Comparative Study of Two Extensions of Heston Stochastic Volatility Model
aa r X i v : . [ q -f i n . M F ] D ec Comparative Study of Two Extensions of HestonStochastic Volatility Model
Gifty [email protected] R. [email protected]. [email protected] of Applied Mathematics, Delhi Technological University,Delhi(India)-110042
Abstract
In the option valuation literature, the shortcomings of one factorstochastic volatility models have traditionally been addressed byadding jumps to the stock price process. An alternate approach inthe context of option pricing and calibration of implied volatility isthe addition of a few other factors to the volatility process. Thispaper contemplates two extensions of the Heston stochastic volatilitymodel. Out of which, one considers the addition of jumps to thestock price process (a stochastic volatility jump diffusion model) andanother considers an additional stochastic volatility factor varyingat a different time scale (a multiscale stochastic volatility model).An empirical analysis is carried out on the market data of optionswith different strike prices and maturities, to compare the pricingperformance of these models and to capture their implied volatilityfit. The unknown parameters of these models are calibrated usingthe non-linear least square optimization. It has been found that themultiscale stochastic volatility model performs better than the Hestonstochastic volatility model and the stochastic volatility jump diffusionmodel for the data set under consideration.Keywords. Stochastic volatility; Multiscale stochastic volatility; Meanreversion; Option pricing; Time scales; Jump diffusion
The derivative pricing model proposed by Black and Scholes[5] assumesthe volatility to be constant and asset log-return distribution as Gaussian.Empirically, the volatility is not constant but it smiles and the log-returndistributions are non-Gaussian in nature characterised by heavy tails andhigh peaks. A wide range of research has been done to improve uponclassical Black-Scholes model. The model has been extended to include eitherconstant volatility with jumps(e.g. jump diffusion (JD) models of Merton[15]1nd Kou[13]) or to consider volatility itself as a continuous time stochasticprocess(e.g. stochastic volatility models given by Hull and White[11],Scott[17], Wiggins[21], Stein and Stein[20], Heston[10] and Ball and Roma[3],etc.). Stochastic volatility models allow the volatility to fluctuate randomlyand are able to incorporate many stylized facts about volatility namelyvolatility smile and skew, mean reversion and leverage to name a few.In the single factor stochastic volatility models, Heston stochasticvolatility model is most popular as it gives a fast and easily implemented semiclosed form solution for the European options and is relatively economicalfrom the computational point of view. Despite its success and popularity, ithas some shortcomings. The model is unable to fit implied volatility acrossall strikes and maturities particularity for the options with short expiry [12].Also, Shu and Zhang[19] obtained that the Heston model overprices out-of-money (OTM) and short-term options and it underprices in-the-money(ITM) options.In the option valuation literature, the shortcomings of one factor stochasticvolatility models have traditionally been addressed by adding jumps to thestock price process(e.g. stochastic volatility jump diffusion (SVJ) modelsof Bates[4], Scott[18] and Pan[16], etc.). Jumps are added to the stockprice dynamics of a stochastic volatility model which improve its pricingperformance for the short-term options [2]. An alternate approach is theconsideration of multiscale stochastic volatility (MSV) models to address theshortcomings of one factor stochastic volatility models (see [9, 14]). In thesemodels, volatility is driven by several factors varying at different time scales.Alizadeh et al.[1] found the evidence of two factors of volatility with onehighly persistent factor and other quickly mean reverting factor. Extendingthis idea, Fouque et al.[8] proposed a two factor stochastic volatility modelwith one fast mean reverting factor and another slowly varying factor.Christoffersen et al.[6] empirically showed that the two-factor models improveone factor models in the term structure dimension as well as in the moneynessdimension.As both type of models (SVJ or MSV) are the extensions of classical singlefactor stochastic volatility models, this motivated us to study and comparethese two approaches in context of Heston stochastic volatility model. Forthis, we have considered two extensions of Heston stochastic volatility model.One is the stochastic volatility jump diffusion model proposed by Yan andHanson[22] which is an extension of Heston stochastic volatility model byadding jumps to the stock price process with log-uniformly distributed jump2mplitude. The another model is the multiscale stochastic volatility modelproposed by Fouque and Lorig[7], in which a fast mean-reverting factoris additionally considered in the framework of Heston stochastic volatilitymodel. These two models are compared with each other, and also withthe Heston stochastic volatility model using S&P 500 index options data.Firstly, the model parameters are calibrated using non-linear least squareoptimization. Then the models’ fit to the market implied volatility is capturedagainst log moneyness at different time to maturity. The mean relative errorof models’ prices with market data is also calculated. We have obtained thatthe multiscale stochastic volatility model performs better than the other twomodels.The rest of the paper is organised as follows: The underlying modelshas been explained in Section 2. The empirical analysis has been conductedin Section 3, where the calibration of the models’ parameters, models’ fitto market implied volatility and mean relative error of model prices withmarket data has been reported and the results obtained are discussed. Theconclusion has been given in Section 4.
