Decomposition formula for rough Volterra stochastic volatility models
Raul Merino, Jan Pospíšil, Tomáš Sobotka, Tommi Sottinen, Josep Vives
DDecomposition formula for rough Volterrastochastic volatility models
Raúl Merino , Jan Pospíšil ∗ , Tomáš Sobotka ,Tommi Sottinen , 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 2732/8, 301 00 Plzeň, Czech Republic, Department of Mathematics and Statistics, University of Vaasa,P.O. Box 700, FIN-65101 Vaasa, Finland, VidaCaixa S.A., Investment Risk Management Department,C/Juan Gris, 2-8, 08014 Barcelona, Spain.
Received 1 August 2019
Abstract
The research presented in this article provides an alternative option pricing approachfor a class of rough fractional stochastic volatility models. These models are increasinglypopular between academics and practitioners due to their surprising consistency with financialmarkets. However, they bring several challenges alongside. Most noticeably, even simplenon-linear financial derivatives as vanilla European options are typically priced by means ofMonte-Carlo (MC) simulations which are more computationally demanding than similar MCschemes for standard stochastic volatility models.In this paper, we provide a proof of the prediction law for general Gaussian Volterraprocesses. The prediction law is then utilized to obtain an adapted projection of the fu-ture squared volatility – a cornerstone of the proposed pricing approximation. Firstly, adecomposition formula for European option prices under general Volterra volatility models isintroduced. Then we focus on particular models with rough fractional volatility and we derivean explicit semi-closed approximation formula. Numerical properties of the approximationfor a popular model – the rBergomi model – are studied and we propose a hybrid calibrationscheme which combines the approximation formula alongside MC simulations. This schemecan significantly speed up the calibration to financial markets as illustrated on a set of AAPLoptions.
Keywords : Volterra stochastic volatility; rough volatility; rough Bergomi model; optionpricing; decomposition formula
MSC classification : 60G22; 91G20; 91G60
JEL classification : G12; C58; C63 ∗ Corresponding author, [email protected] a r X i v : . [ q -f i n . P R ] A ug Introduction
It is well known that the main issue of the Black-Scholes model lies in its assumptions aboutvolatility of the modelled asset. Opposed to the model assumptions, the realized volatility timeseries tend to cluster, depend on the spot asset level and certainly they do not take a constantvalue within a reasonable time-frame (see e.g. Cont (2001)).To deal with the aforementioned inconsistencies, stochastic volatility (SV) models have beenproposed originally by Hull and White (1987) and later e.g. by Heston (1993). These modelsdo not only assume that the asset price follows a specific stochastic process, but also that theinstantaneous volatility of asset returns is of random nature as well. Especially, the latter approachby Heston became popular in the eyes of both practitioners and academics. Several modifications ofthis model have been proposed over the last 20 years: models with jump-diffusion dynamics (Bates1996; Duffie, Pan, and Singleton 2000), with time-dependent parameters (Mikhailov and Nögel2003; Elices 2008; Benhamou, Gobet, and Miri 2010), with fractionally scaled volatility (Comteand Renault 1998; Alòs, León, and Vives 2007; El Euch and Rosenbaum 2019) and models withseveral aspects combined (Pospíšil and Sobotka 2016; Baustian, Mrázek, Pospíšil, and Sobotka2017).The original pricing approach of Heston (1993) was several times revisited, e.g by Lewis (2000),Attari (2004) and Albrecher, Mayer, Schoutens, and Tistaert (2007) with focus on semi-closed formFourier transform solutions by Kahl and Jäckel (2006), Alfonsi (2010) with respect to Monte-Carlosimulation techniques and last but not least by Alòs (2012) who introduced an analytical approachto option pricing approximation. This approach improves on the techniques introduced by Hulland White (1987) and shows how an adaptive projection of future volatility can be used to priceEuropean options under Heston (1993) model. Many other papers generalized this idea, see e.g.Alòs, de Santiago, and Vives (2015); Merino and Vives (2015), or recently also a paper by Merino,Pospíšil, Sobotka, and Vives (2018). In this article, we revisit the latter approach and we come upwith the approximation technique for SV models with volatility process driven by the fractionalBrownian motion. This includes exponential rough fractional volatility models introduced byGatheral, Jaisson, and Rosenbaum (2014, 2018).Although many SV models have been proposed since the original Hull and White (1987) model,it seems that none of them can be considered as the universal best market practice approach.Several models might perform well for calibration to complex volatility surfaces, but can sufferfrom over-fitting or they might not be robust in the sense described by Pospíšil, Sobotka, andZiegler (2018). Also a model with a good fit to implied volatility surface might not be in-line withthe observed time-series properties.Pioneers of the fractional SV models – Comte and Renault (1998), see also Comte, Coutin,and Renault (2012) – assumed the so-called Hurst parameter ranged within H ∈ (1 / , whichimplies that the spot variance evolution is represented by a persistent process, i.e. it would havea long-memory property. In Alòs, León, and Vives (2007), a mean reverting fractional stochas-tic volatility model with H ∈ (0 , was presented. Gatheral, Jaisson, and Rosenbaum (2018)and Bayer, Friz, and Gatheral (2016) came up with a more detailed analysis of rough fractionalvolatility models which should be consistent with market option prices (Bayer, Friz, and Gatheral2016), with realized volatility time series and also they could provide superior volatility predictionresults to several other models (Bennedsen, Lunde, and Pakkanen 2017). An approach consider-ing a two factor fractional volatility model, combining a rough term ( H < ) and a persistentone ( H > ), was presented in Funahashi and Kijima (2017). Recently, also an approximation fortarget-volatility options under log-normal fractional SABR model was studied by Alòs, Chatterjee,Tudor, and Wang (2019) who use the Malliavin caluculus techniques to derive the decompositionformula. In parallel, short-term at-the-money asymptotics for a class of stochastic volatility mod-els were studied by El Euch, Fukasawa, Gatheral, and Rosenbaum (2019) who use the Edgeworthexpansion of the density of an asset price. Named after a hydrologist Harold Edwin Hurst, for more information on fractional Brownian motion, seeSection 4.3 and e.g. the article by Mandelbrot and Van Ness (1968)
Let S = ( S t , t ∈ [0 , T ]) be a strictly positive asset price process under a market chosen risk neutralprobability measure P that follows the model: d S t = rS t d t + σ t S t (cid:16) ρ d W t + (cid:112) − ρ d ˜ W t (cid:17) , (1)where S is the current price, r ≥ is the interest rate, W t and ˜ W t are independent standardWiener processes defined on a probability space (Ω , F , P ) and ρ ∈ ( − , . In the following, wewill denote by F W and F ˜ W the filtrations generated by W and ˜ W respectively. Moreover, wedefine F := F W ∨ F ˜ W . The volatility process σ t is a square-integrable process assumed to beadapted to the filtration generated by W and its trajectories are assumed to be a.s. càdlàg andstricly positive a.e.. Note that ρ is the correlation between the price and the volatility processes.Without any loss of generality, it will be convenient in the following sections to make thechange of variable X t = log S t , t ∈ [0 , T ] , and write d X t = (cid:18) r − σ t (cid:19) d t + σ t (cid:16) ρ d W t + (cid:112) − ρ d ˜ W t (cid:17) . (2)Recall that Z := ρW + (cid:112) − ρ ˜ W is a standard Wiener process.The following notation will be used throughout the paper: • 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 ± ( y ) = 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 ) + ] where E t is the conditional expectation respect to the σ − algebra F t . • Recall that from the Feynman-Kac formula for the model (2), the operator L y := ∂ t + 12 y ∂ x + (cid:18) r − y (cid:19) ∂ x − r (3)satisfies L y BS ( t, x, y ) = 0 . • We define the operators
Λ := ∂ x , Λ n := ∂ nx , Γ := (cid:0) ∂ x − ∂ x (cid:1) and Γ = Γ ◦ Γ . In particular,for the Black-Scholes formula, using straightforward calculations, we get: 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 R t := 18 E t (cid:34)(cid:90) Tt d (cid:104) M, M (cid:105) u (cid:35) and U t := ρ E t (cid:34)(cid:90) Tt σ u d (cid:104) M, W (cid:105) u (cid:35) , where (cid:104)· , ·(cid:105) denotes the quadratic covariation process and M is the F− martingale defined by M t := (cid:90) T E t (cid:2) σ s (cid:3) d s. (4) In this section, we provide an insight on a generic decomposition formula based on the workof Alòs (2012), Merino and Vives (2015) and Merino, Pospíšil, Sobotka, and Vives (2018). Inparticular, we recover the results for a generic stochastic volatility model presented in Merino,Pospíšil, Sobotka, and Vives (2018).It is well known that if the stochastic volatility process is independent of the price process, thepricing formula of a vanilla European call option is given by V t = E t [ BS ( t, X t , ¯ σ t )] where ¯ σ t is the so called average future variance that is defined by ¯ σ t := 1 T − t (cid:90) Tt σ u d u. Naturally, ¯ σ t is called the average future volatility .4e consider the adapted projection of the future variance a t := (cid:90) Tt E t [ σ u ] d u (5)and the average future variance as v t := E t (¯ σ t ) = a t T − t to obtain a decomposition of V t in terms of v t . This idea switches an anticipative problem relatedto the anticipative process ¯ σ t into a non-anticipative one related to the adapted process v t .Taking into account M defined in (4), we can write d v t = 1 T − t (cid:2) d M t + (cid:0) v t − σ t (cid:1) d t (cid:3) . In this paper, we will utilize the following lemma which is proved in Alòs (2012), p. 406.
Lemma 3.1.
