Decomposition formula for jump diffusion models
aa r X i v : . [ q -f i n . P R ] J un Decomposition formula for jump diffusionmodels
Raúl Merino , Jan Pospíšil ∗ , Tomáš Sobotka , and Josep Vives Facultat de Matemàtiques i Informàtica, Universitat de Barcelona,Gran Via 585, 08007 Barcelona, Spain, NTIS - New Technologies for the Information Society, Faculty of Applied Sciences,University of West Bohemia, Univerzitní 8, 301 00 Plzeň, Czech Republic, VidaCaixa S.A., Investment Risk Management Department,C/Juan Gris, 2-8, 08014 Barcelona, Spain.
Received 2 March 2018Revised 28 September 2018Accepted 1 October 2018
Abstract
In this paper we derive a generic decomposition of the option pricing formula for modelswith finite activity jumps in the underlying asset price process (SVJ models). This is anextension of the well-known result by Alòs (2012) for Heston (1993) SV model. Moreover,explicit approximation formulas for option prices are introduced for a popular class of SVJmodels - models utilizing a variance process postulated by Heston (1993). In particular, weinspect in detail the approximation formula for the Bates (1996) model with log-normal jumpsizes and we provide a numerical comparison with the industry standard - Fourier transformpricing methodology. For this model, we also reformulate the approximation formula in termsof implied volatilities. The main advantages of the introduced pricing approximations aretwofold. Firstly, we are able to significantly improve computation efficiency (while preservingreasonable approximation errors) and secondly, the formula can provide an intuition on thevolatility smile behaviour under a specific SVJ model.
Keywords : option pricing; stochastic volatility models; jump diffusion models; impliedvolatility
MSC classification : 60G51; 91G20; 91G60
JEL classification : G12; C58; C63
The main problem of the Black-Scholes option pricing model is the assumption of constant volatil-ity for the underlying stock price process. In practice, this model is used as a marking model toquote implied volatilities instead of traded option prices. Contrary to the model assumptions, theimplied volatilities observed in the vanilla option markets are not flat - they typically exhibit anon-zero skew and a convex smile-like shape in the moneyness dimension. To correctly capturethe shape of implied volatility surfaces, various stochastic volatility (SV) models were developed.These models assume that not only the spot prices are stochastic, but also their volatility is driven ∗ Corresponding author, [email protected] Preprint of an article published in International Journal of Theoretical and Applied FinanceVol. 21, No. 8 (2018) 1850052, DOI 10.1142/S0219024918500528c (cid:13) y a suitable stochastic process. Another way how to deal with drawbacks of the Black-Scholesmodel is to add a jump term to the stock price process. This results into the jump diffusionsetting, which was originally studied by Merton (1976). In this article, we build an option priceapproximation framework for a popular class of financial models that utilize both of the aforemen-tioned ideas. Hence, the main objects of our study are stochastic volatility jump diffusion (SVJ)models.The first SVJ model is credited to Bates (1996) who incorporated a stochastic variance processpostulated by Heston (1993) alongside Merton (1976) - style jumps. The variance of stock pricesfollows a CIR process (Cox, Ingersoll, and Ross 1985) and the stock prices themselves are assumedto be of a jump diffusion type with log-normal jump sizes. In particular, this model should improvethe market fit for short-term maturity options, while the original Heston (1993) approach wouldoften need unrealistically high volatility of variance parameter to fit reasonably well the short-term smile (Bayer, Friz, and Gatheral 2016; Mrázek, Pospíšil, and Sobotka 2016). An SVJ modelwith a non-constant interest rate was introduced by Scott (1997). Several other authors studiedSVJ models that have a different distribution for jump sizes, e.g. Yan and Hanson (2006) utilizedlog-uniform jump amplitudes.Naturally, one can extend SVJ models by adding jumps into the variance process (e.g. a modelintroduced by Duffie, Pan, and Singleton (2000)). However, based on several empirical studies,these models tend to overfit market prices and despite having more parameters than the orig-inal Bates (1996) model they might not provide a better calibration errors (see e.g. Gatheral(2006)). Another way to improve standard SV models might be to introduce time-dependentmodel parameters. The Heston (1993) model with time-dependent parameters was studied byMikhailov and Nögel (2003) for piece-wise constant parameters, by Elices (2008) for a linear de-pendence and a more general modification was introduced by Benhamou, Gobet, and Miri (2010).These approaches involve several additional parameters and might also suffer from overfitting.Moreover, Bayer, Friz, and Gatheral (2016) mentioned that these models do not fully comply withproperties of observable market data - a general overall shape of the volatility surface typicallydoes not change in time and hence the option prices should be derived using a time-homogeneousstochastic process.The valuation of derivatives under these more complex models is, of course, a more elabo-rate task compared to the standard Black-Scholes model. Many authors have introduced semi-closed form formulas using various transformation techniques of the pricing partial (integro) dif-ferential equations, to name a few: Heston (1993), Bates (1996), Scott (1997), Lewis (2000),Albrecher, Mayer, Schoutens, and Tistaert (2007), Baustian, Mrázek, Pospíšil, and Sobotka (2017)and many others. Although transform pricing methods are typically efficient tools to evaluate non-path dependent derivatives, they do not provide any intuition on the smile behavior. Moreover,calibration routines utilizing these methods lead typically to non-convex optimization problems(see e.g. Mrázek, Pospíšil, and Sobotka (2016)).Other authors considered approximation techniques that were pioneered by Hull and White(1987). In the last years, the Hull and White (1987) pricing formula was reinvented using tech-niques of the Malliavin calculus, because a future average volatility that is used in the for-mula is a non adapted stochastic process. In Alòs (2006), Alòs, León, and Vives (2007) andAlòs, León, Pontier, and Vives (2008), a general jump diffusion model with no prescribed volatilityprocess is analyzed. There have been several extensions thereof, e.g. by assuming Lévy processesin Jafari and Vives (2013), see also the survey in Vives (2016).In Alòs (2012), a new approach of dealing with the Hull and White formula and the Hestonmodel has been proposed. The main idea of this approach is to use an adapted projection forthe future volatility. The formula provides a valuable intuition on the behavior of smiles andterm structures under the Heston model. This is not a purely theoretical result - it can sig-nificantly fasten/improve the calibration process by providing a good initial guess by analyticalcalibration or by specifying a region where calibrated parameters should lie in as it is done inAlòs, de Santiago, and Vives (2015). In Merino and Vives (2015), the idea of Alòs (2012) hasbeen used to find a general decomposition formula for any stochastic volatility process satisfyingbasic integrability conditions. 2n the present paper, we apply the same set of ideas and we extend them to the domain of SVJmodels with finite activity jumps. This should serve not only to find a more efficient way to pricevanilla options compared to transform pricing methods (see Section 5), but as a side product weprovide a similar intuition of the smile behavior for the studied SVJ model.In particular, we start by finding a generic decomposition formula for a vanilla call optionprice and an approximation for both the price and implied volatility under a specific SVJ model.Explicit pricing formulas are provided for one of the most popular SVJ models - Heston (1993) typemodels with compound Poisson process in the stock price evolution. To assess the accuracy andefficiency of the newly derived solution, we perform a numerical comparison for the Bates (1996)model (i.e. log-normal jump sizes) alongside its Fourier transform pricing formula introduced byBaustian, Mrázek, Pospíšil, and Sobotka (2017).The structure of the paper is as follows. In Section 2, we give basic preliminaries and ournotation related to SVJ models. This notation will be used throughout the paper without beingrepeated in particular theorems, unless we find useful to do so in order to guide the reader throughthe results. In Sections 3 and 4, we derive decomposition formulas for SV and SVJ models,respectively, generalizing the decomposition formula obtained by Alòs (2012). Newly obtaineddecomposition is rather versatile since it does not need to specify the underlying volatility process.Particular approximation formulas for several SVJ models are presented in Section 5 alongside thenumerical comparison for the Bates (1996) model. The decomposition result in terms of impliedvolatilities is introduced in Section 6. A discussion of the results is provided in Section 7 andtechnical error estimates are presented in A.
