Econometria dei mercati nanziari c.a. A.A Scopes of Part I. 1.a. Prices and returns of nancial assets: denitions

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1 Econometria dei mercati nanziari c.a. A.A Scopes of Part I 1.a. Prices and returns of nancial assets: denitions 1.b. Three stylized facts about asset returns 1.c. Which (time series) model for nancial asset returns? Luca Fanelli University of Bologna luca.fanelli@unibo.it

2 1a Prices and returns of nancial assets We denote by P t the price at time t of any nancial asset. We tipically have in mind the prices of equities (stock prices), but P t can also be a stock index (e.g. S&P 500,...) t will typically denote days or weeks, rarely months.

3 7 6 UNICREDIT Figure 1. UNICREDIT, daily basis, 24/03/ /03/2010

4 DOWJONES Figure 2. DOWJONES, daily basis, 24/03/ /03/2010

5 We denote by p t :=log(p t ). The log transformation changes only the scale of the price, not its pattern over time P_UNICREDIT Figure 3. Log price of UNICREDIT, daily basis, 24/03/ /03/2010

6 P_DOWJONES Figure 4. DOWJONES, daily basis, 24/03/ /03/2010

7 The simple return of an asset price is given (assuming that there are no dividends) by R t := P t P t 1 P t 1 := P t P t 1 or R t :=100 P t P t 1 P t 1 : The sign of R t says wheter the price of the asset has increased or decreased from time t 1 to time t, or whether the stock index has increased or decreased from time t 1 to time t. The magnitude of change. P t P t 1 reveals the amount of the Media constantly inform us about this! An useful denition is the one of log-return: r t := log(p t =P t 1 ):= log(p t ) log(p t 1 ):=p t p t 1 :=p t :

8 20 15 R_UNICREDIT Figure 5. Log return of UNICREDIT, daily basis, 24/03/ /03/2010

9 12 8 R_DOWJONES Figure 6. Log return of DOWJONES, daily basis, 24/03/ /03/2010

10 Is there a link between R t and r t? r t := log(p t =P t 1 ) log P t P t 1 + P t 1 P t 1 = log 1 + P! t P t 1 P t 1 = log 1 + P! t P t 1! = log (1 + R t ) : It is known that for 0<x<0:10, log (1 + x) x, hence r t := log (1 + R t ) R t is a good approximation on condition that returns are not too high or low. During this course we will deal with log-returns, except where explicitly stated otherwise. Given the substantial equivalence seen above, we will use the word `returns' also for log-returns.

11 Any trader would be happy if he/she could forecast, at time T (today), the unknown quantity r T +1 (the return of tomorrow) perfectly. The knowledge of r T +1 would provide the trader with the information about the the (log)price at time T +1: r T +1 := log(p T +1 ) unknown at time T log(p T ) : known at time T Obviously, nobody can forecast returns perfectly. Since r T +1 is a random variable at time T, we can only make probabilistic (thus uncertain) statements about r T +1. The mission would be reducing our uncertainty about r T +1 as much as possible. We need a stochastic model. Econometrics provides stochastic models.

12 Suppose the trader has its own stochastic model for asset returns. If the model predicts that it is likely that r T +1 :=r > 0, then it is conveniet to take a long position, i.e. to buy the asset today (or to keep it in the portfolio) and sell it tomorrow. Conversely, if the model predicts that it is likely that r T +1 :=r < 0, then it is conveniet to take a short position. A short position consists in acquiring the right property of the asset from someone that has already purchased it and selling the asset today at the price of today with the obligation of paying it tomorrow at the price of tomorrow (`vendita allo scoperto'). However, it is not only important the knowlege of the sign (r T +1 >0/r T +1 <0) and magnitude (jrj is high/low) of the forecasted return, but it is also crucial to known how much risk is associated with the asset.

13 If the model predicts that tomorrow the market is going to be extremely volatile (high risk), then he/she will take more risk compared to the case in which the market is less volatile (low risk). Financial economics (and econometrics) is an history about balancing return and risk: higher returns are associated with higher risk and lower returns with lower risk. How to reach the optimal position?

