A decomposition formula for fractional Heston jump diffusion models
aa r X i v : . [ q -f i n . P R ] J u l A DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMPDIFFUSION MODELS
MARC LAGUNAS-MERINO AND SALVADOR ORTIZ-LATORRE
Abstract.
We present an option pricing formula for European options in a stochasticvolatility model. In particular, the volatility process is defined using a fractional integralof a diffusion process and both the stock price and the volatility processes have jumps inorder to capture the market effect known as leverage effect. We show how to computea martingale representation for the volatility process. Finally, using Itô calculus forprocesses with discontinuous trajectories, we develop a first order approximation formulafor option prices. There are two main advantages in the usage of such approximatingformulas to traditional pricing methods. First, to improve computational efficiency, andsecond, to have a deeper understanding of the option price changes in terms of changesin the model parameters. Introduction
Classical stochastic volatility models, where the volatility also follows a diffusion process,have been proven to be capable of reproducing some important features of the impliedvolatility together with its variation with respect to the strike price. These features areusually described through the smile or skew, see [13]. One of the main downsides fromthese class of models is their inability to explain what is known as the term structure ofthe skew, i.e. the dependence of the skew to the time to maturity.For instance, it can be observed in [14], how a decrease of the smile amplitude when timeto maturity increases, turns out to be much slower than it should be according to standardstochastic volatility models. On the other hand, the observed short-time implied volatilityskew slope tends to infinity as time to maturity tends to zero, whereas this limit is aconstant under classical stochastic volatility models.On one hand, the long-memory features for the volatility process can be achieved by theintroduction of fractional noises with a Hurst parameter
H > / in the volatility process,as introduced by Comte and Renault in [7] and deeply studied in [4]. This allows to endowthe volatility with high persistence in the long-run, showing the steepness of long-termvolatility smiles without over increasing the short-run persistence. On the other hand,it was proved in [3] that these models fail to describe the short-time behavior of impliedvolatility. In order to overcome this limitation, we present one of the possible approaches,consisting in adding a jump term to both the stock price and the volatility processeswith a correlation factor between both jumps in order to reproduce the so-called leverageeffect . This results into a combination between the fractional setup of Alòs and Yang [4]and the jump diffusion framework, first studied by Merton in [21]. In the present work,we build an option price approximation methodology for a combination of the fractional Date : July 29, 2020.We are grateful for the financial support from Department of Mathematics, University of Oslo. This is a well known effect observed in most markets that shows how most measures of volatility of anasset are negatively correlated with the returns of that asset. and the jump diffusion models previously mentioned, which capture the short-time andlong-time behavior of implied volatility. Hence, we will study what we will call fractionalstochastic volatility jump diffusion (FSVJJ) models with both jumps in the price andvolatility processes.The first stochastic volatility models with jumps were introduced by Bates in [6] and wereachieved by incorporating jumps to stochastic variance processes, previously introducedby Heston (1993) in [16]. In our case of study, the variance of stock prices follows acombination of a CIR process [9], a fractional integral of the stochastic term in the CIRprocess and a jump term driven by a Lévy-type process. Adding the jump frameworkto the model should improve the market fit for short-term maturity options, overcomingthe original problem of the Heston stochastic volatility model. The last would requireunrealistically high values for the vol-of-vol parameter in order to obtain a reasonable fit ofshort-term smiles. An alternative approach to model short-term smiles is the use of roughfractional volatility models, see for instance [14, 11].Pricing derivatives under stochastic volatility jump diffusion models involves, naturally,an extra degree of complexity compared to the standard Black-Scholes pricing framework.This has motivated the development of approximating formulas in the literature such as[17, 1, 2, 15, 20]. These formulas provide good intuition on the behavior of the smiles anda better understanding of the effects of changes in the model parameters onto the price of aderivative. Despite not being closed pricing formulae, they bring clarity to the practitionerto understand the effects of model parameters in the option price. As well as, speed up thecalibration process as proved in [15]. We will use this idea to find a general decompositionformula for a fractional Heston model with jumps in the price and the volatility processesunder basic integrability conditions. In a recent paper [19], the authors find decompositionformulas in the setup of rough volatility models.This paper is organized as follows. First, we introduce in Section 2 a detailed description ofthe model that will be used throughout the article. Section 3 is devoted to introduce somepreliminary concepts and notation needed in later parts of our study. Then, in Section4 we present an exact expansion formula for option pricing in terms of the Black-Scholesformula adjusted by extra terms that depend on the future expected volatility, the jumpsand correlation parameters. We provide a martingale representation of future expectedvolatility in Section 5, by means of the Clark-Ocone-Haussmann formula and end thesection providing its dynamics. We conclude by developing a first order approximatingformula of a call option under our fractional Heston jump diffusion model. Finally, weprovide an appendix in Section 7 with some additional technical results.2.
A Fractional Heston Model with Jumps in the Stock Log-Prices andVolatility (FSVJJ) Model
Let
T > be fixed time horizon and let (Ω , F , Q ) be a complete probability space. Assumethat in (Ω , F , Q ) there are defined W = { W } t ∈ [0 ,T ] and ˜ W = n ˜ W o t ∈ [0 ,T ] , two independentstandard Brownian motions. Moreover, assume that in (Ω , F , Q ) there is defined a Lévysubordinator J = { J } t ∈ [0 ,T ] which is independent of the Brownian motions W and ˜ W .We define the filtration F = {F t } t ∈ [0 ,T ] to be the minimal augmented filtration generated Subordinators are Lévy processes with increasing paths. Alternatively, they are Lévy processes withfinite variation paths and postive jumps. These type of processes are often used in Lévy-based financialmodels. See [8].
DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 3 by W, ˜ W and J . We assume that J is a Lévy subordinator with generating triplet (0 , ℓ, γ ) ,that is, E (cid:2) e iyJ t (cid:3) = exp (cid:18) t (cid:18) iγy + Z ∞ (cid:0) e iyz − − z {
Finally, we will consider a fractional volatility model with only positive jumps built ona combination of the processes
Y, Z and a fractional integral of Z . Let us denote thisfractional volatility process with jumps by σ = (cid:8) σ t (cid:9) t ∈ [0 ,T ] , where(2.3) σ t , U t + c νZ t + c νI H − Z t + c ηJ t , t ∈ [0 , T ] , where H ∈ (cid:0) , (cid:1) , c , c , c , η ≥ and I H − is the left-sided fractional Riemann-Liouvilleintegral of the path Z · ( ω ) of order H − on [0 , T ] . Recall that, given f ∈ L ([0 , T ]) , theleft-sided fractional Riemann-Liouville integral of f of order α ∈ R + on [0 , T ] is defined as(2.4) (cid:0) I α f (cid:1) ( t ) ,