Let ≤ t ≤ u ≤ T and G t := F t ∨ F WT . For every n ≥ , there exists C = C ( n ) suchthat | E ( Λ n Γ BS ( u, X u , v u ) | G t ) | ≤ C ( a u ) − ( n +1) , where a t is defined by (5) . Remark 3.2.
It is easy to see that the previous Lemma holds for put options and for several othernon-path dependent options as well (e.g. Gap options).
Now we use a generic decomposition formula proved in Merino, Pospíšil, Sobotka, and Vives(2018).
Theorem 3.3 (Generic decomposition formula) . Let B t be a continuous semimartingale withrespect to the filtration F W , let A ( t, x, y ) be a C , , ([0 , T ] × [0 , ∞ ) × [0 , ∞ )) function and let v t and 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 the following way: E t (cid:104) e − r ( T − t ) A ( T, X T , v T ) B T (cid:105) = A ( t, X t , v t ) B t + E t (cid:34)(cid:90) Tt e − r ( u − t ) ∂ y A ( u, X u , v u ) B u T − u (cid:0) v u − σ u (cid:1) d u (cid:35) + E t (cid:34)(cid:90) Tt e − r ( u − t ) A ( u, X u , v u ) d B u (cid:35) + 12 E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:0) ∂ x − ∂ x (cid:1) A ( u, X u , v u ) B u (cid:0) σ u − v u (cid:1) d u (cid:35) + 12 E t (cid:34)(cid:90) Tt e − r ( u − t ) ∂ y A ( u, X u , v u ) B u T − u ) d (cid:104) M · , M · (cid:105) u (cid:35) + ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ∂ x,y A ( u, X u , v u ) B u σ u T − u d (cid:104) W · , M · (cid:105) u (cid:35) + ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ∂ x A ( u, X u , v u ) σ u d (cid:104) W · , B · (cid:105) u (cid:35) + E t (cid:34)(cid:90) Tt e − r ( u − t ) ∂ y A ( u, X u , v u ) 1 T − u d (cid:104) M · , B · (cid:105) u (cid:35) . roof. See the proof of Theorem 3.1 in Merino, Pospíšil, Sobotka, and Vives (2018).Using the previous decomposition formula, we find that
Corollary 3.4 (BS decompostion formula) . Under assumptions of the Theorem 3.3, we can obtaina decomposition of European option price V t as: V t = BS ( t, X t , v t )+ ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W · , M · (cid:105) u (cid:35) + 18 E t (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M · , M · (cid:105) u (cid:35) = BS ( t, X t , v t ) + ( I ) + ( II ) Proof.
Using Theorem 3.3 with A ( t, X t , v t ) = BS ( t, X t , v t ) and B ≡ , the proof follows in astraightforward way.The terms ( I ) and ( II ) are not easy to evaluate. Therefore, it becomes important to findsimpler approximations to ( I ) and ( II ) and estimate the error terms. In order to find theseapproximations, we are going to apply Theorem 3.3 to find a decomposition formula for the terms ( I ) and ( II ) . Using A ( t, X t , v t ) = ΛΓ BS ( t, X t , v t ) and B t = U t = ρ E t (cid:34)(cid:90) Tt σ u d (cid:104) W · , M · (cid:105) u (cid:35) a decomposition of the term ( I ) can be found, and using A ( t, X t , v t ) = Γ BS ( t, X t , v t ) and B t = R t = 18 E t (cid:34)(cid:90) Tt d (cid:104) M · , M · (cid:105) u (cid:35) a decomposition of the term ( II ) is obtained.After that process, we can approximate the price of a call option by V t = BS ( t, X t , v t )+ ΛΓ BS ( t, X t , v t ) U t + Γ BS ( t, X t , v t ) R t + (cid:15) t . where (cid:15) t denotes error terms. Terms of (cid:15) t under a general setting for σ t are provided in AppendixA. We note that the error term will depend on the assumed volatility dynamics. In this section, we apply the generic decomposition formula to model (2) with general Volterravolatility process defined as σ t := g ( t, Y t ) , t ≥ , (6)6here g : [0 , + ∞ ) × R (cid:55)→ [0 , + ∞ ) is a deterministic function such that σ t belongs to L (Ω × [0 , + ∞ )) and Y = ( Y t , t ≥ is the Gaussian Volterra process Y t = (cid:90) t K ( t, s ) d W s , (7)where K ( t, s ) is a kernel such that for all t > t (cid:90) K ( t, s ) d s < ∞ , (A1)and F Yt = F Wt . (A2)Let r ( t, s ) := E [ Y t Y s ] , t, s ≥ , (8)denote the autocovariance function of process Y t and r ( t ) := r ( t, t ) = E [ Y t ] , t ≥ , (9)be the variance function (i.e. the second moment).Extending the Theorem 3.1 in Sottinen and Viitasaari (2017) enables us to rephrase the adaptedprojection of the future squared volatility. Theorem 4.1 (Prediction law for Gaussian Volterra processes) . Let ( Y t , t ≥ be the Gaus-sian Volterra process (7) satisfying assumptions (A1) and (A2) . Then, the conditional process ( Y u |F t , ≤ t ≤ u ) is Gaussian with F u -measurable mean function ˆ m t ( u ) := E t [ Y u ] = (cid:90) t K ( u, s ) d W s , (10) and deterministic covariance function ˆ r ( u , u | t ) := E t [( Y u − ˆ m t ( u )) ( Y u − ˆ m t ( u ))]= r ( u , u ) − (cid:90) t K ( u , v ) K ( u , v ) d v (11) for u , u ≥ t .Proof. Let ≤ t ≤ u . Then ˆ m t ( u ) = E t [ Y u ] = E (cid:34)(cid:90) u K ( u, s ) d W s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F Wt (cid:35) = (cid:90) t K ( u, s ) d W s and ˆ r ( u , u | t ) = E [( Y u − ˆ m t ( u )) ( Y u − ˆ m t ( u )) | F Wt ]= E (cid:34)(cid:18)(cid:90) u K ( u , v ) d W v − (cid:90) t K ( u , v ) d W v (cid:19) · (cid:18)(cid:90) u K ( u , v ) d W v − (cid:90) t K ( u , v ) d W v (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F Wt (cid:35) = E (cid:34)(cid:90) u t K ( u , v ) d W v (cid:90) u t K ( u , v ) d W v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F Wt (cid:35) (cid:90) u ∧ u t K ( u , v ) K ( u , v ) d v = r ( u , u ) − (cid:90) t K ( u , v ) K ( u , v ) d v. In the upcoming sections, we will denote ˆ r ( u | t ) := ˆ r ( u, u | t ) .Under the general volatility process (6), we have v t = 1 T − t (cid:90) Tt E t (cid:2) g ( u, Y u ) (cid:3) d u and the martingale M t = (cid:90) T E t (cid:2) g ( u, Y u ) (cid:3) d u. In the upcoming lemma, we express the conditional expectation of the future squared volatilityin terms of the mean function ˆ m t ( u ) .Let us denote: F ( t, ˆ m t ( u )) := E t (cid:2) g ( u, Y u ) (cid:3) , Lemma 4.2 (Auxiliary terms in the decomposition formula for the general volatility model) . Let ≤ t ≤ u and F ( t, ˆ m t ( u )) = E t (cid:2) g ( u, Y u ) (cid:3) , then d F ( t, ˆ m t ( u )) = (cid:18) ∂ F ( t, ˆ m t ( u )) + 12 ∂ F ( t, ˆ m t ( u )) K ( u, t ) (cid:19) d t + ∂ F ( t, ˆ m t ( u )) d ˆ m t ( u ) , (12) d (cid:104) M · , W · (cid:105) t = (cid:90) T ∂ F ( t, ˆ m t ( u )) K ( u, t ) d u d t, (13) d (cid:104) M · , M · (cid:105) t = (cid:90) T (cid:90) T ∂ F ( t, ˆ m t ( u )) ∂ F ( t, ˆ m t ( u )) ·· K ( u , t ) K ( u , t ) d u d u d t. (14) Proof.
