PRESENTATION

1. PRESENTATION

2. THE ACADEMY OF ECONOMIC STUDIES BUCHAREST DOCTORAL SCHOOL OF FINANCE AND BANKING PRICING OPTIONS WITH JUMPS IN RETURNS AND VOLATILITIES STUDENT : TOADER CRISTINA ALEXANDRA SUPERVISOR : PROF.UNIV.DR. MOISA ALTAR

3. Dissertation paper outline:- The importance of pricing options with jumps in the pricing kernel and correlated jumps in asset returns and volatilities.-The aims of the paper -The models -The parameters evaluation -The options evaluation-Empirical analysis -Conclusions-References

4. This model is of interest since: 1-the model uses an discret time equation (GARCH) .2-it uses only the historical prices3-it has a continuous time limit and the convergence of the return series towards processes leads to processes that have diffusion and jump elements in either or both returns and volatilities4-it is a generalization of the tipical GARCH option pricing models with normal innovations and is better in remouving more of the biases in option prices relative to the models with conditional normality 5-it allows to price options when the underlying asset’s innovations may be far from normal and when the volatility is stochastic.6-by incorporating jumps in returns and volatilities it can better capture kurtosis and skewness in the time series dynamic and better describe option prices and volatility smile 7-empirical research has shown that models which describe returns by a jump-diffusion process with volatility caracterized by a correlated diffusive stochastic process are incapable in capturing empirical features of equity index returns or option prices.For example , Bates(2000) , Bates(1997), Bates(1990) , Pan(2000) , Eraker(2004) ,Eraker, Johannes, Polson(2002) , Bates(1996) , has shown that models which describe returns by a jump diffusion process with no jumps in volatility can’t capture empirical features of equity index returns and option prices .Bates (2000), Pan(2002) show that while jumps in the return process can explain large daily shocks , these are transient and with no lasting effect on future returns . But , with volatility being diffusive , changes occur gradually and with hight persistence. These models are unlikely to generate clustering of large returns associated with temporarily high level of volatility , a feature that is displayed by the data .In contrast, in this model the jumps add conditional skewness and kurtosis to the daily innovations , rather than providing large shocks .

5. The aims of the paper :-to present the models : Duan (2007 ), Heston –Nandi (2000) , Merton and Black Scholes .-to estimate the parameters for each model .-to show the skewness and kustosis for each model and the influence of those in the pricing.-to evaluate the options : the options using Duan are evaluated by simulation and the model Heston Nandi using a closed-formula . To compare those models of evaluation I compared Black –Scholes using a fomula and simulation.-to compare the models , and to show that the model Duan that incorporate jumps in pricing kernel and correlated in asset prices and volatilities is the most performant .-for comparation someone uses the evaluation errors .

6. J. Duan , P.Ritchken and Z. Sun model ( “Jump starting GARCH : Pricing and Hedging Options with Jumps in Returns and Volatilities” -2007 )We have a discret-time economy for a period [0, T] where uncertainty is defined on a complete probability space with filtration where contains all P -null sets .The joint dynamics of the asset price , and the pricing kernel is:where mt is the marginal utility of consumption at date t .St is the total payout , consisting of prices and dividends.The dynamic of the pricing kernel is given by : Nt is distribuited as a poisson random variable with parameter lambda .The random variables are independent for .The asset price follow the process :where Nt is the same Poisson random variable as in the pricing kernel .The local variance of the logaritmic returns for t , wiewed from t-1 is:

7. Under measure P , the dynamics of the asset price is :where :Lema:Q is a probability measure For any Ft, Zt:Under measure Q , the dynamics of the asset price is distributionally equivalent to : where:

8. Under measure Q ,ht has the form :The process is stationary if The Heston Nandi model:The spot asset price St, follows the following process over time steps of length :where r: continuously compounded interest rate z: standard normal disturbanceh : the conditional variance of the log-return :alpha: determines the kurtosis of the distributiongamma : appears in asymetric influence of shocks and

9. The stationarity of the model is if :In the case of risk neutral distribution :where:The value of a Call option obeys the Black-Scholes-Rubinstein formula.The conditional generating function is:The caracteristic function is :The genereting function: The european Call option is worth :

10. For the parameters evaluation we use maxim likelihood estimation and for the maximisation of the log-likelihood function we use the downhill simplex.The conditional probability function is for Duan: where is the normal density function with mean and varance: The loglikelihood function is :The loglikelihood function is for Heston :where:

11. The option evaluation for Duan:The Monte Carlo Method is based on:If we consider n+1 trading points where then the dynamic of the asset price is :

12. We will consider :and we will obtain:where: process wiener . standard normal random variable, independent of wiener process and poisson process . goes to a poisson process

13. Parameters estimation results of the Duan model :beta_0 beta_1 beta_2 c delta mu gamma lam 0.6155e-7 0.798 0.1208 0.6847 -3.980e-4 -0.0136 3.883 4.254 loglk = 2836.57variance of y_t = 1.478e-4variance of h = 6.787e-11unit root condition = 0.976spot volatility = 0.015long run = 0.000002596 skewness_th = -0.00499kurtosis_th = 3.683724skewness_emp = -0.06832kurtosis_emp = 3.21074659 Parameters estimation results of the Heston model :alpha beta gamma lambda omega loglikelihood0.6337e-6 0.7616 605.38 -0.178816 0.0066e-6 3176.733139variance of e_t = 1.472858e-004variance of h = 5.828640e-009skew_z: -0.081kurt_z: 3.010UNIT ROOT condition = 0.994spot volatility = 0.114long run volt = 0.162570gamma risk-neutral= 605.380544-0.178816 +0.5 = 605.7persistece: beta=0.761631 foarte persistent reaction: alpha*gamma^2=0.633758e-6*605.7*605.7=0.232 is not very strong .=605.

14. Parameters estimation of the Merton model:Beta_0 beta_1 beta_2 c delta mu gamma lambda loglike67.551e-7 0 0 0 -0.1332 0.0360 2.1128 4.4486 2764.891571variance of y_t = 1.472801e-004variance of h = 5.812950e-014spot volatility = 0.041skewness_th =0.02250skewness_emp =0.2336kurtosis_th =3.61127kurtosis_emp =4.74423 k=0.9gamma_opt=1The results in option pricing: The continuously compunded Treasury Bill are :t = 0.00277 0.0833 0.25/ 0.5 1rate =0.0214 0.02229 0.0239 0.0264 0.02831 The errors :

26. Conclusions • This paper presents an analytical comparison beetween the model Heston Nandi (2000) and Duan (2007) . • The model Heston Nandi presents a closed form evaluation solution for option values when the variance of the spot returns follows a GARCH(p, q) process and is correlated with assets returns . The model with a single lag converge s to Heston(1993) continuous-time stochastic volatility model as the observation interval shrinks . • The literature shows that the results of the discrete-time model are very close with those of the continuous-time model . The advantage of the discret model is that it can be easily estimated only from the history of asset prices . • Like the model presented by Duan, the model can use and option prices to estimate the parameters of the model. More than that, we can make an actualization of the parameters for the out of sample period , so the results will be much more fitted with the options on the market . What is important in this model is that it has the ability to simultaneously capture the path –dependence in volatility and the correlation of volatility with assets returns . • The model Duan extend the model Duan(1995) and generalize the GARCH models with stochastic volatilities and normal innovations , and it is a generalization for Heston model too. Here it incorporate non-normal innovations in returns and volatilities and in this way the options are priced in the presence of skewed and leptokurtic innovations . In the paper we can see that the model Merton performed the worst , the conclusion of that beeing that including jumps and stochastic volatility , the model perform better because it will allow fat tails and highter kurtosis that explaine the time series of the S&P 500 . • One could use the model Heston Nandi with multiple lags , to incorporate fat tailed distributions of the one step daily return or can use intra-day data . • If we introduce more dependence in the dynamics of the pricing kernel , it could allow greater skewness and kurtosis in return distributions over longer time horizons .

27. One limit of the model is that the option evaluation by simulation is computational very burdensome in a long time series of options records . • Another problem is the evaluation of many parameters that can take a lot of time because the convergence is very sensible of each small modification of any initial value for parameters unlike the Heston Nandi model or Merton when this problem do not exist . So , a big number of parameters can be a very big problem . • Even the model is very complicated , a closed-form option valuation formula can be usefull because , the evaluation with a such method do not depend of any independent random variables and is more sure the evaluation . • What I can propose is the research in obtaining of a option valuation formula , then the lambda mean of the jump can be a stochastic variable and again , the interest rate can develop a stochastic process . • The model can again have a pure poisson process for very large movement on the market . Because , a poisson variable suppose that cannot be predicted , and here the mean that is known of the jump imply a some level of prediction . • The poisson process depend of the the mean of the jump and the time period . And the number of the jumps is smaller , if the time period is smaller . • What Duan, Ritchken, Sun brings here is the necesity of incorporating jumps , and the necesity of non-normality of the conditional local returns .

