Supplemental material for: Dynamic jump intensities and risk premiums: evidence from S&P500 returns and options

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1 Sulemental material for: Dynamic jum intensities and risk remiums: evidence from S&P5 returns and otions Peter Christo ersen University of Toronto, CBS and CREATES Kris Jacobs University of Houston and Tilburg University Chayawat Ornthanalai Georgia Institute of Technology December 6, In this sulement we rovide four sets of additional results. In Section we derive continuous-time limits of our models and comare them to existing models in the literature. In Section we rovide more detail on the risk remiums in our model. In Section 3 we resent the a ne comonent GARCH model and rovide estimation results. Finally, in Section 4 we show autocorrelation lots for the models return innovations. Model comarison using continuous-time limits In this section we derive the continuous-time limits for our model. Denoting the sie of the time ste by, our log return dynamic and the variance and jum intensity rocesses can be written as follows R (t + ) = r + ( =)h (t + ) + ( y )h y (t + ) + (t + ) + y (t + ) and h (t + ) = w () + b () h (t) + a () h (t) ((t) c () h (t)) + d () (y(t) e ()) h y (t + ) = w y () + b y () h y (t) + a y () h (t) ((t) c y () h (t)) + d y () (y(t) e y ()) :

2 where (t+) N(; h (t+)) and y (t + ) J h y (t + ) ; ; : Note that we write the GARCH arameters as a function of the time interval in order to study their convergence as the time interval shrinks to ero. Before deriving the limits, we need to de ne the variance and jum intensity as er unit time (t) = h (t) =; and y (t) = h y (t) =. Similarly, we exress the normal and jum innovation (t) and y (t) as functions of the time interval by letting (t) (t) (t; ) and y(t) = y(t; ); () where (t; ) N(; ) and y(t; ) J ( y (t); () ; ()) for () = = and () = = : It is easy to rove that y(t) and y(t; ) in () are identically distributed by looking at their moment generating functions. Using this arameteriation, we can write the return dynamics as R (t + ) r = ( ) (t + ) + ( y ) y (t + ) () + (t + ) (t + ; ) + y(t + ; ): Similarly, the GARCH dynamics for the variance and jum intensity can be written as (t + ) = w () + d () y (t + ) = w y () + d y (). Convergence of the model + b () (t) + a () y(t; ) e () + b y () y (t) + a y () y(t; ) ey () (t; ) c () (t) (3) (t; ) c y () (t) (4) There are various ways to obtain continuous-time limits as the time interval shrinks to ero. We follow the aroach of Heston and Nandi () to obtain the limits for the GARCH dynamics, and of Duan, Ritchken and Sun ( 6) to obtain the limit for the Comound Poisson rocess. First, consider the limit for the variance of the Normal innovation, (t + ). We let the

3 GARCH arameters in (3) be a function of the time interval according to w () = 4 ; a () = 4 ; c () = d () = ; e () = ; b () = : Substituting the above relations into (3) and eliminating the exectations and quadratic variations that converge in robability to ero gives (t + ) (t) = + ( (t)) (t) (t; ) + y(t; ) : We now consider the convergence of (t; ) and y(t; ): Notice that as the time interval shrinks to ero we have the following convergence (t; ) =) dw (t); where W (t) is a Wiener rocess. Note that the sign in front of dw (t) does not imact its limit since the Wiener increment is symmetric. For the convergence of the Comound Poisson rocess, we use the results in Duan, Ritchken and Sun (6) y(t; ) =) X N(t) + dn(t) y(t; ) =) X N(t) + dn(t); (5) where dn(t) is a Poisson counting rocess with intensity y (t) and the X i s reresent a sequence of indeendent standard normal random variables that are indeendent of W (t) and N(t). Note that a more roer notation for y (t) is y (t ), where t indicates the left continuous convergence of the limit. However for simlicity, we dro the t notation. We further note that for small dt, X N(t) + is normally distributed with mean and variance : Letting Q(t) be distributed as N ; ; the variance dynamic of the GARCH model therefore converges weakly to d (t) = ( (t)) dt + (t)dw (t) + Q(t) dn(t) Similarly, the return dynamic converges weakly to d ln S (t) = r + ( ) (t) + ( y ) y (t) dt + (t)dw (t) + Q(t)dN(t) Next, consider the continuous-time limits of the jum intensity in (4). We let the GARCH arameters be functions of the time interval according to w y () = y y 4 y ; a y () = 4 y ; c y () = y d y () = y ; e y () = y ; by () = y : 3