1Γ ( α ) Z t f ( u ) ( t − u ) α − du. On the other hand let ρ ∈ [ − , , ρ ≤ and consider a Lévy model for the dynamics ofthe stock log-price in the time interval [0 , T ] given by the following equation X t = x + rt − Z t σ s ds + Z t σ s (cid:18) ρ dW s + q − ρ dB s (cid:19) + ρ ηJ t , where r is the risk-free rate. For the process e − rt e X t to be a local martingale, we mustassume, see Corollary 5.2.2 in [5], that ρ ηγ + Z (cid:0) e ρ ηz − − ρ ηz { Following similar ideas to the ones found in [4], we will extend the decomposition formulato a fractional Heston model with infinite activity jumps in both prices and volatility. Itis well known that V t , the value at time t of a derivative whose payoff is h ( X T ) , is givenby the risk neutral pricing formula V t ( h ) = e − r ( T − t ) E [ h ( X T ) | F t ] . We now proceed to introduce some definitions and notations which will be used throughoutthe paper: • We will denote E t [ · ] , E [ · | F t ] . DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 5 • Let BS ( t, x, σ ) denote the price of a plain vanilla European call option under theclassical Black-Scholes pricing formula with constant volatility σ , stock log-price x ,strike price K , time to maturity T − t , and constant interest rate r . In this case, BS ( t, x, σ ) = e x Φ ( d + ) − Ke − r ( T − t ) Φ ( d − ) , where Φ denotes the standard normal cumulative probability function and d ± isdefined as d ± , x − ln K − r ( T − t ) σ √ T − t ± σ √ T − t. • In our setting, the price of a call option at time t is given by V t = e − r ( T − t ) E t h(cid:0) e X T − K (cid:1) + i . • We recall from the Feynman-Kac formula for the continuous version of the model (2 . , the operator L σ , ∂ t + 12 σ ∂ xx + (cid:18) r − σ (cid:19) ∂ x − r. Note that L σ BS ( · , · , σ ) = 0 by construction. • We will use an adapted projection of the future average variance defined by(3.1) v t , T − t Z Tt E t (cid:2) σ s (cid:3) ds, to obtain a decomposition of V t in terms of v t . This idea, used in [2], switches ananticipative problem into a non-anticipative one, related to the adapted process v t . • We define M t , R T E t (cid:2) σ s (cid:3) ds . Notice then that the projected future averagevariance can be written as v t = T − t (cid:16) M t − R t σ s ds (cid:17) . Recall that, by definition, M is a martingale with respect to the filtration generated by W and J , is also F ˜ W -independent and its dynamics is given by dM t = νA ( T, t ) q ¯ σ t dW t + c ηdJ t , as it is later proved in Proposition 9. It will also be useful to introduce M c , thecontinuous part of the process M , with dynamics given by dM ct = νA ( T, t ) q ¯ σ t dW t . • Let { X t } t ∈ [0 ,T ] and { Y t } t ∈ [0 ,T ] be two Itô-Lévy processes given by the followingdynamics dX t = α x ( t ) dt + β x ( t ) dW t + Z ∞ γ x ( t, z ) ˜ N ( dt, dz ) ,dY t = α y ( t ) dt + β y ( t ) dW t + Z ∞ γ y ( t, z ) ˜ N ( dt, dz ) . Given a function F ∈ C , , ([0 , T ] × R × R ) , we define ∆ x F ( t, X t − , Y t − ) , F ( t, X t − + γ x ( t, z ) , Y t − ) − F ( t, X t − , Y t − ) , ∆ y F ( t, X t − , Y t − ) , F ( t, X t − , Y t − + γ y ( t, z )) − F ( t, X t − , Y t − ) , ∆ x F ( t, X t − , Y t − ) , ∆ x F ( t, X t − , Y t − ) − γ x ( t, z ) ( ∂ x F ) ( t, X t − , Y t − ) , ∆ y F ( t, X t − , Y t − ) , ∆ y F ( t, X t − , Y t − ) − γ y ( t, z ) ( ∂ y F ) ( t, X t − , Y t − ) . DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 6 • The introduction of the following differential operators is very convenient for no-tational purposes, and both expressions will be used indistinctly throughout thearticle. Λ , ∂ x , Γ , (cid:0) ∂ x − ∂ x (cid:1) ; Γ = Γ ◦ Γ = (cid:0) ∂ x − ∂ x + ∂ x (cid:1) . • Given two continuous semimartingales X and Y , we define the following processes L [ X, Y ] t , E t (cid:20)Z Tt σ u d [ X, Y ] u (cid:21) ,D [ X, Y ] t , E t (cid:20)Z Tt d [ X, Y ] u (cid:21) , for t ∈ [0 , T ] . Since the derivatives of BS ( t, X t , v t ) are not bounded, we will make use of an approximat-ing argument. Consider the approximation v δt of v t for a fixed δ > , given by(3.2) v δt , s T − t (cid:18) δ + Z Tt E t [ σ s ] ds (cid:19) = s T − t (cid:18) δ + M t − Z t σ s ds (cid:19) . The following proposition shows how the dynamics of this process is obtained. Proposition 1. For fixed δ > , let v δ be the approximation of the adapted projection ofthe average future volatility process, defined in (3 . . Then, the dynamics of v δ is given by dv δt = ( v δt ) − σ t v δt ( T − t ) dt + dM ct v δt ( T − t ) − d [ M c , M c ] t (cid:0) v δt (cid:1) ( T − t ) (3.3) + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ℓ ( dz ) dt + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ˜ N ( dt, dz ) , where g δ ( t, m, y ) = q T − t ( δ + m − y ) . Proof. Define the realized variance as Y t , R t σ s ds for every t ∈ [0 , T ] . Note that v δt = g δ ( t, M t , Y t ) , where the dynamics of Y and M are given by dY t = σ t dt,dM t = νA ( T, t ) q σ t dW t + Z ∞ c ηz ˜ N ( dt, dz ) . Now following [22], we can apply the multidimensional Itô formula for Itô-Lévy processesto obtain dv δt = dg δ ( t, M t , Y t )= ∂g δ ∂t ( t, M t , Y t ) dt + ∂g δ ∂m ( t, M t , Y t ) dM ct + ∂g δ ∂y ( t, M t , Y t ) dY t + 12 ∂ g δ ∂m ( t, M t , Y t ) d [ M c , M c ] t + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ℓ ( dz ) dt + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ˜ N ( dt, dz )= ( v δt ) v δt ( T − t ) dt + dM ct v δt ( T − t ) − σ t v δt ( T − t ) dt − d [ M c , M c ] t (cid:0) v δt (cid:1) ( T − t ) DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 7 + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ℓ ( dz ) dt + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ˜ N ( dt, dz )= ( v δt ) − σ t v δt ( T − t ) dt + dM ct v δt ( T − t ) − d [ M c , M c ] t (cid:0) v δt (cid:1) ( T − t ) + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ℓ ( dz ) dt + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ˜ N ( dt, dz ) , where γ m ( t, z ) , c ηz in the expression for ∆ m g δ ( t, M t − , Y t − ) . (cid:3) Remark . It will be useful in further results to write down the dynamics for the continuousparts of X t and v δt , respectively as dX ct = (cid:18) r − ζ ( ρ , η ) − σ t (cid:19) dt + σ t (cid:18) ρ dW t + q − ρ d ˜ W t (cid:19) , (3.4) d (cid:16) v δt (cid:17) c = (cid:20) ( v δt ) − σ t v δt ( T − t ) + Z ∞ ∆ m g δ ( t, M t − , Y t − ) ℓ ( dz ) (cid:21) dt (3.5) + dM ct v δt ( T − t ) − d [ M c , M c ] t (cid:0) v δt (cid:1) ( T − t ) . General expansion formulas In this section we present the main result of the paper. We provide an exact expansionformula for option pricing in terms of the Black-Scholes formula adjusted by extra termsdepending on the future expected variance, the Lévy measure of the jumps and correlationparameters. Theorem 3. Let B = { B t , t ∈ [0 , T ] } be a continuous semimartingale with respect to thefiltration F W ∨ F J , let A ( t, x, y ) be a C , , ([0 , T ] × R × R ) function such that L y A = (cid:18) ∂ t + 12 y ∂ xx + (cid:18) r − y (cid:19) ∂ x − r (cid:19) A = 0 , and let v t and M t be defined as in the previous section. Then, for every t ∈ [0 , T ] , theexpectation of e − rT A ( T, X T , v T ) B T can be written as follows: e − r ( T − t ) E t [ A ( T, X T , v T ) B T ]= A ( t, X t , v t ) B t − ζ ( ρ , η ) E t (cid:20)Z Tt e − rs ∂ x A ( s, X s , v s ) B s ds (cid:21) + 12 E t (cid:20)Z Tt e − r ( s − t ) (cid:0) ∂ x − ∂ x (cid:1) A ( s, X s , v s ) B s (cid:0) σ s − v s (cid:1) ds (cid:21) + 12 E t (cid:20)Z Tt e − r ( s − t ) ∂ y A ( s, X s , v s ) B s (cid:20) v s − σ s v δs ( T − s ) + 2 Z ∞ ∆ m g δ ( s, M s − , Y s − ) ℓ ( dz ) (cid:21) ds (cid:21) − E t (cid:20)Z Tt e − r ( s − t ) ∂ y A ( s, X s , v s ) B s (cid:20) d [ M c , M c ] s v s ( T − s ) (cid:21)(cid:21) + E t (cid:20)Z Tt e − r ( s − t ) A ( s, X s , v s ) dB s (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) ∂ xy A ( s, X s , v s ) B s σ s v s ( T − s ) d [ W, M c ] s (cid:21) DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 8 + 18 E t (cid:20)Z Tt e − r ( s − t ) ∂ y A ( s, X s , v s ) B s d [ M c , M c ] s v s ( T − s ) (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) ∂ x A ( s, X s , v s ) σ s d [ W, B ] s (cid:21) + q − ρ E t (cid:20)Z Tt e − r ( s − t ) ∂ x A ( s, X s , v s ) σ s d h ˜ W , B i s (cid:21) + E t (cid:20)Z Tt e − r ( s − t ) ∂ y A ( s, X s , v s ) d [ M c , B ] s v s ( T − s ) (cid:21) + E t (cid:20)Z Tt Z ∞ e − r ( s − t ) B s (cid:2) ∆ x A ( s, X s − , v s − ) + ∆ y A ( s, X s − , v s − ) (cid:3) ℓ ( dz ) ds (cid:21) . Proof. Let F (cid:0) t, X t , v δt (cid:1) , e − rt A (cid:0) t, X t , v δt (cid:1) B t , and apply again, the multidimensional Itôformula for Lévy processes to F (cid:0) t, X t , v δt (cid:1) . To do so, we will consider the continuous partsof X t and v δt , respectively given by equations (3 . and (3 . . Therefore we can write theItô formula in its integral version over the time interval [ t, T ] as follows: e − rT A (cid:16) T, X T , v δT (cid:17) B T = e − rt A (cid:16) t, X t , v δt (cid:17) B t − r Z Tt e − rs A (cid:16) s, X s , v δs (cid:17) B s ds + Z Tt e − rs ∂ s A (cid:16) s, X s , v δs (cid:17) B s ds + Z Tt e − rs ∂ x A (cid:16) s, X s , v δs (cid:17) B s dX cs + Z Tt e − rs ∂ y A (cid:16) s, X s , v δs (cid:17) B s d (cid:16) v δs (cid:17) c + Z Tt e − rs A (cid:16) s, X s , v δs (cid:17) dB s + 12 Z Tt e − rs ∂ x A (cid:16) s, X s , v δs (cid:17) B s d [ X c , X c ] s + Z Tt e − rs ∂ xy A (cid:16) s, X s , v δs (cid:17) B s d h X c , (cid:16) v δ (cid:17) c i s + 12 Z Tt e − rs ∂ y A (cid:16) s, X s , v δs (cid:17) B s d h(cid:16) v δ (cid:17) c , (cid:16) v δ (cid:17) c i s + Z Tt e − rs ∂ x A (cid:16) s, X s , v δs (cid:17) d [ X c , B ] s + Z Tt e − rs ∂ y A (cid:16) s, X s , v δs (cid:17) d h(cid:16) v δ (cid:17) c , B i s + Z Tt Z ∞ e − rs B s h ∆ x A (cid:16) s, X s − , v δs − (cid:17) + ∆ y A (cid:16) s, X s − , v δs − (cid:17)i ℓ ( dz ) ds + Z Tt Z ∞ e − rs B s h ∆ x A (cid:16) s, X s − , v δs − (cid:17) + ∆ y A (cid:16) s, X s − , v δs − (cid:17)i ˜ N ( ds, dz ) . Now, recalling the definitions of dX ct and d (cid:0) v δt (cid:1) c , the fact that d [ X c , X c ] s = σ s ds,d h X c , (cid:16) v δ (cid:17) c i s = σ s v δs ( T − s ) (cid:18) ρ d [ W, M c ] s + q − ρ d h ˜ W , M c i s (cid:19) ,d h(cid:16) v δ (cid:17) c , (cid:16) v δ (cid:17) c i s = d [ M c , M c ] s v δs ) ( T − s ) , DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 9 d [ X c , B ] s = σ s (cid:18) ρ d [ W, B ] s + q − ρ d h ˜ W , B i s (cid:19) ,d h(cid:16) v δ (cid:17) c , B i s = d [ M c , B ] s v δs ( T − s ) , and the independence between M and ˜ W , we can rewrite e − rT A (cid:0) T, X T , v δT (cid:1) B T as e − rT A (cid:16) T, X T , v δT (cid:17) B T = e − rt A (cid:16) t, X t , v δt (cid:17) B t + Z Tt e − rs (cid:18) ∂ s + 12 σ s ∂ x + (cid:18) r − σ s (cid:19) ∂ x − r (cid:19) A (cid:16) s, X s , v δs (cid:17) B s ds − ζ ( ρ , η ) Z Tt e − rs ∂ x A (cid:16) s, X s , v δs (cid:17) B s ds + Z Tt e − rs ∂ x A (cid:16) s, X s , v δs (cid:17) B s (cid:20) σ s (cid:18) ρ dW s + q − ρ d ˜ W s (cid:19)(cid:21) + 12 Z Tt e − rs ∂ y A (cid:16) s, X s , v δs (cid:17) B s (cid:20) ( v δs ) − σ s v δs ( T − s ) + 2 Z ∞ ∆ m g δ ( s, M s − , Y s − ) ℓ ( dz ) (cid:21) ds + 12 Z Tt e − rs ∂ y A (cid:16) s, X s , v δs (cid:17) B s (cid:20) dM cs v δs ( T − s ) (cid:21) − Z Tt e − rs ∂ y A (cid:16) s, X s , v δs (cid:17) B s " d [ M c , M c ] s ( v δs ) ( T − s ) + Z Tt e − rs A (cid:16) s, X s , v δs (cid:17) dB s + ρ Z Tt e − rs ∂ xy A (cid:16) s, X s , v δs (cid:17) B s σ s v δs ( T − s ) d [ W, M c ] s + 12 Z Tt e − rs ∂ y A (cid:16) s, X s , v δs (cid:17) B s d [ M c , M c ] s v δs ) ( T − s ) + ρ Z Tt e − rs ∂ x A (cid:16) s, X s , v δs (cid:17) σ s d [ W, B ] s + q − ρ Z Tt e − rs ∂ x A (cid:16) s, X s , v δs (cid:17) σ s d h ˜ W , B i s + Z Tt e − rs ∂ y A (cid:16) s, X s , v δs (cid:17) d [ M c , B ] s v δs ( T − s )+ Z Tt Z ∞ e − rs B s h ∆ x A (cid:16) s, X s − , v δs − (cid:17) + ∆ y A (cid:16) s, X s − , v δs − (cid:17)i ℓ ( dz ) ds + Z Tt Z ∞ e − rs B s h ∆ x A (cid:16) s, X s − , v δs − (cid:17) + ∆ y A (cid:16) s, X s − , v δs − (cid:17)i ˜ N ( ds, dz ) . We can identify in the previous expression, the operator L σ s . We know from [18], that thefollowing is true, L σ s = L v δs + 12 (cid:18) σ s − (cid:16) v δs (cid:17) (cid:19) (cid:0) ∂ x − ∂ x (cid:1) , where L v δs A = 0 . Therefore, multiplying by e rt , taking conditional expectation and letting δ ց , combined with the use of the dominated convergence theorem, we obtain resultfollows, ending the proof. (cid:3) DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 10 If we assume additional properties on the function A one can simplify the formula givenin the previous theorem. In the following corollary, we assume certain relationship of thepartial derivative with respect to y and the first and second order partial derivatives withrespect to x . This relation is often referred in the literature as the Delta-Gamma-Vegarelationship and it is satisfied by the Black-Scholes function. Corollary 4. Let the function A and the process B , be defined as in Theorem 3. Supposethat the function A satisfies the Delta-Gamma-Vega relationship given by (4.1) ∂ y A ( t, x, y ) 1 y ( T − t ) = (cid:0) ∂ xx − ∂ x (cid:1) A ( t, x, y ) . Then, for every t ∈ [0 , T ] , the following formula holds: e − r ( T − t ) E t [ A ( T, X T , v T ) B T ]= A ( t, X t , v t ) B t − ζ ( ρ , η ) E t (cid:20)Z Tt e − rs Λ A ( s, X s , v s ) B s ds (cid:21) + E t (cid:20)Z Tt e − r ( s − t ) Γ A ( s, X s , v s ) B s (cid:20) v s ( T − s ) Z ∞ ∆ m g ( s, M s − , Y s − ) ℓ ( dz ) (cid:21) ds (cid:21) + E t (cid:20)Z Tt e − r ( s − t ) A ( s, X s , v s ) dB s (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) ΛΓ A ( s, X s , v s ) B s σ s d [ W, M c ] s (cid:21) + 18 E t (cid:20)Z Tt e − r ( s − t ) Γ A ( s, X s , v s ) B s d [ M c , M c ] s (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) Λ A ( s, X s , v s ) σ s d [ W, B ] s (cid:21) + q − ρ E t (cid:20)Z Tt e − r ( s − t ) Λ A ( s, X s , v s ) σ s d h ˜ W , B i s (cid:21) + 12 E t (cid:20)Z Tt e − r ( s − t ) Γ A ( s, X s , v s ) d [ M c , B ] s (cid:21) + E t (cid:20)Z Tt Z ∞ e − r ( s − t ) B s (cid:2) ∆ x A ( s, X s − , v s − ) + ∆ y A ( s, X s − , v s − ) (cid:3) ℓ ( dz ) ds (cid:21) . Proof. If (4 . holds, then it is trivial to see that ∂ xy A ( t, x, y ) = y ( T − t ) (cid:0) ∂ xx − ∂ xx (cid:1) A ( t, x, y ) ,∂ yy A ( t, x, y ) = y ( T − t ) y ( T − t ) (cid:0) ∂ xx − ∂ x (cid:1) A ( t, x, y )+ y ( T − t ) (cid:0) ∂ xx − ∂ x (cid:1) A ( t, x, y ) . Therefore, by replacing (4 . together with the previous equalities in Theorem 3, the resultis straightforward. (cid:3) The next result yields an analogous formula to Proposition 9 in [4], that contains thecontinuous part of the formula and the discontinuous terms coming from the jumps assumedin the model. DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 11 Theorem 5. Assume the model given by equations 2.3, such that kθ ≥ ν , (cid:18) − c − c T α α Γ( α ) (cid:19) ≥ , and let A ( t, X t , v t ) = BS ( t, X t , v t ) and B t ≡ . Then, for every t ∈ [0 , T ] , Then we have V t = BS ( t, X t , v t ) − ζ ( ρ , η ) E t (cid:20)Z Tt e − rs Λ BS ( s, X s , v s ) ds (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) σ s d [ W, M c ] s (cid:21) + 18 E t (cid:20)Z Tt e − r ( s − t ) Γ BS ( s, X s , v s ) d [ M c , M c ] s (cid:21) + E t (cid:20)Z Tt e − r ( s − t ) Γ BS ( s, X s , v s ) (cid:20) v s ( T − s ) Z ∞ ∆ m g ( s, M s − , Y s − ) ℓ ( dz ) (cid:21) ds (cid:21) + E t (cid:20)Z Tt Z ∞ e − r ( s − t ) (cid:2) ∆ x BS ( s, X s − , v s − ) + ∆ y BS ( s, X s − , v s − ) (cid:3) ℓ ( dz ) ds (cid:21) , BS ( t, X t , v t ) − ζ ( ρ , η ) Z Tt e − rs Λ BS ( s, X s , v s ) ds + ( I ) + ( II ) + ( III ) + ( IV ) . (4.2) Proof. The result is trivially achieved by replacing A ( t, X t , v t ) = BS ( t, X t , v t ) and B t ≡ in Corollary 4, and noticing that V t = e − r ( T − t ) E t (cid:2) BS (cid:0) T, X T , v δT (cid:1)(cid:3) . (cid:3) Martingale Representation of the future expected volatility This section is devoted to derive an expression for the dynamics of the integrated projectedfuture variance M . In the next proposition we show that, under certain conditions on theparameters, the variance process is bounded away from zero (lower bounded by a strictlypositive function). Proposition 6. Consider α ∈ (cid:0) , (cid:1) and T ≥ . Assuming that kθ ≥ ν and (cid:18) − c − c T α α Γ( α ) (cid:19) ≥ . Then for all < t < T (5.1) σ t ≥ ¯ σ e − κt + θ (1 − e − κt ) (cid:18) − c − c T α α Γ( α ) (cid:19) a.s. Proof. We know by definition of the volatility equation given by (2 . , that σ t = U t + c νZ t + c ν α ) Z t ( t − r ) α − Z r dr + c ηJ t . Knowing that jumps are only positive and that c ≥ , η > , we can lower bound thevolatility process by σ t ≥ U t + c νZ t + c ν α ) Z t ( t − r ) α − Z r dr. DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 12 Back to the Heston’s volatility process, we know that it has to be positive, therefore νZ t > − U t = − ( θ + e − κt (¯ σ − θ )) for all initial condition ¯ σ . And letting ¯ σ → we havethat νZ t ≥ − θ (1 − e − κt ) a.s. Now we can write σ t ≥ U t − c θ (1 − e − κt ) − c θ Γ( α ) Z t ( t − r ) α − (1 − e − κr ) dr ≥ U t − c θ (1 − e − κt ) − c θ (1 − e − κt )Γ( α ) Z t ( t − r ) α − dr = U t − c θ (1 − e − κt ) − c θ (1 − e − κt ) t α α Γ( α )= ¯ σ e − κt + θ (1 − e − κt ) (cid:18) − c − c t α α Γ( α ) (cid:19) ≥ ¯ σ e − κt + θ (1 − e − κt ) (cid:18) − c − c T α α Γ( α ) (cid:19) . And it is a positive quantity since we have by hypothesis that (cid:18) − c − c T α α Γ( α ) (cid:19) ≥ . (cid:3) Now as we have defined v t = T − t (cid:16) M t − R t σ s ds (cid:17) , when applying the Itô Lemma we needto compute dv t which implies computing dM t , for M t = R T E t (cid:2) σ s (cid:3) ds . The computationof this last derivative, needs to be done by means of Clark-Ocone-Haussmann formula andMalliavin calculus techniques. A good reference for this topic is, for instance, [22]. Proposition 7. We have that σ t ∈ D , N and D Ns,z σ t = c ηz. Proof. Note that σ t = Y t + c νZ t + c ν α ) R t ( t − r ) α − Z r dr + c ηJ t . Now we have that D Ns,z σ t = D Ns,y (cid:18) Y t + c νZ t + c ν α ) Z t ( t − r ) α − Z r dr + c ηJ t (cid:19) = D Ns,z c ηJ t , and since J t is a pure jump Lévy process which can be represented as J t = − ζ ( ρ , η ) t + R t R ∞ z ˜ N ( ds, dz ) , following [22], we have that D Ns,z σ t = D Ns,z c ηJ t = D Ns,z c η (cid:18) − ζ ( ρ , η ) t + Z t Z ∞ z ˜ N ( ds, dz ) (cid:19) = D Ns,z (cid:18)Z t Z ∞ c ηz ˜ N ( ds, dz ) (cid:19) = c ηz. (cid:3) Proposition 8. Assume that kθ ≥ ν . Then, we have that σ t ∈ D , W and (5.2) D Ws σ t = c D Ws ¯ σ t + c Γ( α ) Z ts ( t − r ) α − D Ws ¯ σ r dr. Proof. Starting from the definition of σ t , we have that D Ws σ t = D Ws (cid:0) Y t + c νZ t + c νI α Z t + c ηJ t (cid:1) = c νD Ws Z t + c I α νD Ws Z t . DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 13 We have to note that D Ws J t = 0 and that D Ws ¯ σ t = νD Ws Z t , hence we can rewrite theprevious expression as D Ws σ t = c D Ws ¯ σ t + c Γ( α ) Z ts ( t − r ) α − D Ws ¯ σ r dr. Now, for instance, we have to solve D Ws ¯ σ t . To achieve this we will make use of Theorem 2.1in [10] where in our particular case µ ( s, ¯ σ t ) , κ ( θ − ¯ σ t ) and σ ( s, ¯ σ t ) , ν p ¯ σ t . Therefore,we have that by replacing and doing some basic algebraic manipulations we obtain thefollowing expression. D Ws ¯ σ t = σ exp (cid:26)Z ts (cid:20) ∂ µ − µ∂ σσ − 12 ( ∂ σ ) σ − ∂ σσ (cid:21) ( u, ¯ σ u ) du (cid:27) = ν q ¯ σ t exp (cid:26)Z ts (cid:20) − κ − (cid:18) κθ − ν (cid:19) σ u (cid:21) du (cid:27) = ν q ¯ σ t f ( t, s ) , where f ( t, s ) , exp nR ts h − κ − (cid:16) κθ − ν (cid:17) σ u i du o . (cid:3) Proposition 9. Assume that kθ ≥ ν and (cid:16) − c − c T α α Γ( α ) (cid:17) ≥ . Then, (5.3) dM t = νA ( T, t ) q ¯ σ t dW t + c ηdJ t . Proof. Using the Clark-Ocone-Haussmann formula we have(5.4) σ t = E (cid:2) σ t (cid:3) + Z t E s (cid:2) D Ws σ t (cid:3) dW s + Z t Z ∞ E s (cid:2) D Ns,z σ t (cid:3) ˜ N ( ds, dz ) . Now we will treat the second addend using 5.2 and doing some algebraic manipulations,to obtain the following relationship Z t E s (cid:2) D Ws σ t (cid:3) dW s = Z t E s (cid:20) c D Ws ¯ σ t + c Γ( α ) Z ts ( t − r ) α − D Ws ¯ σ r dr (cid:21) dW s = Z t (cid:20) c E s (cid:2) D Ws ¯ σ t (cid:3) + c Γ( α ) Z ts ( t − r ) α − E s (cid:2) D Ws ¯ σ r (cid:3) dr (cid:21) dW s . Remembering from proposition (8) that D Ws ¯ σ t = νD Ws Z t we can rewrite the previousequation as Z t E s (cid:2) D Ws σ t (cid:3) dW s = Z t (cid:20) c ν E s (cid:2) D Ws Z t (cid:3) + c ν Γ( α ) Z ts ( t − r ) α − E s (cid:2) D Ws Z r (cid:3) dr (cid:21) dW s = Z t (cid:20) c ν exp {− κ ( t − s ) } + (cid:18) c ν Γ( α ) Z ts ( t − r ) α − exp {− κ ( t − s ) } dr (cid:19)(cid:21) p ¯ σ s dW s , as Z t = R t e − κ ( t − s ) p ¯ σ s dW s . Finally if we set a ( t, s ) := (cid:20) c ν exp {− κ ( t − s ) } + (cid:18) c ν Γ( α ) Z ts ( t − r ) α − exp {− κ ( t − s ) } dr (cid:19)(cid:21) p ¯ σ s , we can write R t E s (cid:2) D Ws σ t (cid:3) dW s in a more compact way as Z t E s (cid:2) D Ws σ t (cid:3) dW s = Z t a ( t, s ) dW s . DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 14 We will continue by treating the third addend in equation 5.4, using proposition (7) . Z t Z ∞ E s (cid:2) D Ns,z σ t (cid:3) ˜ N ( ds, dz ) = Z t Z ∞ c ηz ˜ N ( ds, dz ) , which is a martingale process. Therefore, recalling that we defined M t = R T E t (cid:2) σ s (cid:3) ds , wecan write dM t as dM t = (cid:18)Z Tt a ( r, t ) dr (cid:19) dW t + c ηd ˜ J t = ν Z Tt c exp {− κ ( r − t ) } p ¯ σ r dr dW t + c ν Γ( α ) Z Tt Z rt ( r − u ) α − exp {− κ ( u − t ) } p ¯ σ r dudr dW t + c ηd ˜ J t . Using Fubini’s Theorem and integrating the expression we obtain dM t = ν Z Tt c exp {− κ ( u − t ) } q ¯ σ t dr + c ν Γ( α ) Z rt Z Tt ( r − u ) α − exp {− κ ( u − t ) } q ¯ σ t drdu dW t + c ηd ˜ J t = ν Z Tt c exp {− κ ( u − t ) } q ¯ σ t dr dW t + c να Γ( α ) Z rt ( T − u ) α exp {− κ ( u − t ) } q ¯ σ t du dW t + c ηd ˜ J t = ν (cid:18)Z Tt (cid:18) c α Γ( α ) ( T − u ) α + c (cid:19) exp {− κ ( u − t ) } du (cid:19) q ¯ σ t dW t + c ηd ˜ J t . Defining A ( T, t ) , R Tt (cid:16) c α Γ( α ) ( T − u ) α + c (cid:17) exp {− κ ( u − t ) } du, we can rewrite the pre-vious expression as dM t = νA ( T, t ) q ¯ σ t dW t + c ηd ˜ J t . (cid:3) Call Price Approximations in the Fractional Heston Model with Jumps This section is aimed at providing an approximation formula for the pricing of vanillaoptions presented in Theorem 5, where we assumed the FSVJJ model presented in previoussections. In particular, we will deduce a first order approximation formula with respectto the vol-of-vol parameter ν , and the jump parameter η . In order to do so, we willneed to introduce a series of technical lemmas. These will help us bound the conditionalexpectation of the integrated future variance, needed to bound the error terms of theapproximating formula. Lemma 10. The following results regarding some of the expressions considered in previoussections hold. (1) A ( T, t ) = R Tt (cid:16) c Γ( α ) ( T − u ) α + c (cid:17) e − κ ( u − t ) du .(2) E t (cid:2) ¯ σ s (cid:3) = θ + (cid:0) ¯ σ t − θ (cid:1) e − κ ( s − t ) = ¯ σ t e − κ ( s − t ) + θ (cid:0) − e − κ ( s − t ) (cid:1) (3) E t h σ s p ¯ σ s i = ¯ σ t e − κ ( s − t ) + θ (cid:0) − e − κ ( s − t ) (cid:1) + O (cid:0) ν + η (cid:1) . (4) dM ct = νA ( T, t ) p ¯ σ t dW t . DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 15 (5) d [ W, M c ] t = νA ( T, t ) p ¯ σ t dt .