28. References • Amin.K and V.Ng (1993) , “ARCH Processes and Option Valuation” , working paper, University of Michigan • Amin .I and Ng.K (1993) “Option Valuation with Systematic Stochastic Volatility “ The journal of Finance , 48, 881-910 • Bates.D , (1991), “The Crash of ‘87 : What It Expected ? The evidence from options Markets “ The Journal of Finance , 46, 1009-1044 • Bates.D, (1996) , Jumps and Stochastic Volatility :Exchange Rate Processes Implicit in Deutsche Mark Options “ The Review of Financial Studies ,9,69-1007 • Bakshi.G and C.Cao and Z.Chen (1997) “Empirical Performance of Alternative Option Pricing Models “ Journal of Finance , 53, 499-547. • Ball.A and Torous.N (1985) ,”On Jumps in Common Stocks and their Impact on Call Option Pricing” , The Journal of Finance , 40, 155-173 . • Black.F and M. Scholes , (1973) , “The pricing of Options and Corporate Liabilities” , • Journal of Political Economy • Bollerslev, T., (1986) “Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics , 31, 307-327 • Brennan .M, (1979) , “The pricong of Contigent Claims in Discrete Time Models”, Journal of Finance , 34, 53-68. • Christoffersen , P. and K. Jacobs (2004) “Option Valuation with Conditional Skewness, forthcoming “ Journal of Econometrics • Christoffesen, P. and K. Jacobs (2004) “Whitch GARCH Model for Option Valuation?” Management Science, 50, 1204-1221. • Cox.J and S.Ross (1981), “A Re-Examination of Traditional Hypotheses about the Term Structure of Interest Rates “ , The Journal of Finance , 36, 769-799. • Cox.J and S.Ross , (1976) , “The Valuation of Options for Alternative Stochastic Processes” , Journal of Financial Economics , 3, 145-166. • Derman.E and Kani , 1994, “Riding on the Smile”, Risk, 7, 32-39 • Duan.J , (1995) “The GARCH Option Pricing Model” , Mathematical Finance , 5, 13-32

29. Duan.J , (1996) “Cracking the Smile”, Risk, 9, 55-59. • Duan.J .(1997) “ Augmented GARCH(p,q) Process and its Diffusion Limit” Journal of Econometrics , 79, 97-127 • Duan.J , P.Ritchken and Z.Sun (2007) , “Jump Starting GARCH:Pricing and Hedging Options with Jumps in Returns and Volatilities” • Duan.J , G.Gauthier and J Simonato (1999) “An Analytical Approximation for the GARCH Option Pricing Model” Journal of Computational Finance , 2, 75-116 . • Duan.J, P.Ritchken and Z.Sun (2005) “Approximating GARCH-Jump Models , Jump –Diffusion Processes , and Option Pricing “ , Mathematical Finance , forthcoming. • Dumas.B , J.Fleming and R Whaley (1998) “Implied Volatility Functions: Empirical Tests , Journal of Finance, 53, 2059-2106. • Foster , Douglas , and Daniel Nelson (1994) , “Asymptotic Filtering Theory for Univariate ARCH Models “, Econometrica , 62, 1-41 . • Heston .S , (1993) “A Closed-Form Solution for Options with Stochastic Volatility , with Applications to Bond and Currency Options “ , Review of Financial Studies, 6, 327-343 .Heston.S and S. Nandi (2000) , “A Closed Form GARCH Option Pricing Model “ , Rewiev of Financial Studies , 13, 585-625. • Longstaff.F (1995) , “Option Pricong and the Martingale Restriction”, , Review of Financial Studies ,4, 1091-1224. • Hull .J, (1998) “Options , Futures and Other Derivatives (3rd ed), Prentice –Hall , New Jersey • Merton.F and B.Scholes (1973) , “The Pricing of Options and Corporate Liabilities”, The Journal of Political Economy , 3, 637-654. • Odegaard B., (2007) , “Financial Numerical Recipes in C++” • Press.H,A. Teokolsky, T.Vetterling and P.Flannery (2002) , Numerical Recipes in C , The art of Scientific Computing (2.nd ed) ,Cambridge University Press • Rubinstein.M (1994) , “Implied Binomial Trees” , “Journal of Finance”, 69, 771-818. • Scott.L (1987) “Option Pricing When the Variance Changes Randomly: Theory , Estimation and An Application “, Journal of Financial and Quantitative Analysis, 22, 419-438 .