Let S = { S t , t ∈ [0 , T ] } be a strictly positive price process under a market chosen risk neutralprobability that follows the model: dS t = rS t dt + σ t S t (cid:16) ρdW t + p − ρ d ˜ W t (cid:17) + S t − dZ t , (1)where S is the current price, W and ˜ W are independent Brownian motions, r is the interest rate, ρ ∈ ( − , is the correlation between the two Brownian motions and Z t = Z t Z R ( e y −
1) ˜ N ( ds, dy ) where N and ˜ N denote the Poisson measure and the compensated Poisson measure, respectively.We can associate to measure N a compound Poisson process J , independent of W and ˜ W , withintensity λ ≥ and jump amplitudes given by random variables Y i , independent copies of a randomvariable Y with law given by Q . Recall that this compound Poisson process can be written as J t := Z t Z R yN ( ds, dy ) = n t X i =1 Y i , where n t is a λ − Poisson process. Denote by k := E Q ( e Y − . Without any loss of generality, it will be convenient in the following sections, to use as under-lying process, the log-price process X t = log S t , t ∈ [0 , T ] , that satisfies dX t = (cid:18) r − λk − σ t (cid:19) dt + σ t (cid:16) ρdW t + p − ρ d ˜ W t (cid:17) + dJ t . (2)We introduce also the corresponding continuous process, d ˜ X t = (cid:18) r − λk − σ t (cid:19) dt + σ t (cid:16) ρdW t + p − ρ d ˜ W t (cid:17) . (3)3he volatility process σ is a square-integrable process assumed to be adapted to the filtrationgenerated by W and J and its trajectories are assumed to be a.s. square integrable, càdlàg andstrictly positive a.e. Remark 2.1.
Observe that this is a very general stochastic volatility model. We can consider thefollowing particular cases: • If σ is constant and we have finite activity jumps, we have a generic jump-diffusion modelas for example the Merton model. In the particular case of σ = 0 we have an exponentialLévy model. • If we assume no jumps, that is λ = 0 , we have a generic stochastic volatility diffusion model.This is the case treated in Merino and Vives (2015). • If in addition ρ = 0 we have a generalization of different non correlated stochastic volatil-ity diffusion models as Hull and White (1987), Scott (1987), Stein and Stein (1991) orBall and Roma (1994). • If we assume no correlation but presence of jumps we cover for example the Heston-Koumodel (e.g. see Gulisashvili and Vives (2012)), or any uncorrelated model with the additionof finite activity Lévy jumps on the price process. • Finally, if we have no jumps and σ is constant, we have the classical Osborne-Samuelson-Black-Scholes model. The following notation will be used throughout the paper: • We denote by F W , F ˜ W and F N the filtrations generated by the independent processes W , ˜ W and J respectively. Moreover, we define F := F W ∨ F ˜ W ∨ F N . • We will denote by BS ( t, x, y ) the price of a plain vanilla European call option under theclassical Black-Scholes model with constant volatility y , current log stock price x , time tomaturity τ = T − t , strike price K and interest rate r . In this case, BS ( t, x, y ) = e x Φ( d + ) − Ke − rτ Φ( d − ) , where Φ( · ) denotes the cumulative distribution function of the standard normal law and d ± = x − ln K + ( r ± y ) τy √ τ . • In our setting, the call option price is given by V t = e − rτ E t [( e X T − K ) + ] . • Recall that from the Feynman-Kac formula for the model (3), the operator L σ := ∂ t + 12 σ t ∂ x + (cid:18) r − λk − σ t (cid:19) ∂ x − r (4)satisfies L σ BS ( t, ˜ X t , σ t ) = 0 . • We define the operators
Λ := ∂ x , Γ := (cid:0) ∂ x − ∂ x (cid:1) and Γ = Γ ◦ Γ . In particular, for theBlack-Scholes formula we obtain: Γ BS ( t, x, y ) := e x y √ πτ exp (cid:18) − d ( y )2 (cid:19) , ΛΓ BS ( t, x, y ) := e x y √ πτ exp (cid:18) − d ( y )2 (cid:19) (cid:18) − d + ( y ) y √ τ (cid:19) , Γ BS ( t, x, y ) := e x y √ πτ exp (cid:18) − d ( y )2 (cid:19) d ( y ) − yd + ( y ) √ τ − y τ . We define p n ( λT ) as the Poisson probability mass function with intensity λT . I.e. p n takesthe following form: p n ( λT ) := e − λT ( λT ) n n ! . In this section, following the ideas of Alòs (2012), see also Merino and Vives (2015), we extend thedecomposition formula to a generic stochastic volatility model. We recall that the formula is validwithout having to specify the underlying volatility process explicitly, which enables us to obtain avery flexible decomposition formula. The formula proved in Alòs (2012) is the particular case ofthe Heston model.It is well known that if the stochastic volatility process is independent of the price process,then the pricing formula of a plain vanilla European call is given by V t = E t [ BS ( t, S t , ¯ σ t )] where ¯ σ t is the so called average future variance and it is defined by ¯ σ t := 1 T − t Z Tt σ s ds. Naturally, ¯ σ t is called the average future volatility, see Fouque, Papanicolaou, and Sircar (2000),page 51.The idea used in Alòs (2012) consists of using an adapted projection of the average futurevariance v t := E t (¯ σ t ) = 1 T − t Z Tt E t [ σ s ] ds to obtain a decomposition of V t in terms of v t . This idea switches an anticipative problem relatedwith the anticipative process ¯ σ t into a non-anticipative one related to the adapted process v t .We define M t = Z T E t (cid:2) σ s (cid:3) ds, (5)and hence dv t = 1 T − t (cid:2) dM t + (cid:0) v t − σ t (cid:1) dt (cid:3) . Recall that M is a martingale with respect the filtration generated by W and J .The following processes will play an important role in a generic decomposition formula thatwill be introduced in this section. Let R t = 18 E t "Z Tt d [ M, M ] u (6)and U t = ρ E t "Z Tt σ u d [ W, M ] u , (7)where [ · , · ] denotes the quadratic covariation process.Now we prove a generic version of Theorem 2.2 in Alòs (2012) which will be useful for ourproblem. 5 heorem 3.1 (Generic decomposition formula) . Let B t be a continuous semimartingale withrespect to the filtration F t , let A ( t, x, y ) be a C , , ([0 , T ] × [0 , ∞ ) × [0 , ∞ )) function and let v t , M t be defined as above. Then we are able to formulate the expectation of e − rT A ( T, ˜ X T , v T ) B T in thefollowing way: E h e − rT A ( T, ˜ X T , v T ) B T i = A (0 , ˜ X , v ) B + E "Z T e − ru ∂ y A ( u, ˜ X u , v u ) B u T − u (cid:0) v u − σ u (cid:1) du + E "Z T e − ru A ( u, ˜ X u , v u ) dB u + 12 E "Z T e − ru (cid:0) ∂ x − ∂ x (cid:1) A ( u, ˜ X u , v u ) B u (cid:0) σ u − v u (cid:1) du + 12 E "Z T e − ru ∂ y A ( u, ˜ X u , v u ) B u T − u ) d [ M, M ] u + ρ E "Z T e − ru ∂ x,y A ( u, ˜ X u , v u ) B u σ u T − u d [ W, M ] u + p − ρ E "Z T e − ru ∂ x,y A ( u, ˜ X u , v u ) B u σ u T − u d [ ˜ W , M ] u + ρ E "Z T e − ru ∂ x A ( u, ˜ X u , v u ) σ u d [ W, B ] u + p − ρ E "Z T e − ru ∂ x A ( u, ˜ X u , v u ) σ u d [ ˜ W , B ] u + E "Z T e − ru ∂ y A ( u, ˜ X u , v u ) 1 T − u d [ M, B ] u . Proof.
