Tempered Stable Processes with Time Varying Exponential Tails
aa r X i v : . [ q -f i n . C P ] A ug Tempered Stable Processes with Time VaryingExponential Tails
Young Shin Kim
Associate ProfessorCollege of Business, Stony Brook University,100 John S. Toll Drive, Stony Brook, NY 11794, USATel: +1 (631) 632-7171e-mail: [email protected]
Kum-Hwan Roh Associate ProfessorDepartment of Mathematics, Hannam UniveristyDeajeon, Koreae-mail: [email protected]
Raphael Douady
Research ProfessorUniversity of Paris I: Panth`eon-SorbonneFrancee-mail: [email protected] Kum-Hwan Roh gratefully acknowledges the support of Basic Science Research Program through theNational Research Foundation of Korea (NRF) grant funded by the Korea government [Grant No. NRF-2017R1D1A3B03036548]. empered Stable Processes with Time VaryingExponential Tails August 26, 2020
Abstract
In this paper, we introduce a new time series model having a stochastic exponential tail.This model is constructed based on the Normal Tempered Stable distribution with a time-varying parameter. The model captures the stochastic exponential tail, which generates thevolatility smile effect and volatility term structure in option pricing. Moreover, the modeldescribes the time-varying volatility of volatility. We empirically show the stochastic skew-ness and stochastic kurtosis by applying the model to analyze S&P 500 index return data. Wepresent the Monte-Carlo simulation technique for the parameter calibration of the model forthe S&P 500 option prices. We can see that the stochastic exponential tail makes the modelbetter to analyze the market option prices by the calibration.
Key words:
Option pricing, Stochastic exponential tail, Volatility of volatility. Normal tem-pered stable distribution, L´evy process
The tempered stable process is popularly used as an option pricing model overcoming draw-backs of Black-Scholes model (see Barndorff-Nielsen and Levendorskii (2001), Barndorff-Nielsen and Shephard(2001), Carr et al. (2002), and Kim (2005)) since the class of tempered stable processes are semi-martingale and has exponential tails which are fatter than Gaussian distribution. Moreover, thetempered stable option price model explains the volatility smile and skew effect since its tails arefat and asymmetric. However, the class of tempered stable models’ independent and stationary in-crements fails to capture the stochastic volatility, stochastic skewness, and stochastic kurtosis in themarket. In this paper, we construct a new market model that has stochastic exponential tails. Using2he stochastic exponential tails, the new model can capture more stochastic properties observed inthe market, including stochastic skewness and kurtosis, and volatility of volatility (vol-of-vol).Managing volatility and vol-of-vol are important issues in portfolio and risk managementand derivative pricing. Since they are not directly observed in the market, VIX index (CBOE(2009)) and VVIX index (CBOE (2012)) are provided for measuring the volatility and vol-of-vol of U.S. stock market, respectively. Academically, ARCH and GARCH models by Engle(1982) and Bollerslev (1986). The implied volatility extracted from the Black-Scholes model(Black and Scholes (1973)) has been popularly used to observe the volatility.Applying the ARMA-GARCH model to empirical daily log-returns of a stock or an index, wecan see that the residual distribution still has fat-tails and asymmetricity (see Kim et al. (2008,2011)). In order to capture those the fat-tails and skewness of the residual distribution, ARMA-GARCH model with the standard normal tempered stable innovation distribution (ARMA-GARCH-NTS model) was studied in risk management and portfolio management in many literatures includ-ing Kim (2015), Anand et al. (2016, 2017) and Kurosaki and Kim (2018). The normal temperedstable (NTS) distribution was presented in finance by Barndorff-Nielsen and Levendorskii (2001)and Barndorff-Nielsen and Shephard (2001) to describe the fat-tail and skewness of asset