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# Applied Statistics IV

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Fourth Session, MSc 4th Year

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• As we have b, we can replace it in the equation of the regression line
• Rf will be used in the backward inductionDrift is cancelled through risk neutral probability
• Rf will be used in the backward inductionDrift is cancelled through risk neutral probability
• Rf will be used in the backward inductionDrift is cancelled through risk neutral probability
• V(X/n)=1/n^2 * np(1-p)
• 0.52+- 1.96*0.04669
• S/SQRT(6)*2.571 +- Actual mean
• ### Applied Statistics IV

1. 1. Q1 2012 ESGF 4IFM Q1 2012 Vincent JEANNIN – ESGF 4IFM vinzjeannin@hotmail.com Applied Statistics 1
2. 2. Interim Exam Sum Up Reminders of last session Capital Asset Pricing Model Thinking algorithmic ESGF 4IFM Q1 2012 • • • • vinzjeannin@hotmail.com Summary of the session (est. 4.5h) 2
3. 3. ESGF 4IFM Q1 2012 1 vinzjeannin@hotmail.com Interim Exam Sum-Up 3
4. 4. When E is minimal? When partial derivatives i.r.w. a and b are 0 Attention, logarithms are not additive! vinzjeannin@hotmail.com Minimising residuals ESGF 5IFM Q1 2012 Two parameters to estimate: • Intercept α • Gradient β 4
5. 5. Change the variable Z=ln(X) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Solution? 5
6. 6. vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Leads easily to the intercept 6
7. 7. 7 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
8. 8. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 We have and Finally… 8
9. 9. Z=ln(X) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Don’t forget… 9
10. 10. Accept or reject the regression? vinzjeannin@hotmail.com Hedging is linear… ESGF 5IFM Q1 2012 No forecast possible (one particular stock against the market) Check correlation and R Squared 10 Check the normality of residuals
11. 11. 11 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
12. 12. N(-1,2) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 2 N(0,1) 12
13. 13. Let’s build a tree with 5 steps, with S=104.57, σ=10%, 1 year to maturity 125.05 119.58 114.45 109.35 104.57 100 104.57 95.62 114.45 109.35 109.35 104.57 100 91.44 119.58 95.62 87.44 100 91.44 83.62 vinzjeannin@hotmail.com 130.77 ESGF 4IFM Q1 2012 3 13
14. 14. 130.77 119.58 20 19.58 109.35 5 100 5 91.44 0 83.62 vinzjeannin@hotmail.com • Pay off capped to 20 • Pay off between 100 inclusive and 109.35 inclusive: 5.00 ESGF 4IFM Q1 2012 Last node value 0 14
15. 15. vinzjeannin@hotmail.com Final Value 12.50 15 ESGF 4IFM Q1 2012
16. 16. What is the new price of the Call (initial price \$8.00) if S moves up \$2.5 with delta=0.5525 and a gamma of 0.0222, volatility moves up 1.75 point with a 0.8422 Vega, r moves up 1.2 basis point with Rho=178.5448 and placing you 3 days after with a final Theta of -0.9723? 10.73 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 4 16
17. 17. Random walk! Past series has no importance! Trial s Independents! vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 5 17
18. 18. Reminder of the last session ESGF 4IFM Q1 2012 Multiple regression vinzjeannin@hotmail.com More than one explanatory variables R-Square is very often very poor Extension APT 18 “Pure” factors
19. 19. • • • • • Corruption: current corruption CorruptionPrediction: future corruption School: level of education GDP: GDP Distortion: how badly policies are run vinzjeannin@hotmail.com Let’s discuss… ESGF 4IFM Q1 2012 Ratio Investment / GDP , World Bank, developing countries 19
20. 20. Be logic vinzjeannin@hotmail.com • General to specific: this starts off with a comprehensive model, including all the likely explanatory variables, then simplifies it. • Specific to general: this begins with a simple model that is easy to understand, then explanatory variables are added to improve the model’s explanatory power. ESGF 4IFM Q1 2012 How to find the right model? Have the best R-Squared 20 Not over complicate
21. 21. 3 steps Identify Fit Forecast vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 What is a model? 21
22. 22. Trend Seasonality Residual vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 3 components 22
23. 23. ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com Variation (price or percentage is a differentiation) 23 Series with stationarity much easier to modelise
24. 24. On the values On the residuals vinzjeannin@hotmail.com Most cases you will find autocorrelation ESGF 4IFM Q1 2012 Once the series is stationary, look for autoccorrelation 24
25. 25. Parameters of the model White noise vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Auto Regressive model AR(n) 25
26. 26. f ( x) n! x! ( n x p (1 p) (n x) x )! Large sample: Normal Distribution N np , np (1 p) n is the size of the sample, x, the number individuals with the particular characteristic vinzjeannin@hotmail.com Small sample: Binomial Distribution ESGF 4IFM Q1 2012 Estimations 26
27. 27. Estimate a proportion Normal approximation Standardisation possible Normal approximation works only if vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Binomial Distribution 27
28. 28. Easy solve! vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Let’s look for p with a 95% confidence interval 28
29. 29. vinzjeannin@hotmail.com 95% confidence interval ESGF 4IFM Q1 2012 52 Heads out of 100 toss… 29
30. 30. Mean estimation vinzjeannin@hotmail.com Student’s Statistic ESGF 4IFM Q1 2012 Mean has a Student’s distribution Degree of freedom 30 n-1
31. 31. SD: DF: S: ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com IPO Premiums IPO1 / 12% IPO2 / 15% IPO3 / 13% IPO4 / 18% IPO5 / 20% IPO6 / 5% t: 31
32. 32. vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Is Martingale safe? 32
33. 33. How many portfolio can be built? How to chose the weights? Using Variance/Covariance Matrix to select the portfolio Optimisation of either the risk or the return vinzjeannin@hotmail.com 5 stocks available ESGF 5IFM Q1 2012 Capital Asset Pricing Model 33
34. 34. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Infinite number of long only portfolios 34
35. 35. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Would you buy just Air Liquide? 35
36. 36. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 You’d only invest on the so called Efficient Frontier 36
37. 37. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 For a particular return, you take the lowest risk 37 For a particular risk, you take the highest return
38. 38. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Unless there’s a risk free rate 38
39. 39. Straight forward, mean is linear, weighted average ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com For a particular combination you need to calculate the expected return 39
40. 40. We already know ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com For a particular combination you need to calculate the variance (or SD) Not enough, need the general case for a bigger number of assets 40
41. 41. No linear formula to select the good one Need a computer and algorithms vinzjeannin@hotmail.com Millions of portfolio ESGF 5IFM Q1 2012 Thinking Algorithmic 41
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