Applying the Itô formula to the process e − rt A ( t, ˜ X t , v t ) B t we obtain: e − rT A ( T, ˜ X T , v T ) B T = A (0 , ˜ X , v ) B − r Z T e − ru A ( u, ˜ X u , v u ) B u du + Z T e − ru ∂ t A ( u, ˜ X u , v u ) B u du + Z T e − ru ∂ x A ( u, ˜ X u , v u ) B u d ˜ X u + Z T e − ru ∂ y A ( u, ˜ X u , v u ) B u dv u + Z T e − ru A ( u, ˜ X u , v u ) dB u + 12 Z T e − ru ∂ x A ( u, ˜ X u , v u ) B u d [ ˜ X, ˜ X ] u + 12 Z T e − ru ∂ y A ( u, ˜ X u , v u ) B u d [ v , v ] u + Z T e − ru ∂ x,y A ( u, ˜ X u , v u ) B u d [ ˜ X, v ] u Z T e − ru ∂ x A ( u, ˜ X u , v u ) d [ ˜ X, B ] u + Z T e − ru ∂ y A ( u, ˜ X u , v u ) d [ v , B ] u . In the next step we apply the Feynman-Kac operator with volatility v t , alongside the definitionof M t . After algebraic operations, we retrieve e − rT A ( T, ˜ X t , v T ) B T = A (0 , ˜ X , v ) B + 12 Z T e − ru ∂ x A ( u, ˜ X u , v u ) B u ( v u − σ u ) du + Z T e − ru ∂ x A ( u, ˜ X u , v u ) B u σ u ( ρdW u + p − ρ d ˜ W u )+ Z T e − ru ∂ y A ( u, ˜ X u , v u ) B u T − u dM u + Z T e − ru ∂ y A ( u, ˜ X u , v u ) B u T − u (cid:0) v u − σ u (cid:1) du + Z T e − ru A ( u, ˜ X u , v u ) dB u + 12 Z T e − ru ∂ x A ( u, ˜ X u , v u ) B u (cid:0) σ u − v u (cid:1) du + 12 Z T e − ru ∂ y A ( u, ˜ X u , v u ) B u T − u ) d [ M, M ] u + ρ Z T e − ru ∂ x,y A ( u, ˜ X u , v u ) B u σ u T − u d [ W, M ] u + p − ρ Z T e − ru ∂ x,y A ( u, ˜ X u , v u ) B u σ u T − u d [ ˜ W , M ] u + ρ Z T e − ru ∂ x A ( u, ˜ X u , v u ) σ u d [ W, B ] u + p − ρ Z T e − ru ∂ x A ( u, ˜ X u , v u ) σ u d [ ˜ W , B ] u + Z T e − ru ∂ y A ( u, ˜ X u , v u ) 1 T − u d [ M, B ] u . After applying expectations on both sides of the equation, we end up with the statement of thetheorem.
In the previous section, we have given a general decomposition formula that can be used forstochastic volatility models with continuous sample paths. In this section, we are going to extendthe previous decomposition to the case of a general jump diffusion model with finite activity jumps.The main idea, like the one used in Merino and Vives (2017), is to adapt the pricing processin a way to be able to apply the decomposition technique effectively. In our case, this wouldtranslate into conditioning on the finite number of jumps n T . If we denote J n = P ni =0 Y i , usingthe integrability of Black-Scholes function, we can obtain the following conditioning formula forEuropean options with payoff at maturity T : BS ( T, X T , v T ) . V = e − rT E [ BS ( T, X T , v T )] e − rT + ∞ X n =0 p n ( λT ) E " BS T, ˜ X T + n T X i =0 Y i , v T ! (cid:12)(cid:12)(cid:12) n T = n = e − rT + ∞ X n =0 p n ( λT ) E h BS (cid:16) T, ˜ X T + J n , v T (cid:17)i = e − rT ∞ X n =0 p n ( λT ) E h E J n h BS ( T, ˜ X T + J n , v T ) ii = e − rT ∞ X n =0 p n ( λT ) E h G n ( T, ˜ X T , v T ) i . where G n ( T, ˜ X T , v T ) := E J n h BS ( T, ˜ X T + J n , v T ) i . We have switched our problem from a jump diffusion model with stochastic volatility to anotherone with no jumps. Combining the generic SV decomposition formula (from Theorem 3.1) andconditioning on the number of jumps we obtain a corner-stone for our approximation.
Corollary 4.1 (SVJ decomposition formula) . Let X t be a log-price process (2) , G n be the previ-ously defined function. Then we can express the call option fair value V using the Poisson massfunction p n and a martingale process M t (defined by (5) ). In particular, V = ∞ X n =0 p n ( λT ) G n (0 , ˜ X , v )+ 18 ∞ X n =0 p n ( λT ) E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u + ρ ∞ X n =0 p n ( λT ) E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] u . Proof.
We apply Theorem 3.1 to A ( t, ˜ X t , v t ) := G n ( t, ˜ X t , v t ) and B t ≡ . Note that ∂ σ BS ( t, x, σ ) = ( T − t )2 (cid:0) ∂ x − ∂ x (cid:1) BS ( t, x, σ ) and ∂ σ BS ( t, x, σ ) = ( T − t ) (cid:0) ∂ x − ∂ x (cid:1) BS ( t, x, σ ) . Then, the corollary follows immediately. Note that in order to apply the Itô formula to function G n we need to use a mollifier argument as it is done in Merino and Vives (2015). Remark 4.2.
For clarity, in the following we will refer to terms of the previous decomposition as V = ∞ X n =0 p n ( λT ) G n (0 , ˜ X , v ) + ∞ X n =0 p n ( λT ) [( I n ) + ( II n )] . To compute the above expression can be cumbersome. The main idea is to find an alternativeformula such that the main terms are easier to be computed while paying the price by havingmore terms in the formula. Fortunately, in many cases these new terms can be neglected asapproximation error. The size of the error depends on the model and whether we are focusing onshort or long time dynamics.The following lemma is proved in Alòs (2012), p. 406; and will help us to derive bounds on theerror terms that appear in the main result of this paper - a computationally suitable decompositionformula for generic finite activity SVJ models. 8 emma 4.3.