returns.The standard NTS (stdNTS) distribution is a special case of the NTS distribution with zero meanand unit variance (See Rachev et al. (2011)).The volatility clustering and fat-tailed asymmetric distribution have been studied in option pric-ing in literature. The L´evy process model, stochastic volatility model, and GARCH model wereintroduced to overcome the drawback of Black-Scholes option pricing model. For instance, theL´evy stable model was applied to the option pricing in Hurst et al. (1999) and Carr and Wu (2003).The tempered stable option pricing models were discussed in Boyarchenko and Levendorski˘i (2002)and Carr et al. (2002). Stochastic volatility model was applied to option pricing in Heston (1993),and stochastic volatility model with the L´evy driving process has been studied in Carr et al. (2003).The discrete-time volatility clustering effect was considered for option pricing by taking GARCH3odel in Duan (1995). GARCH option pricing model with non-Gaussian tempered stable innova-tion was studied by Kim et al. (2010) and regime-switching tempered stable model were appliedto the option pricing in Kim et al. (2012). The skewness and kurtosis were used in addition tovolatility for option pricing in Aboura and Maillard (2016). Moreover, L´evy process model withlong-range dependence was presented in Kim et al. (2019).While the stochastic volatility and volatility clustering were studied, the term structure of vol-of-vol was studied for VIX and VVIX derivatives pricing. For example, the class of L´evy OrnsteinUhlenbeck process is used for modeling vol-of-vol in Mencia and Sentana (2013) and the Hestonstyle term structure of vol-of-vol has been presented in Huang et al. (2018), and Branger et al. (2018). Also, Fouque and Saporito (2018) considered the Heston style volatility model for thevolatility together with the dependence feature between VIX and S&P 500 index.In this paper, we will discuss two empirical properties of skewness and excess kurtosis: (1)the residual distribution of S&P 500 index daily return has negative skewness and large excesskurtosis. Moreover, the absolute value of skewness is increasing then the excess kurtosis is risingtogether. (2) Skewness and excess kurtosis of S&P 500 index daily return distribution are notconstant but time-varying. We will present a new advanced model named the Stochastic Tail NTS(StoT-NTS) model to describe those two properties. In order to construct the model, we takeARMA-GARCH-NTS model and apply a simple time series model to one shape parameter ofstdNTS distribution. After constructing the model, a parameter estimation method for the StoT-NTS model will be provided. Using the model, one can capture the time-varying vol-of-vol onstock or index return process. After the model construction, we apply the model to option pricing.We discuss the Monte-Carlo simulation algorithm for European option pricing on the StoT-NTSmodel. To verify the model’s performance, we calibrate parameters of the model using the S&P500 index option prices. As mentioned in the previous paragraph, Aboura and Maillard (2016)considers the skewness and excess kurtosis in option pricing, while we consider a parametric modelwith stochastic skewness and stochastic kurtosis in option pricing in this paper. The StoT-NTS4ption pricing model can extract the structure of invisible time-varying vol-of-vol in the marketoption prices.The remainder of this paper is organized as follows. The NTS distribution is discussed inSection 2. In Section 3, we present the stochastic properties of skewness and excess kurtosis ofthe residual distribution for ARMA-GARCH model and empirical study using the S&P 500 indexdaily return data. The StoT-NTS model is constructed in this section and shows the model hastime-varying vol-of-vol. The option pricing model on the StoT-NTS model is discussed in Section4. The Monte-Caro algorithm and model calibration are also provided in this section. Finally,Section 5 concludes.
