The Normal Inverse Gaussian GARCH Model

Consider a one-period rate of return for the underlying asset and let S_t be the asset price at time t and ht be the conditional variance of the log return over time interval [t-1, t]. Assume price process under the physical measure P can be modeled as following:

mean zero and variance h ; r is the constant one-period risk-free interest rate and _t λ could be interpreted as “excess return with respect to a risk-free investment per unit of risk” or called the unit risk premium.

Traditionally,ε_t is assumed normally distributed with mean zero and varianceσ². Duan (1995) assumes that ε_t conditional on the information set Ω is normally _t distributed with mean zero and varianceh , which follows a _t GARCH p q process of ( , ) Bollerslev (1986) under measure P. We assumes that ε_t conditional on the information set Ω is normal inverse Gaussian with mean zero and variance _t−1 h , which _t follows a nonlinear asymmetric GARCH (NGARCH) model proposed by Engle and Ng (1993) under P-measure as below:

ε

_t Ω ∼_t−1 ^PNIG h

(

_t,0,0,

η )

(2.14) ^{ht = +w α ε}

(

^{t−1−θ ht−1}

)

^2+βht−1

where θ is a non-negative parameter to capture the negative correlation between the innovations of asset return and its conditional volatility. The parametric restrictions ofw>0,α ≥0,β≥ ensures that conditional volatility is non-negative. And, to ensure 0 that the unconditional variance of asset return is bounded (or covariance stationarity) the parameter restriction^α

(

^1+θ2

)

+ < is required. From (2.16) the unconditional variance ^{β 1} can be derived:

^ω (

¹⁻

^α

⁽¹⁺

^θ

²⁾⁻

^β )

^. (2.15)

Combining (2.13) and (2.14), we label this model as NIG-NGARCH (1, 1). Ifθ = , we 0 call the model as NIG-GARCH (1,1). By taking the limit of η approach infinity, the GARCH (1,1) option pricing model of Duan (1995) is a limiting case of NIG-GARCH(1,1) and the NGARCH (1,1) option pricing model is also a limiting case of NIG-NGARCH (1,1). The Black-Scholes model is also a special case of NIG-GARCH (1,1) as η approach infinity, α = and0 β = . 0

3. The GARCH-Type Option Pricing Model

There are two kinds of discrete time GARCH models to value the European option.

One is the nonanalytical solution for option values which use Monte-Carlo simulation to compute the option prices. Another is the closed-form solution which uses the inversion formula of Heston and Nandi (2000) in terms of the characteristic function. The former is usually used to compute the option price in empirical work but it takes more time than the latter. The GARCH –type option pricing models are presented in next two sections.

3.1 The NIG-GARCH Option Pricing Model

In order to develop the GARCH option pricing model, Duan(1995) introduced the locally risk-neutral valuation relationship (LRNVR) as a counterpart of the conventional risk-neutral argument extended in Rubinstein(1976) and Brennan (1979).

From another economically meaningful assumption thatS_t Ω essentially does not allow _t−1 the arbitrage profits, which we can just profit with risk free rate or can be written

as

(

1

)

1

r

t t t

E S Ω− =S e− . Under this risk-neutralized probability measure Q, the price process in (2.13) and (2.14) becomes varianceh . (3.1) is risk-neutral and can easily be showed in appendix C. The risk-neutral _t NIG-NGARCH (1,1) option pricing model contains six parameters η , ω , α , β , θ , λ . From (3.1) we can easily derive the following terminal asset price:

The principle of no arbitrage (or called the law of one price) indicates that a European call option with strike price K and expiring at maturity T must have the following value at time t:

C

^NIG-NG_t

= e

^{−r(T t)−}

E

_Q

⎡ ⎣ max (S

_T

− K , 0) Ω ⎤

_t

⎦ .

^(3.4)

where E_Q[] denotes the expectation under the risk-neutral distribution. By letting η approach infinity, from (3.1) to (3.4) also converge to GARCH option pricing model of Duan under measure Q. Since the option price can’t be derived analytically, Monte Carlo simulations are used to compute the approximate option prices.

3.2 The Heston-Nandi GARCH Option Pricing Model

Another famous discrete-time pricing model is the Heston-Nandi GARCH option valuation model (2000). They developed a closed-form solution for European option values in GARCH model which simultaneously captures path dependence in volatility (volatility clustering) and the negative correlation of volatility with asset returns (leverage effects). Considering the Heston-Nandi GARCH (1,1) process that asset price process under measure P is defined as

1 one-period continuously compounding interest rate. The parameter γ controls the skewness or the asymmetry of the distribution of the log returns. Under the risk-neutralized probability measure Q, the asset price process in (3.5) become

^*t

If the characteristic function of the log spot price is f( φi ), then a European call option with strike price K and expiring at maturity T is worth the following value at time t:

In GARCH models, the most commonly used method in estimating the vector of unknown parameters,

θ

, is the maximum likelihood estimates (MLE).