Let ≤ t ≤ s ≤ T and G t := F t ∨ F WT . For every n ≥ , there exists C = C ( n ) suchthat (cid:12)(cid:12)(cid:12) E (cid:16) Λ n Γ BS (cid:16) s, ˜ X s , v s (cid:17)(cid:12)(cid:12)(cid:12) G t (cid:17)(cid:12)(cid:12)(cid:12) ≤ C Z Ts E s (cid:0) σ θ (cid:1) dθ ! − ( n +1) . Theorem 4.4 (Computationally suitable SVJ decomposition) . Let X t be a log-price process (2) and G n be the previously defined function. Then we can express the call option fair value V usingthe Poisson probability mass function p n and processes R t , U t defined by (6) and (7) , respectively.In particular, V = ∞ X n =0 p n ( λT ) G n (0 , ˜ X , v )+ ∞ X n =0 p n ( λT )Γ G n (0 , ˜ X , v ) R + ∞ X n =0 p n ( λT )ΛΓ G n (0 , ˜ X , v ) U + ∞ X n =0 p n ( λT )Ω n where Ω n are error terms fully derived in Appendix A.1.Proof. We use Theorem 3.1 iteratively for the following choices of A ( t, X t , v t ) :(I): A ( t, X t , v t ) := Γ G n ( t, ˜ X t , v t ) and B t := R t = 18 E t "Z Tt d [ M, M ] u . (II): A ( t, X t , v t ) := ΛΓ G n ( t, ˜ X t , v t ) and B t := U t = ρ E t "Z Tt σ u d [ W, M ] u . and then the statement follows immediately. See also the terms in Appendix A.1.As we will illustrate in the upcoming sections for Heston-type SVJ models - this formula canbe efficiently evaluated, while the neglected error terms do not significantly limit a practical useof the formula. The main ingredients, to get SVJ approximate pricing formula, are expressionsfor R , U and G n (0 , ˜ X , v ) . Now we provide some insight how the latter term can be expressedunder various jump-diffusion settings. Remark 4.5.
In particular, we have a closed formula for a log-normal jump diffusion model (e.g.Bates (1996) SVJ model): G n (0 , ˜ X , v ) = BS , ˜ X , r v + n σ J T ! where we modified the risk-free rate used in the Black-Scholes formula to r ∗ = r − λ (cid:16) e µ J + σ J − (cid:17) + n µ J + σ J T . very similar formula for the Merton case is deduced by Hanson (2007). More details will followin the next sections. Under general (finite-activity) jump diffusion settings, we will need to solve Z R BS (cid:16) , ˜ X + y, v (cid:17) f J n ( y ) dy where f J n = ( f ∗ nY )( y ) is the convolution of the law of n jumps.Here we provide a list of known results for various popular models. • Kou (2002) double exponential model: f ∗ ( n ) ( u ) = e − η u n X k =1 P n,k η k k − u k − { u ≥ } + e − η u n X k =1 Q n,k η k k − − u ) k − { u< } where P n,k = n − X i = k (cid:18) n − k − i − k (cid:19)(cid:18) ni (cid:19)(cid:18) η η + η (cid:19) i − k (cid:18) η η + η (cid:19) n − i p i q n − i for all ≤ k ≤ n − , and Q n,k = n − X i = k (cid:18) n − k − i − k (cid:19)(cid:18) ni (cid:19)(cid:18) η η + η (cid:19) n − i (cid:18) η η + η (cid:19) i − k p n − i q i for all ≤ k ≤ n − . In addition, P n,n = p n and Q n,n = q n . • Yan and Hanson (2006) model uses log-uniform jump sizes and hence the density is of theform (Killmann and von Collani 2001): f ∗ ( n ) ( u ) = P ˜ n ( n,u ) i =0 ( − i ( ni ) ( u − na − i ( b − a )) n − ( n − b − a ) n if na ≤ u ≤ nb otherwise.where ˜ n ( n, u ) := h u − nab − a i is the largest integer less than u − nab − a . In this section, we apply the previous generic results to derive a pricing formula for SVJ modelswith the Heston variance process. The aim is not to provide pricing solution for all known/studiedmodels, but rather to detail the derivation for a selected model and comment on possible extensionto different models. I.e. we focus on models with dynamics satisfying the following stochasticdifferential equations dX t = (cid:18) r − λk − σ t (cid:19) dt + σ t (cid:16) ρdW t + p − ρ d ˜ W t (cid:17) + dJ t (8) dσ t = κ (cid:0) θ − σ t (cid:1) dt + ν q σ t dW t (9)where σ , κ , θ , ν are positive constants satisfying the Feller condition κθ ≥ ν . The process σ t represents an instantaneous variance of the price at time t , θ is a long run average level of thevariance, κ is a rate at which σ t reverts to θ and, last but not least, ν is a volatility of volatilityparameter. We will distinguish between the two cases: • either jump amplitudes follow a Gaussian process (Bates (1996) model), • or they are driven by other models, e.g. a log-uniform process (Yan and Hanson (2006)model). 10 .1 Approximation of the SVJ models of the Heston type For a standard Heston model, we have the following results, see Alòs, de Santiago, and Vives(2015):
Lemma 5.1.
Assume the standard notation from the previous sections alongside specific defini-tions. Define ϕ ( t ) := R Tt e − κ ( z − t ) dz. We have the following results:1. For s ≥ t we have E t ( σ s ) = θ + ( σ t − θ ) e − κ ( s − t ) = σ t e − κ ( s − t ) + θ (1 − e − κ ( s − t ) ) , so, in particular, this quantity is bounded below by σ t ∧ θ and above by σ t ∨ θ .2. E t (cid:16)R Tt σ s ds (cid:17) = θ ( T − t ) + σ t − θκ (cid:0) − e − κ ( T − t ) (cid:1) . dM t = νσ t (cid:16)R Tt e − κ ( u − t ) du (cid:17) dW t = νκ σ t (cid:0) − e − κ ( T − t ) (cid:1) dW t . U t := ρ E t (cid:16)R Tt σ s d h M, W i s (cid:17) = ρ ν R Tt E t (cid:0) σ s (cid:1) (cid:16)R Ts e − κ ( u − s ) du (cid:17) ds = ρν κ n θκ ( T − t ) − θ + σ t + e − κ ( T − t ) (cid:0) θ − σ t (cid:1) − κ ( T − t ) e − κ ( T − t ) (cid:0) σ t − θ (cid:1)o . R t := E t (cid:16)R Tt d h M, M i s (cid:17) = ν R Tt E t (cid:0) σ s (cid:1) (cid:16)R Ts e − κ ( u − s ) du (cid:17) ds = ν κ ( θ ( T − t ) + (cid:0) σ t − θ (cid:1) κ (cid:16) − e − κ ( T − t ) (cid:17) − θκ (cid:16) − e − κ ( T − t ) (cid:17) − (cid:0) σ t − θ (cid:1) ( T − t ) e − κ ( T − t ) + θ κ (cid:16) − e − κ ( T − t ) (cid:17) + (cid:0) σ t − θ (cid:1) κ (cid:16) e − κ ( T − t ) − e − κ ( T − t ) (cid:17)) . dU t = ρν (cid:16)R Tt e − κ ( z − t ) ϕ ( z ) dz (cid:17) σ t dW t − ρν ϕ ( t ) σ t dt ,7. dR t = ν (cid:16)R Tt e − κ ( z − t ) ϕ ( z ) dz (cid:17) σ t dW t − ν ϕ ( t ) σ t dt . Furthermore, the following lemma is proved in Alòs, de Santiago, and Vives (2015).
Lemma 5.2.
Let all the objects be well defined as above, then for a standard Heston model wehave that(i) R Ts E s ( σ u ) du ≥ θκ (cid:16)R Ts e − κ ( u − s ) du (cid:17) ,(ii) R Ts E s (cid:0) σ u (cid:1) du ≥ σ s (cid:16)R Ts e − κ ( u − s ) du (cid:17) . Remark 5.3.
We can utilize these equalities to get analogue results for Theorem 4.4. The Ω n terms can be founded in Appendix A.2. Now we have all the tools needed to introduce the main practical result - pricing formula11 orollary 5.4 (Heston-type SVJ pricing formula) . Let G n (0 , ˜ X , v ) takes the expression as inRemark 4.5 for a particular jump-type setting, let R = ν κ ( θT + (cid:0) σ − θ (cid:1) κ (cid:0) − e − κT (cid:1) − θκ (cid:0) − e − κT (cid:1) − (cid:0) σ − θ (cid:1) T e − κT + θ κ (cid:0) − e − κT (cid:1) + (cid:0) σ − θ (cid:1) κ (cid:0) e − κT − e − κT (cid:1)) and let U = ρν κ (cid:8) θκT − θ + σ + e − κT (cid:0) θ − σ (cid:1) − κT e − κT (cid:0) σ − θ (cid:1)(cid:9) . Then the European option fair value is expressed as V = ∞ X n =0 p n ( λT ) G n (0 , ˜ X , v )+ ∞ X n =0 p n ( λT )Γ G n (0 , ˜ X , v ) R + ∞ X n =0 p n ( λT )ΛΓ G n (0 , ˜ X , v ) U + ∞ X n =0 p n ( λT )Ω n where Ω n are error terms detailed in Appendix A.2 . The upper bound for any Ω n is given by Ω n ≤ ν ( | ρ | + ν ) (cid:18) r ∧ ( T − t ) (cid:19) Π( κ, θ ) where Π( κ, θ ) is a positive function. Therefore, the total error Ω = ∞ X n =0 p n ( λT )Ω n is bounded by the same constant.Proof. We plug-in the Heston volatility model dynamics into Theorem 4.4. Using the integrabilityof the Black-Scholes function, Fubini Theorem and the fact that the upper bound of Lemma 4.3does not depend on the log spot price, the upper bound can be used for every G n function. UsingLemma 5.1 and Lemma 5.2 we prove the corollary. The whole proof is in Appendix A.3. Remark 5.5 (Approximate fractional SVJ model) . For the model introduced by Pospíšil and Sobotka(2016) one can derive a very similar decomposition as in Corollary 5.4. In fact, only the terms R and U have to be changed while the other terms remain the same. In this section, we compare the newly obtained approximation formula for option prices underBates (1996) model (i.e. log-normal jump sizes alongside Heston model’s instantaneous variance)with the market standard approach for pricing European options under SVJ models - the Fourier-transform based pricing formula. The comparison is performed with two important aspects inmind: 12
Strike price O p t i o np r i ce Call option prices under Bates (1996) model
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Strike price -7 -6 -5 -4 A b s o l u t ee rr o r s Absolute errors in log scale Figure 1: Approximation and reference prices for ρ = − . , ν = 5% and τ = 0 . . • practical precision of the pricing formula when neglecting the total error term Ω , • efficiency of the formula expressed in terms of the computational time needed for particularpricing tasks.In particular, we utilize a semi-closed form solution with one numerical integration as a refer-ence price (Baustian, Mrázek, Pospíšil, and Sobotka 2017) alongside a classical solution derived byBates (1996) . The numerical integration errors according to Baustian, Mrázek, Pospíšil, and Sobotka(2017) should be typically well beyond − , hence we can take the numerically computed pricesas the reference prices for the comparison.Due to the theoretical properties of the total error term Ω , we illustrate the approximationquality for several values of ρ and ν while keeping other parameters fixed .In Figure 1, we inspect a mode of low volatility of the spot variance ν and low absolute valueof the instantaneous correlation ρ between the two Brownian motions. The errors for an optionprice smile that corresponds to τ = 0 . are within − − − range, while slightly better absoluteerrors were obtained at-the-money. Increasing either the absolute value of ρ or volatility ν should,in theory, worsen the computed error measures. However, if only one of the values is increased weare still able to keep the errors below − in most of the cases, see Figure 2.Last but not least, we illustrate the approximation quality for parameters that are not wellsuited for the approximation. This is done by setting ν = 50% , correlation ρ = − . and asmile with respect to τ = 3 . The obtained errors are depicted by Figure 3. Despite the values ofparameters, the shape of the option price curve remains fairly similar to the one obtained by amore precise semi-closed formula.Main advantage of the proposed pricing approximation lies in its computational efficiency –which might be advantageous for many tasks in quantitative finance that need fast evaluationof derivative prices. To inspect the time consumption we set up three pricing tasks. We use With a slight modification mentioned in Gatheral (2006) to not suffer the "Heston trap" issues. The considered model and market parameters take the following values: S = 100 ; r = 0 . ; τ = 0 . ; v = 0 . ; κ = 1 . ; θ = 0 . ; λ = 0 . ; µ J = − . ; σ J = 0 . . Strike price O p t i o np r i ce Call option prices under Bates (1996) model
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Strike price -5 -4 -3 A b s o l u t ee rr o r s Absolute errors in log scale Figure 2: Approximation and reference prices for ρ = − . , ν = 5% and τ = 0 . .a batch of call options with different strikes and times to maturities that involves all typesof options . In the first task, we evaluate prices for the batch with respect to (uniformly)randomly sampled parameter sets. This should encompass a similar number of price evaluationsas a market calibration task with a very good initial guess. Further on, we repeat the sametrials only for and parameter sets, to mimic the number of evaluations for a typicallocal-search calibration and a global-search calibration respectively, for more information aboutcalibration tasks see e.g. Mikhailov and Nögel (2003) and Mrázek, Pospíšil, and Sobotka (2016).The obtained computational times are listed in Table 1. Unlike the formulas with numericalintegration, the proposed approximation has almost linear dependency of computational time onthe number of evaluated prices. Also the results vary based on the randomly generated parametervalues for numerical schemes much more than for the approximation – this is caused by adaptivityof numerical quadratures that were used . The newly proposed approximation is typically × fastercompared to the classical two integral pricing formula and the computational time consumptiondoes not depend on the model- nor on market-parameters. In the above section, we have computed a bound for the error between the exact price and theapproximated pricing formula for the SVJ models of the Heston type. Now, we are going toderive an approximation of the implied volatility surface alongside the corresponding ATM impliedvolatility profiles. These approximations can help us to understand the volatility dynamics ofstudied models in a better way. It includes OTM, ATM, ITM options with short-, mid- and long-term times to maturities For both Baustian, Mrázek, Pospíšil, and Sobotka (2017) and Gatheral (2006) formulas we use an adaptiveGauss-Kronrod(7,15) quadrature. Strike price O p t i o np r i ce Call option prices under Bates (1996) model
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Strike price -3 -2 -1 A b s o l u t ee rr o r s Absolute errors in log scale Figure 3: Approximation and reference prices for ρ = − . , ν = 50% and τ = 3 . The price of an European call option with strike K and maturity T is an observable quantitywhich will be referred to as P obs = P obs ( K, T ) . Recall that the implied volatility is defined as thevalue I ( T, K ) that satisfies BS (0 , S , I ( T, K )) = P obs . Using the results from the previous section, we are going to derive an approximation to theimplied volatility as in Fouque, Papanicolaou, Sircar, and Solna (2003). We use the idea to expandthe implied volatility function I ( T, K ) with respect to two scales. For illustration of the idea, werecall that according to asymptotic sequences { δ k } ∞ k =0 , { ǫ k } ∞ k =0 converging to are considered.Thus, we can write f = f , + δf , + ǫf , + O (( δ + ǫ ) ) , for a particular function f . Let ǫ = ρν and δ = ν , then we expand I ( T, K ) with respect to thesetwo scales as I ( T, K ) = v + ρνI ( T, K ) + ν I ( T, K ) + O (( ρν + ν )) . We will denote by ˆ I ( T, K ) = v + ρνI ( T, K ) + ν I ( T, K ) the approximation to the impliedvolatility and by ˆ V (0 , x, v ) the approximation to the option price which was obtained in Corollary5.4. We know that according to Corollary 5.4: ˆ V (0 , x, v ) = ∞ X n =0 p n ( λT ) BS (0 , x + J n , v )+ ∞ X n =0 p n ( λT )Γ BS (0 , x + J n , v ) R + ∞ X n =0 p n ( λT )ΛΓ BS (0 , x + J n , v ) U . † [sec] Speed-up factorApproximation formula .