Let ≤ t ≤ u and X t ( u ) = E t (cid:2) g ( u, Y u ) (cid:3) . (15)Theorem 4.1 implies that X t ( u ) = (cid:90) R g ( u, z ) ˆ ϕ t ( u, z ) d z, where ˆ ϕ t ( u, z ) = 1 (cid:112) π ˆ r ( u | t ) exp (cid:26) −
12 ( z − ˆ m t ( u )) ˆ r ( u | t ) (cid:27) (16)is a Gaussian density function with stochastic mean ˆ m t ( u ) and deterministic variance ˆ r ( u | t ) . Tocalculate the quadratic variation, we note that ˆ ϕ t ( u, z ) = f ( t, ˆ m t ( u )) , (17)where f ( t, m ) = 1 (cid:112) π ˆ r ( u | t ) exp (cid:26) −
12 ( z − m ) ˆ r ( u | t ) (cid:27) . (18)Since d (cid:104) ˆ m · ( u ) (cid:105) t = K ( u, t ) d t,
8e retrieve the following expression by Itô’s formula, d f ( t, ˆ m t ( u )) = (cid:18) ∂ f ( t, ˆ m t ( u )) + 12 ∂ f ( t, ˆ m t ( u )) K ( u, t ) (cid:19) d t + ∂ f ( t, ˆ m t ( u )) d ˆ m t ( u ) , and consequently, d (cid:104) f ( · , ˆ m · ( u )) (cid:105) t = ( ∂ f ( t, ˆ m t ( u )) K ( u, t )) d t. Due to: ∂ f ( t, ˆ m t ( u )) = z − ˆ m t ( u )ˆ r ( u | t ) ˆ ϕ t ( u, z ) , we obtain d (cid:104) ˆ ϕ · ( u, z ) (cid:105) t = (cid:18) z − ˆ m t ( u )ˆ r ( u | t ) ˆ ϕ t ( u, z ) K ( u, t ) (cid:19) d t. More generally, we have d (cid:104) ˆ ϕ · ( u, z ) , ϕ · ( u, z ) (cid:105) t = ( z − ˆ m t ( u ))( z − ˆ m t ( u ))ˆ r ( u | t ) ˆ ϕ t ( u, z ) ˆ ϕ t ( u, z ) K ( u, t ) d t, and consequently, d (cid:104) X · ( u ) (cid:105) t = d (cid:28)(cid:90) R g ( u, z ) ˆ ϕ ( u, z ) d z , (cid:90) R g ( u, z ) ˆ ϕ ( u, z ) d z (cid:29) t = (cid:90) (cid:90) R g ( u, z ) g ( u, z ) d (cid:104) ˆ ϕ ( u, z ) , ˆ ϕ ( u, z ) (cid:105) t d z d z = (cid:90) (cid:90) R g ( u, z ) g ( u, z )( ˆ m t ( u ) − z )( ˆ m t ( u ) − z ) ·· ˆ ϕ t ( u, z ) ˆ ϕ t ( u, z ) (cid:18) K ( u, t )ˆ r ( u | t ) (cid:19) d z d z d t. Let F = F ( t, m ) be a C , -function of time t and “spot” m = ˆ m t ( u ) of the prediction martingale.Because of the filtrations are the same, we have, in general, d (cid:104) F ( · , ˆ m · ( u )) , W · (cid:105) t = ∂ F ( t, ˆ m t ( u )) K ( u, t ) d t. Now we set F ( t, ˆ m t ( u )) = X t ( u ) = (cid:90) R g ( u, z ) ˆ ϕ t ( u, z ) d z, and applying the Itô formula, we obtain (12). Moreover, d (cid:104) M · , W · (cid:105) t = d (cid:42)(cid:90) T F ( · , ˆ m · ( u )) d u, W · (cid:43) t = (cid:90) T d (cid:104) F ( · , ˆ m · ( u )) , W · (cid:105) t d u = (cid:90) T ∂ F ( t, ˆ m t ( u )) K ( u, t ) d u d t and d (cid:104) M · , M · (cid:105) t = d (cid:42)(cid:90) T F ( · , ˆ m · ( u )) d u, (cid:90) T F ( · , ˆ m · ( u )) d u (cid:43) t (cid:90) T (cid:90) T d (cid:104) F ( · , ˆ m · ( u )) , F ( · , ˆ m · ( u )) (cid:105) t d u d u = (cid:90) T (cid:90) T ∂ F ( t, ˆ m t ( u )) ∂ F ( t, ˆ m t ( u )) K ( u , t ) K ( u , t ) d u d u d t. Assume now that X t is the log-price process (2) with σ t being the exponential Volterra volatilityprocess σ t = g ( t, Y t ) = σ exp (cid:26) ξY t − αξ r ( t ) (cid:27) , t ≥ , (19)where ( Y t , t ≥ is the Gaussian Volterra process (7) satisfying assumptions (A1) and (A2), r ( t ) is its autocovariance function (9), and σ > , ξ > and α ∈ [0 , are model parameters. Lemma 4.3 (Auxiliary terms in the decomposition formula for the exponential Volterra volatilitymodel) . Let σ t be as in (19) and ≤ t ≤ u . Then F ( t, ˆ m t ( u )) = σ exp (cid:8) ξ ˆ m t ( u ) + 2 ξ ˆ r ( u | t ) − αξ r ( u ) (cid:9) , (20) ∂ F ( t, ˆ m t ( u )) = 2 ξF ( t, ˆ m t ( u )) , (21) d (cid:104) M · , W · (cid:105) t = 2 σ ξ (cid:90) T exp (cid:8) ξ ˆ m t ( u ) + 2 ξ ˆ r ( u | t ) − αξ r ( u ) (cid:9) K ( u, t ) d u d t, (22) d (cid:104) M · , M · (cid:105) t = 4 σ ξ (cid:90) T (cid:90) T exp { ξ ( ˆ m t ( u ) + ˆ m t ( u )) } ·· exp (cid:8) ξ (ˆ r ( u | t ) + ˆ r ( u | t )) (cid:9) ·· exp (cid:8) − αξ ( r ( u ) + r ( u )) (cid:9) ·· K ( u , t ) K ( u , t ) d u d u d t. (23) Proof.
Let ˆ ϕ t ( u, z ) be given by (16). Then F ( t, ˆ m t ( u )) = (cid:90) R g ( u, z ) ˆ ϕ t ( u, z ) d z, = σ e − αξ r ( u ) (cid:90) R e ξz (cid:112) π ˆ r ( u | t ) exp (cid:18) −
12 ( z − ˆ m t ( u )) ˆ r ( u | t ) (cid:19) d z. It is now easy to calculate the partial derivative ∂ F . We get ∂ F ( t, ˆ m t ( u )) = σ e − αξ r ( u ) (cid:90) R e ξz (cid:112) π ˆ r ( u | t ) exp (cid:18) −
12 ( z − ˆ m t ( u )) ˆ r ( u | t ) (cid:19) z − ˆ m t ( u )ˆ r ( u | t ) d z. Changing variables v = z − ˆ m t ( u ) √ ˆ r ( u | t ) and d z = (cid:112) ˆ r ( u | t ) d v , we obtain ∂ F ( t, ˆ m t ( u )) = σ e − αξ r ( u ) (cid:112) ˆ r ( u | t ) e ξ ˆ m t ( u ) (cid:90) R e ξ √ ˆ r ( u | t ) v vφ ( v ) d v = σ e − αξ r ( u ) (cid:112) ˆ r ( u | t ) e ξ ˆ m t ( u ) E (cid:16) e ξ √ ˆ r ( u | t ) Z Z (cid:17) , where Z ∼ N (0 , . Using formula E [ Ze αZ ] = αe α , we get ∂ F ( t, ˆ m t ( u )) = 2 σ ξ exp (cid:8) ξ ˆ m u ( u ) + 2 ξ ˆ r ( u | t ) − αξ r ( u ) (cid:9) = 2 ξF ( t, ˆ m t ( u )) . Remaining formulas follow accordingly. 10 emark 4.4.
Using that F ( t, ˆ m t ( u )) = E t (cid:2) σ u (cid:3) , it is straightforward to see that d M t = 2 ξ (cid:32)(cid:90) Tt E t (cid:2) σ u (cid:3) K ( u, t ) d u (cid:33) d W t , (24) d (cid:104) M · , W · (cid:105) t = 2 ξ (cid:90) T E t (cid:2) σ u (cid:3) K ( u, t ) d u d t, (25) d (cid:104) M · , M · (cid:105) t = 4 ξ (cid:90) T (cid:90) T E t (cid:2) σ u (cid:3) E t (cid:2) σ u (cid:3) K ( u , t ) K ( u , t ) d u d u d t. (26) Lemma 4.5.
Let σ t be as in (19) and ≤ t ≤ u . Then, we can re-write F ( t, ˆ m t ( u )) as E t (cid:2) σ u (cid:3) = σ t exp (cid:26) − αξ ( r ( u ) − r ( t )) + 2 ξ (cid:90) t ( K ( u, z ) − K ( t, z )) dW z + 2 ξ ˆ r ( u | t ) (cid:27) . (27) Moreover, we also have the following equalities E t (cid:20) σ u exp (cid:26) ξ (cid:90) u ( K ( s, z ) − K ( u, z )) dW z (cid:27)(cid:21) = σ t exp (cid:110) − αξ ( r ( u ) − r ( t )) + ξ (cid:90) t (2 K ( s, z ) + K ( u, z ) − K ( t, z )) dW z + ξ (cid:90) ut (2 K ( s, z ) + K ( u, z )) dz (cid:111) and E t (cid:20) σ u exp (cid:26) ξ (cid:90) u ( K ( s, z ) + K ( v, z ) − K ( u, z )) dW z (cid:27)(cid:21) = σ t exp (cid:110) − αξ ( r ( u ) − r ( t )) + 2 ξ (cid:90) t ( K ( s, z ) + K ( v, z ) − K ( t, z )) dW z + 2 ξ (cid:90) ut ( K ( s, z ) + K ( v, z )) dz (cid:111) . Proof.
The calculations to obtain these statements are straightforward.
Proposition 4.6 (Terms in the approximation formula for the exponential Volterra volatilitymodel) . Let σ t be as in (19) and ≤ t ≤ u . Then U t = ρξσ t (cid:90) Tt (cid:90) T exp (cid:26) − αξ ( r ( u ) − r ( t )) + ξ (cid:90) t (2 K ( s, z ) + K ( u, z ) − K ( t, z )) d W z (cid:27) · exp (cid:110) ξ (cid:90) ut (2 K ( s, z ) + K ( u, z )) d z − αξ ( r ( s ) − r ( u )) + 2 ξ ˆ r ( s | u ) (cid:111) · K ( s, u ) d s d u (28) and R t = 12 ξ σ t (cid:90) Tt (cid:90) T (cid:90) T exp (cid:110) − αξ ( r ( s ) + r ( v ) − r ( t )) + 2 ξ (ˆ r ( s | u ) + ˆ r ( v | u ))+ 2 ξ (cid:90) t ( K ( s, z ) + K ( v, z ) − K ( t, z )) d W z + 2 ξ (cid:90) ut ( K ( s, z ) + K ( v, z )) d z (cid:111) · K ( s, u ) K ( v, u ) d s d v d u (29)11 n particular, U = ρξσ (cid:90) T (cid:90) T exp (cid:110) ξ (cid:90) u [2 K ( s, z ) + K ( u, z )] d z (cid:111) ·· exp (cid:110) ξ ˆ r ( s | u ) − αξ r ( u ) − αξ r ( s ) (cid:111) K ( s, u ) d s d u (30) and R = 12 σ ξ (cid:90) T (cid:90) T (cid:90) T exp (cid:110) ξ (cid:90) u [ K ( s, z ) + K ( v, z )] d z (cid:111) ·· exp (cid:110) ξ (ˆ r ( s | u ) + ˆ r ( v | u )) − αξ ( r ( s ) + r ( v )) (cid:111) ·· K ( s, u ) K ( v, u ) d s d v d u. (31) Proof.