14 1b Three stylized facts about asset returs We have understood that we have to treat asset returns as random variables and thus we need the aid of probability and statistical analysis. Then the rst question to address should be: which are the salient properties observed in asset returns? If we capture and understand these features, indeed, we can try understanding which is the `best' stochastic model to put forth. General properties that are expected to be present in any set of returns are called stylized facts. Assume that t measures days or weeks (sometimes months). Then there are three important properties that are found in almost all sets of daily returns obtained from a few year of prices:

15 Three stylized facts 1. The distribution of returns is not Gaussian: r t N(; 2 ): 2. There is almost no correlation between returns for dierent days: Corr(r t ; r t ) = 0, :=1,2,... Recall that given two scalar random variables X; Y, then Cov(X; Y ) Corr(X; Y ):= [V ar(x)] 1=2 [V ar(y )] 1=2 Cov(X; Y ):=E [(X E(X))(Y E(Y ))] :=E(XY ) E(X)E(Y ):

16 3. Positive and strong dependence between absolute retunrs on nearby day and likewise for squared returns: Corr(jr t j ; jr t Corr(r 2 t ; r2 t j) > 0, :=1,2,..., max ) > 0, :=1,2,..., max This property explains the volatility clustering phenomenon.

17 An important result of probability theory says that if the random variables in the sequence x 1 ; x 2 ; :::; x n are independent (i.e. if their joint probability distribution is obtained as the product of the marginal probabiloity distributions), then the random variables in the sequence or in the sequence are independent as well. (x 1 ) 2 ; (x 2 ) 2 ; :::; (x n ) 2 jx 1 j ; jx 2 j ; :::; jx n j

18 The three stylized facts tell us that asset returns are generated by random variables which are not independent over time (albeit their correlation is typically low). Recall: Cov(X; Y ) = 0 ; ( X and Y stochastically independent!!!!

19 The fact that asset returns are not independent over time makes sense because out intuition suggests that what happens today in the market is somehow related to what happened yesterday and what will happen tomorrow! The three stylized facts can all be explained by changes through time in volatility (volatilityvariance). This will lead us to consider a class of models in which the volatility of asset returns will be explicitly modelled.

20 Focus on the distribution of returns - It is approximately symmetric - it has fat tails (leptokurtic) - it has high peaks Series: R_UNICREDIT Sample 24/03/ /03 /2010 Observations 2610 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque Bera Probability Series: R_DOWJONES Sample 24/03/ /03 /2010 Observations 2610 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque Bera Probability

21 Consider the scalar random variable X N(; 2 ) and its density: 1 f(x):= (2) 1=2 e 1 2 2(x )2 : It is known that in this case: skewness: S(X):=E (X ) 3 3 = E[(X )3 ] 3 = 0 kurtosis: K(X):=E (X ) 4 4 = E[(X )4 ] 4 = 3 The quantity K(X) 3 is called `excess' kurtosis (it takes the Gaussian as benchmark). Asset returns display excess kurtosis and mild asymmetry.

22 Imagine to have T observations (we will call these observations time series) r 1 ; ::::; r T relative to the random variable r t that generates the returns. Assume that a set of regularity conditions that we will examine in the next slides hold. Under these regularity conditions, we can estimate the moments of r t, hence S(r t ) and K(r t ).

23 In particular, ^ r := 1 T TX t=1 r t estimates r :=E(r t ) ^ 2 r:= 1 T 1 ^ 3 r:= 1 T 1 ^ 4 r := 1 T 1 TX t=1 TX t=1 TX t=1 (r t ^ r ) 2 estimates 2 r:=e[(r t r ) 2 ] (r t ^) 3 estimates 3 r:=e[(r t r ) 3 ] (r t ^) 4 estimates 4 r:=e[(r t r ) 4 ] therefore ^S(r t ):= ^3 r (^ r ) 3, ^K(r t ):= ^4 r (^ 2 r )2:

24 Alternative distribution A satisfactorily alternative probability distribution for daily returns must have high kurtosis and can be approximately symmetric. There are many possible alternative leptokurtic distributions that can be used in place of the Gaussian, e.g. the Student-t We consider a distribution that is consistent with our purposes.

25 Imagine that! t is a random scalar variable and that r t j! t N(; g(! t )), 2 t :=g(! t ) where g() is a function that takes values in R + and! t is called mixing variable. Thus 2 t variance. = V a( r t j! t ) = g(! t ) is a conditional In this case, the marginal distribution of r t is not Gaussian but is Mixed Gaussian (it is a mixture of Gaussian) and is possible to prove that it is leptokurtic.