97 3 . × .
03 2 . × .
67 2 . × Baustian, Mrázek, Pospíšil, and Sobotka (2017) .
09 1 . × .
28 1 . × .
95 2 . × Gatheral (2006) . - . - . - † The results were obtained on a PC with Intel Core i7-6500U CPU and 8 GB RAM.
To simplify the notation, we define γ n := d ( x, r, σ ) − d ( x + J n , r, σ )2 and D ( x, J n , σ, T ) := E J n (cid:20) e J n + γ n σT (cid:18) − d + ( x + J n , r, σ ) σ √ T (cid:19)(cid:21) ,D ( x, J n , σ, T ) := E J n (cid:20) e J n + γ n σ T (cid:16) d ( x + J n , r, σ ) − σd + ( x + J n , r, σ ) √ T − (cid:17)(cid:21) . Using the fact that ∂ σ BS ( t, x, σ ) = e x e − d ( σ ) / √ T − t √ π , we can re-write the approximated price as ˆ V (0 , x, v ) = ∞ X n =0 p n ( λT ) BS (0 , x + J n , v )+ ∂ σ BS ( v ) ∞ X n =0 p n ( λT ) D ( x, J n , σ, T ) U + ∂ σ BS ( v ) ∞ X n =0 p n ( λT ) D ( x, J n , v , T ) R . where we write BS ( v ) as a shorthand for BS (0 , x, v ) . Consider now the Taylor expansion of BS (0 , x, I ( T, K )) around v : BS (0 , x, I ( T, K )) = BS ( v ) + ∂ σ B ( v )( ρνI ( T, K ) + ν I ( T, K ) + · · · )+ 12 ∂ σ BS ( v )( ρνI ( T, K ) + ν I ( T, K ) + · · · ) + · · · = BS ( v ) + ρν∂ σ BS ( v ) I ( T, K ) + ν ∂ σ BS ( v ) I ( T, K ) + · · · . Noticing that BS ( v ) = ∞ X n =0 p n ( λT ) BS (0 , x + J n , v ) ˆ V (0 , x, v ) = BS (0 , x, I ( T, K )) , we obtain ˆ I ( T, K ) := ρνI ( T, K ) = U ∞ X n =0 p n ( λT ) D ( x, J n , v , T ) , (10) ˆ I ( T, K ) := ν I ( T, K ) = R ∞ X n =0 p n ( λT ) D ( x, J n , v , T ) . (11)Hence, we have the following approximation of implied volatility ˆ I ( T, K ) = v + U ∞ X n =0 p n ( λT ) D ( x, J n , v , T )+ R ∞ X n =0 p n ( λT ) D ( x, J n , v , T ) . In particular, when we look at the ATM curve, we have that ˆ I AT M ( T ) = v + U ∞ X n =0 p n ( λT ) E J n (cid:20) e J n + γ n v T (cid:18) − J n T v (cid:19)(cid:21) − R ∞ X n =0 p n ( λT ) E J n (cid:20) e J n + γ n v T (cid:18)
14 + 1 v T − J n v T (cid:19)(cid:21) . Remark 6.1.
When T converges to , the dynamics of the model is the same as in the Hestonmodel. This is due to the behavior of the Poisson process when T ↓ . The Bates model is a particular example of SVJ model of the Heston type. The fact that jumpsare also log-normal makes the model more tractable. In this section, we will adapt the genericformulas to this particular case. In this model, after each jump, the drift- and volatility-likeparameters will change. We define ˜ v ( n )0 = r v + n σ J T as the new volatility and ˜ r n = r − λ (cid:16) e µ J + σ J − (cid:17) + n µ J + σ J T as the new drift. The parameter n is the number of realized jumps, µ J and σ J are the jump-sizeparameters and λ is the jump intensity. For simplicity, we denote: c n := − λ (cid:16) e µ J + σ J − (cid:17) + n µ J + σ J T .