We have that U t = ρ E t (cid:34)(cid:90) Tt σ u d (cid:104) M · , W · (cid:105) u (cid:35) = ρξ E t (cid:34)(cid:90) Tt σ u (cid:32)(cid:90) T E u (cid:2) σ s (cid:3) K ( s, u ) d s (cid:33) d u (cid:35) = ρξ (cid:90) Tt E t (cid:34) σ u (cid:32)(cid:90) T E u (cid:2) σ s (cid:3) K ( s, u ) d s (cid:33)(cid:35) d u = ρξ (cid:90) Tt (cid:90) T E t (cid:20) σ u exp (cid:110) − αξ ( r ( s ) − r ( u )) + 2 ξ (cid:90) u ( K ( s, z ) − K ( u, z )) d W z + 2 ξ ˆ r ( s | u ) (cid:111) K ( s, u ) (cid:21) d s d u = ρξ (cid:90) Tt (cid:90) T E t (cid:20) σ u exp (cid:110) ξ (cid:90) u ( K ( s, z ) − K ( u, z )) d W z (cid:111)(cid:21) · exp (cid:110) − αξ ( r ( s ) − r ( u )) + 2 ξ ˆ r ( s | u ) (cid:111) K ( s, u ) d s d u = ρξσ t (cid:90) Tt (cid:90) T exp (cid:26) − αξ ( r ( u ) − r ( t )) + ξ (cid:90) t (2 K ( s, z ) + K ( u, z ) − K ( t, z )) d W z (cid:27) · exp (cid:110) ξ (cid:90) ut (2 K ( s, z ) + K ( u, z )) d z − αξ ( r ( s ) − r ( u )) + 2 ξ ˆ r ( s | u ) (cid:111) K ( s, u ) d s d u Similarly, we have that R t = 18 E t (cid:34)(cid:90) Tt d (cid:104) M · , M · (cid:105) u (cid:35) = 12 ξ E t (cid:90) Tt (cid:32)(cid:90) T E u (cid:2) σ s (cid:3) K ( s, u ) d s (cid:33) d u = 12 ξ (cid:90) Tt E t (cid:32)(cid:90) T E u (cid:2) σ s (cid:3) K ( s, u ) d s (cid:33) d u = 12 ξ (cid:90) Tt E t (cid:34)(cid:32)(cid:90) T (cid:90) T E u (cid:2) σ s (cid:3) E u (cid:2) σ v (cid:3) K ( s, u ) K ( v, u ) d s d v (cid:33)(cid:35) d u = 12 ξ (cid:90) Tt E t (cid:34)(cid:90) T (cid:90) T σ u K ( s, u ) K ( v, u ) exp (cid:8) − αξ ( r ( s ) + r ( v ) − r ( u )) ξ (cid:90) u ( K ( s, z ) + K ( v, z ) − K ( u, z )) d W z + 2 ξ (ˆ r ( s | u ) + ˆ r ( v | u )) (cid:27) d s d v (cid:21) d u = 12 ξ (cid:90) Tt (cid:90) T (cid:90) T E t (cid:20) σ u exp (cid:26) ξ (cid:90) u ( K ( s, z ) + K ( v, z ) − K ( u, z )) d W z (cid:27)(cid:21) exp (cid:8) − αξ ( r ( s ) + r ( v ) − r ( u )) + 2 ξ (ˆ r ( s | u ) + ˆ r ( v | u )) (cid:9) K ( s, u ) K ( v, u ) d s d v d u = 12 ξ σ t (cid:90) Tt (cid:90) T (cid:90) T exp (cid:110) − αξ ( r ( s ) + r ( v ) − r ( t )) + 2 ξ (ˆ r ( s | u ) + ˆ r ( v | u ))+ 2 ξ (cid:90) t ( K ( s, z ) + K ( v, z ) − K ( t, z )) d W z + 2 ξ (cid:90) ut ( K ( s, z ) + K ( v, z )) d z (cid:111) · K ( s, u ) K ( v, u ) d s d v d u. For the exponential Volterra volatility model we can determine an upper error bound for theprice approximation in the following way.
Theorem 4.7 (Upper error bound for the exponential Volterra volatility model) . Let X t be alog-price process (2) with σ t being the exponential Volterra volatility process (19) . Then we canexpress the call option fair value V t using the processes R t , U t from Proposition 4.6. In particular, V t = BS ( t, X t , v t )+ ΛΓ BS ( t, X t , v t ) U t + Γ BS ( t, X t , v t ) R t + (cid:15) t , where (cid:15) t are error terms of order O (cid:16)(cid:0) ξ + ρξ (cid:1) (cid:17) .Proof. Note that using (24) we have that d (cid:104) M, M (cid:105) t = 4 ξ (cid:32)(cid:90) Tt E t (cid:2) σ u (cid:3) K ( u, t ) d u (cid:33) d t, d (cid:104) M, W (cid:105) t = 2 ξ (cid:32)(cid:90) Tt E t (cid:2) σ u (cid:3) K ( u, t ) d u (cid:33) d t. Applying the Jensen’s inequality to (27), we can see that a t ≥ σ t ( T − t ) exp (cid:40) T − t (cid:90) Tt (cid:2) − αξ ( r ( u ) − r ( t )) + 2 ξ ( ˆ m t ( u ) − ˆ m t ( t )) + 2 ξ r ( u | t ) (cid:3) d u (cid:41) . Then, it is easy to find that T − ta t ≤ σ exp (cid:40) − ξ ˆ m t ( t ) + αξ r ( t ) − T − t (cid:90) Tt (cid:2) − αξ ( r ( u ) − r ( t )) + 2 ξ ( ˆ m t ( u ) − ˆ m t ( t )) + 2 ξ r ( u | t ) (cid:3) d u (cid:41) where the exponent − ξ ˆ m t ( t ) + αξ r ( t ) − T − t (cid:90) Tt (cid:2) − αξ ( r ( u ) − r ( t )) + 2 ξ ( ˆ m t ( u ) − ˆ m t ( t )) + 2 ξ r ( u | t ) (cid:3) du. is a Gaussian process. Therefore a t has finite moments of all orders.13sing the error terms specified in the Appendix A and Lemma 3.1, we find the followingdecompositions for each term (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E t (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M, M (cid:105) u (cid:35) − Γ BS ( t, X t , v t ) R t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 3 a u + 3 a u + 1 a u (cid:19) R u d (cid:104) M, M (cid:105) u (cid:35) (32) + Cρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 2 a u + 1 a u (cid:19) R u σ u d (cid:104) W, M (cid:105) u (cid:35) + Cρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 1 a u (cid:19) σ u d (cid:104) W, R (cid:105) u (cid:35) + C E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 2 a u + 1 a u (cid:19) d (cid:104) M, R (cid:105) u (cid:35) . and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W, M (cid:105) u (cid:35) − ΛΓ BS ( t, X t , v t ) U t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 2 a u + 1 a u (cid:19) U u d (cid:104) M, M (cid:105) u (cid:35) (33) + Cρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 1 a u (cid:19) U u σ u d (cid:104) W, M (cid:105) u (cid:35) + Cρ E t (cid:34)(cid:90) Tt e − r ( u − t ) a u σ u d (cid:104) W, U (cid:105) u (cid:35) + C E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 1 a u (cid:19) d (cid:104) M, U (cid:105) u (cid:35) . Since all previous conditional expectations are obviously finite, the error order is O (cid:16)(cid:0) ξ + ρξ (cid:1) (cid:17) .Further, we express differentials with respect to the n th -power of the exponential Volterravolatility process when Y t is a semimartingale. Lemma 4.8.
Let σ t be as in (19) and Y t a semimartingale. Let n ≥ , we have that d σ nt = σ nt K ( t, t ) (cid:104) nξ d W t + n ξ K ( t, t ) ( n − α ) d t (cid:105) . (34) Proof.
The formula is an immediate consequence of the Itô formula.
Lemma 4.9.
