. A path with 400 steps

48. Infinity Steps