(6) d [ M c , M c ] t = ν A ( T, t )¯ σ t dt .(7) L [ W, M c ] t = ν R Tt A ( T, s ) E t hp σ s ¯ σ s i ds .(8) D [ M c , M c ] t = ν R Tt A ( T, s ) E t (cid:2) ¯ σ s (cid:3) ds. (9) dL [ W, M c ] t = d (cid:18) E t (cid:20)Z Tt σ s d [ W, M c ] s (cid:21)(cid:19) = ν (cid:18)Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) ¯ σ t dW t − ν ¯ σ t A ( T, t ) dt + O (cid:0) ν + νη (cid:1) . (10) dD [ M c , M c ] t = d (cid:18) E t (cid:20)Z Tt d [ M c , M c ] s (cid:21)(cid:19) = ν (cid:18)Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) ¯ σ t dW t − ν ¯ σ t A ( T, t ) dt. Proof. The proof of this result can be found in the subsection 7.1 of the Appendix. (cid:3) Lemma 11. Let ≤ t < s ≤ T and G t , F t ∨ F WT ∨ F J T . Then for every n ≥ , thereexists C = C ( n, ρ ) such that | E [ ∂ nx G ( s, X s + a, v s ) | G t ] | ≤ C (cid:18)Z Ts E s [ σ θ ] dθ (cid:19) − ( n +1) , where a ≥ , and G ( t, x, y ) , (cid:0) ∂ x − ∂ x (cid:1) BS ( t, x, y ) .Proof. By means of simple computations we know that G ( s, X s + a, v s ) = Ke − r ( T − s ) φ (cid:16) X s + a − µ − , v s √ T − s (cid:17) , where µ − , ln ( K ) − (cid:0) r − v s / (cid:1) ( T − s ) and φ is the normal density distribution function.Notice that v s √ T − s = qR Ts E s (cid:2) σ θ (cid:3) dθ . The properties of φ under the derivation sign,allow us to write the following equality ∂ nx G ( s, X s + a, v s ) = ( − n Ke − r ( T − s ) ∂ nµ − φ (cid:16) X s + a − µ − , v s √ T − s (cid:17) . Now we consider the conditional expectation with respect to the filtration G t that allow usto know the trajectories of the instantaneous variance and the jump process up to time T .Therefore, we can write the following,(6.1) E [ ∂ nx G ( s, X s + a, v s ) | G t ] = ( − n Ke − r ( T − s ) ∂ nµ − E h φ (cid:16) X s + a − µ − , v s √ T − s (cid:17) | G t i . The law of X s given G t is a normal random variable with mean ˜ µ , X t + Z st (cid:18) r − σ θ (cid:19) dθ + ρ η (cid:16) ˜ J s − ˜ J t (cid:17) − ζ ( ρ , η ) ( s − t ) + ρ Z st σ θ dW θ , and variance ˜Σ , (cid:0) − ρ (cid:1) R st σ θ dθ = (cid:0) − ρ (cid:1) R st E s (cid:2) σ θ (cid:3) dθ . Now, we can compute theconditional expectation as follows, E h φ (cid:16) X s + a − µ − , v s √ T − s (cid:17) | G t i = Z R φ (cid:16) x + a − µ − , v s √ T − s (cid:17) f X ( x ) dx DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 16 = Z R φ (cid:16) x + a − µ − , v s √ T − s (cid:17) φ (cid:16) x − ˜ µ, ˜Σ (cid:17) dx = Z R " v s p π ( T − s ) exp ( − (cid:18) x + a − µ − v s √ T − s (cid:19) ) p π ˜Σ exp − x − ˜ µ p ˜Σ ! dx. We know that the product of two Gaussian probability density functions, results in theexpression given by equation (7 . from proposition (17) in the Appendix 7. Therefore, wecan rewrite the previous conditional expectation as E h φ (cid:16) X s + a − µ − , v s √ T − s (cid:17) | G t i = 1 r π (cid:16) ˜Σ + v s ( T − s ) (cid:17) exp − (˜ µ + a − µ − ) (cid:16) ˜Σ + v s ( T − s ) (cid:17) × Z R r π ˜Σ v s ( T − s )˜Σ+ v s ( T − s ) exp − (cid:16) x − ˜ µv s ( T − s )+( µ − − a ) ˜Σ˜Σ+ v s ( T − s ) (cid:17) ˜Σ v s ( T − s )˜Σ+ v s ( T − s ) dx = 1 r π (cid:16) ˜Σ + v s ( T − s ) (cid:17) exp − (˜ µ + a − µ − ) (cid:16) ˜Σ + v s ( T − s ) (cid:17) = φ (cid:18) ˜ µ + a − µ − , q ˜Σ + v s ( T − s ) (cid:19) . The last equality results from the fact that the integral of the Gaussian density equals toone.Putting this result in (6 . , we have | E [ ∂ nx G ( s, X s , v s ) | G t ] | = (cid:12)(cid:12)(cid:12)(cid:12) ( − n Ke − r ( T − s ) ∂ nµ − φ (cid:18) ˜ µ + a − µ − , q ˜Σ + v s ( T − s ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . Now, notice that | ∂ nx φ ( x, σ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) x n σ n +1 e − x σ (cid:12)(cid:12)(cid:12)(cid:12) , for all x ∈ R , σ ∈ R + and n ≥ . Let C ( n ) = n + 1 be a positive constant, and d = σ . Then, trivially the following holds | ∂ nx φ ( x, σ ) | ≤ x C e − dx . Note that the function ψ ( x ) = x C e − dx has a global maximum at x = ± q C d and therefore, | ψ ( x ) | ≤ ψ (cid:18)q C d (cid:19) ≤ Cσ ( n +1) . Therefore, since σ = q ˜Σ + v s ( T − s ) , we can write | E [ ∂ nx G ( s, X s , v s ) | G t ] |≤ C (cid:16) ˜Σ + v s ( T − s ) (cid:17) − ( n +1) ≤ C (cid:18)(cid:0) − ρ (cid:1) Z st E s (cid:2) σ θ (cid:3) dθ + Z Ts E s (cid:2) σ θ (cid:3) dθ (cid:19) − ( n +1) ≤ C (cid:18)(cid:0) − ρ (cid:1) Z Tt E s (cid:2) σ θ (cid:3) dθ − (cid:0) − ρ (cid:1) Z Ts E s (cid:2) σ θ (cid:3) dθ + Z Ts E s (cid:2) σ θ (cid:3) dθ (cid:19) − ( n +1) DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 17 ≤ C (cid:18)(cid:0) − ρ (cid:1) Z Tt E s (cid:2) σ θ (cid:3) dθ + ρ Z Ts E s (cid:2) σ θ (cid:3) dθ (cid:19) − ( n +1) ≤ C (cid:18)Z Ts E s (cid:2) σ θ (cid:3) dθ (cid:19) − ( n +1) . (cid:3) Lemma 12. Assume that kθ > ν and let ϕ ( t ) , R Tt e − κ ( z − t ) dz = κ (cid:0) − e − κ ( T − t ) (cid:1) .Then, for all ≤ s < t ≤ T (1) ( i ) R Ts E s (cid:2) ¯ σ u (cid:3) du ≥ ¯ σ s ϕ ( s ) ,(2) ( ii ) R Ts E s (cid:2) ¯ σ u (cid:3) du ≥ θκ ϕ ( s ) . Proof. From statement 2 in Lemma (10) we know that E t (cid:2) ¯ σ s (cid:3) = ¯ σ t e − κ ( s − t ) + θ (cid:0) − e − κ ( s − t ) (cid:1) .Now ( i ) results from lower bounding the conditional expectation by considering only thefirst term of the equality and integrating in the interval [ s, T ] as follows, Z Ts E s (cid:2) ¯ σ u (cid:3) du ≥ Z Ts ¯ σ s e − κ ( u − s ) du ≥ ¯ σ s κ (cid:16) − e − κ ( T − s ) (cid:17) . In order to prove ( ii ) , we lower bound E t (cid:2) ¯ σ s (cid:3) ≥ θκϕ ( s ) and integrate in the interval [ s, T ] .Therefore, we can write Z Ts E s (cid:2) ¯ σ u (cid:3) ≥ θκ ϕ ( s ) . (cid:3) Lemma 13. Let g ( t, m, y ) , q m − yT − t . Then, the following inequality holds, (cid:12)(cid:12) ∂ m g ( t, M t , Y t ) (cid:12)(cid:12) ≤ (cid:0) ¯ σ t ϕ ( t ) (cid:1) − √ T − t . Proof. By a simple calculation, we know that ∂ m g ( t, m, y ) = − 14 ( T − t ) (cid:18) m − yT − t (cid:19) − . Recalling the definition of processes M t and Y t , the following holds (cid:12)(cid:12) ∂ m g ( t, M t , Y t ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − 14 ( T − t ) (cid:18) M t − Y t T − t (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 14 √ T − t ( M t − Y t ) − = 14 √ T − t (cid:18)Z Tt E t (cid:2) σ s (cid:3) ds (cid:19) − . From Lemma (12) ( i ) , we know that R Tt E t (cid:2) σ u (cid:3) du ≥ ¯ σ t ϕ ( t ) , finishing the proof. (cid:3) DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 18 Theorem 14. (1st order approximation formula). Fix T > . Assume the model 2.3,where the volatility process σ = { σ s , s ∈ [0 , T ] } satisfies the conditions kθ > ν and (cid:16) − c − c T α α Γ( α ) (cid:17) > C for some positive constant C . Then V t , e − r ( T − t ) E t h(cid:0) e X T − K (cid:1) + i = BS ( t, X t , v t ) + ρ BS ( t, X t , v t ) L [ W, M c ] t + E t (cid:20)Z Tt Z ∞ e − r ( s − t ) (cid:16) ρ L [ W, M c ] s (cid:17) ∆ x ΛΓ BS ( s, X s − , v s − ) ℓ ( dz ) ds (cid:21) + ǫ t , where ǫ t is the error term and satisfies | ǫ t | ∈ O (cid:0) ν + η (cid:1) . Proof. This proof relies on applying Corollary 4 iteratively to the different terms appearingin the call price formula given by equation (4 . in Theorem 5. This way, the resultingformula will only contain terms of order O (cid:0) ν + η (cid:1) which will be incorporated into theerror term.Note that we will omit the term − ζ ( ρ , η ) E t hR Tt e − rs Λ A ( s, X s , v s ) B s ds i in the applica-tion of Corollary 4 and treat it as part of the error term in the approximating formula.Since E t h(cid:12)(cid:12)(cid:12)R Tt e − rs Λ A ( s, X s , v s ) B s ds (cid:12)(cid:12)(cid:12)i < + ∞ and from Lemma 15 we know that ζ ( ρ , η ) = Z ∞ ( e ρ ηz − − ρ ηz ) ℓ ( dz )= ρ η Z ∞ Z z e λρ ηz (1 − λ ) dλℓ ( dz ) ≤ ρ η Z ∞ z ℓ ( dz ) , where we have used that ¸ ρ ≤ so e λρ ηz (1 − λ ) ≤ , and we also have that R ∞ z ℓ ( dz ) < ∞ . (cid:3) • Step 1: Applying Corollary (4) to term ( I ) in equation (4 . with A ( t, X t , v t ) =ΛΓ BS ( t, X t , v t ) and B t = ρ L [ W, M c ] t and recalling that B T = 0 by definition,this gives ( I ) = ρ BS ( t, X t , v t ) L [ W, M c ] t (6.2) + ρ E t (cid:20)Z Tt e − r ( s − t ) Λ Γ BS ( s, X s , v s ) L [ W, M c ] s σ s d [ W, M c ] s (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) L [ W, M c ] s d [ M c , M c ] s (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) Λ Γ BS ( s, X s , v s ) σ s d [ W, L [ W, M c ]] s (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) d [ M c , L [ W, M c ]] s (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) L [ W, M c ] s × v s ( T − s ) Z ∞ ∆ m g ( s, M s − , Y s − ) ℓ ( dz ) ds (cid:21) DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 19 + ρ E t (cid:20)Z Tt Z ∞ e − r ( s − t ) L [ W, M c ] s × (cid:2) ∆ x ΛΓ BS ( s, X s − , v s − ) + ∆ y ΛΓ BS ( s, X s − , v s − ) (cid:3) ℓ ( dz ) ds (cid:3) (6.3) , ( I.I ) + ( I.II ) + . . . + ( I.V II ) . (6.4) Notice also, that we can apply Lemma 10 since we are working under the fractionalHeston model with jumps. Therefore the previous equation can be rewritten as ( I ) = ρ ν BS ( t, X t , v t ) (cid:18)Z Tt A ( T, s ) E t hp σ s ¯ σ s i ds (cid:19) + ρ ν E t (cid:20)Z Tt e − r ( s − t ) Λ Γ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) E s hp σ z ¯ σ z i dz (cid:19) A ( T, s ) σ s p ¯ σ s ds (cid:21) + ρ ν E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) E s hp σ z ¯ σ z i dz (cid:19) A ( T, s )¯ σ s ds (cid:21) + ρ ν E t (cid:20)Z Tt e − r ( s − t ) Λ Γ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) e − κ ( z − t ) dz (cid:19) σ s ¯ σ s ds (cid:21) + ρ ν E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) e − κ ( z − t ) dz (cid:19) A ( T, s )¯ σ s ds (cid:21) + ρ ν E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, u ) E s hp σ u ¯ σ u i du (cid:19) × v s ( T − s ) Z ∞ ∆ m g ( s, M s − , Y s − ) ℓ ( dz ) ds (cid:21) + ρ ν E t (cid:20)Z Tt Z ∞ e − r ( s − t ) (cid:18)Z Ts A ( T, u ) E s hp σ u ¯ σ u i du (cid:19) × (cid:2) ∆ x ΛΓ BS ( s, X s − , v s − ) + ∆ y ΛΓ BS ( s, X s − , v s − ) (cid:3) ℓ ( dz ) ds (cid:3) . The terms ( I.II ) . . . ( I.V II ) , belong to the error term O (cid:0) ν + η (cid:1) . We will provethe previous statement for the terms ( I.III ) and ( I.V ) , the proof for the rest ofthe terms is analogous. ( I.III ) + ( I.V )= ρ ν E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) E s hp σ z ¯ σ z i dz (cid:19) A ( T, s )¯ σ s ds (cid:21) + ρ ν E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) e − κ ( z − t ) dz (cid:19) A ( T, s )¯ σ s ds (cid:21) . Using the fact that A ( T, z ) is a decreasing function, defining a s , v s √ T − s , andusing the inequality from Lemma (11) , we can upper bound the previous expressionby | ( I.III ) + ( I.V ) |≤ C ρ ν E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 2 a s + 1 a s (cid:19) (cid:18)Z Ts E s hp σ z ¯ σ z i dz (cid:19) A ( T, s )¯ σ s ds (cid:21) + C ρ ν E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 1 a s (cid:19) A ( T, s )¯ σ s ds (cid:21) . DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 20 Remember that it follows from Lemma (10) that R Ts E s hp σ z ¯ σ z i dz ≤ R Ts E s (cid:2) ¯ σ z (cid:3) dz + C ( T − s ) ν = a s + C ( T − s ) ν , and from Lemma (12) ( i ) , that a s ϕ ( s ) ≥ ¯ σ s . Hence | ( I.III ) + ( I.V ) |≤ C ρ ν E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 2 a s + 1 a s (cid:19) a s ϕ ( s ) A ( T, s ) ds (cid:21) + C ρ ν E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 2 a s + 1 a s (cid:19) a s ϕ ( s ) C ( T − s ) A ( T, s ) ds (cid:21) + C ρ ν E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 4 a s (cid:19) A ( T, s ) a s ϕ ( s ) ds (cid:21) ≤ C ρ ν E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 6 a s + 1 (cid:19) A ( T, s ) ϕ ( s ) ds (cid:21) + C ρ ν E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 2 a s + 1 a s (cid:19) a s ϕ ( s ) C ( T − s ) A ( T, s ) ds (cid:21) . From ( ii ) in Lemma (12) , we know that a s ≥ q θκ ϕ ( s ) . Note also that ϕ ( s ) ≤ κ for all s ∈ [0 , T ] . Taking into account that A ( T, s ) ≤ ϕ ( s ) (cid:16) c Γ( α ) ( T − t ) α + c (cid:17) ,we can finally upper bound the previous sum as follows | ( I.III ) + ( I.V ) | ≤ C ρ ν E t (cid:20)Z Tt e − r ( s − t ) ds (cid:21) . The same reasoning applies to obtain an upper bound of terms ( I.II ) , ( I.IV ) .Notice the term ( I.I ) , depends linearly on ν and therefore, it is part of the first orderapproximation formula. We will now provide upper bounds for the discontinuousterm ( I.V I ) . We start applying Lemma (15) to the functions G ( x ) = g ( s, x, Y s − ) , ∆ G ( M s − , c ηz ) = c η z Z ∂ m g ( s, M s − + λc ηz, Y s − ) (1 − λ ) dλ,G ( x ) = ΛΓ BS ( s, X s − , x )∆ G ( v s − , c ηz ) = c η z Z ∂ v ΛΓ BS ( s, X s − , v s − + λc ηz ) (1 − λ ) dλ. Given that the terms ∆ G i are proportional to η , all we need to prove is that theintegrals in the term ( I.V I ) have an upper bound, in order to properly justify thatthey belong to the error term. – Using the inequality from Lemma (11) , we can upper bound the term ( I.V I ) as follows | ( I.V I ) | ≤ C ρ c η z E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 1 a s (cid:19) L [ W, M c ] s × v s ( T − s ) Z ∞ Z ∂ m g ( s, M s − + λc ηz, Y s − ) (1 − λ ) dλℓ ( dz ) ds (cid:21) . From Lemma (13) we can upper bound (cid:12)(cid:12) ∂ m g ( s, M s − + λc ηz, Y s − ) (cid:12)(cid:12) . We alsoknow that L [ W, M c ] s = ν R Ts A ( T, r ) E s hp σ r ¯ σ r i dr ≤ νA ( T, s ) R Ts E s hp σ r ¯ σ r i dr ,as A ( T, t ) is a decreasing function of t ∈ [0 , T ] . From Lemma (10) followsthat R Ts E s hp σ r ¯ σ r i dr ≥ R Ts E s (cid:2) ¯ σ r (cid:3) dr , a s , and from Lemma (12) ( i ) , that DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 21 a s ϕ ( s ) ≥ ¯ σ s . Therefore we can upper bound the term ( I.V I ) as follows | ( I.V I ) | ≤ C ρ c η z E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 1 a s (cid:19) L [ W, M c ] s × v s ( T − s ) Z ∞ (cid:20)Z √ T − s (cid:0) ¯ σ s ϕ ( s ) (cid:1) − (1 − λ ) dλ (cid:21) ℓ ( dz ) ds (cid:21) ≤ C ρ c η z E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 1 a s (cid:19) L [ W, M c ] s v s √ T − s (cid:0) ¯ σ s ϕ ( s ) (cid:1) − ds (cid:21) ≤ C ρ c η z E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 1 a s (cid:19) L [ W, M c ] s (cid:0) a s (cid:1) − ds (cid:21) ≤ C ρ ν c η z E t (cid:20)Z Tt e − r ( s − t ) (cid:18) a s + 1 a s (cid:19) a s ϕ ( s ) (cid:0) a s (cid:1) − A ( T, s ) ds (cid:21) ≤ C ρ ν c η z E t (cid:20)(cid:18) c Γ ( α ) ( T − t ) α + c (cid:19) Z Tt e − r ( s − t ) (cid:18) a s + 1 a s (cid:19) ds (cid:21) . Again, from ( ii ) in Lemma (12) , we know that a s ≥ q θκ ϕ ( s ) Replacing it inthe previous inequality and taking into account that ϕ ( s ) ≤ κ for all s ∈ [0 , T ] ,proves that the term ( I.V I ) ∈ O (cid:0) η (cid:1) . – We can rewrite the term ( I.V II ) recalling ∆ G and the Delta-Gamma-Vegarelationship given by equation (7 . . If we also make use of Lemma (11) anduse the same bounding techniques we have already been using along the proof,we can rewrite the term as follows, ( I.V II ) = ρ E t (cid:20)Z Tt Z ∞ e − r ( s − t ) L [ W, M c ] s ∆ x ΛΓ BS ( s, X s − , v s − ) ℓ ( dz ) ds (cid:21) + ρ c η E t (cid:20)Z Tt Z ∞ e − r ( s − t ) L [ W, M c ] s × (cid:20) z Z ΛΓ BS ( s, X s − , v s − + λc ηz ) ( v s − + λc ηz ) ( T − s ) (1 − λ ) dλ (cid:21) ℓ ( dz ) ds (cid:21) ≤ ρ E t (cid:20)Z Tt Z ∞ e − r ( s − t ) L [ W, M c ] s ∆ x ΛΓ BS ( s, X s − , v s − ) ℓ ( dz ) ds (cid:21) + C νρ c η E t (cid:20)Z Tt e − r ( s − t ) (cid:18) A ( T, s ) ϕ ( s ) (cid:19) (cid:18) a s + 2 a s + 1 a s (cid:19) × Z ∞ h ( v s − + λc ηz ) ( T − s ) i ℓ ( dz ) ds (cid:21) . The bounds used in the proof of the previous term