4 Substituting these into (4) and eliminating all the exectations and quadratic variations that converge in robability to ero gives y (t + ) y (t) = y y + y y (t) + y y y (t) y y (t) (t; ) + y y(t; ) : Using the results for the limits of the Normal innovation that (t; ) =) dw (t); and for the Comound Poisson rocess in (5), we obtain the following weak convergence result for the jum intensity d y (t) = y y + y y (t) + y y y (t) dt + y y (t)dw (t) + y Q(t) dn(t):. Summary We have shown that under the arametric assumtions made the continuous-time limit of our model is given by d ln S (t) = r + ( ) (t) + ( y ) y (t) dt + (t)dw (t) + Q(t)dN(t) (6) d (t) = + ( (t)) dt + (t)dw (t) + Q(t) dn(t) (7) d y (t) = y y + y y (t) + y y y (t) dt (8) + y y (t)dw (t) + y Q(t) dn(t) where W (t) is a Wiener rocess, N t is a Poisson counting rocess with intensity y (t); and Q(t) N ; reresents the stochastic jum sie that is indeendent of W (t) and N(t): We can write the above limits more comactly by using vector notation d ln S (t) = d (t) d y (t) = r + ( ) (t) + ( y ) y (t) + ( (t)) y y + y y (t) + y y y (t) dt + (t)dw (t) + Q(t)dN(t) " dt +.3 The Santa-Clara and Yan model () y y y # (t)dw (t) Q(t) dn(t) The model of Santa-Clara and Yan (, henceforth SCY) falls in the class of quadratic models. The return dynamic follows d ln S (t) = r + (t) (t) y (t) dt + (t)dw s (t) + Q(t)dN(t) 4

5 where W (t) is a Wiener rocess and N(t) is a Poisson rocess with intensity y (t): The equity remium term in the SCY model is denoted as (t) : Because the model is quadratic, the equity remium is a non-linear function of (t) and y (t): The SCY model arametries the variance and the jum intensity as follows (t) = Z ; y (t) = Y where Z and Y are latent state variables with the following dynamics dz(t) ( = Z(t)) dy (t) y y Y (t) dt + y dw (t) dw Y (t) The three Wiener rocesses in the SCY model are arameteried with the following correlation matrix.4 Model comarison = s sy s y sy y When comaring the continuous-time limit of our model to the SCY model, the similarity is that they both allow for dynamic volatility and dynamic jum intensity. As far as modeling di erences are concerned, our model has both advantages and disadvantages comared to the SCY model. The quadratic nature of the SCY model is an imortant advantage. Our model falls in the a ne class, and in that sense it is closer to the class of continuous-time a ne SVJ models. On the other hand, the SCY model as is standard does not allow for jums in volatility C A : and jums in the jum intensity whereas our model does. If we shut down the jums in volatility and jums in jum intensity in our model, that is we set, = and y = in (7) and (8), then our model falls under the a ne jum-di usion class d ln S (t) = d (t) d y (t) = r + ( ) (t) + ( y ) y (t) dt + (t)dw (t) + Q(t)dN(t) ( (t)) y y (t) + y y y (t) dt + (t)dw (t): y y This restricted version of our model is similar to the a ne jum-di usion models studied in several aers in the continuous-time literature, but it does not nest the quadratic jumdi usion model of SCY. 5