Firstly, the two models to be considered for the empirical analysis has beenexplained.
Yan and Hanson [22] proposed a SVJ model which considers the log-uniformdistribution of the jump amplitudes in the stock price process. The model isexplained below:Let X t be the stock price at time t whose dynamics under the risk-neutralprobability measure P ∗ is dX t = X t (( r − λJ ) dt + p V t dW xt + J ( U ) dN t ) (1)where r is the risk free interest rate and J ( U ) is the Poisson jump-amplitudewith mean J . The variance V t follows the CIR process given by dV t = κ ( θ − V t ) dt + σ p V t dW vt (2)3ith κ as the rate of mean-reversion, θ as the long-run mean value and σ asthe volatility of variance. The condition 2 κθ ≥ σ must be satisfied to ensurethe positivity of the process (2). W xt and W vt are the standard Brownianmotions for the stock price process and the volatility process respectivelywith correlation E [ dW xt .dW vt ] = ρ xv dtU is the amplitude mark process which is assumed to be uniformly distributedwith density ϕ U ( u ) = ( n − m if m ≤ u ≤ n, U = ln( J ( U ) + 1) N t is the standard Poisson jump counting process with jump intensity λ , J ( U ) dN t is the Poisson sum which is given as J ( U ) dN t = dN t X i =1 J ( U i )here U i is the i th jump-amplitude random variable and J , the mean of jump-amplitude J , is given as J = E [ J ( U )] = e n − e m n − m − . Under this model, the pricing formula for the European call option, interms of log stock price s = ln( x ), is given as: C svj = e s P ( s, v, t, K, T ) − Ke − r ( T − t ) P ( s, v, t, K, T ) (3)where v = V t is the variance at time t , T is the maturity time, K is the strikeprice and r is the risk free interest rate. The subscript svj in the price C svj is just to specify the price obtained from SVJ model. The same conventionis also followed for Heston and MSV model.For j = 1 , P j ( s, v, t, K, T ) = 12 + 1 π Z ∞ Re (cid:20) e − iφ ln K f j ( s, v, t, φ, T ) iφ (cid:21) dφ (4)4here the characteristic function f j of P j is f j ( s, v, t, φ, t + τ ) = e A (1 j ) ( τ,φ )+ A (2 j ) ( τ,φ ) v + iφs + β j ( τ ) (5)with τ = T − t and β j ( τ ) = rτ δ j, ; δ j, = 1 for j = 2 and 0 for j = 1. Theother terms are A (1 j ) ( τ, φ ) = rφiτ − ( λJ iφ + λJ δ j, + rδ j, ) τ + λτ (cid:20) e ( iφ + δ j, ) n − e ( iφ + δ j, ) m ( n − m )( iφ + δ j, ) − (cid:21) + A ′ (1 j ) ( τ, φ ) (6) A (2 j ) ( τ, φ ) = b j − ρσφi + d j σ (cid:18) − e d j τ − g j e d j τ (cid:19) (7)and A ′ (1 j ) ( τ, φ ) = κθσ (cid:20) ( b j − ρσφi + d j ) τ − (cid:18) − g j e d j τ − g j (cid:19)(cid:21) (8)with g j = b j − ρσφi + d j b j − ρσφi − d j d j = q ( ρσφi − b j ) − σ (2 α j φi − φ )and α = 12 , α = − , b = κ − ρσ, b = κ (9)The unknown parameters of this model are κ, θ, σ, ρ, v, λ, m and n .After the SVJ model, the MSV model of Fouque and Lorig [7] is givenbelow. Fouque and Lorig [7] extended the Heston stochastic volatility model to aMSV model by considering an additional fast mean-reverting volatility factorin the Heston stochastic volatility model. This model is given below.Under P ∗ , the dynamics of stock price X t is given as dX t = rX t dt + η t X t dW xt (10)5ere η t = √ V t f ( Y t ). Y t and V t are respectively the fast and the slow scalefactors of volatility with their dynamics given as dY t = V t ǫ ( m − Y t ) dt + µ √ r V t ǫ dW yt (11)and dV t = κ ( θ − V t ) dt + σ p V t dW vt (12) W xt , W yt and W vt are the standard Brownian motions for the stock priceprocess and for the fast and the slow factors of volatility respectively with E ( dW xt .dW yt ) = ρ xy dt, E ( dW xt .dW vt ) = ρ xv dt, and E ( dW yt .dW vt ) = ρ yv dt . ρ xy , ρ xv and ρ yv are constants which satisfy ρ xy < , ρ xv < , ρ yv < ρ xy + ρ xv + ρ yv − ρ xy ρ xv ρ yv < Y t follows the OU process with the mean-reversion rate V t /ǫ and volatility of volatility parameter µ √ q V t ǫ . ǫ > Y t is fast mean-reverting towards its long-run mean m . The slow volatility factor V t , as already explained for SVJ model, is thesquare root process. It slowly reverts to its long-run mean θ .Fouque and Lorig[7] used the perturbation technique to obtain theexpression for European call option prices. The asymptotic expansion ofprice in powers of √ ǫ is given as C ǫmsv ( x, y, v, t ) = C + √ ǫC + ǫC + ... (13)They obtained the first order approximation to the price of the Europeancall option as C ǫmsv ( x, v, t ) ≈ C ( x, v, t ) + √ ǫC ( x, v, t )This price approximation is clearly independent of the fast factor ofvolatility and depends only on the slow volatility factor v . The approximatedprice is perturbed around the Heston price C at the effective correlation ρ xv < f > , where < f > is the average of f ( y ) with respect to long-rundistribution of the volatility factor Y t .The first order approximation term C is C = e s Q ( s, v, t, K, T ) − Ke − r ( T − t ) Q ( s, v, t, K, T ) (14)where s = ln x . For j = 1 , Q j ( s, v, t, K, T ) = 12 + 1 π Z ∞ Re (cid:20) e − iφ ln K q j ( s, v, t, φ, T ) iφ (cid:21) dφ (15)6he characteristic function q j of Q j is q j ( s, v, t, φ, t + τ ) = ( κθ ˆ q ( τ, φ ) + v ˆ q ( τ, φ ))( e A ′ (1 j ) ( τ,φ )+ A (2 j ) ( τ,φ ) v + iφs ) (16)here ˆ q ( τ, φ ) = Z τ ˆ q ( z, φ ) dz, ˆ q ( τ, φ ) = Z τ B ( z, φ ) e A (3 j ) ( τ,φ,z ) dz with A (3 j ) ( τ, φ, z ) = ( b j − ρσφi + d j ) 1 − g j d j g j ln (cid:18) g j e d j τ − g j e d j z − (cid:19) B ( τ, φ ) = − ( V A (2 j ) ( τ, φ )(2 α j φi − φ ) + V A j ) ( τ, φ )( φi ) + V (2 α j φ i + φ )+ V A (2 j ) ( τ, φ )( − φ )) (17)All the other terms are already given in Eq.