Let σ t be as in (19) and Y t is a semimartingale. We can calculate d U t and d R t .In order to simplify the notation, we define the two following functions ϕ ( t, s, x, T ) := exp (cid:26) − αξ ( r ( x ) − r ( u )) + ξ (cid:90) u (2 K ( s, z ) + K ( x, z ) − K ( u, z )) d W z (cid:27) ·· exp (cid:110) ξ (cid:90) xu (2 K ( s, z ) + K ( x, z )) d z − αξ ( r ( s ) − r ( x )) + 2 ξ ˆ r ( s | x ) (cid:111) , ( t, s, v, x, T ) := exp (cid:110) − αξ ( r ( s ) + r ( v ) − r ( t )) + 2 ξ (ˆ r ( s | x ) + ˆ r ( v | x ))+ 2 ξ (cid:90) t ( K ( s, z ) + K ( v, z ) − K ( t, z )) d W z + 2 ξ (cid:90) xt ( K ( s, z ) + K ( v, z )) d z (cid:111) . Then d U t = ρξσ t K ( t, t ) (cid:20) ξ d W t + 32 ξ K ( t, t ) (3 − α ) d t (cid:21) (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) K ( s, x ) d s d x − ρξσ t (cid:90) T ϕ ( t, s, t, T ) K ( s, x ) d s d t + ρξσ t (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) (cid:110) αξ d r ( t ) + ξ (2 K ( s, t ) + K ( x, t ) − K ( t, t )) d W t + 12 ξ (2 K ( s, t ) + K ( x, t ) − K ( t, t )) d t − ξ K ( s, t ) + K ( x, t )) d t (cid:111) K ( s, x ) d s d x. and d R t = 12 ξ σ t K ( t, t ) (cid:2) ξ d W t + 2 ξ K ( t, t ) (4 − α ) d t (cid:3)(cid:90) Tt (cid:90) T (cid:90) T Φ( t, s, v, x, T ) · K ( s, x ) K ( v, x ) d s d v d x − ξ σ t (cid:90) T (cid:90) T Φ( t, s, v, t, T ) K ( s, t ) K ( v, t ) d s d v d t + 12 ξ σ t (cid:90) Tt (cid:90) T (cid:90) T Φ( t, s, v, x, T ) K ( s, x ) K ( v, x ) (cid:110) αξ d r ( t ) + 2 ξ ( K ( s, t ) + K ( v, t ) − K ( t, t )) d W t + 2 ξ ( K ( s, t ) + K ( v, t ) − K ( t, t )) dt − ξ ( K ( s, t ) + K ( v, t )) d t (cid:111) d s d v d x We define the following auxiliary function ζ ( t, T ) := (cid:90) Tt E t (cid:2) σ z (cid:3) K ( z, t ) d z. Then, it is easier to see that the covariations are the following d (cid:104) U, W (cid:105) t = 3 ρξ σ t K ( t, t ) (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) K ( s, x ) d s d x d t + ρξ σ t (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) (2 K ( s, t ) + K ( x, t ) − K ( t, t )) K ( s, x ) d s d x d t, d (cid:104) U, M (cid:105) t = 6 ρξ σ t K ( t, t ) ζ ( t, T ) (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) K ( s, x ) d s d x d t + 2 ρξ σ t ζ ( t, T ) (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) (2 K ( s, t ) + K ( x, t ) − K ( t, t )) K ( s, x ) d s d x d t, d (cid:104) R, W (cid:105) t = 2 ξ σ t K ( t, t ) (cid:90) Tt (cid:90) T (cid:90) T ψ ( t, s, v, x, T ) · K ( s, x ) K ( v, x ) d s d v d x d t + ξ σ u (cid:90) Tu (cid:90) T (cid:90) T ψ ( t, s, v, x, T ) K ( s, x ) K ( v, x ) ( K ( s, u ) + K ( v, u ) − K ( u, u )) d s d v d x d t nd d (cid:104) R, M (cid:105) t = 4 ξ σ t K ( t, t ) ζ ( t, T ) (cid:90) Tt (cid:90) T (cid:90) T ψ ( t, s, v, x, T ) · K ( s, x ) K ( v, x ) d s d v d x d t + 2 ξ σ t ζ ( t, T ) (cid:90) Tt (cid:90) T (cid:90) T ψ ( t, s, v, x, T ) K ( s, x ) K ( v, x ) ( K ( s, t ) + K ( v, t ) − K ( t, t )) d s d v d x d t. Proof.
Now, we can re-write U t as U t = ρξσ t (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) K ( s, x ) d s d x and R t as R t = 12 ξ σ t (cid:90) Tt (cid:90) T (cid:90) T ψ ( t, s, v, x, T ) · K ( s, x ) K ( v, x ) d s d v d x. We have that d U t = ρξ d σ t (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) K ( s, x ) d s d x − ρξσ t (cid:90) T ϕ ( t, s, t, T ) K ( s, x ) d s d t + ρξσ t (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) (cid:110) αξ d r ( t ) + ξ (2 K ( s, t ) + K ( x, t ) − K ( t, t )) d W t + ξ (2 K ( s, t ) + K ( x, t ) − K ( t, t )) d t − ξ K ( s, t ) + K ( x, t )) d t (cid:111) K ( s, x ) d s d x. Using Lemma 4.8, we obtain d U t = ρξσ t K ( t, t ) (cid:20) ξ d W t + 32 ξ K ( t, t ) (3 − α ) d t (cid:21) (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) K ( s, x ) d s d x − ρξσ t (cid:90) T ϕ ( t, s, t, T ) K ( s, x ) d s d t + ρξσ t (cid:90) Tt (cid:90) T ϕ ( t, s, x, T ) (cid:110) αξ d r ( t ) + ξ (2 K ( s, t ) + K ( x, t ) − K ( t, t )) d W t + 12 ξ (2 K ( s, t ) + K ( x, t ) − K ( t, t )) d t − ξ K ( s, t ) + K ( x, t )) d t (cid:111) K ( s, x ) d s d x. We have that d R t = 12 ξ d σ t (cid:90) Tt (cid:90) T (cid:90) T Φ( t, s, v, x, T ) · K ( s, x ) K ( v, x ) d s d v d x − ξ σ t (cid:90) T (cid:90) T Φ( t, s, v, t, T ) K ( s, t ) K ( v, t ) d s d v d t + 12 ξ σ t (cid:90) Tt (cid:90) T (cid:90) T Φ( t, s, v, x, T ) K ( s, x ) K ( v, x ) (cid:110) αξ d r ( t ) + 2 ξ ( K ( s, t ) + K ( v, t ) − K ( t, t )) d W t ξ ( K ( s, t ) + K ( v, t ) − K ( t, t )) dt − ξ ( K ( s, t ) + K ( v, t )) d t (cid:111) d s d v d x. Using Lemma 4.8, we obtain d R t = 12 ξ σ t K ( t, t ) (cid:2) ξ d W t + 2 ξ K ( t, t ) (4 − α ) d t (cid:3)(cid:90) Tt (cid:90) T (cid:90) T Φ( t, s, v, x, T ) · K ( s, x ) K ( v, x ) d s d v d x − ξ σ t (cid:90) T (cid:90) T Φ( t, s, v, t, T ) K ( s, t ) K ( v, t ) d s d v d t + 12 ξ σ t (cid:90) Tt (cid:90) T (cid:90) T Φ( t, s, v, x, T ) K ( s, x ) K ( v, x ) (cid:110) αξ d r ( t ) + 2 ξ ( K ( s, t ) + K ( v, t ) − K ( t, t )) d W t + 2 ξ ( K ( s, t ) + K ( v, t ) − K ( t, t )) dt − ξ ( K ( s, t ) + K ( v, t )) d t (cid:111) d s d v d x. Remark 4.10.