Assuming _t ^t

t

z = h

ε are independently and identically distributed standardized

innovations with common density function f which has mean zero and variance one. The log-likelihood function can be expressed as

(

t

) ( )

t

where

^θ

=

( ω α β λ θ η

^{, , , , ,}

)

for NIG-NGARCH (1,1). If the density function f is

The parameters θ can be estimated by using numerical methods such as non-linear minimization as well.

4. Empirical study

4.1 Empirical study on stock index

We can check the adequacy of a fitted GARCH-type model by examining the series of standardized innovations

{ }

zt . First, using the Ljung-Box (LB)-Q statistic to check for serial correlation (autocorrelation) on the series of

{ }

zt and conditional heteroscedasticity (or called (G)ARCH effects) on the series of

{ }

^z2t . The null hypothesis of the former is

“no autocorrelation” and the latter is “no ARCH effects”. Second, using QQ-plot to check the distribution assumption. In addition, we can also use Jarque-Bera (JB) test to check the normal assumption and Kolmogorov-Smirnov (KS) test to check the NIG assumption on the series of

{ }

zt .

We use daily returns on S&P 500 index from July 1, 1988 to June 28, 1991 to estimate the models’ parameters under measure P. Table 1 shows the maximum likelihood estimates and some characteristics of the GARCH-type models. The annualized volatility (252days) of NGARCH (1,1) and NIG-NGARCH (1,1) is

(

²

)

252ω 1−α θ(1+ )−β and the Heston-Nandi GARCH(1,1) is ^{252(ω α+ ) 1}

(

^{−αγ2−β}

)

^.

All the GARCH models imply a 14% annualized volatility and Ljung-Box-Q statistic for testing autocorrelation and GARCH effect shows those models are adequate for the data at the 5% significance level.

Figure 1 shows the QQ-plots of the six GARCH-type models on standardized innovations. We can find that the NIG-GARCH and NIG-NGARCH option pricing model can almost capture the empirical data. However, the GARCH models of Duan (1995) and Heston and Nandi (2000) can not capture the empirical data on the fatter tails which often observed in financial time series. Further evidence can be seen from the testing results in Table 1. The value of the Kolmogorov-Smirnov test statistic on the

NIG-GARCH and NIG-NGARCH are 0.025 and 0.028 with p-values 0.741 and 0.602 which do not reject the null hypothesis of NIG at the 5% significance level. The p-value of the Jarque-Bera test on GARCH model with normal assumption almost equal to zero that also rejects the null hypothesis of normality at the 5% significance level. Therefore, our model can well fit the daily returns on S&P 500 index.

4.2 Empirical study on Options

The previous section has already showed the empirical results on index data. We proceed to investigate the performance of these models on option pricing. Special attention is paid to the out-of-sample performance. For this empirical study, S&P 500 (SPX) index call options which are European-type and traded on the Chicago Board Options Exchange (CBOE) are considered. Because of the active market, S&P 500 index options have been included as illustrations in many researches, including Bakshi, Cao and Chen (1997), Dumas, Fleming and Whaley (1998), Heston and Nandi (2000) and so on.

The data set here is identical to that of Bakshi, Cao and Chen (1997). We use call option contracts on 235 trading days from January 8, 1991 to December 31, 1991 and apply the same filters of Bakshi, Cao and Chen (1997). Table 2 gives the basic description on the options data including the average call option prices and the number of call option contracts across days to maturity (DTM) and moneyness. We restrict the maturities of option contract between 7 and 80 days and define the moneyness as S K where S is the S&P 500 index level and K is the strike price. Our call option data set contains 4324 option contracts and the average option price is $13.51. In next section we will discuss the interesting question on how to examine the models’ performance in out-of-sample analysis.

We analyze the out-of-sample performance by the following steps. First, the parameters of all the models are estimated from the previous three years’ S&P 500 index and forecast the next one month call option prices. Second, we roll the parameters every month and repeat step 1 twelve times then we can get the one year forecasted model option prices. We note that the parameters in one month (about 20 trading days) do not

change significantly if the stock market structure does not vary largely. Finally, the forecasted option prices are plugged into the Black-Scholes formula to get the model-implied volatility. Then three loss functions in terms of the root mean square error (RMSE), the mean absolute percentage error (MAPE), and the mean absolute error of log-implied volatility (log-IVMAE) are calculated to evaluate the performance of different option pricing models. The loss functions mentioned above are defined as:

(

model

)

²

where Cmodel and Cobs are the model forecasted and data observed call prices, respectively, and n is the total number of call option contracts, σmodel=BS⁻¹

