1. Risk Management Stochastic Finance 2004 Autumn School João Duque September 2004

2. Risk Management Risk (latu sensus) = Uncertainty (strictu sensus) = Quantified Uncertainty Management = Decision Making

3. Which one is a Stock Price?

4. And now?

5. Stock Prices’ Properties Historical prices are useless when forecasting future prices (weak form of efficiency); Prices of financial assets are supposed to be positive or null (never negative); Forecasting financial prices is hard, since they tend to be strongly random; Financial price returns seem random but they seem to respect two empirical governing rules: i) stock price return is directly proportional to time ii) volatility is directly proportional to the square root of time.

6. Market Efficiency – Roberts (1967) Strong Form Prices reflect all price sensitive information Semistrong Form Prices reflect all publc price sensitive information Weak Form Prices reflect all past recorded price sensitive information

7. Stock Prices’ Randomness Lo, Mamaysky e Wang, Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation, J. of F. (2000) We find that certain technical patterns, when applied to many stocks over many time periods, do provide incremental information, especially for Nasdaq stocks. Although this does not necessarily imply that technical analysis can be used to generate “excess” trading profits...

8. Stock Price returns proportional to time Empirical Experiment: Collected a data set (time series) of 10 years of daily prices for some individual stocks, as well as for some stock indices. Prices in their original currency were collected on a daily basis (closing prices) from June 24 th 1994, to June 24 th 2004. Prices of individual stocks were adjusted for dividends, stock splits and other events that may affect stock price returns.

9. Stock Price returns proportional to time Individual Stocks: BCP - the private Portuguese leading bank; IBM, Coca Cola, Pepsico, General Motors - all from the US; BT Group and British Airways - from the UK; BASF - from Germany; Nestle - from Switzerland, and; Nokia – from Finland

10. Stock Price returns proportional to time Individual Stocks: PSI 20 – composed of the 20 most liquid stocks of Euronext Lisbon; I BEX 35 – composed of the 35 most liquid stocks of the Madrid Stock Exchange; D J EuroStoxx – which combines prices from the 50 biggest European companies; FTSE 100 – composed of the top 100 stocks traded in the London Stock Exchange; S&P 500 – composed of the most liquid and significant blue chip stocks of the US market; Nasdaq 100 – composed of the top 100 most promising but risky stocks of the US market.

11. Stock Price returns proportional to time Started by computing the daily continuously compounded rate of return Also computed the stock price (index) returns for larger time periods: week, month, quarter, semester, year.

12. Stock Price returns proportional to time Followed by computing the average stock price (index) return

13. Table 1-A Average stock price returns for individual stocks estimated on a basis of different time window returns

14. Table 1-B Average stock price returns for stock indices estimated on a basis of different time window returns

15. Table 2-A Relative size of the average rate of return of stock i estimated on a basis of different time window returns

16. Table 2-B Relative size of the average rate of return of index i estimated on a basis of different time window returns

17. Table 3- Relative size of average rates of return estimated on a basis of different time window returns That is, the average rate of return of a stock or an index tends to be directly proportional to the time used to estimate the rates of return.

18. Volatility proportional to the square root of time. We also computed different standard deviations based on different time window returns. The standard deviation for each series of stock (index) returns was computed using the following equation: k denotes the type of data used (daily, weekly, monthly, etc.)

19. Table 4-A Historical volatility for individual stocks estimated on a basis of different time window returns

20. Table 4-B Historical volatility for individual stock indices estimated on a basis of different time window returns

21. Table 5-A Relative size of historical volatility of stock i estimated on a basis of different time window returns

22. Table 5-B Relative size of historical volatility of index i estimated on a basis of different time window returns

23. Table 6- Relative size of average historical volatility relative to its daily historical volatility estimated on a basis of different time window returns  Financial price returns seem random but they seem to respect two empirical governing rules: i) stock price return is directly proportional to time ii) volatility is directly proportional to the square root of time.

24. A First Attempt to Model the Empirical Findings Assume that a variable z starts at time t with the value z(t) and that it will change as a result of the elapsing of time. Changes on variable z are equal to the differences on that variable taken at two consecutive moments of time.