26 The mixing variable! t can been associated with observable variables such as trading volumes, the number of transactions, the news that aect the market, etc. We will consider models of the type above in which the changes of 2 t :=g(! t) are explained by a particular class of time series model. In particular, we consider: 2 t ARCH or GARCH processes

27 Focus on the correlation of returns Imagine to have T observations (we will call these observations time series) r 1 ; ::::; r T relative to the random variable r t that generates the returns. Assume that a set of regularity conditions that we will examine in the next slides hold.

28 Then the quantity Cov(r t ; r t ) Corr(r t ; r t ):= V ar(r t ) 1=2 V ar(r t ) 1=2 = Cov(r t; r t ) V ar(r t ), = 1; 2; :::; max can be estimated with 1 P Tt=+1 Corr(r \ T 1 (r t ^ r )(r t ^ r ) t ; r t ):= P Tt=1 (r t ^ r ) 2 where max << T: 1 T 1 = 1; 2; :::; max

29 Let ():=Corr(r t ; r t ), = 1; 2; ::: The estimated ^():= \ Corr(rt ; r t ), = 1; 2; :::; max can be used to test the hypothesis that the returns are genrated by an independent and identically distributed process, i.e.: against H 0 : ():=0; = 1; 2; ::: H 1 : () 6= 0 for at least one : Indeed, it can be shown that under the above mentioned regularity conditions and under H 0 : ^()! D N(0; 1):

30 More specically, consider the two hypotheses H 0 : H 1 : (1):=0 & (2):=0 &... & (k):=0 () 6= 0 for at least one 2 [1; k]; It is then possible to show that: Q k :=T kx =1 [^()] 2! D 2 (k), k = 1; 2; ::: where Q k is known as the Q-statistic.

31 Focus on the squared correlation of returns Imagine to have T observations (we will call these observations time series) r 1 ; ::::; r T relative to the random variable r t that generates the returns. Assume that a set of regularity conditions that we will examine in the next slides hold.

32 Then the quantity Corr(r 2 t ; r 2 t Cov(rt 2 ):= ; r2 t ) V ar(rt 2)1=2 V ar(rt 2 )1=2 = Cov(r2 t ; r2 t ) V ar(r 2 t ), = 1; 2; :::; max can be estimated by considering the time series of squared returns: (r 1 ) 2 ; ::::; (r T ) 2 and applying the same formula as above, obtaining \ Corr(r 2 t ; r2 t ):= 1 T 1 P Tt=+1 h r 2 t ( 1 T P Tt=1 r 2 t )i h r 2 t ( 1 T 1 T 1 P Tt=1 h r 2 t ( 1 T P Tt=1 r 2 t )i 2 P Tt=1 r 2 t )i = 1; 2; :::; max where max << T.

33 1.c. Which (time series) model for nancial asset returns? We know that if we want to model and forecast asset returns we need a probabilistic (stochastic) approach. Henceforth, given a nancial asset, we shall treat the observations r 1 ; :::; r T as nite realization of a stochastic process fr t g (innite sequence of random variables). (We shall formalize the notion of stochastic process in the next set of slides).

34 We have seen that asset retuns display systematically the following features: - non normality and in particular fat tails; - mild correlation - strong positive correlation of squared (or absolute value) returns.

35 In light of the tree stylized facts, a possible candidate model that captures all three properties above might be something like r t = + u t, u t = t " t (i) :=E(r t ); (ii) t is a positive random variable ( t > 0 a.s.) that changes over time t according to some rule to be specied; (iii) n 2 t o is stationary with E( 4 t ) < 1 and Corr( 2 t ; 2 t ) > 0, = 1; 2; ::: (iv) f" t g iidn(0; 1) or f" t g iidstudent-t(n), n>2 (v) f t g and f" t g are stochastically independent, that means that for each integer n > 0 the random vectors 0 are independent. 1. n 1 C A and 0 " 1. " n 1 C A

36 Observe that the points (ii)-(iii) are consistent with many possible models in which t varies over time. The implications of our candidate specication are: 1. r t is distributed as a Mixture of Gaussian (Mixed Gaussian) hence K(r t ) > 3: 2. Corr(r t ; r t )=0, = 1; 2; ::: 3. Corr(rt 2; r2 t )>0, = 1; 2; ::: In Part I of this course we will deal with variations of the model above. In particular, in our setup, point (ii) will be specialized by means of a class of time series models known as Autoregressive Conditional Heteroskedasticity (ARCH) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH). Before coming to this class of model we need some preliminary work.