(8) to Eq.(9). The unknownparameters of this model are κ, θ, σ, ρ, v, V , V , V and V .In the next section, the empirical analysis is conducted to compare thesemodels. For the empirical analysis, the data of S&P 500 index options is consideredfrom January 4 , Data sharing is not applicable to this article as no new data were created or analyzedin this study. .1 Calibration of Model Parameters The unknown parameters of Heston stochastic volatility model, SVJ modeland MSV model are calibrated using the data of S&P 500 index options. Let M , M and M denote the parameter sets of unknown parameters of Heston,SVJ and MSV model respectively , such that M = ( ζ , ρ, σ, θ, v ) M = ( ζ , ρ, σ, θ, v, λ, m, n ) M = ( ζ , ρ, σ, θ, v, V , V , V , V ) (18)here all of these unknown parameters are already mentioned in Section 2except ζ , which is obtained from the condition 2 κθ ≥ σ of the CIR process(2) such that ζ = κ − σ θ , ζ ≥ κ = ζ + σ θ is obtained from the calibrated values of ζ , σ and θ .These parameters are calibrated by non-linear least square optimizationusing MATLAB 2012b . The objective function is defined as:∆ h ( M ) = X j X i ( j ) ( C mkt ( T j , K i ( j ) ) − C h ( T j , K i ( j ) , M )) (19)∆ svj ( M ) = X j X i ( j ) ( C mkt ( T j , K i ( j ) ) − C svj ( T j , K i ( j ) , M )) (20)∆ msv ( M ) = X j X i ( j ) ( C mkt ( T j , K i ( j ) ) − C msv ( T j , K i ( j ) , M )) (21)where C mkt ( T j , K i ( j ) ) is the market price of call option with maturity T j . Foreach expiration T j , the available collection of strike prices is K i ( j ) . Similarly,for a particular value of T j and K i ( j ) , C h ( T j , K i ( j ) , M ), C svj ( T j , K i ( j ) , M ) and C msv ( T j , K i ( j ) , M ) are the prices of the European call options with expirationdate T j and exercise price K i ( j ) , calculated from the Heston stochasticvolatility model with parameter set M , SVJ model with the parameter set M and MSV model with the parameter set M respectively.The optimal set of parameters M ∗ , M ∗ and M ∗ is obtained which satisfies∆ h ( M ∗ ) = min M (∆ h ( M ))∆ svj ( M ∗ ) = min M (∆ svj ( M ))∆ msv ( M ∗ ) = min M (∆ msv ( M )) (22)8irstly, the optimal parameter set for the Heston stochastic volatilitymodel, M ∗ , is calibrated. Once the M ∗ is obtained, the initial iteration forSVJ model is taken as ( M ∗ , , − . , .
01) with the lower and upperbounds for the last three components as (1 , − ,
0) and (100 , , M ∗ , . , , ,
0) with the lower and upper bounds for last fourcomponents as ( − . , − . , − . , − .