Note that the Theorem 4.7 requires that the conditional expectations of (32) and(33) be finite. In the case where σ t is as in (19) and Y t is a semimartingale, it easy to see thatthese conditions are met using Lemma 4.9. Let us now focus on a very important example of Gaussian Volterra processes, the fractionalBrownian motion (fBm), which is a process with a Hurst parameter H ∈ (0 , , with covariancefunction r ( t, s ) := E [ B Ht B Hs ] = 12 (cid:0) t H + s H − | t − s | H (cid:1) , t, s ≥ , (35)and, in particular, with variance r ( t ) := r ( t, t ) = t H , t ≥ . (36)One of the first applications of fractional Brownian motion is credited to Hurst (1951) whomodelled the long term storage capacity of reservoirs along the Nile river. However, the idea ofthis concept goes back to Kolmogorov (1940), who studied Wiener spirals and some other curvesin Hilbert spaces. Later, Lévy (1953) used the Riemann–Liouville fractional integral to define theprocess as ˜ B Ht := 1Γ( H + 1 / t (cid:90) ( t − s ) H − / d W s , where H may be any positive number. This type of integral turned out to be ill-suited to ap-plications of fractional Brownian motion because of its over-emphasis on the origin for manyapplications. In their highly cited work, Mandelbrot and Van Ness (1968) introduced the Weyl’srepresentation of the fractional Brownian motion: B Ht := 1Γ( H + 1 / Z t + t (cid:90) ( t − s ) H − / d W s , (37)where Z t := (cid:90) −∞ (cid:104) ( t − s ) H − / − ( − s ) H − / (cid:105) d W s W t is the standard Wiener process. Nowadays, the most widely used representation of fBmis the one by Molchan and Golosov (1969) B Ht := t (cid:90) K H ( t, s ) d W s , (38)where K H ( t, s ) := C H (cid:34)(cid:18) ts (cid:19) H − ( t − s ) H − − (cid:18) H − (cid:19) s H − (cid:90) ts z H − ( z − s ) H − d z (cid:35) (39) C H := (cid:115) H Γ (cid:0) − H (cid:1) Γ (cid:0) H + (cid:1) Γ (2 − H ) . To understand the connection between Molchan-Golosov and Mandelbrot-Van Ness representationsof fBm we refer readers to the paper by Jost (2008).Despite of the above mentioned arguments, Alòs, Mazet, and Nualart (2000) proposed toconsider a process ˆ B t = (cid:82) t ( t − s ) H − / d W s instead of B Ht in fractional stochastic calculus, since Z t has absolutely continuous trajectories. Since B Ht is not a semimartingale, the process ˆ B t =Γ( H + 1 / B Ht − Z t is also not a semimartingale. Later on, Thao (2006) introduced the so calledapproximate fractional Brownian motion process as ˆ B εt = t (cid:90) ( t − s + ε ) H − / d W s , H ∈ (0 , , H (cid:54) = 12 , ε > , and showed that for every ε > the process ˆ B εt is a semimartingale and it converges to ˆ B t in L (Ω) when ε tends to zero. This convergence is uniform with respect to t ∈ [0 , T ] (Thao 2006,Theorem 2.1).Let us now consider the exponential fractional volatility process σ t := σ exp (cid:26) ξB Ht − αξ r ( t ) (cid:27) , t ≥ , (40)where ( B Ht , t ≥ is one of the above mentioned representations of fBm. We are especiallyinterested in the "rough" models, i.e. when H < / . In this case, we call the model (1) withvolatility process (40) the rough fractional stochastic volatility model ( α RFSV). Note that if α = 1 ,we get the rBergomi model (Bayer, Friz, and Gatheral 2016; Gatheral, Jaisson, and Rosenbaum2018), if α = 0 , we get the original exponential fractional volatility model. Values of α between 0and 1 give us a new degree of freedom that can be viewed as a weight between these two models. Example 4.11 (Volatility driven by the approximate fractional Brownian motion) . Let us considermodel (2) with volatility process σ t = σ exp (cid:26) ξ ˜ B t − αξ r ( t ) (cid:27) , where ˜ B t = (cid:90) t ˜ K ( t, s ) d W s and ˜ K ( t, s ) = √ H ( t − s + ε ) H − / , s ≤ t, ε ≥ , H ∈ (0 , . hen r ( t, s ) = (cid:90) t ∧ s ˜ K ( t, v ) ˜ K ( s, v ) d v,r ( t ) = (cid:90) t ˜ K ( t, v ) d v = 2 H (cid:90) t ( t − v + ε ) H − d v = ( t + ε ) H − ε H . Note that if ε = 0 , we get exactly the variance r ( t ) = t H , that it is the variance of the standardfractional Brownian motion. Further we have ˆ r ( t | u ) = r ( t ) − (cid:90) u ˜ K ( t, v ) d v = r ( t ) − H (cid:90) u ( t − v + ε ) H − d v = ( t − u + ε ) H − ε H and thus U = ρσ ξ √ H (cid:90) T (cid:90) T exp (cid:110) ξ H (cid:90) u [( u − v + ε ) H − / + 2( s − v + ε ) H − / ] d v (cid:111) ·· exp (cid:110) ξ [( s − u + ε ) H − ε H ] (cid:111) ·· exp (cid:110) − αξ [( u + ε ) H + 2( s + ε ) H − ε H ] (cid:111) ·· ( s − u + ε ) H − / d s d uR = σ ξ H (cid:90) T (cid:90) T (cid:90) T exp (cid:110) ξ H (cid:90) u [( t − v + ε ) H − / + ( t − v + ε ) H − / ] d v (cid:111) ·· exp (cid:110) ξ [( t − u + ε ) H + ( t − u + ε ) H − ε H ] (cid:111) ·· exp (cid:110) − αξ [( t + ε ) H + ( t + ε ) H − ε H ] (cid:111) ·· ( t − u + ε ) H − / ( t − u + ε ) H − / d t d t d u. Example 4.12 (Volatility driven by the standard Wiener process) . If in the previous Example 4.11we take H = 1 / and ε = 0 , we get model (2) with exponential Wiener volatility process σ t = σ exp (cid:26) ξ ˜ W t − αξ r ( t ) (cid:27) , (41) where ˜ W t = (cid:90) t ˜ K ( t, s ) d W s is the standard Wiener process, i.e. where ˜ K ( t, s ) = { s ≤ t } . In this case, we have that v t = σ t (2 − α ) ξ ( T − t ) (cid:2) exp { (2 − α ) ξ ( T − t ) } − (cid:3) ,r ( t, s ) = (cid:90) t ∧ s ˜ K ( t, v ) ˜ K ( s, v ) d v = t ∧ s,r ( t ) = (cid:90) t ˜ K ( t, v ) d v = t. Define φ ( t, T , α ) := (cid:90) Tt exp (cid:8) (2 − α ) ξ ( s − t ) (cid:9) d s. (42)19 t is easy to see that d M t = 2 ξσ t d W t φ ( t, T , α ) , and thus U = ρσ ξ (cid:90) T (cid:90) T exp (cid:110) ξ (cid:90) u [ { v ≤ u } + 2 · { v ≤ s } ] d v (cid:111) ·· exp (cid:110) ξ ( s − u ) − αξ u − αξ s (cid:111) { u ≤ s } d s d u = ρσ ξ (cid:90) T (cid:90) T exp (cid:110) ξ u (cid:111) exp (cid:110) ξ [(4 − α ) s − (4 + α ) u ] (cid:111) { u ≤ s } d s d u (43) = 2 ρσ − α )(3 − α )(5 − α ) ξ (cid:20) − α ) exp (cid:110) ξ (3 − α ) T (cid:111) − − α ) exp (cid:110) ξ (2 − α ) T (cid:111) + 5 − α (cid:21) (44) and R = 12 σ ξ (cid:90) T (cid:90) T (cid:90) T exp (cid:110) ξ (cid:90) u [ { v ≤ t } + { v ≤ t } ] d v (cid:111) ·· exp (cid:110) ξ [ t + t − u ] − αξ [ t + t ] (cid:111) { u ≤ t } { u ≤ t } d t d t d u = 12 σ ξ (cid:90) T (cid:90) T (cid:90) T exp (cid:110) ξ u (cid:111) exp (cid:110) ξ [ t + t − u ] − αξ [ t + t ] (cid:111) { u ≤ t } { u ≤ t } d t d t d u (45) = σ − α ) (4 − α )(6 − α ) ξ (cid:104) (2 − α ) exp (cid:110) − α ) ξ T (cid:111) − (4 − α )(6 − α ) exp (cid:110) − α ) ξ T (cid:111) + 8(4 − α ) exp (cid:110) (2 − α ) ξ T (cid:111) − − α ) (cid:105) . (46) For a model without exponential drift ( α = 0 ) these formulas simplify to U = ρσ ξ (cid:20) (cid:110) ξ T (cid:111) − (cid:110) ξ T (cid:111) + 5 (cid:21) R = σ ξ (cid:104) exp (cid:110) ξ T (cid:111) − (cid:105) (cid:104) exp (cid:110) ξ T (cid:111) + 3 (cid:105) and for the classical Bergomi model ( α = 1 ) we get U = ρσ ξ (cid:104) exp (cid:110) ξ T (cid:111) − (cid:110) ξ T (cid:111) + 2 (cid:105) R = σ ξ (cid:104) exp (cid:110) ξ T (cid:111) −
15 exp (cid:110) ξ T (cid:111) + 24 exp (cid:110) ξ T (cid:111) − (cid:105) . For matter of convenience, we define the functions ψ ( t, T , α ) = (cid:90) Tt exp (cid:8) (8 − α ) ξ ( s − t ) (cid:9) (cid:2) exp (cid:8) (2 − α ) ξ ( T − s ) (cid:9) − (cid:3) d s (47) and ζ ( t, T , α ) = (cid:90) Tt exp (cid:26)
12 (9 − α ) ξ ( s − t ) (cid:27) (cid:2) exp (cid:8) (2 − α ) ξ ( T − s ) (cid:9) − (cid:3) d s. (48)20 e can re-write U t and R t as U t = ρσ t (2 − α ) ξ ζ ( t, T , α ) (49) and R t = σ t − α ) ξ ψ ( t, T , α ) . (50) It is easy to find the d U t and d R t , d U t = ρ d σ t (2 − α ) ξ ζ ( t, T , α ) + ρσ t (2 − α ) ξ ζ (cid:48) ( t, T , α ) d t (51) = ρ (cid:0) ξσ t d W t + (18 − α ) ξ σ d t (cid:1) (2 − α ) ξ ζ ( t, T , α ) + ρσ t (2 − α ) ξ ζ (cid:48) ( t, T , α ) d t (52) and d R t = d σ t − α ) ξ ψ ( t, T , α ) + σ t − α ) ξ ψ (cid:48) ( t, T , α ) d t (53) = 4 ξσ t d W t + 2(8 − α ) σ t d t − α ) ξ ψ ( t, T , α ) + σ t − α ) ξ ψ (cid:48) ( t, T , α ) d t. (54) Remark 4.13.
We can do a Taylor expansion of U and R to understand better their dependen-cies, doing that we obtain U ∼ ρξT σ t (cid:32)
12 + 112 (13 − α ) ξ T + 196 ( α (19 α − ξ T − α −
13) ( α (13 α −
70) + 97) ξ T + O (cid:0) T (cid:1)(cid:33) and R ∼ ξ T σ t (cid:18) −
16 ( α −
2) + 124 ( α − α − ξ T − α − (cid:0) α − α + 140 (cid:1) ξ T + O (cid:0) T (cid:1)(cid:19) . Theorem 4.14 (Decomposition formula for exponential Wiener volatility model) . Let X t bethe log-price process (2) with σ t being the exponential Wiener volatility process defined in (41) .Assuming without any loss of generality that the options starts at time 0, then we can express thecall option fair value V using the processes U , R from (44) and (46) respectively. In particular, V = BS (0 , X , v )+ ΛΓ BS (0 , X , v ) U + Γ BS (0 , X , v ) R + (cid:15). where (cid:15) denotes error terms and for α ≥ , | (cid:15) | is at most of the order Cξ ( √ T + ρξ ) T / Π( α, T, ξ, ρ ) . The exact bound is given in Appendix B. roof. The detailed proof is given in Appendix B, where we also examine the order of magnitudeby the first Taylor term of the integrals.