As a consequence, we have that d ± (cid:16) x, ˜ r n , ˜ v ( n )0 (cid:17) = x − ln K + ˜ r n T ˜ v ( n )0 √ T ± ˜ v ( n )0 √ T . Following the steps done in the generic formula, we can define the variables D B, (cid:16) x, ˜ r n , ˜ v ( n )0 , T (cid:17) = e γ n ˜ v ( n )0 T − d + (cid:16) x, ˜ r n , ˜ v ( n )0 (cid:17) ˜ v ( n )0 √ T , B, (cid:16) x, ˜ r n , ˜ v ( n )0 , T (cid:17) = e γ n ˜ v ( n )0 T d (cid:16) x, ˜ r n , ˜ v ( n )0 (cid:17) − ˜ v ( n )0 d + (cid:16) x, ˜ r n , ˜ v ( n )0 (cid:17) √ T − (cid:16) ˜ v ( n )0 (cid:17) T . It follows that ˆ I B, ( T, K ) = ρνI B, ( T, K ) = U ∞ X n =0 p n ( λT ) D B, (cid:16) x, ˜ r n , ˜ v ( n )0 , T (cid:17) , (12) ˆ I B, ( T, K ) = ν I B, ( T, K ) = R ∞ X n =0 p n ( λT ) D B, (cid:16) x, ˜ r n , ˜ v ( n )0 , T (cid:17) . (13)The approximation of the implied volatility surface has the following shape ˆ I B ( T, K ) = v + U ∞ X n =0 p n ( λT ) e γ n ˜ v ( n )0 T − d + (cid:16) x, ˜ r n , ˜ v ( n )0 (cid:17) ˜ v ( n )0 √ T + R ∞ X n =0 p n ( λT ) e γ n ˜ v ( n )0 T d (cid:16) x, ˜ r n , ˜ v ( n )0 (cid:17) − ˜ v ( n )0 d + (cid:16) x, ˜ r n , ˜ v ( n )0 (cid:17) √ T − (cid:16) ˜ v ( n )0 (cid:17) T . In particular, the ATM implied volatility curve under the studied model takes the form: ˆ I AT MB ( T ) = v + U ∞ X n =0 p n ( λT ) e γ ATMBatesn ˜ v ( n )0 T − c n (cid:16) ˜ v ( n )0 (cid:17) − R ∞ X n =0 p n ( λT ) e γ ATMBatesn ˜ v ( n )0 T
14 + 1 (cid:16) ˜ v ( n )0 (cid:17) T − c n (cid:16) ˜ v ( n )0 (cid:17) where γ AT MBatesn = − c n T + c n T (cid:16) ˜ v ( n )0 (cid:17) . In the previous section we have compared the approximation and semi-closed form formulas foroption prices under Bates (1996) model. For this model, we also illustrate the approximationquality in terms of implied volatilities.Because there is no exact closed formula for implied volatilities under the studied model, we takeas a reference price the one obtained by means of the complex Fourier transform (Baustian, Mrázek, Pospíšil, and Sobotka2017). Once we have computed the prices we use a numerical inversion to obtain the desired im-plied volatilities.As previously, we start by comparing implied volatilities for well-suited parameter sets. Theillustration in Figure 4 is obtained by setting ρ = − . , ν = 5% and other parameters as in Section5.2. Typically, for a well-suited parameter set, the absolute approximation errors stay within therange − − − .Even for not entirely well-suited parameters we are able to obtain reasonable errors especiallyfor ATM options, see Figures 5 and 6. In the mode of high volatility ν of the variance processand high absolute value of the instantaneous correlation ρ , the curvature of the smile is not fullycaptured. However, the errors are typically well below − even in this adverse setting.18 Strike price I m p li e d v o l a t ili t y Implied volatilities under Bates (1996) model
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Strike price -8 -7 -6 -5 A b s o l u t ee rr o r s Absolute errors in log scale Figure 4: Approximation and reference implied volatilities for ρ = − . , ν = 5% and τ = 0 . .
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Strike price -6 -5 -4 -3 A b s o l u t ee rr o r s Absolute errors in log scale Figure 5: Approximation and reference implied volatilities for ρ = − . , ν = 5% and τ = 0 . .19 Strike price I m p li e d v o l a t ili t y Implied volatilities under Bates (1996) model
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Strike price -5 -4 -3 -2 A b s o l u t ee rr o r s Absolute errors in log scale Figure 6: Approximation and reference implied volatilities for ρ = − . , ν = 50% and τ = 3 . The aim of the paper was to derive a generic decomposition formula for SVJ option pricing modelswith finite activity jumps. In Section 4 we derived this decomposition by extending the resultsobtained by Alòs (2012) for Heston (1993) SV model. Newly obtained decomposition is ratherversatile since it does not need to specify the underlying volatility process and only commonintegrability and specific sample path properties are required.Particular approximation formulas for several SVJ models were presented in Section 5 togetherwith the numerical comparison for the Bates (1996) model for which we showed that the newlyproposed approximation is typically three times faster compared to the classical two integral semi-closed pricing formula. Moreover, its computational time does not depend on the model parametersnor on market data. The biggest advantage of the proposed pricing approximation therefore liesin its computational efficiency, which is advantageous for many tasks in quantitative finance suchas calibration to real market data that can lead to an extensive number of formula evaluations forSVJ models. On the other hand, general decomposition formula allowed us to understand the keyterms contributing to the option fair value under specific models and hence this theoretical resulthas also its practical impact.In Section 6, we have obtained an approximated volatility surface under SVJ models and weprovided a boundary case simplification for ATM options. In particular, we have studied theapproximation in the Bates (1996) model case. A numerical comparison of this approximation isalso presented.Although the generic approach covers various interesting SVJ models, there are other modelsthat do not fit into the general structure described in Section 2. For these models, such asBarndorff-Nielsen and Shephard (2001) model or infinite activity jumps models, we still might beable to derive a similar decomposition, that was beyond the scope of the present paper. Newlyobtained results therefore give suggestions on how to derive approximation formulas for othermodels. 20 cknowledgements
This work was partially supported by the GACR Grant GA18-16680S Rough models of fractionalstochastic volatility. Computational resources were provided by the CESNET LM2015042 and theCERIT Scientific Cloud LM2015085, provided under the programme "Projects of Large Research,Development, and Innovations Infrastructures".The work of Josep Vives is partially supported by Spanish grant MEC MTM 2016-76420-P.
A Appendices
In the following appendices we obtain the error terms of the decomposition in Theorem 4.4 (Ap-pendix A.1), the same formulas for the SVJ model of the Heston type (Appendix A.2) and upperbounds for those terms using Corollary 5.4 (Appendix A.3).
A.1 Decomposition formulas in the general model
In this section, we obtain the error terms for a general model.
A.1.1 Decomposition of the term ( I n ) The term I can be decomposed by E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − Γ G n (0 , ˜ X , v ) R = 18 E "Z T e − ru Γ G n ( u, ˜ X u , v u ) R u d [ M, M ] u + ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) R u σ u d [ W, M ] u + ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, R ] u + 12 E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, R ] u . A.1.2 Decomposition of the term ( II n ) The term II can be decomposed by ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] u − ΛΓ G n (0 , ˜ X , v ) U = 18 E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) U u d [ M, M ] u + ρ E "Z T e − ru Λ Γ G n ( u, ˜ X u , v u ) U u σ u d [ W, M ] u + ρ E "Z T e − ru Λ Γ G n ( u, ˜ X u , v u ) σ u d [ W, U ] u + 12 E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) d [ M, U ] u . .2 Decomposition formulas in the general model for the SVJ modelsof the Heston type In this section, we obtain the error terms for the SVJ models of the Heston type.
A.2.1 Decomposition of the term ( I n ) in the SVJ models of the Heston type The term I can be decomposed by E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − ν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds ! = ν E "Z T e − ru Γ G n ( u, ˜ X u , v u ) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + ρν E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + ρν E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! σ u du + ν E "Z T e − ru Γ G n ( u, ˜ X u , v u ) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! ϕ ( u ) σ u du . A.2.2 Decomposition of the term ( II n ) in the SVJ models of the Heston type The term II can be decomposed by ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] u − ρν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds ! = ρν E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + ρ ν E "Z T e − ru Λ Γ G n ( u, ˜ X u , v u ) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + ρ ν E "Z T e − ru Λ Γ G n ( u, ˜ X u , v u ) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! σ u du + ρν E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! σ u ϕ ( u ) du . A.3 Upper-Bound of decomposition formulas in the SVJ models of theHeston type
In this section, we obtain the upper-bounds for the SVJ models of the Heston type.