Let α ∈ (0 , , θ, γ > , and µ, β ∈ R . Let T be a positive random variable whose character-istic function φ T is equal to φ T ( u ) = exp (cid:18) − θ − α α (cid:0) ( θ − iu ) α − θ α (cid:1)(cid:19) . (1)The random variable T is referred to as Tempered Stable Subordinator . The normal temperedstable (NTS) random variable X with parameters ( α , θ , β , γ, µ ) is defined as X = µ − β + β T + γ √T W, (2)5here W ∼ N (0 , is independent of T , and we denote X ∼ NTS ( α , θ , β , γ, µ ) . The character-istic function (Ch.F) of X is given by φ NT S ( u ) = E [ e iuX ]= exp ( µ − β ) iu − θ − α α (cid:18) θ − iβu + γ u (cid:19) α − θ α !! . The first four moments of X are as follows: • Mean: E [ X ] = µ • Variance: var ( X ) = γ + β (cid:18) − α θ (cid:19) • skewness: S ( X ) = β (2 − α ) (6 γ θ − αβ + 4 β ) √ θ (2 γ θ − αβ + 2 β ) / • Excess kurtosis: K ( X ) = (2 − α ) ( α β − αβ − αβ γ θ + 24 β + 48 β γ θ + 12 γ θ )2 θ (2 γ θ − αβ + 2 β ) Hence, if µ = 0 and γ = q − β (cid:0) − α θ (cid:1) with | β | < q θ − α then ǫ ∼ NTS ( α , θ , β , γ, µ ) has zeromean and unit variance. Put β = B q θ − α for B ∈ ( − , , then | β | < q θ − α and γ = √ − B .Then the Ch.F of ǫ equals to φ ǫ ( u ) = E [ e iuǫ ]= exp − iuB r θ − α − θ − α α θ − iuB r θ − α + u (cid:0) − B (cid:1)! α − θ α In this case ǫ is referred to as the standard NTS random variable with parameters ( α, θ ; B ) , and wedenote ǫ ∼ stdNTS ( α, θ ; B ) . The Ch.F is denoted by φ stdNT S ( u ; α, θ ; B ) = φ ǫ ( u ) . For ǫ, we have S ( ǫ ) = r − α θ B (cid:18) − B ) + 4 − α − α B (cid:19) (3) The standard NTS distribution is defined by the NTS distribution with µ = 0 and γ = q − β (cid:0) − α θ (cid:1) underthe condition that | β | < q θ − α and denoted to stdNTS ( α, θ, β ) , in many literature including Kim and Kim (2018),Anand et al. (2016), Anand et al. (2017), and Kim et al. (2015). In this paper, we change the parameterization for theconvenience. K ( ǫ ) = (2 − α )2 θ ( α − α − (cid:18) B − α (cid:19) + (cid:18) (24 − α ) (cid:18) B − α (cid:19) + 3(1 − B ) (cid:19) (1 − B ) ! . (4)Suppose that α and θ are fixed then we have a function B ( S ( ǫ ) , K ( ǫ )) , θ > . We can easily check the following facts: • if B = 0 S ( ǫ ) = 0 and K ( ǫ ) = 32 θ (2 − α ) . • if B = ± then γ = √ − B = 0 , and hence S ( ǫ ) = ± (4 − α ) p θ (2 − α ) and K ( ǫ ) = ( α − α − θ (2 − α ) . For example, • if α = 1 . and θ = 1 . , then S ( ǫ ) ∈ [ − . , . and K ( ǫ ) ∈ [0 . , . . • if α = 0 . and θ = 3 , then S ( ǫ ) ∈ [ − . , . and K ( ǫ ) ∈ [0 . , . .Other example cases of the function are presented in the Figure 1. The points of ( S ( ǫ ) , K ( ǫ )) aresmoothly connected parabolic curve for B ∈ [ − , . B Taking the ARMA(1,1)-GARCH(1,1) model as y t +1 = c + ay t + bσ t ǫ t + σ t +1 ǫ t +1 σ t +1 = κ + ξσ t ǫ t + ζ σ t , Skewness E xc e ss K u r t o s i s = 1.8, = 1.5 = 1.8, = 3 = 0.8, = 1.5 = 0.8, = 3 Figure 1: Graph of skewness to Excess kurtosis for ǫ ∼ stdNTS ( α, θ ; B ) with ( α, θ ) ∈ { (1 . , . , (1 . , , (0 . , . , (0 . , } and B ∈ [ − , .we assume that ǫ t ∼ stdNTS ( α, θ ; B ) . Then we obtain the ARMA-GARCH-NTS model. Supposethat the parameter α and θ are fixed real numbers, and parameter B is replaced to a random variable,then we obtain a new time series model. In this paper, we assume that • ( ǫ t ) t =1 , , ··· is not i.i.d, but ǫ t | t − ∼ stdNTS ( α, θ ; B t ) , • and ( B t ) t =1 , , ··· is given by a ARIMA(1,1,0) model as follows: B t +1 = B t + ∆B t +1 ∆B t +1 = a + a ∆B t + σ Z Z t +1 , where a , a ∈ R , | a | < , σ Z > , and ( Z t ) t =1 , , ··· is i.i.d with Z t ∼ N (0 , .This time series model is referred to as the Stochastic Tails ARMA-GARCH-NTS model or shortlythe
