Pensions,Lotteries,Financial Markets: Measuring Statistical Risk

Transcription

1 Pensions,Lotteries,Financial Markets: Wolfgang Härdle Humboldt-Universität zu Berlin Center for Applied Statistics and Economics

2 Pension Systems 2 How risky are the demographics? Germany male under 20 male betw een male over 65 female under 20 female betw een female over 65 male under 20 male betw een male over 65 female under 20 female betw een female over

3 Pension Systems 3 How risky are the demographics? USA male under 20 male betw een male over 65 female under 20 female betw een female over male under 20 male betw een male over 65 female under 20 female betw een female over

4 4 Basis for rational decisions Dynamic data visualization Fast computing of different scenarios

5 5 Demographic Risk Management Population Dynamics Government Pensions Until 2030 relative to 2006 premium rate rises up to 30% costsriseby50% Private Life Insurance

6 Private Life Insurance as Solution? 6 Premium depends on future life expectancies Mortality deviated dramatically from forecasts Estimation of Cohort Life Expectancy at 65 Male Female DAV 1994 R DAV 2004 R DAV 1994 R DAV 2004 R For years 24 years 25 years 27 years For years 30 years 28 years 34 years

7 7 Demographic Risk Path-breaking technological or medical innovation Financial disaster for retirees Huge costs for the pay-as-you-go social system Systematic risk for capital markets

8 Lotteries 8 Are there winning numbers? How much cash predicts the theory? What are the odds?

9 Some history 9 Genova: 5 out of 90 for the city council Casanova: lotto in France French court discovers income source

10 Lotteries 10 What are the odds? s possible numbers choose r from s number of possibilities: s r = s! r!(s r)! D, CZ s=49, r= A, CH s=45, r=

11 Lotteries 11 Pascal s Triangle Binomial Coefficients Wikipedia

12 Lotteries 12 Are there winning numbers? (1, 8, 15, 22, 29, 36); (19, 27, 29, 31, 38, 44); (2, 15, 14, 1, 16, 1, 18, 20, 5)= (B,O,N,A,P,A,R,T,E)? popular 9 frequency=3.1% unpopular 43 frequency=1.4%

13 Lotteries 13 Χ²-test for uniform distribution Czech Lotto

14 Lotteries 14 Monty Hall problem Wikipedia

15 Lotteries 15 1/3 1/3 1/3 a) b) 2/3 2/3 Wikipedia

16 Lotteries Monty Hall Problem 16 Wikipedia

17 17 Numbers below 31 have lower payment Test on uniform distribution (quality control) Behavioral Finance (irrational decisions)

18 18 Financial Markets How volatile is a portfolio? Risk Management Option pricing

19 Financial Markets 19

20 20 Financial Markets

21 21 Financial Markets

22 Financial Markets 22

23 Financial Markets 23 Strike Price

24 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-2 A first approach on option pricing: the binary one-period model There exist different possibilities of investment: zerobond with interest rate zero S 0 the current value of the stock S T the value of the stock at time T could be S u or S d (P[S T = S u ] = p, P[S T = S d ] = 1 p) C 0 (C T ) the price of European call at time 0 (T ) with strike price K ECON BOOT CAMP

25 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-3 Model Structure S 0 C 0 p S T = S u C T = S T K 1 p S T = S d C T = 0 ECON BOOT CAMP S u K S d

26 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-4 Important point: x (number of stock) and y (amount of the zerobond) can be chosen in a way that ensures the same return as the owner of the option: xs u + y = r u = S T K (1) xs d + y = r d = 0 (2) This portfolio strategy (x, y) is called hedge strategy. Solution of (1) - (2) x = r u r d S u S d, (3) y = S u r d S d r u S u S d (4) ECON BOOT CAMP

27 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-5 Perfect Market What is a fair price C 0 of an option? Compare the return of the option with the return of hedge strategy. option: Pay today C 0, get at time T the return r u or r d respectively. buy x stocks and invest y into the zerobond: Pay today xs 0 + y, get at time T the return xs u + y or xs d + y respectively. ECON BOOT CAMP

28 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-6 You can get the same financial product (payment of r u or r d ) for the price C 0 (option price) or for the price: xs 0 + y = S 0 r u r d S u S d + S u r d S d r u S u S d (5) = (S u S 0 )r d + (S 0 S d )r u S u S d (6) No Arbitrage (no free lunch): There never exists the same financial instrument with two different prices. Conclusion: C 0 = (S u S 0 )r d + (S 0 S d )r u S u S d (7) The price C 0 does not depend on p (probability of rising prices of the share). ECON BOOT CAMP

29 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-7 Interpretation with C 0 = (S u S 0 )r d + (S 0 S d )r u S u S d (8) = S u S 0 S u S d r d + S 0 S d S u S d r u (9) = (1 q)r d + qr u (10) q = S 0 S d S u S d (11) = C 0 is the expected value of C T with respect to the probability measure defined by q. ECON BOOT CAMP

30 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-8 Interpretation of q Expectation of the share price S T under q: (1 q)s d + qs u = S u S 0 S u S d S d + S 0 S d S u S d S u (12) = S 0 (13) q was chosen such, that the expected return of the share is 0. (in general equal to the interest rate) Martingale Look for a stochastic model where the return of all shares is 0. Under this stochastic model we calculate the expected option pay-off. This expectation is the fair price of an option. This stochastic model does not correspond to the (approximative) true model. p q! (14) ECON BOOT CAMP

31 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-9 Summary of the One-Period-Model two different approaches of calculation for the option price calculation of the hedge strategy, i.e. solve a system of linear equations calculation of the expected option payment in the Martingale-model both approaches require a perfect market (existence of a hedge strategy) option price does not depend on p ECON BOOT CAMP

32 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-10 Example: the one period model K = 270, r = 5% Duplicating portfolio (1) S T = 300, C T = x + y = 30 (2) S T = 250, C T = 0 250x + y = 0 It follows: x = 0.6, y = 150 Thus, selling 150 bonds and buying 0.6 of the stock duplicates the payoff of the call. Therefore the price of the call is: = ECON BOOT CAMP

33 Y Distribution comparison NIG Laplace Normal Cauchy X The Basics of Option Management 1-11 Example: Martingale measure approach Under the measure q, S has to be a martingale: S T S 0 = 270 = E q 1 + r = q q = 0.67 Price of the call: = Both approaches provide the same results. + (1 q) ECON BOOT CAMP

34 Financial Markets 24 Implied Binomial Tree Stock Prices Arrow-Debreu Prices T=1, n = 2

35 Financial Markets 25 Strike Price: 90 EUR Payoff for a Call option: C( 90, 1) = ( ) ( ) =

36 26 Financial Markets Villa in Hirschgarten Appartment in Kreuzberg

37 27 Real Estate Markets Credit Scoring Real estate valuation

38 Real Estate Markets 28 Berlin Notes: Observations (1991q1-2007q2).

39 Real Estate Markets 29 Steglitz-Zehlendorf Schweizer Viertel Notes: 1837 Observations (1991q1-2007q2).

40 30 Valuation in the presence of uncertainty Volatility prognosis Transparency for developers and investors