25. A First Attempt to Model the Empirical Findings A first attempt to model changes in variable z can be given by the following equation: Properties: Values of dz for any two different non-overlapping periods of time are independent (as a result of  being independent and identically distributed for any concretisation)  Variable z is said to follow a Wiener Process

26. A First Attempt to Model the Empirical Findings If instead of a continuous time process where we continuously observe variable z we assume that observations of z occur regularly every  t (time interval) Let’s go model it!

27. A First Attempt to Model the Empirical Findings Fulfilling Properties: Property I - Property II - ? Property III - Property IV - ? The model doesn’t fit!

28. A Second Attempt to Model the Empirical Findings Generalized Wiener Process It adds two characteristics to the Wiener Process: It adds a trend to the random part of the process It includes an amplifier / reducer within the random part of the process in order to adjust the process to the specific properties of each stock Let’s go model it!

29. A Second Attempt to Model the Empirical Findings Fulfilling Properties: Property I - Property II - ? Property III - Property IV - The model doesn’t fit!

30. A Third Attempt to Model the Empirical Findings Ito Process Let’s go model it!

31. A Third Attempt to Model the Empirical Findings Fulfilling Properties: Property I - Property II - Property III - Property IV - The model fits! =>

32. Plausible Questions 1)Why do we need models that do not help to forecast prices and the only thing they do is to increase the feeling of price uncertainty? 2)Can we make any money out of these models? How? 3)How is possible to profit from some piece of knowledge (model) that only re- emphasises our ignorance about price formation?

33. Good Applications Pricing Derivative instruments Developing arbitrage strategies

34. Is the Model Right? No! Skewness Kurtosis Volatility clusters Other stochastic variables Volatility Interest rates Dividends Jumps...

35. Table 7 – Kurtosis and skewness of daily continuously compounded rates of return for a database collected from June 24 th 1994 to June 25 th 2004

36. 20-days rolling historical volatility

38. A Simple Binomial Model A stock price is currently $20 In three months it will be either $22 or $18 Stock Price = $22 Stock Price = $18 Stock price = $20

39. Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option A 3-month call option on the stock has a strike price of 21.

40. Consider the Portfolio:long  shares short 1 call option Portfolio is riskless when 22  – 1 = 18  or  = 0.25 22  – 1 18  Setting Up a Riskless Portfolio

41. Valuing the Portfolio (Risk-Free Rate is 12%) The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22  0.25 – 1 = 4.50 The value of the portfolio today is 4.5 e – 0.12  0.25 = 4.3670

42. Valuing the Option The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.25  20 ) The value of the option is therefore 0.633 (= 5.000 – 4.367 )

43. Generalization A derivative lasts for time T and is dependent on a stock S0uƒuS0uƒu S 0 d ƒ d S0ƒ0S0ƒ0

44. Generalization Consider the portfolio that is long  shares and short 1 derivative The portfolio is riskless when S 0 u  – ƒ u = S 0 d  – ƒ d or S 0 u  – ƒ u S 0 d  – ƒ d S 0 – f 0

45. Generalization Value of the portfolio at time T is S 0 u  – ƒ u Value of the portfolio today is (S 0 u  – ƒ u )e –rT Another expression for the portfolio value today is S 0  – f 0 Hence ƒ 0 = S 0  – ( S 0 u  – ƒ u )e –rT

46. Generalization Substituting for  we obtain ƒ = [ p ƒ u + (1 – p )ƒ d ]e –rT where

47. Risk-Neutral Valuation ƒ 0 = [ p ƒ u + (1 – p )ƒ d ]e -rT The variables p and (1  – p ) can be interpreted as the risk-neutral probabilities of up and down movements The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate S0u ƒuS0u ƒu S0d ƒdS0d ƒd S0ƒ0S0ƒ0 p (1  – p )

48. Irrelevance of Stock’s Expected Return When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant

49. f t = value of the derivative instrument at time t. S t = value of the underlying instrument at time t. K = exercise price of the derivative instrument. Using Monte Carlo Simulation Call Option Put Option