(

Cmodel, , , ,S K T r

)

,

( )

1 , , , ,

obs BS Cobs S K T r

σ = ⁻ , andBS⁻¹is the inverse of the Black-Scholes formula with respect to σ, S the stock index, K the strike price, T the time-to-maturity, and r the riskfree interest rate. The loss functions in (4.1) and (4.2) are the most commonly used in out-of-sample performance. However, there are some drawbacks in using the call price to compare the model performance. Using the RMSE loss function indeed puts much more weight on expensive options, such as in-the-money or long time-to-maturity call option contracts. On the other hand, using MAPE loss function puts much more weight on cheap options, such as out-of-money or short time-to-maturity call option contracts. Since the market convention is to quote in terms of volatility, some literature proposes alternative loss function to compare the model with the implied Black-Scholes volatility, such as PAN (2002), Christoffersen and Jacobs (2004).

Table 3 shows the out-of-sample RMSE loss function across days to maturity and moneyness for NIG-NGARCH, NGARCH and HN-GARCH models during the

out-of-sample period from January 8, 1991 to December 31, 1991. The overall RMSE are 0.9964, 1.0010, and 1.0500 for the NIG-NGARCH, NGARCH and HN-GARCH, respectively. Through the overall RMSE we can know which model is best, but it does not reveal too much information. So, we report the RMSE by moneyness category which has some interesting results. First, the NIG-NGARCH model appears to have the smallest valuation error for at-the-money call options, NGARCH model has the smallest valuation error for (deep) out-of-the-money call options and the HN-GARCH model has the smallest valuation error for (deep) in-the-money call options. Second, the performance of HN-GARCH is worst for (deep) out-of-the-money call options and the NGARCH is worst for (deep) in-the-money call options. Generally specking, the NIG-NGARCH model’s performance is superior to other GARCH models for at-the-money calls and does not perform too badly for in-the-money or out-of-the-money calls. It is also true for the mean absolute percentage error (MAPE) and the mean absolute error of log-implied volatility (log- IVMAE) reported in Table 4 and Table 5, respectively.

Figure 2 displays the data observed and model forecasted implied volatility where the horizontal axis is the number of call option contracts and the vertical axis is the annualized volatility (252 days). It shows that the model forecasted implied volatility of the NIG-NGARCH and the NGARCH can almost capture the data observed implied volatility. However, the forecasted implied volatility of the HN-GARCH model does not vary much over the number of call option contracts. As mentioned before, the GARCH model can capture the volatility clustering of return for the underlying asset. We also see that the forecasted implied volatility of nonanalytical solution GARCH models (NIG-NGARCH and NGARCH models) can fit the observed implied volatility of call option data. The HN-GARCH can capture path dependence in volatility of return, but it can not fit the observed implied volatility of call option data that indicates that there may be some question in this closed-form model.

5. Conclusion

This thesis proposes a model combining the GARCH option valuation and innovations for the asset returns following the NIG distribution. The model is called NIG-GARCH and will include the GARCH option pricing model of Duan (1995) as a limiting case. In addition, we also consider the “leverage effects” in the model, called NIG-NGARCH, which can explain the asymmetric effect of stock volatility. There are two results in our empirical studies. First, empirical results on index shows that our models can adequately fit the daily returns on S&P 500 index because the distribution assumption of normal inverse Gaussian has fatter tails than the Gaussian and can be used to explain excess kurtosis. Second, empirical results on call option shows that in out-of-sample performance our model is superior to the GARCH models of Duan (1995) and Heston and Nandi (2000) for at-the-money calls and does not perform too badly for in-the-money or out-of-the-money calls. Besides, we also find that although nonanalytical solution GARCH models take a lot of time in computing, they can fit the observed implied volatility of call option data much better than the closed-form models.

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Appendix A. Tables

Table 1 Maximum likelihood estimates and Model characteristics

ω 1.10E-06 3.80E-06 6.30E-07 3.50E-06 5.30E-13 6.10E-13 α 0.0194 0.0233 0.0218 0.0307 1.10E-06 7.00E-07

persistence 0.9859 0.9505 0.992 0.9545 0.9854 0.9906 Annualized

Volatility 0.1388 0.1395 0.1412 0.1388 0.1395 0.1374 JB test 59.63 Maximum likelihood estimates of all the GARCH (1,1) models using the daily returns on S&P 500 index from July 1,1988 to June 28,1991. The sample size of observations is equal to 756. JB test = Jarque-Bera test, KS test = Kolmogorov-Smirnov (KS) test, LB-Qz (p) = Ljung-Box Q test on autocorrelation for p lagged standardized innovations, LB-Qz² (q) = Ljung-Box Q test on GARCH effects for q lagged squared standardized innovations. The parenthesis gives the p-value of all the tests.