6 Risk remium structure: comarison with existing studies The framework in this aer leads to a simle a ne seci cation of the equity remium in the variance of the normal innovation and the jum intensity. That is, log E t [ex (R t+ )] = h ;t+ + y h y;t+ where h ;t+ and y h y;t+ reresent the normal risk remium and the jum risk remium, resectively. We now show that the resulting decomosition of the equity remium is consistent with existing studies such as Pan (), Broadie, Chernov and Johannes (7) and Liu, Pan and Wang (5). Lemma Using the EMM in Proosition and the risk-neutral dynamics in Proosition, the conditional equity remium for the return dynamic in (.) can be written as where h y;t+ h ;t+ + h y;t+ h y;t+ (9) e + h y;t+ and h y;t+ e + h y;t+ are the comensators to the jum comonent under the risk-neutral and hysical measure resectively. h i Proof. Taking the logarithm of E St+ t S t = e r+h ;t++ yh y;t+ yields the conditional equity remium h ;t+ + y h y;t+. We are left to show that y h y;t+ = h y;t+ h y;t+: Using the EMM condition y e + e y + y e ( +y) + = ; we can write y h y;t+ = e + h y;t+ e ( +y) + h y;t+ e y + y = h y;t+ h y;t+ where the second line in the above equation follows from! h y;t+ = h y;t+ ex y + y and = + y () which was shown in Proosition Using an equilibrium model with a reresentative agent, Liu, Pan and Wang (5) also obtain exression (9) for the equity remium. They also refer to h ;t+ as the normal risk 6

7 remium and y h y;t+ = h y;t+ h y;t+ as the jum risk remium. 3 The a ne comonent GARCH model Christo ersen, Jacobs, Ornthanalai and Wang (8) develo an a ne comonent GARCH model, and demonstrate its suerior otion ricing erformance relative to the benchmark a ne Heston-Nandi GARCH(,) rocess. The richer volatility dynamics enable the comonent model to cature atterns in long-maturity as well as in short-maturity otions. The return dynamic of the comonent GARCH is identical to that of the GARCH(,) model. It is given by R t+ ln(s t+ =S t ) = r + ( =) h t+ + t+ () where t+ N (; h t+ ). The variance dynamic of the comonent GARCH model however consists of two comonents: the short-run comonent h t+ ; and the long-run comonent q t+ h t+ = q t+ + (h t q t ) + t h t h t h t t q t+ =! + q t + ' t h t h t h t : t () To rice otions, we need the risk-neutralied version of the comonent GARCH model. The risk-neutral comonent GARCH dynamic is given by R t+ ln(s t+ =S t ) = r + h t+ = q t+ + (h t q t ) + h t t h t q t+ =! + q t + ' h t t h t h t+ + t+ (3) t h t t h t ; and where the risk-neutral arameters are de ned as = + + ' ; = + + ', and i = i + ; for i = ; : We have estimated the comonent GARCH using the data set used in our aer. Table S contains the estimation results. Panel A shows the MLE estimates on returns only, and Panel B shows the MLE estimates obtained using returns and otions jointly. Comaring the Otion IVRMSE of 5.5% in Panel A with the results in Table of the aer, we see that the comonent GARCH model outerforms the DVCJ and CVDJ models, 7