A.3.1 Upper-Bound of the term ( I n ) in the SVJ models of the Heston type We can re-write the decomposition formula as E "Z T e − r ( u − t ) Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − ν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds ! ν E "Z T e − ru (cid:0) ∂ x − ∂ x + 3 ∂ x − ∂ x (cid:1) Γ G n ( u, ˜ X u , v u ) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + ρν E "Z T e − ru (cid:0) ∂ x − ∂ x + ∂ x (cid:1) Γ G n ( u, ˜ X u , v u ) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + ρν E "Z T e − ru (cid:0) ∂ x − ∂ x (cid:1) Γ G n ( u, ˜ X u , v u ) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! σ u du + ν E "Z T e − ru (cid:0) ∂ x − ∂ x + ∂ x (cid:1) Γ G n ( u, ˜ X u , v u ) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! ϕ ( u ) σ u du . Applying Lemma 4.3 and defining a u := v u √ T − u , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − ν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ν E "Z T e − ru (cid:18) a u + 3 a u + 3 a u + 1 a u (cid:19) v u ( T − u ) ϕ ( u ) σ u du + C | ρ | ν E "Z T e − ru (cid:18) a u + 2 a u + 1 a u (cid:19) v u ( T − u ) ϕ ( u ) σ u du + C | ρ | ν E "Z T e − ru (cid:18) a u + 1 a u (cid:19) σ u ϕ ( u ) du + C ν E "Z T e − ru (cid:18) a u + 2 a u + 1 a u (cid:19) ϕ ( u ) σ u du . Now, using Lemma 5.2 (ii), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − ν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ν E "Z T e − ru (cid:18) a u + 3 a u + 3 a u + 1 a u (cid:19) v u ( T − u ) ϕ ( u ) du + C | ρ | ν E "Z T e − ru (cid:18) a u + 2 a u + 1 a u (cid:19) v u ( T − u ) ϕ ( u ) du + C | ρ | ν E "Z T e − ru (cid:18) a u + 1 a u (cid:19) v u ( T − u ) ϕ ( u ) du + C ν E "Z T e − ru (cid:18) a u + 2 a u + 1 a u (cid:19) ϕ ( u ) v u ( T − u ) du . Finally, applying Lemma 5.2 (i), we find that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − ν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ν E "Z T e − ru √ θκ √ θκ + 6 θκ + 3 √ κ √ θκ + 1 κ ! du + C | ρ | ν E "Z T e − ru θκ + 2 √ κ √ θκ + 1 κ ! du C | ρ | ν E "Z T e − ru θκ + √ κ √ θκ ! du + C ν E "Z T e − ru √ θκ √ θκ + 4 θκ + √ κ √ θκ ! du . Then we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − ν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ν √ θκ √ θκ + 6 θκ + 3 √ κ √ θκ + 1 κ ! Z T e − ru du ! + C | ρ | ν θκ + 2 √ κ √ θκ + 1 κ ! Z T e − ru du ! + C | ρ | ν θκ + √ κ √ θκ ! Z T e − ru du ! + C ν √ θκ √ θκ + 4 θκ + √ κ √ θκ ! Z T e − ru du ! . Using the fact that R Tt e − ru ds ≤ r ∧ T , we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − ν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ν ( | ρ | + ν ) (cid:18) r ∧ T (cid:19) Π ( κ, θ ) where Π is a positive function. A.3.2 Upper-Bound of the term ( II n ) in the SVJ models of the Heston type We can re-write the decomposition formula as ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] u − ρν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds ! = ρν E "Z T e − ru (cid:0) ∂ x − ∂ x + ∂ x (cid:1) Γ G n ( u, ˜ X u , v u ) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + ρ ν E "Z T e − ru (cid:0) ∂ x − ∂ x (cid:1) Γ G n ( u, ˜ X u , v u ) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + ρ ν E "Z T e − ru ∂ x Γ G n ( u, ˜ X u , v u ) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! σ u du + ρν E "Z T e − ru (cid:0) ∂ x − ∂ x (cid:1) Γ G n ( u, ˜ X u , v u ) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! σ u ϕ ( u ) du Applying Lemma 4.3 and defining a u := v u √ T − u , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] u − ρν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C | ρ | ν E "Z T e − ru (cid:18) a u + 2 a u + 1 a u (cid:19) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + C ρ ν E "Z T e − ru (cid:18) a u + 1 a u (cid:19) Z Tu E u (cid:0) σ s (cid:1) ϕ ( s ) ds ! σ u ϕ ( u ) du + C ρ ν E "Z T e − ru a u Z Tu e − κ ( z − u ) ϕ ( z ) dz ! σ u du + C | ρ | ν E "Z T e − ru (cid:18) a u + 1 a u (cid:19) Z Tu e − κ ( z − u ) ϕ ( z ) dz ! σ u ϕ ( u ) du . Using Lemma 5.2 (ii), then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] u − ρν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ρ | ν E "Z T e − ru (cid:18) a u + 2 a u + 1 a u (cid:19) v u ( T − u ) ϕ ( u ) du + C ρ ν E "Z T e − ru (cid:18) a u + 1 a u (cid:19) v u ( T − u ) ϕ ( u ) du + C ρ ν E "Z T e − ru a u ϕ ( u ) v u ( T − u ) du + C | ρ | ν E "Z T e − ru (cid:18) a u + 1 a u (cid:19) ϕ ( u ) v u ( T − u ) du . Finally, applying Lemma 5.2 (i), we find that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] cu − ρν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ρ | ν E "Z T e − ru θκ + 2 √ κ √ θκ + 1 κ ! du + C ρ ν E "Z T e − ru √ √ θκ + 1 κ ! du + C ρ ν E "Z T e − ru √ √ θκ du + C | ρ | ν E "Z T e − ru θκ + √ κ √ θκ ! du . Then we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] cu − ρν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C | ρ | ν θκ + 2 √ κ √ θκ + 1 κ ! Z T e − ru du ! + C ρ ν √ √ θκ + 1 κ ! Z T e − ru du ! C ρ ν √ √ θκ Z T e − ru du ! + C | ρ | ν θκ + √ κ √ θκ ! Z T e − ru du ! . Using the fact that R Tt e − ru ds ≤ r ∧ T , we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] cu − ρν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ρ | ν ( | ρ | + ν ) (cid:18) r ∧ T (cid:19) Π ( κ, θ ) where Π is a positive function. A.3.3 Upper-Bound for the terms ( I n ) and ( II n ) in the SVJ models of the Hestontype We have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E "Z T e − ru Γ G n ( u, ˜ X u , v u ) d [ M, M ] u − ν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E "Z T e − ru ΛΓ G n ( u, ˜ X u , v u ) σ u d [ W, M ] cu − ρν G n (0 , ˜ X , v ) Z T E (cid:0) σ s (cid:1) ϕ ( s ) ds !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ν ( | ρ | + ν ) (cid:18) r ∧ T (cid:19) Π ( κ, θ ) + | ρ | ν ( | ρ | + ν ) (cid:18) r ∧ T (cid:19) Π ( κ, θ ) ≤ ν ( | ρ | + ν ) (cid:18) r ∧ T (cid:19) Π( κ, θ ) . where function Π is the maximum of functions Π and Π .26 eferences Albrecher, H., Mayer, P., Schoutens, W., and Tistaert, J. (2007).
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