Table 2 Basic description of Call Options data Panel A. Average Call Option Prices

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.91 1.34 2.98 2.07 0.95<S / K<0.99 1.60 3.42 6.21 3.87 0.99<S / K<1.01 4.82 8.41 12.09 8.42 1.01<S / K<1.05 12.96 16.16 19.53 16.17

1.05<S / K 25.47 27.65 30.70 27.87

All 11.79 13.25 15.49 13.51

Panel B. Number of Call Options

moneyness DTM<20 20<DTM<50 50<DTM<80 All

<0.95 2 84 70 156

[0.95,0.99) 210 712 339 1261

[0.99,1.01) 164 354 163 681

[1.01,1.05) 299 658 286 1243

>1.05 216 542 225 983

All 891 2350 1083 4324

The table shows the basic description of call option contracts written on the S&P 500 index for a total 235 trading days from January 8, 1991 to December 31, 1991. Panel A describes the average call option prices and Panel B describes the number of call options across days to maturity (DTM) and moneyness (S/K) where S/K is defined as the S&P500 index value over the option strike price.

Table 3 Out-of-Sample Performance with loss function RMSE Panel A. NIG-NGARCH Model RMSE

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.6704 0.7767 1.1945 0.9853 0.95<S / K<0.99 0.5439 1.1522 1.5337 1.1963 0.99<S / K<1.01 0.6531 1.1815 1.3820 1.1338 1.01<S / K<1.05 0.5240 0.8985 1.1383 0.8897 1.05<S / K 0.4200 0.7021 0.8749 0.6970 All 0.5331 0.9809 1.2717 0.9964

Panel B. NGARCH Model RMSE

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.6802 0.7394 1.2058 0.9761 0.95<S / K<0.99 0.5216 1.1345 1.5252 1.1821 0.99<S / K<1.01 0.6651 1.2011 1.3819 1.1461 1.01<S / K<1.05 0.5319 0.9508 1.1289 0.9164 1.05<S / K 0.4147 0.7126 0.8727 0.7015 All 0.5322 0.9928 1.2666 1.0010

Panel C. Heston-Nandi Model RMSE

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.4360 1.0603 1.6100 1.3308 0.95<S / K<0.99 0.5957 1.3006 1.7922 1.3703 0.99<S / K<1.01 0.6687 1.1686 1.4306 1.1434 1.01<S / K<1.05 0.5132 0.8197 1.0462 0.8190 1.05<S / K 0.4327 0.6599 0.7379 0.6371 All 0.5479 1.0198 1.3723 1.0500 The table shows the root mean square error (RMSE) of call option across days to maturity (DTM) and moneyness (S/K). Panel A, B, C stand for the models of NIG-NGARCH, NGARCH and HN-GARCH, respectively.

Table 4 Out-of-Sample Performance with loss function MAPE Panel A. NIG-NGARCH Model MAPE

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.0067 0.6192 0.4377 0.5391

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.0068 0.5887 0.4426 0.5250

Panel C. Heston-Nandi Model MAPE

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.0042 0.8587 0.5909 0.7333 The table shows the mean absolute percentage error (MAPE) of call option contracts across days to maturity (DTM) and moneyness (S/K). Panel A, B, C stand for the models of NIG-NGARCH, NGARCH and HN-GARCH, respectively.

Table 5 Out-of-Sample Performance with loss function log-IVMAE Panel A. NIG-NGARCH Model log-IVMAE

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.2362 0.1523 0.1522 0.1542

Panel B. NGARCH Model log-IVMAE

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.2409 0.1460 0.1539 0.1519

Panel C. Heston-Nandi Model log-IVMAE

moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.1575 0.2026 0.2015 0.2015 The table shows the mean absolute error of log-implied volatility (log-IVMAE) of call option contracts across days to maturity (DTM) and moneyness (S/K). Panel A, B, C stand for the models of NIG-NGARCH, NGARCH and HN-GARCH, respectively.

Table 6 Four characteristics for IG and NIG distributions

Appendix B. Figures

Figure 1 QQ-plot of all the GARCH (1,1) models

The figure shows QQ-plot of the standardized innovations for daily returns on S&P 500 index from July 1, 1988 to June 28, 1991.

Figure 2 Implied volatility of data observed and model forecasted

The figure shows the implied volatility of data observed and model forecasted where obs is data observed implied volatility, NIGNG, NG and HN are model implied volatilities of NIG-NGARCH, NGARCH and HN-GARCH, respectively.

Figure 3a NIG densities with different σ

The figure shows NIG densities with β= 0,μ= 0,η= 3 andσ= 1, 2, 3 (from inner to outer).

Figure 3b NIG densities with different β

Figure 3b NIG densities with different β

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