Remark 4.15.
It is worth to mention that the order of the error bound from Theorem 4.14 isbetter than the general estimate from Theorem 4.7, where the time dependency is not considered.To get finer estimates also for the exponential fractional model (case H (cid:54) = 1 / ), a proof similarto the one in Appendix B would have to be performed with more complicated but still tractablecalculations. Example 4.16 (Volatility driven by the standard fractional Brownian motion) . Let us considera model with volatility process σ t = exp { ξB Ht − αξ r ( t ) } , where B Ht is the standard fractional Brownian motion as defined in (38) , i.e. with the Molchan-Golosov kernel (39) . Then, the formulas for U and R are given in Proposition 4.6 with particularkernel (39) , autocovariance function (36) and ˆ r ( t, s | u ) as in Theorem 4.1. In this case, we donot give the formulas for U and R after substituting the Molchan-Golosov kernel, since theseformulas are too long. However, it is worth to mention that the formulas are explicit and numericalevaluation requires only the computation of some multiple Gaussian integrals. In this section, we focus on numerical aspects of the introduced approximation formula. Wedetail on its numerical implementation and a comparison with the Monte Carlo (MC) simulationframework introduced by Bennedsen, Lunde, and Pakkanen (2017) will be provided.In the second part of this section, we also introduce two interesting outcome analysis forrBergomi model. In particular, we show how the model can be efficiently calibrated using theapproximation formula to short maturity smiles. We remark that classical SV models (e.g. Hestonmodel) might fail to fit the short term smiles, unless they exploit high volatility of volatility levelsfor which they would be typically inconsistent with the long term skew of the volatility surface.In what follows, we will inspect the approximation quality for rBergomi model and time tomaturity / volatility of volatility ξ scenarios. Based on the nature of error terms (see AppendixA) those two factors should play prominent role when it comes to approximation quality. We note that for the models studied in this paper, we have obtained either a semi-closed formor analytical formula for standard Wiener case (H=0.5). Moreover, for the class of exponentialfractional models – represented by the α RFSV model – we only need to numerically evaluatemultiple integrals in R and U .In our case, this was done using a trapezoid quadrature routine – not necessarily the mostefficient approach, but easy to implement. We used a discretisation of integrands such that thenumerical error doesn’t affect the results in a significant way. I.e. to be lower than standard MCerrors when comparing to simulated prices or lower than the expected approximation error.For benchmarking we use a first-order hybrid MC scheme introduced by Bennedsen, Lunde,and Pakkanen (2017) alongside 50000 MC sample paths. Similarly to the implementation ofthe approximation formula, we remark that this scheme could be also improved as described inMcCrickerd and Pakkanen (2018). Typically we used from 1000 up to 27000 points for 3D integrals. .2 Sensitivity analysis for rBergomi ( α = 1 ) approximation w.r.t. in-creasing ξ and time to maturity τ In this section, we illustrate the approximation quality for European call options under variousmodel regimes / data set properties as described in Table 1. We use option expiries up to 1Y – weare expecting a loss of approximation quality, based on the nature of the approximation formula.Since we utilized a first order approximation arguments with respect to volatility of volatility, weare also expecting more pronounced differences between MC simulations and the formula for largevalues of ξ . Model params Values ξ { , , } σ ρ -20% H α = 1 and ε = 0 . Data andparameter values are: v = 8% , ξ = 10% , ρ = − , H = 0 . )In Figure 1, we illustrate the approximation quality of the rBergomi approximation for low ξ values. We can observe an expected behaviour: a very good match upto 3M expiry and almostlinear deterioration of the quality with increasing τ . Also the approximation formula provides asimilar scale of errors across the tested moneyness.For different moneyness regimes and 1M time to maturity, we obtained the following discrep-ancies between the MC trials and the introduced formula, measured in the relative option fairvalue (FV) : Relative FV is the absolute option fair value divided by the initial spot price. ξ = 10% ξ = 50% ξ = 100%
80% 4.5e-04 -8.1e-05 -0.2992890% 3.9e-04 2.6e-05 -0.02797100% 2.3e-04 7.2e-04 0.95436110% -1.5e-05 -7.7e-05 0.09690120% -1.2e-05 -2.7e-04 -0.46417In the table above, we can see reasonable approximation error measures which fell belowstandard 1 MC error for ξ = 10% and ξ = 50% regimes. Due to the theoretical properties of theapproximation formula, we observe significant deterioration for high volatility of volatility regimes.This also depends on the time to maturity of the approximated option – the shorter maturity wehave, the higher ξ we can allow to obtain reasonable approximation errors (i.e. of the order 1e-04and lower in terms of FV).Although the introduced approximation is typically not suitable for calibrations to the wholevolatility surface – due to the deterioration of approximation quality when increasing time tomaturity – we will illustrate how it can significantly speed-up MC calibration to the providedforward at-the-money (ATMF) backbone. Unlike previous analyses, which were based on artificial data / model parameter values, we inspectan application fo the formula on the calibration to real option market data. In particular, we utilizefour data sets of AAPL options which were analysed in detail by Pospíšil, Sobotka, and Ziegler(2018). Descriptive statistics of the data sets are provided in Table 2. The following calibrationtest trials will be considered. • Calibration to short maturity smiles:This should illustrate how well the model can fit short maturity smiles using the introducedapproximation formula without exploiting too high volatility of volatility values ( ξ ). Foreach data set we selected the shortest maturity slice with more than one traded option. Thevalues were not interpolated by any model, i.e. we calibrated to discrete close mid-prices oftraded options. We also confirm, that both MC simulation and the formula reprice the smilewith the final calibrated parameters without significant differences. • Hybrid calibration to the ATMF backbone:In the second trial, we calibrate to each data set ATMF backbone. We note that becausewe have only a discrete set of traded options we might not have for each maturity an optionwith strike equal to the corresponding forward. Hence, we take an option with the closeststrike to the forward value for each expiry. We use the proposed approximation formula onlyfor τ < . , for longer time to maturities we price by MC simulations.In both cases, the calibration routine was formulated as a standard least-square optimizationproblem. I.e. to obtain calibrated parameters, we numerically evaluated ˆΘ = arg min f (Θ) = arg min N (cid:88) i =1 [ Mid i − rBergomi i (Θ)] , (55)where N is the total number of contracts for the calibration, Mid i is the mid price of the i th optionand rBergomi i (Θ) represents the corresponding model price based on parameter set Θ . The modelprice is either obtained by the approximation formula or by means of MC simulations otherwise.24 ata Table 2: Data on AAPL options used in calibration trialsThe optimization is performed using Matlab’s local search trust region optimizer which also needsan initial guess to start with.All following results will be quoted in relative FV: e.g. rBergomi i (Θ) /S and also differencesbetween market and the calibrated model will be denoted using this measure. For the calibrationto the whole surface of European options, errors in FV below . are typically considered tobe acceptable, whereas anything exceeding difference is considered as a significant modelinconsistency.Firstly, we display the results for the short-maturity calibration. In Figure 2, we illustratethat even with a not well suited initial guess for Data ξ values compared to other three calibration trials and also slightlylonger maturity – the data set th May (Data τ < . we will use the approximation formula and MC simulationsotherwise. For the calibrated parameters, we will also measure the time spent computing FVsby each pricer. For completeness, we remark that both implementations are not perfect. I.e. forMC simulations under the rBergomi model, the "turbocharging" improvements were introducedby McCrickerd and Pakkanen (2018) recently. On the other hand, numerical integrations inthe approximation could be performed by some adaptive quadrature and could be vectorized toimprove the computation speed.In Figure 4, we illustrate calibration fit to the ATMF backbone of the option price surface. Weconclude that we have retrieved similar errors for both the prices computed using the proposedapproximation and the longer maturity option prices quantified by MC simulations. The final fitof the calibrated model (recomputed by MC simulations) is very good, especially considering thatthe studied model has only 4 parameters. Moreover, only a fraction of the time spent by MCpricer was needed to computed all FV using the approximation formula. In particular, . ofthe pricing time we were computing MC simulation estimates of FVs. We also note that / oftotal evaluations were computed by the approximation formula. Excluding any data loading / manipulation routines. a) Comparison of calibrated rBergomi and market data (Data Calibrated params: σ = 3 . ξ = 39 . ρ = − . H = 0 . MSE = 0.0883
Figure 2: Calibration results for rBergomi - short maturity smiles (Data