StoT-NTS model. 8ote that, the conditional skewness of σ t ǫ t is given as S ( σ t ǫ t |F t − ) = S ( ǫ t |F t − )= r − α θ B t (cid:18) − B t ) + 4 − α − α B t (cid:19) by (3). Moreover, the conditional variance of variance for σ t +1 ǫ t +1 isvar ( var ( σ t +1 ǫ t +1 |F t ) |F t − ) = ξ ( σ t |F t − ) E [ ǫ t |F t − ]= ξ ( κ + ξσ t − ǫ t − + ζ σ t − ) K ( ǫ t |F t − ) . Since ǫ t | t − ∼ stdNTS ( α, θ ; B t ) , we obtainvar ( var ( σ t +1 ǫ t +1 |F t ) |F t − )= (2 − α )2 θ ξ ( κ + ξσ t − ǫ t − + ζ σ t − ) × ( α − α − (cid:18) B t − α (cid:19) + (cid:18) (24 − α ) (cid:18) B t − α (cid:19) + 3(1 − B t ) (cid:19) (1 − B t ) ! , by (4). Hence, the StoT-NTS process captures the time varying skewness and time varying vol-of-vol for the random variable B t . We estimate model parameters using S&P 500 index daily log-return data. ARMA(1,1)-GARCH(1,1) parameters are estimated for every 3,607 working days between December 26, 2003to June 1, 2018. In each estimation, we use 1,000 historical log-returns by the current day. Forexample, • at December 26, 2003, we estimate those parameters using 1,000 daily log-returns fromJanuary 4, 2000 to December 26, 2003, 9 date S k e w ne ss / / / / / / / / / / / / / / / / / / / / -0.500.511.522.533.54 date E x K u r t o s i s / / / / / / / / / / / / / / / / / / / / Figure 2: Time series of empirical skewness and excess kurtosis for each residual sets R , R , · · · , R . • at June 1, 2018, we estimate those parameters using 1,000 daily log returns from May 7,2014 to June 1, 2018.Then we obtain 3,607 residual sets. Each residual set contains 1,000 elements extracted fromthe estimation. Let R , R , · · · , R be those residual sets. For instance, R is the residualset extracted from the ARMA(1,1)-GARCH(1,1) estimation at December 26, 2003, and R isthe residual set extracted from the estimation at June 1, 2018. We calculate empirical skewnessS ( R t ) and empirical excess kurtosis K ( R t ) for R t ∈ { R , R , · · · , R } . Then we obtain theskewness time series ( S ( R t )) t =1 , , ··· , and excess kurtosis time series ( K ( R t )) t =1 , , ··· , , whichare presented in Figure 2. Moreover, we plot pairs of excess kurtosis and skewness ( S ( R t ) , K ( R t )) for t ∈ { , , · · · , } as Figure 3. We found that, negative skewness leads large excess kurtosis,and small excess kurtosis follows zero skewness. α and θ We fit α and θ of the stdNTS process as follows: • Select one α ∈ (0 , and one θ > . Let M α,θ = { ( S ( ǫ ) , K ( ǫ )) | ǫ ∼ stdNTS ( α, θ ; B ) for B ∈ [ − , } . 10igure 3: Dots are empirical excess kurtosis values and their corresponding empirical skewnessvalues for set of residuals R t ∈ { R , R , · · · , R } . The solid curve is the curve of excesskurtosis and skewness for ǫ ∼ stdNTS ( α, θ ; B ) with estimated parameters α = 1 . and θ =1 . . • Applying interpolation for M α,θ , we define a function f α,θ from skewness to excess kurtosis.That is, f α,θ ( S ( ǫ )) = K ( ǫ ) for ( S ( ǫ ) , K ( ǫ )) ∈ M α,θ . • Find optimal ( α ∗ , θ ∗ ) minimize the square error for the empirical data as ( α ∗ , θ ∗ ) = arg min ( α,θ ) T X t =1 [ f α,θ ( S ( R t )) − K ( R t ))] /T where T = 3607 .By the fitting method, we obtained ( α ∗ , θ ∗ ) = (1 . , . , and the solid curve in Figure 3 isthe function f α ∗ ,θ ∗ . 11 date B t / / / / / / / / / / / / / / / / / / / / Figure 4: The time series of the estimated B t for each residual set in { R , R , · · · , R } ( B t ) t ≥ We fix parameters α = 1 . and θ = 1 . , and fit parameter B of stdNTS to the dailyresidual set R t for t ∈ { , , · · · , } . In this parameter fit, we find the empirical cdf F empt usingKS-density for R t and find B using the least square curve fit as B t = arg min B X x k ∈ R t ( F ( x k ; α, θ, B ) − F empt ( x k )) where F ( x ; α, θ, B ) is the CDF of stdNTS ( α, θ ; B ) . Figure 4 presents the time series of the es-timated B t for daily residual R t with t ∈ { , , · · · , } . Figure 5 has two plates. The upperplate exhibits the empirical skewness time series and the skewness time series of stdNTS( α , θ ; B t ),and the bottom plate provides the empirical excess kurtosis time series and the excess kurtosis ofstdNTS( α , θ ; B t ), where α = 1 . , θ = 1 . and B t in Figure 4.We apply the ARIMA(1,1,0) model to the time series ( B t ) t ≥ given in Figure 4 as ∆B t +1 = c B + a B ∆B t + σ B Z, p -value c B − . . − . . a B − . . − .