also apply here, ending theproof that shows the term ( I.V II ) can be written as ( I.V II ) = ρ E t (cid:20)Z Tt Z ∞ e − r ( s − t ) L [ W, M c ] s ∆ x ΛΓ BS ( s, X s − , v s − ) ℓ ( dz ) ds (cid:21) + O (cid:0) η (cid:1) This concludes the proof to show that ( I ) = ρ ν BS ( t, X t , v t ) (cid:18)Z Tt A ( T, s ) E t hp σ s ¯ σ s i ds (cid:19) + ρ ν E t (cid:20)Z Tt Z ∞ e − r ( s − t ) (cid:18)Z Ts A ( T, u ) E s hp σ u ¯ σ u i du (cid:19) ∆ x ΛΓ BS ( s, X s − , v s − ) ℓ ( dz ) ds (cid:21) + O (cid:0) v + η (cid:1) . DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 22 • Step 2: Applying Corollary 4 to term ( II ) in equation (4 . with A ( t, X t , v t ) =Γ BS ( t, X t , v t ) and B t = D [ M c , M c ] t and recalling that B T = 0 by definition,we have ( II ) = 18 Γ BS ( t, X t , v t ) D [ M c , M c ] t + ρ E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) D [ M c , M c ] s σ s d [ W, M c ] s (cid:21) + 164 E t (cid:20)Z Tt e − r ( s − t ) Γ BS ( s, X s , v s ) D [ M c , M c ] s d [ M c , M c ] s (cid:21) + ρ E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) σ s d [ W, D [ M c , M c ]] s (cid:21) + 116 E t (cid:20)Z Tt e − r ( s − t ) Γ BS ( s, X s , v s ) d [ M c , D [ M c , M c ]] s (cid:21) + 18 E t (cid:20)Z Tt e − r ( s − t ) Γ BS ( s, X s , v s ) D [ M c , M c ] s × v s ( T − s ) Z ∞ ∆ m g ( s, M s − , Y s − ) ℓ ( dz ) ds (cid:21) + 18 E t (cid:20)Z Tt Z ∞ e − r ( s − t ) D [ M c , M c ] s × (cid:2) ∆ x Γ BS ( s, X s − , v s − ) + ∆ y Γ BS ( s, X s − , v s − ) (cid:3) ℓ ( dz ) ds (cid:3) . Applying Lemma 10, the previous equation is rewritten as ( II ) = ν BS ( t, X t , v t ) Z Tt A ( T, s ) E t (cid:2) ¯ σ s (cid:3) ds + ρ ν E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) E s (cid:2) ¯ σ z (cid:3) dz (cid:19) A ( T, s ) σ s p ¯ σ s ds (cid:21) + ν E t (cid:20)Z Tt e − r ( s − t ) Γ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) E s (cid:2) ¯ σ z (cid:3) dz (cid:19) A ( T, s )¯ σ s ds (cid:21) + ρ ν E t (cid:20)Z Tt e − r ( s − t ) ΛΓ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) e − κ ( z − t ) dz (cid:19) σ s ¯ σ s ds (cid:21) + ν E t (cid:20)Z Tt e − r ( s − t ) Γ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, z ) e − κ ( z − t ) dz (cid:19) A ( T, s )¯ σ s ds (cid:21) + ν E t (cid:20)Z Tt e − r ( s − t ) Γ BS ( s, X s , v s ) (cid:18)Z Ts A ( T, u ) E s (cid:2) ¯ σ u (cid:3) du (cid:19) × v s ( T − s ) Z ∞ ∆ m g ( s, M s − , Y s − ) ℓ ( dz ) ds (cid:21) + ν E t (cid:20)Z Tt Z ∞ e − r ( s − t ) (cid:18)Z Ts A ( T, u ) E s (cid:2) ¯ σ u (cid:3) du (cid:19) × (cid:2) ∆ x Γ BS ( s, X s − , v s − ) + ∆ y Γ BS ( s, X s − , v s − ) (cid:3) ℓ ( dz ) ds (cid:3) . Observe that all the terms in ( II ) can be incorporated into the error term, i.e. ( II ) = O (cid:0) ν (cid:1) . This is due to the dependency of each term on higher order of ν and the fact that all terms can be upper bounded following the reasoning weprovided for terms ( I.III ) and ( I.V ) . • Step 3: Proving the term ( III ) in equation (4 . belongs to the error term O (cid:0) η (cid:1) ,is analogous to the discussion we did for the term ( I.V I ) . DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 23 • Step 4: Again, an analogous discussion as the one performed in the study of ( I.V II ) applies here, showing that ( IV ) = E t (cid:20)Z Tt Z ∞ e − r ( s − t ) ∆ x ΛΓ BS ( s, X s − , v s − ) ℓ ( dz ) ds (cid:21) + O (cid:0) η (cid:1) . Appendix In this appendix we gather additional technical lemmas. Lemma 15. Let G ∈ C ( R ) and consider the expression ∆ G ( x, h ) defined as ∆ G ( x, h ) , G ( x + h ) − G ( x ) − hG ′ ( x ) . Then, the following equality holds ∆ G ( x, h ) = h Z G ′′ ( x + λh ) (1 − λ ) dλ. (7.1) Proof. It is Taylor’s Theorem with integral remainder. (cid:3) Proposition 16. (Delta-Gamma-Vega Relationship) Let BS ( t, x, y ) = e x Φ ( d + ) − e − r ( T − t ) K Φ ( d − ) , where Φ denotes the cumulative distribution function of a standard normal distribution and d ± = x − ln K + (cid:16) r ± y (cid:17) ( T − t ) y √ T − t ; d + = d − + y √ T − t. Then for every t ∈ [0 , T ] , the following formula, known as the Delta-Gamma-Vega rela-tionship, holds. (7.2) ∂ y BS ( t, x, y ) 1 y ( T − t ) = (cid:0) ∂ xx − ∂ x (cid:1) BS ( t, x, y ) . Proof. We start by computing the log-Delta. ∂ x BS ( t, x, y ) = e x Φ ( d + ) . Now the log-Gamma is computed as ∂ xx BS ( t, x, y ) = e x Φ ( d + ) + e x φ ( d + ) y √ T − t . Therefore, we have that (cid:0) ∂ xx − ∂ x (cid:1) BS ( t, x, y ) = e x φ ( d + ) y √ T − t . On the other hand, the Vega is derived as follows ∂ y BS ( t, x, y ) = e x φ ( d + ) ∂ y d + − e − r ( T − t ) Kφ ( d − ) ∂ y d − = e x φ ( d + ) h ∂ y d − + ∂ y (cid:16) y √ T − t (cid:17)i − e − r ( T − t ) Kφ ( d − ) ∂ y d − = e x φ ( d + ) ∂ y (cid:16) y √ T − t (cid:17) = e x φ ( d + ) √ T − t. We will name log-Delta the change in the option price with respect to the change in the underlyingasset log-price. This is a different sensitivity of what is commonly referred to as the Delta, i.e. the changein the option price with respect to the change in the underlying asset price. DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 24 The relationship follows trivially from these computations. (cid:3) Proposition 17. Let φ ( x, σ ) denote the density function of the normal law with mean zeroand standard deviation σ . Then, for µ , µ ∈ R and σ , σ strictly positive real numberswe have (7.3) φ ( x − µ , σ ) φ ( x − µ , σ ) = φ x − µ σ + µ σ σ + σ , s σ σ σ + σ ! φ (cid:18) µ − µ , q σ + σ (cid:19) . Proof. This proof results from basic algebraic manipulations. Note that we can triviallywrite the product of densities as follows φ ( x − µ , σ ) φ ( x − µ , σ ) = 1 p πσ exp ( − ( x − µ ) σ ) p πσ exp ( − ( x − µ ) σ ) = 1 p πσ σ π exp ( − ( x − µ ) σ − ( x − µ ) σ ) = 1 p πσ σ π exp ( − σ ( x − µ ) − σ ( x − µ ) σ σ ) . Since σ + σ > we can rewrite the previous expression as φ ( x − µ , σ ) φ ( x − µ , σ ) = 1 r π σ σ σ + σ (cid:0) σ + σ (cid:1) π × exp − σ σ + σ ( x − µ ) − σ σ + σ ( x − µ ) σ σ σ + σ . Notice from the previous expression that s π σ σ σ + σ (cid:0) σ + σ (cid:1) π = s π σ σ σ + σ q π (cid:0) σ + σ (cid:1) , therefore, replacing this in the product of the two pdf’s and considering that x = e ln x , oneobtains that φ ( x − µ , σ ) φ ( x − µ , σ )= 1 r π σ σ σ + σ exp ln 1 q π (cid:0) σ + σ (cid:1) exp − σ σ + σ ( x − µ ) − σ σ + σ ( x − µ ) σ σ σ + σ r π σ σ σ + σ exp − 12 ln (cid:0) π (cid:0) σ + σ (cid:1)(cid:1) + − σ σ + σ ( x − µ ) − σ σ + σ ( x − µ ) σ σ σ + σ . Focusing on the exponential term, we will perform some further algebraic manipulationson it as follows. − 12 ln (cid:0) π (cid:0) σ + σ (cid:1)(cid:1) + − σ σ + σ ( x − µ ) − σ σ + σ ( x − µ ) σ σ σ + σ DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 25 = − (cid:16) σ σ σ + σ (cid:17) ln (cid:0) π (cid:0) σ + σ (cid:1)(cid:1) − σ σ + σ (cid:0) x − xµ + µ (cid:1) − σ σ + σ (cid:0) x − xµ + µ (cid:1) σ σ σ + σ = − σ − σ σ + σ x − x (cid:16) − µ σ − µ σ σ + σ (cid:17) + (cid:18) − µ σ − µ σ − σ σ ln ( π ( σ + σ )) σ + σ (cid:19) σ σ σ + σ = − x + 2 x (cid:16) − µ σ − µ σ σ + σ (cid:17) − (cid:18) µ σ + µ σ + σ σ ln ( π ( σ + σ )) σ + σ (cid:19) σ σ σ + σ . Now, one can write the previous equality as a second order polynomial of the form − x +2 Ax − A − C = − ( x − A ) − C , where A and C are given by A = µ σ + µ σ σ + σ ,C = σ σ (cid:0) σ + σ (cid:1) ( µ − µ ) + σ σ σ + σ ln (cid:0) π (cid:0) σ + σ (cid:1)(cid:1) . Equiped with this, we can now rewrite the product of the two Gaussian densities as φ ( x − µ , σ ) φ ( x − µ , σ ) = 1 r π σ σ σ + σ exp − ( x − A ) − C σ σ σ + σ , = φ x − A, s σ σ σ + σ ! exp − C σ σ σ + σ . We will focus on the second exponential term to further expand it. exp − C σ σ σ + σ = exp − σ σ ( σ + σ ) ( µ − µ ) + σ σ σ + σ ln (cid:0) π (cid:0) σ + σ (cid:1)(cid:1) σ σ σ + σ = 1 q π (cid:0) σ + σ (cid:1) exp ( − ( µ − µ ) (cid:0) σ + σ (cid:1) ) = φ (cid:18) µ − µ , q σ + σ (cid:19) . This ends the proof. (cid:3) Proof of Lemma 10. We will only prove by order statements 3, 9 and 10, as therest are trivially deduced from the definitions and results provided in previous sections. • In order to see that E t h σ s p ¯ σ s i = ¯ σ t e − κ ( s − t ) + θ (cid:0) − e − κ ( s − t ) (cid:1) + O (cid:0) ν + η (cid:1) weconsider the process u r , s ( U s + νZ sr ) (cid:18) U s + c νZ sr + c νI H − Z sr + c ηJ r (cid:19) , where U t , θ + e − κt (cid:0) ¯ σ − θ (cid:1) , DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 26 Z sr , Z r e − κ ( s − u ) p ¯ σ u dW u ,I H − Z sr = 1Γ (cid:0) H − (cid:1) Z r Z u ( s − u ) H − du = 1Γ (cid:0) H − (cid:1) Z r (cid:18)Z u e − κ ( u − v ) p ¯ σ v dW v (cid:19) ( s − u ) H − du, r ∈ [0 , s ] . Note that u s = p σ s ¯ σ s , Z ss = Z s , s ∈ [0 , T ] and I H − Z ss = I H − Z s . ApplyingFubini, we can write I H − Z sr = 1Γ (cid:0) H − (cid:1) Z r (cid:18)Z rv e − κ ( u − v ) ( s − u ) H − du (cid:19) p ¯ σ v dW v = Z r ψ ( r, s, v ) p ¯ σ v dW v , where ψ ( r, s, v ) = 1Γ (cid:0) H − (cid:1) Z rv e − κ ( u − v ) ( s − u ) H − du = e − κ ( s − v ) Γ (cid:0) H − (cid:1) Z s − rs − v e κw w H − dw. Next, consider the process Π sr = c νZ sr + c νI H − Z sr = c ν Z r e − κ ( s − u ) p ¯ σ u dW u + c ν Z r ψ ( r, s, u ) p ¯ σ u dW u = Z r ζ ( r, s, u ) p ¯ σ u dW u , where ζ ( r, s, u ) = c νe − κ ( s − u ) + c νψ ( r, s, u ) . Note that ζ ( r, s, r ) = c νe − κ ( s − r ) + c νψ ( r, s, r ) = c νe − κ ( s − r ) , and ∂ ζ ( r, s, u ) = c ν∂ ψ ( r, s, u ) = c ν Γ (cid:0) H − (cid:1) e − κ ( r − u ) ( s − r ) H − . Therefore, d Π sr = (cid:16) c νe − κ ( s − u ) + c νψ ( r, s, u ) (cid:17) p ¯ σ r dW r + c ν Γ (cid:0) H − (cid:1) Z r e − κ ( r − u ) ( s − r ) H − p ¯ σ u dW u ! dr. Consider the following process Θ sr = Π sr + c ηJ r . We have that dZ sr = e − κ ( s − r ) p ¯ σ r dW r ,d Θ sr = d Π sr + c ηdJ r ,d Θ s,cr = d Π sr ,d h Z s , Z s i r = e − κ ( s − r ) ¯ σ r dr,d h Z r , Θ s,c i r = n c νe − κ ( s − u ) + c νψ ( r, s, u ) o e − κ ( s − r ) ¯ σ r dr, DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 27 d h Θ s,c , Θ s,c i t = (cid:16) c νe − κ ( s − u ) + c νψ ( r, s, u ) (cid:17) ¯ σ r dr. Next, we can apply Itô formula to f ( Z sr , Θ sr ) , where f is the function defined by f ( x, y ) , p ( U s + νx ) ( U s + y ) . We have that ∂ f ( x, y ) = ν (cid:18) U s + yU s + νx (cid:19) / , ∂ f ( x, y ) = 12 (cid:18) U s + νxU s + y (cid:19) / ,∂ f ( x, y ) = − ν U s + y ) / ( U s + νx ) − / ,∂ f ( x, y ) = ν U s + y ) − / ( U s + νx ) − / ,∂ f ( x, y ) = − 14 ( U s + y ) − / ( U s + νx ) / . Note that f ( Z ss , Θ ss ) = p ¯ σ s σ s = s ( U s + νZ s ) (cid:18) U s + c νZ s + c νI H − Z s + c ηJ s (cid:19) . Now, an application of the Itô formula yields the following expression f ( Z ss , Θ ss )= f ( Z s , Θ s ) + Z s ∂ f ( Z sr , Θ sr ) dZ sr + Z s ∂ f ( Z sr , Θ sr ) d Θ s,cr + Z s ∂ f ( Z sr , Θ sr ) d h Z s , Z s i r + Z s ∂ f ( Z sr , Θ sr ) d h Z s , Θ s,c i r + Z s ∂ f ( Z sr , Θ sr ) d h Θ s,c , Θ s,c i r + Z s Z ∞ ∆ y f (cid:0) Z sr , Θ sr − (cid:1) ˜ N ( dr, dz ) + Z s Z ∞ ∆ y f (cid:0) Z sr , Θ sr − (cid:1) ℓ ( dz ) dr = f ( Z s , Θ s ) + ν Z s (cid:18) U s + Θ sr U s + νZ sr (cid:19) / e − κ ( s − r ) p ¯ σ r dW r + ν Z s (cid:18) U s + νZ sr U s + Θ sr (cid:19) / h(cid:16) c e − κ ( s − u ) + c ψ ( r, s, u ) (cid:17) p ¯ σ r dW r + c Γ (cid:0) H − (cid:1) Z r e − κ ( r − u ) ( s − r ) H − p ¯ σ u dW u ! dr − ν Z s ( U s + Θ sr ) / ( U s + νZ sr ) − / e − κ ( s − r ) ¯ σ r dr + ν Z s ( U s + Θ sr ) − / ( U s + νZ sr ) − / n c e − κ ( s − u ) + c ψ ( r, s, u ) o e − κ ( s − r ) ¯ σ r dr − ν Z s ( U s + Θ sr ) − / ( U s + νZ sr ) / (cid:16) c e − κ ( s − u ) + c ψ ( r, s, u ) (cid:17) ¯ σ r dr + Z s Z ∞ ∆ y f (cid:0) Z sr , Θ sr − (cid:1) ˜ N ( dr, dz ) + Z s Z ∞ ∆ y f (cid:0) Z sr , Θ sr − (cid:1) ℓ ( dz ) dr. Now, taking expectations and writing the terms with ν as an error term of order O (cid:0) ν (cid:1) , we can rewrite the previous equation as E [ f ( Z ss , Θ ss )] = E [ f ( Z s , Θ s )] DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 28 + c ν (cid:0) H − (cid:1) E "Z s Z r (cid:18) U s + νZ sr U s + Θ sr (cid:19) / e − κ ( r − u ) ( s − r ) H − p ¯ σ u dW u dr + E (cid:20)Z s Z ∞ ∆ y f (cid:0) Z sr , Θ sr − (cid:1) ℓ ( dz ) dr (cid:21) + O (cid:0) ν (cid:1) . Again, using Fubini and noting the integrand inside the expectation for the con-tinuous term is square integrable, we obtain E [ f ( Z ss , Θ ss )] = E [ U s ] + E (cid:20)Z s Z ∞ ∆ y f (cid:0) Z sr , Θ sr − (cid:1) ℓ ( dz ) dr (cid:21) + O (cid:0) ν (cid:1) , where f ( Z s , Θ s ) = U s . Finally, it only remains to notice that the term ∆ y f (cid:0) Z sr , Θ sr − (cid:1) = ∂ f ( Z ss , Θ ss ) c η z ∈ O (cid:0) η (cid:1) , allowing us to write E [ f ( Z ss , Θ ss )] = E hp σ s ¯ σ s i = E [ U s ] + O (cid:0) ν + η (cid:1) . • In order to see that dL [ W, M c ] t = ν (cid:16)R Tt A ( T, s ) e − κ ( s − t ) ds (cid:17) ¯ σ t dW t − ν ¯ σ t A ( T, t ) dt + O (cid:0) ν + νη (cid:1) dt , we can compute the following: dL [ W, M c ] t = d (cid:18) ν Z Tt A ( T, s ) E t hp σ s ¯ σ s i ds (cid:19) = d (cid:18) ν Z Tt A ( T, s ) h ¯ σ t e − κ ( s − t ) + θ (cid:16) − e − κ ( s − t ) (cid:17) + O (cid:0) ν + η (cid:1)i ds (cid:19) = d (cid:18) ν ¯ σ t Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) + d (cid:18) νθ Z Tt A ( T, s ) (cid:16) − e − κ ( s − t ) (cid:17) ds (cid:19) + d (cid:18) ν Z Tt A ( T, s ) O (cid:0) ν + η (cid:1) ds (cid:19) = νd ¯ σ t Z Tt A ( T, s ) e − κ ( s − t ) ds + ν ¯ σ t d (cid:18)Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) + νθd (cid:18)Z Tt A ( T, s ) (cid:16) − e − κ ( s − t ) (cid:17) ds (cid:19) + ν O (cid:0) ν + η (cid:1) d (cid:18)Z Tt A ( T, s ) ds (cid:19) . Applying Leibniz rule to derivate under the integral sign we can rewrite the previousexpression as dL [ W, M c ] t = νd ¯ σ t Z Tt A ( T, s ) e − κ ( s − t ) ds + ν ¯ σ t (cid:20) − A ( T, t ) dt + κ (cid:18)Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) dt (cid:21) + νθ (cid:20) − κ (cid:18)Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) dt (cid:21) − ν O (cid:0) ν + η (cid:1) A ( T, t ) dt = ν (cid:18)Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) ¯ σ t dW t − ν ¯ σ t A ( T, t ) dt + O (cid:0) ν + νη (cid:1) . • In order to see that dD [ M c , M c ] t = ν (cid:16)R Tt A ( T, s ) e − κ ( s − t ) ds (cid:17) ¯ σ t dW t − ν ¯ σ t A ( T, t ) dt ,we proceed in an analogous way as in the previous set of computations. dD [ M c , M c ] t = d (cid:18) ν Z Tt A ( T, s ) E t (cid:2) ¯ σ s (cid:3) ds (cid:19) DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 29 = ν d (cid:18)Z Tt A ( T, s ) h ¯ σ t e − κ ( s − t ) + θ (cid:16) − e − κ ( s − t ) (cid:17)i ds (cid:19) = ν (cid:20) d (cid:18) ¯ σ t Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) + θd (cid:18)Z Tt A ( T, s ) (cid:16) − e − κ ( s − t ) (cid:17) ds (cid:19)(cid:21) = ν (cid:20) d ¯ σ t Z Tt A ( T, s ) e − κ ( s − t ) ds + ¯ σ t d (cid:18)Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19)(cid:21) + ν θd (cid:18)Z Tt A ( T, s ) (cid:16) − e − κ ( s − t ) (cid:17) ds (cid:19) = ν (cid:18)Z Tt A ( T, s ) e − κ ( s − t ) ds (cid:19) ¯ σ t dW t − ν ¯ σ t A ( T, t ) dt. 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Financ. Econ. ,3(1-2):125–144, 1976.[22] G. Di Nunno, B. Øksendal, and F. Proske. Malliavin calculus for Lévy processes with applications tofinance . Universitext. Springer-Verlag, Berlin, 2009. DECOMPOSITION FORMULA FOR FRACTIONAL HESTON JUMP DIFFUSION MODELS 30 ( Marc Lagunas-Merino ) Department of Mathematics, University of Oslo, P.O. Box 1053, Blin-dern, N–0316 Oslo, Norway E-mail address : [email protected] ( Salvador Ortiz-Latorre ) Department of Mathematics, University of Oslo, P.O. Box 1053, Blin-dern, N–0316 Oslo, Norway E-mail address ::