05) and (0 . , . , . , . log KX ). Themodels fit are compared relative to market implied volatility (MV) data. Itis given in Fig.1 to Fig.3 for time to maturity 30 days, 90 days and 180 daysrespectively.The parameters are calibrated from the whole data but the results aregiven and discussed for the different maturity times, separately. -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Log Moneyness I m p li e d V o l a t ili t y Calibrated Implied Volatilities (time to maturity = 30days)
MV DataHeston FitSVJ FitMSV Fit
Figure 1: Models’ fit to the implied volatilities of S&P 500 index with 30 days tomaturity
Along with this, the mean relative error of the prices obtained fromHeston stochastic volatility model, SVJ model and MSV model, with the9
Log Moneyness I m p li e d V o l a t ili t y Calibrated Implied Volatilities (time to maturity = 90days)
MV DataHeston FitSVJ FitMSV Fit
Figure 2: Models’ fit to the implied volatilities of S&P 500 index with 90 days tomaturity -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Log Moneyness I m p li e d V o l a t ili t y Calibrated Implied Volatilities (time to maturity = 180days)
MV DataHeston FitSVJ FitMSV Fit
Figure 3: Models’ fit to the implied volatilities of S&P 500 index with 180 days tomaturity
For a particular model with price C model ( T j , K i ( j ) , Θ) at different values of T j and K i ( j ) , the mean relative error (MRE) of model price with respect tomarket price, at time to maturity T j is given as M RE ( j ) = 1 N j X i ( j ) | C model ( T j , K i ( j ) , Θ) − C mkt ( T j , K i ( j ) ) | C mkt ( T j , K i ( j ) ) (23)where N j is the different number of call options that has expiry at time T j ,Θ is the optimal parameter set for the given model.The mean relative error of Heston stochastic volatility model, SVJ modeland MSV model is calculated for S&P 500 index data set. The maturity timeis taken from 30 days to 180 days. Corresponding to a particular maturity,the strike prices range from 75% to 125%. The results are given in Table 1.Table 1: The mean relative error of models prices with respect to marketdata. ModelsMaturity Time (T) Heston SVJ MSV30 days 0.0697 0.0499 0.022590 days 0.0874 0.0987 0.0456180 days 0.0284 0.1070 0.0380
Now, we discuss the results obtained in Fig.1 to Fig.3 and in Table1. Fromthe models fit to the implied volatility given in Fig.1 to Fig.3, it is clearlyobservable that the MSV model performs in an improved way in comparisonto Heston stochastic volatility model and the SVJ model. For at-the-money(ATM) and near the money options, all the three models give equivalentresults. The difference is observable for ITM and OTM options.In Fig.1, the maturity time is short, that is 30 days. For such options,the Heston model fit to market implied volatility is not good. This supportsthe empirical findings that the Heston model poorly performs for short term11ptions. The SVJ model performs better than Heston model and MSV modelfor deep ITM options, but as the log moneyness value increases, MSV modeloutperforms both Heston model and SVJ model.In Fig.2, the maturity time is medium, that is 90 days. The impliedvolatility fit of Heston model is improved for the OTM options. For ITMoptions, implied volatility fit of MSV model is better than the impliedvolatility fit of Heston model. The Heston model fit is equivalent to theSVJ model fit to market implied volatility.In Fig.3, at the longer maturity, which is 180 days, all of the three modelsgive almost similar fit for ITM options but for OTM options the Heston modeloutperforms the other two models. The implied volatility fit of MSV model isbetter than the fit of SVJ model to the market implied volatilities. Thus, outof SVJ model and MSV model, the overall fit of MSV model to the marketimplied volatility is better than SVJ model.Additionally, from Table1, the pricing performance of three models iscompared in terms of mean relative error of models prices with the marketoption price data. For the short and medium term options with maturity 30and 90 days respectively, the mean relative error of MSV model is least. Thusthe MSV model performs better than the SVJ model and Heston model inpricing. For maturity time 30 days, SVJ model performs better than Hestonmodel in pricing, but for maturity 90 days, Heston model gives better pricingperformance.For the long term options with maturity 180 days, the MSV modelperforms better than SVJ model and Heston model outperforms the SVJand MSV model.Thus, out of SVJ model and MSV model, the overall pricing performanceof MSV model is better than SVJ model for the data set under consideration.
The two extensions of Heston stochastic volatility model, already proposedin literature, are studied and compared in this paper on the basis of theirfit to the market implied volatility and pricing performance. An empiricalanalysis is conducted on S&P 500 index options data and the results areobtained for all the three models. It has been obtained that for the data setunder consideration, multiscale stochastic volatility performs better than thestochastic volatility jump model. Thus, the inclusion of additional volatility12actor to a stochastic volatility model enhances its fit to the market impliedvolatility and improves its pricing performance in comparison to the additionof jump factors to the underlying stock price process.
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