32 0 σ B . . · − .
55 0 (b)Value Standard Error t-statistic p -value a B − . . − . σ B . . · − .
99 0 where ∆B t +1 = B t +1 − B t . We obtain the ARIMA(1,1,0) parameters as (a) of Table 1. Theconstant c B is not significant at significant level, and hence we can set c B = 0 . The ARparameter a B and the variance σ B are significant. Set the constant c B = 0 , and re-estimateARIMA(1,1,0) we obtain (b) of Table 1 which is similar to (a). We observe the negative ARparameter, that is, ∆B t is mean reverting. Let ( S t ) t ∈{ , , ··· ,T ∗ } be the underlying asset price process and ( y t ) t ∈{ , , , ··· T ∗ } be the underlyingasset log return process ( y = 0 ) with y t = log( S t /S t − ) where T ∗ < ∞ in the time horizon.Under the physical measure P = L T ∗ t =1 P t , ( y t ) t ∈{ , , , ··· T ∗ } is supposed to follow the StoT-NTSmodel: y t +1 = µ t +1 + σ t +1 ǫ t +1 | t µ t +1 = c + ay t + bσ t ǫ t | t − σ t +1 = κ + ξσ t ǫ t | t − + ζ σ t date S k e w ne ss / / / / / / / / / / / / / / / / / / / / stdNTS SkewnessEmpirical Sknewness -101234567 date E xc e ss K u r t o s i s / / / / / / / / / / / / / / / / / / / / stdNTS Excess KurtosisEmpirical Excess Kurtosis Figure 5: Gray curses are time series of empirical skewness and excess kurtosis and blackcurses are time series of stdNTS skewness and excess kurtosis for each residual set in { R , R , · · · , R } . 14here ǫ t +1 | t ∼ stdNTS ( α, θ ; B t +1 ) , with B t +1 = B t + a B ∆B t + σ B Z t +1 , Z t +1 ∼ N (0 , for t ∈ { , , , · · · T ∗ } . Here, T , W and ( Z t ) t ∈{ , , ··· ,T ∗ } are mutually independent, and ǫ and ∆B are real constants. Let ( r t ) t ∈{ , , ··· ,T ∗ } be sequence of the daily risk-free rate of return. Thereis risk-neutral measure Q = L T ∗ t =1 Q t such that • η t +1 | t = λ t +1 + ǫ t + t | t where λ t +1 = µ t +1 − r t +1 + w t +1 σ t +1 with ω t +1 = log (cid:0) φ stdNT S ( α,θ,B t ) ( − iσ t +1 ) (cid:1) • η t +1 | t ∼ stdNTS ( α, θ ; B t ) under the measure Q with B t +1 = B t + a B ∆B t + σ B Z t +1 , Z t +1 ∼ N (0 , , t = 0 , , · · · , T ∗ . Hence we have y t +1 = r t +1 − ω t +1 + σ t +1 η t +1 | t σ t +1 = κ + ξσ t ( η t | t − − λ t ) + ζ σ t which is the risk-neutral price process.Under the risk-neutral measure Q , the underlying asset price is S t = S e P tj =0 y j for t ∈{ , , , · · · , T ∗ } . The European option with a payoff function H ( S ( T )) at the maturity T with t ≤ T ≤ T ∗ is given by E Q (cid:2) e − r ( T − t ) H ( S ( T )) |F t (cid:3) = E Q h e − r ( T − t ) H ( S t e P Tj = t y j ) |F t i . For example, European vanilla call and put price with strike price K and time to maturity T at time t = 0 are C ( K, T ) = E Q h e − rT max { S e P Tj =0 y j − K, } i P ( K, T ) = E Q h e − rT max { K − S e P Tj =0 y j , } i respectively. Assume r t = r and λ t = λ constant, to simplify the model. Let M be the number of scenariosand T be the time to maturity as a positive integer value, say days to maturity. • Step 1Generate a set of uniform random numbers between 0 and 1 ( U (0 , ), and two sets of inde-pendent standard normal ( N (0 , ) random numbers u m,n ∼ U (0 , , x m,n ∼ N (0 , and z m,n ∼ N (0 , for m = { , , · · · , M } and n = { , , · · · , T } . • Step 2Generate the tempered stable subordinator by inverse transform algorithm, as τ m,n = F invT S ( α,θ ) ( u m,n ) where F invT S ( α,θ ) is the inverse CDF of tempered stable subordinator with parameter ( α, θ ) . • Step 3Simulate ( B t ) ≤ t ≤ T as ( B m,n ) m ∈{ , , ··· ,M } ,n ∈{ , , ··· ,T } , where B m,n = B m,n − + a B ( B m,n − − B m,n − ) + σ B z m,n and B m, is B value at current time, and B m, − B m, = 0 .16 Step 4Using (2), we simulate random number ( η t ) ≤ t ≤ T as ( η m,n ) m ∈{ , , ··· ,M } ,n ∈{ , , ··· ,T } , where η m,n = B m,n r θ − α ( τ m,n −