8 but it is outerformed by the DVDJ and DVSDJ models. The comonent GARCH model in which both comonents are Gaussian is thus better than the single factor DVCJ and CVDJ models but worse than the two-factor DVCJ and CVDJ models. The comonent GARCH model also erforms better than the two-factor Gaussian model in Table. For the case of the joint estimation, the total log-likelihood for the comonent GARCH model in Panel B of Table S is 38,77, which is better than the GARCH and two-factor GARCH model in Table 6, but worse than the three jum models, DVCJ, DVDJ and DVSDJ. Therefore, the new jum models erform well relative to the comonent GARCH model in this case too. 4 Autocorrelations of returns and innovations In this section we resent the autocorrelations of the return data, and comare them to the autocorrelations from the model innovations. The return seci cation is R t+ r + h;t+ + ( y ) h y;t+ + t+ + y t+ : (4) Figure S lots the autocorrelation function of daily S&P5 return, R t+, for 96-9, along with the autocorrelation of the daily total residual from our models, de ned as t+ + y t+ = R t+ r h;t+ ( y ) h y;t+ : (5) Figure S also shows two-standard-deviation Bartlett bands. The to left anel in Figure S shows that the daily S&P5 returns dislay no systematic time-series atterns. The rst autocorrelation is ositive, erhas caturing illiquidity in the cash index that we use. The other ve anels in Figure S show that not surrisingly the residuals in the models we estimate largely retain the autocorrelation atterns from the raw returns. 8

9 References Broadie, M., Chernov, M., Johannes, M., 7. Model seci cation and risk remia: evidence from futures otions. Journal of Finance 6, Christo ersen, P., Jacobs, K., Ornthanalai, C., Wang, Y., 8. Otion valuation with longrun and short-run volatility comonents. Journal of Financial Economics 9, Duan, J.C., Ritchken, P., Sun, Z., 6. Aroximating GARCH-jum models, jum-di usion rocesses, and otion ricing. Mathematical Finance 6, -5. Heston, S., Nandi, S.,. A closed-form GARCH otion ricing model. Review of Financial Studies 3, Liu, J., Pan, J., Wang, T., 5. An equilibrium model of rare-event remia and its imlication for otion smirks. Review of Financial Studies 8, Pan, J.,. The jum-risk remia imlicit in otions: evidence from an integrated timeseries study. Journal of Financial Economics 63, 3 5. Santa-Clara, P., Yan, S.,. Crashes, volatility, and the equity remium: lessons from S&P5 otions. Review of Economics and Statistics 9,

10 Samle Autocorrelation Samle Autocorrelation Samle Autocorrelation Figure S: Autocorrelation functions for S&P5 returns and model residuals Data GARCH (,) DVCJ CVDJ DVDJ DVSDJ Lag Lag Notes to Figure: We lot the autocorrelation function (ACF) for the daily S&P5 return, R t+ in the to left anel and the ACF of the model residuals, t+ + y t+ in the other ve anels. We use daily returns for 96-9.

11 Table S: MLE and joint MLE with comonent GARCH model Panel A: Returns MLE (969-9) Panel B: Joint otions & returns MLE (995-9) Parameters Parameters λ.6e+ λ.e+ (9.E-) (5.35E-3) 8.3E-7.4E-6 (6.65E-8) (6.8E-) β 7.5E- β 7.34E- (.8E-) (8.E-8).39E E-6 (.36E-7) (7.6E-8) 4.53E+.66E+ (7.68E+) (3.36E+) (7.64E+) (5.7E+).E E-6 (.E-7) (4.47E-8) 9.9E- 9.86E- (8.E-4).95E-4 Proerties Proerties Persistence.9977 Long-run risk remium 4.6% Percent of annual variance % Average annual volatility 9.6% Average annual volatility 4.97% Percent of annual variance % Log-likelihood 4,457 Total log-likelihood 38,77 Otion IVRMSE 5.5 Otions log-likelihood 4,49 Return log-likelihood 33,784 Notes to Table: We estimate a comonent GARCH model (see Christoffersen el al. (8)) using MLE on returns and joint MLE on Otions and Returns. Panel A reorts MLE on daily S&P 5 daily returns from June 969 to December 9. Panel B resent results jointly fitting daily S&P5 returns and weekly out-of-the money S&P 5 otions with maturity of 5 days or less using data from January 996 to October 9. Under Proerties we reort relevant characteristics imlied by the model from each estimation methdology.

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