1) + x m,n q (1 − B m,n ) τ m,n . • Step 5Generate σ t , σ m,n = q κ + ξσ m,n − ( η m,n − − λ ) + ζ σ m,n − σ m, is the currently observed volatility, and generate y t using σ t by GARCH option pricingmodel as follows y m,n = r − w m,n + σ m,n η m,n , where ω m,n = log (cid:0) φ stdNT S ( α,θ,B m,n ) ( − iσ m,n ) (cid:1) • Step 6The price process is obtained by S m,n = S exp n X j =1 y m,j ! , for m = { , , · · · , M } and n = { , , · · · , T } .For example, let GARCH parameters be κ = 4 . · − , ξ = 0 . , ζ = 0 . ,ARIMA(1,1,0) parameters for ( B t ) be a B = − . , σ B = 0 . and B = − . , andtempered stable subordinator parameters be α = 1 . and θ = 1 . . Set initial values ofreturn, residual and volatility as y = 0 . , ǫ = 3 . and σ = 0 . , respectively. Assumethat r = (1 / , d = 0 , and λ = 0 , and generate the sample path using the algorithm for T = 22 and M = 100 . Then we obtain the sample path of ( S m,n ) for S = 1 , ( σ m,n ) , and ( B m,n ) as Figure 6.The European call option and put option prices with the strike price K and time to maturity T ( S m,n ) , ( σ m,n ) , and ( B m,n ) from left.can calculated by the simulated price process as follows: C ( K, T ) = e − rT M M X m =1 max { S m,T − K, } P ( K, T ) = e − rT M M X m =1 max { K − S m,T , } . We calibrate the StoT-NTS parameters using the S&P 500 index call and put data for the secondWednesday of each month from January 2016 to December 2017. For each calibration date, we usethe GARCH parameters estimated in Section 3.1. Table 2 provides those GARCH parameters, andvolatility ( σ ) and and residual ( ǫ ) observed of each date. Daily risk free rates of return and dailycontinuous dividend rates are also presented in base-point (bp) unit. We calculate other parameters( α , θ , a B , σ B , B , and λ ) for call and put out-of-the-money (OTM) option prices. We generate oneset of uniform random numbers and two sets of standard normal random numbers in Step 1 for M = 10 , and T = 90 , and fix them. After then find parameters such that minimize the meansquare errors between the model price and the market prices: min Θ X K n S , T n < ( C ( K n , T n ) − C market ( K n , T n )) ! , Θ = ( α , θ , a B , σ B , B , λ ) , where S is S&P 500 index price of the given Wednesday and P market ( K n , T n ) and C market ( K n , T n ) are mid-price of observed bid and ask prices for the call andput on the given day with strike price K n and time to maturity T n .For example, Figure 7 exhibits the market prices and calibrated the StoT-NTS model prices forthe OTM call and puts on 5/10/2017. The calibrated GARCH-NTS model prices obtained by thesimulation method are presented in the figure as a benchmark model. The GARCH-NTS model isoption pricing model as y t +1 = r t +1 − ω t +1 + σ t +1 η t +1 | t σ t +1 = κ + ξσ t ( η t | t − − λ t ) + ζ σ t , where η t +1 | t ∼ stdNTS ( α, θ, B ) with constant B ∈ R (See Kim et al. (2010) and Rachev et al. (2011) for more details). Daily risk free rate of return and daily dividend rate of S&P 500 index ofthe day are r = 0 . bp and d = 0 . bp, respectively. GARCH parameters estimated historicalS&P 500 index return by 5/10/2017 are ( ζ , ξ , κ ) = (0 . , . , . · − ) . The volatilityand residual of the day is σ = 0 . and ǫ = 0 . , respectively. Calibrated standard NTSparameters of GARCH-NTS model are ( α, θ, B ) = (0 . , . , − . and λ = 0 . ,while calibrated parameters of the StoT-NTS model are ( α , θ , a B , σ B , B ) = (0 . , . , . , . , − . and λ = 0 . . Other calibrated parameters for the second Wednesdayof each month from January 2016 to December 2017 are presented in Table 3.For the performance analysis, we use four error estimators, the average absolute error (AAE),the average absolute error as a percentage of the mean price (APE), the average relative percentage19rror (ARPE), and the square root of mean square relative error (RMSRE), defined as follows,AAE = N X n =1 | P n − b P n | N ,
APE = AAE P Nn =1 P n N , ARPE = N X n =1 | P n − b P n | N P n , RMSRE = vuut N X n =1 ( P n − b P n ) N P n , where b P n and P n are model prices and observed market prices of options (OTM calls or OTM puts)with strikes K n , time to maturity T n , n ∈ { , . . . , N } , and N is the number of observed prices.Those four error estimators of the GARCH-NTS and the StoT-NTS models are presented in Table4. According to the table, we can see that all the error estimators for the StoT-NTS model are lessthan corresponding error estimators of the GARCH-NTS model except the cases of 01/13/2016and 4/12/2017, on which two model errors are similar.To verify that the StoT-NTS model performs better than the benchmark GARCH-NTS model,we perform the simple hypothesis tests for APE, and ARPE. Since AAE and APE have the samet-statistic values, we do not need to test both, but we present only APE case test. RMSRE is alsoomitted in this hypothesis test, since it is not linear. Instead of the hypothesis “the StoT-NTSmodel performs better than the benchmark GARCH-NTS model”, we use the following equivalenthypothesis: H : µ NT S ≤ µ StoT vs H : µ NT S > µ
StoT or H : µ NT S − µ StoT ≤ vs H : µ NT S − µ StoT > where µ StoT and µ NT S are means of calibration errors for the StoT-NTS model and GARCH-NTSmodel, respectively.Let N be the number of observed prices. Let P n be observed market prices of option and b P StoTn and b P NT Sn be model prices for the StoT-NTS model and the GARCH-NTS model, respectively, for20
400 1600 1800 2000 2200 2400 2600 28000102030405060
Market PriceHuLK ModelNTS Model
Figure 7: OTM call and put option prices and calibration result for 5/10/2017. Circles ( ◦ ) are market prices of calls and puts. Dot( · ) and plus ( + ) marks are calibrated the StoT-NTS model and the GARCH-NTS model prices, respectively. ∈ { , . . . , N } . Then calibration errors are defined by e StoTAP E ( n ) = | P n − b P StoTn | P Nn =1 P n N and e NT SAAE ( n ) = | P n − b P NT Sn | P Nn =1 P n N e StoTARP E ( n ) = | P n − b P StoTn | P n and e NT SAAE ( n ) = | P n − b P NT Sn | P n for APE, and ARPE, respectively. We perform the t-test for the hypothesis test for the followingtwo cases: • For APE, we set µ StoT = E [ e StoTAP E ] and µ NT S = E [ e NT SAP E ] . Note that µ StoT and µ NT S areAPE’s of the StoT-NTS model and the GARCH-NTS model, respectively. • For ARPE, we set µ StoT = E [ e StoTARP E ] and µ NT S = E [ e NT SARP E ] . Note that µ StoT and µ NT S areARPE’s of the StoT-NTS model and the GARCH-NTS model, respectively.The t -statistic and corresponding p -values of the tests in Table 5. Except the date 01/12/2016,01/11/2017, and 04/12/2017, H is rejected for the APE case. Except the date 01/12/2016, H is rejected for the ARPE case. In the bottom line of the table, we present the result of thosehypothesis tests for all market option prices and model prices we observed in this investigation.Considering total samples of every date in this investigation, H is rejected for APE and ARPE. Wecan conclude that the StoT-NTS model calibration performs typically better than the GARCH-NTScalibration in this investigation. In this paper, we present the StoT-NTS model obtained by taking a stochastic process for theparameter B in NTS process. The model has the stochastic exponential tails, and it deduces thestochastic skewness and stochastic kurtosis of the residual of ARMA-GARCH-NTS model, andhence it captures the time-varying vol-of-vol of the stock or index return time series. Throughthe empirical test of S&P 500 index return data, we observe that the skewness of the residual22s typically negative. Also, if the skewness is decreasing, then the excess kurtosis of residual isincreasing. The NTS distribution can describe this phenomenon by controlling the shape parameter B . By applying the ARIMA(1,1,0) model for the parameter B for time t , the StoT-NTS modeldescribes the stochastic skewness and stochastic kurtosis empirically observed in S&P 500 indexreturn data. The StoT-NTS option pricing model is also discussed as an application of the StoT-NTS model. We present the Monte-Carlo simulation technique based on the model and calibratethe model to the S&P 500 option prices observed in the market. In this empirical investigation, theStoT-NTS option pricing model performs mostly better than the benchmark GARCH-NTS optionpricing model, since the former captures the time-varying vol-of-vol in the risk-neutral market, butthe latter does not. References
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Date ζ ξ κ σ ǫ r (bp) d (bp)01/13/2016 . . . · − . − . . . . . . · − . − . . . . . . · − . . . . . . . · − . . . . . . . · − . − . . . . . . · − . . . . . . . · − . . . . . . . · − . − . . . . . . · − . − . . . . . . · − . . . . . . . · − . . . . . . . · − . . . . . . . · − . . . . . . . · − . . . . . . . · − . − . . . . . . · − . − . . . . . . · − . . . . . . . · − . . . . . . . · − . . . . . . . · − . − . . . . . . · − . . . . . . . · − . . . . . . . · − . . . . . . . · − . − . . . (bp = 10 − ) ParametersDate Model α θ a B σ B B or B λ . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . . . − . . . . − . . HuLK-NTS . . − . . − . . . . − . . HuLK-NTS . . . . − . . Date Model AAE APE ARPE RMSRE01/13/2016 NTS . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . . . . . HuLK . . . . APE ARPEDate
N µ
NT S − µ HuLK t -statistic p -value µ NT S − µ HuLK t -statistic p -value01/13/2016 − . − . . − . − . .
308 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
297 0 . ∗∗∗ . . ∗∗∗ . . · −
309 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
345 0 . ∗ . . − . − . .
316 0 . ∗∗∗ . . · − . ∗∗∗ .
346 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
292 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
300 0 . ∗∗∗ . . · − . ∗∗∗ .
288 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
423 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
351 0 . ∗∗∗ . . · − . ∗ . .
301 0 . . . . ∗∗∗ . . · −
203 0 . ∗ . . . ∗∗ . . · −
306 0 . ∗∗∗ . . · − . ∗∗∗ . . · − − . − . . . ∗∗∗ . . · −
259 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
225 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
321 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
279 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
272 0 . ∗∗ . . · − . ∗∗∗ . . · −
269 0 . ∗∗∗ . . · − . ∗∗∗ . . · −
271 0 . ∗∗∗ . . ∗∗∗ . . · −
260 0 . ∗ . . . ∗∗∗ . . · − total . ∗∗∗ . . ∗∗∗ .5084 0