In previous sections we studied an approximation approach for the pricing of European options un-der rough stochastic volatility dynamics. Our approach is based on the option price decompositionresults obtained by Alòs (2012) for the standard Heston SV model and on its recent generalisationto other SV models by Merino, Pospíšil, Sobotka, and Vives (2018). The main contribution ofour research is to derive pricing formulas suitable for various practical applications by applyingthe general decomposition on a class of Volterra volatility models. Our main focus is laid on therough volatility models introduced by Gatheral, Jaisson, and Rosenbaum (2018) and Bayer, Friz,and Gatheral (2016).In particular, a prediction law for Gaussian Volterra processes was proved and an adaptedprojection of future volatility was obtained in Section 4 for the class of exponential Volterravolatility models, where the volatility process can be expressed as a positive L function of thetime and the Volterra process Y t , i.e. σ t = g ( t, Y t ) . We focused on one particular example ofGaussian Volterra processes, namely on a (rough) fractional Brownian motion B Ht . "Roughness"of sample paths is determined by the Hurst parameter, H ∈ (0 , / , where for H = 0 . we wouldrecover a standard Wiener process.The pricing formula for European options, which is numerically tractable, is then derived under26 a) Comparison of calibrated rBergomi and market data (Data Calibrated params: σ = 4 . ξ = 81 . ρ = − . H = 0 . MSE = 13.9538
Figure 3: Calibration results for rBergomi - short maturity smiles (Data α RFSV model, i.e. for σ t = σ exp (cid:26) ξB Ht − αξ r ( t ) (cid:27) , where α ∈ [0 , and r ( t ) is the corresponding autocovariance function defined in Section 4. Thisnewly introduced model seems to have interesting special cases: • for α = 0 it reverts to a simple exponential RFSV model • and for α = 1 to rBergomi model respectively.Both special cases of the α RFSV model are studied and praised for their surprising consistencywith various financial markets in (Bayer, Friz, and Gatheral 2016) who use the the Bergomi–Guyonexpansion to study the ATM skew approximation of the impied volatilities. However, the authorsadmit that “the Bergomi–Guyon expansion does not converge with values of η consistent with theSPX volatility surface, so the Bergomi–Guyon expansion is not useful in practice for calibrationof the rBergomi model.” This motived us to derive an approximation formula that will be usefulin practice for calibration of the studied models to real market data.In Section 5, the newly obtained approximation formula was compared to the MC pricingapproach introduced by Bennedsen, Lunde, and Pakkanen (2017) and McCrickerd and Pakkanen(2018). This enabled us to numerically verify the obtained solution, to quantify its approximation27igure 4: ATMF calibration results when combining approximation formula ( τ < . ) and MCsimulations.errors under various settings and, last but not least, to comment on suitability of the rBergomimodel for calibration tasks to real market data based on AAPL stock options . The followingconclusions were drawn: • The approximation error is well behaved for short maturities (typically for less than 1M)and the error increases with time to maturity and ξ parameter. • For medium-term expiries we are able to obtain well approximated prices only under low ξ regimes. • Although, the approximation under a rough Volterra process involves several numericalintegration procedures, it is much faster than MC simulation approach implemented in thesame environment . For the standard Wiener case (and in particular for the original Bergomimodel), we can have an analytical pricing formula, but also more efficient and well developedMC simulation schemes. • Considering that the modelling approach studied has only few parameters, we were ableto fit the sample market data surprisingly well. For calibration to short maturity smiles,we can use just the approximation formula – this was verified by recomputing calibrationerrors using MC simulations. For calibration to the whole surface, one can utilize a newlyintroduced hybrid scheme which consist of a combination of approximation and simulationtechniques. The idea is quite simple – to use approximation formula for low maturities orfor low ξ values and MC simulations for the remaining computations. Suitability of thisscheme was judged by a simple calibration to an ATMF-like backbone. We retrieved a wellcalibrated model, while saving a significant computational time compared to the calibrationbased on MC simulations only.However, we remark that implementation of the approximation could be further improved –for simplicity we used a simple trapezoidal quadrature to numerically evaluate integrals appearingin U t and R t expressions. These data sets were analysed and described in Pospíšil, Sobotka, and Ziegler (2018) The numerical trials were implemented in MATLAB environment, see . α RFSV model are already introduced in this manuscript.Also to fit some of the pronounced forward variance curves, one might need a more complexterm structure – a simple exponential drift term under the rBergomi model might not be sufficientlyflexible. In this case, one can utilize our approach to obtain the approximation formula for a newrough volatility model. In fact, once the new volatility function g is postulated it should be onlya matter of algebraic operations to obtain the corresponding approximation formula. Funding
The work of Jan Pospíšil was partially supported by the Czech Science Foundation (GAČR) grantno. GA18-16680S “Rough models of fractional stochastic volatility”. The work of Josep Vives waspartially supported by Spanish grant MEC MTM 2016-76420-P.
Acknowledgements
Computational resources were provided by the CESNET LM2015042 and the CERIT ScientificCloud LM2015085, provided under the programme “Projects of Large Research, Development, andInnovations Infrastructures”.
A Decomposition formulas in the general model
In this appendix, we give the error terms for a decomposition of the general model.The term (I) can be decomposed as ρ E (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W, M (cid:105) u (cid:35) − ΛΓ BS ( t, X t , v t ) U t = 18 E (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) U u d (cid:104) M, M (cid:105) u (cid:35) + ρ E (cid:34)(cid:90) Tt e − r ( u − t ) Λ Γ BS ( u, X u , v u ) U u σ u d (cid:104) W, M (cid:105) u (cid:35) + ρ E (cid:34)(cid:90) Tt e − r ( u − t ) Λ Γ BS ( u, X u , v u ) σ u d (cid:104) W, U (cid:105) u (cid:35) + 12 E (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) d (cid:104) M, U (cid:105) u (cid:35) . The term (II) can be decomposed as E (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M, M (cid:105) u (cid:35) − Γ BS ( t, X t , v t ) R t = 18 E (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) R u d (cid:104) M, M (cid:105) u (cid:35) + ρ E (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) R u σ u d (cid:104) W, M (cid:105) u (cid:35) ρ E (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W, R (cid:105) u (cid:35) + 12 E (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M, R (cid:105) u (cid:35) . B Upper-bound for decomposition formula for the exponen-tial Wiener volatility model
In this appendix we obtain the upper-bound for the decomposition formula for the exponentialWiener volatility model in Example 4.12.
B.1 Upper-bound for term (I)
For matter of convenience, we define the function χ ( t, T , α ) := (cid:90) Tt exp (cid:26)
12 (9 − α ) ξ ( s − t ) (cid:27) (cid:2) exp (cid:8) (2 − α ) ξ ( T − s ) (cid:9) − (cid:3) d s. We can rewrite U t as U t = ρσ t (2 − α ) ξ χ ( t, T , α ) . It is easy to find that d U t = ρ d σ t (2 − α ) ξ χ ( t, T , α ) + ρσ t (2 − α ) ξ χ (cid:48) ( t, T , α ) d t = ρ (cid:0) ξσ t d W t + (18 − α ) ξ σ d t (cid:1) (2 − α ) ξ χ ( t, T , α ) + ρσ t (2 − α ) ξ χ (cid:48) ( t, T , α ) d t. If α ≥ , we can find an upper-bound for χ ( t, T , α ) which is χ ( t, T , α ) ≤ (cid:90) Tt exp (cid:26) ξ ( s − t ) (cid:27) (cid:2) exp (cid:8) ξ ( T − s ) (cid:9) − (cid:3) d s = 245 ξ (cid:18) − (cid:8) ξ ( T − t ) (cid:9) + 4 exp (cid:26) ξ ( T − t ) (cid:27) + 5 (cid:19) . We can re-write the decomposition formula as ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W, M (cid:105) u (cid:35) − ΛΓ BS ( t, X t , v t ) U t = 18 E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:0) ∂ x − ∂ x + ∂ x (cid:1) Γ BS ( u, X u , v u ) U u d (cid:104) M, M (cid:105) u (cid:35) + ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:0) ∂ x − ∂ x (cid:1) Γ BS ( u, X u , v u ) U u σ u d (cid:104) W, M (cid:105) u (cid:35) + ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ∂ x Γ BS ( u, X u , v u ) σ u d (cid:104) W, U (cid:105) u (cid:35) + 12 E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:0) ∂ x − ∂ x (cid:1) Γ BS ( u, X u , v u ) d (cid:104) M, U (cid:105) u (cid:35) . a u , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W, M (cid:105) u (cid:35) − ΛΓ BS ( t, X t , v t ) U t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 2 a u + 1 a u (cid:19) U u d (cid:104) M, M (cid:105) u (cid:35) + Cρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 1 a u (cid:19) U u σ u d (cid:104) W, M (cid:105) u (cid:35) + Cρ E t (cid:34)(cid:90) Tt e − r ( u − t ) a u σ u d (cid:104) W, U (cid:105) u (cid:35) + C E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 1 a u (cid:19) d (cid:104) M, U (cid:105) u (cid:35) . Noting that a u = σ u φ / ( u, T, α ) , where φ was defined in (42), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W, M (cid:105) u (cid:35) − ΛΓ BS ( t, X t , v t ) U t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cρξ − α ) E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) σ u φ ( u, T, α ) + 2 σ u φ / ( u, T, α ) + σ u (cid:19) χ ( u, T, α ) d u (cid:35) + Cρ (2 − α ) E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) σ u φ / ( u, T, α ) + σ u φ ( u, T, α ) (cid:19) χ ( u, T, α ) d u (cid:35) + Cρ − α ) E t (cid:34)(cid:90) Tt e − r ( u − t ) σ u φ / ( u, T, α ) χ ( u, T, α ) d u (cid:35) + 3 Cρξ (2 − α ) E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) σ u φ ( u, T, α ) + σ u φ / ( u, T, α ) (cid:19) χ ( u, T, α ) d u (cid:35) . Being σ u the only stochastic component, we can get the expectation inside. Each power of σ u hasa different forward value, in this case, we can bound all terms by exp (cid:8) ξ ( u − t ) (cid:9) . We have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W, M (cid:105) u (cid:35) − ΛΓ BS ( t, X t , v t ) U t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cρξ − α ) (cid:90) Tt e − r ( u − t ) exp (cid:26) ξ ( u − t ) (cid:27) (cid:18) σ t φ ( u, T, α ) + 2 σ t φ / ( u, T, α ) + σ t (cid:19) χ ( u, T, α ) d u + Cρ (2 − α ) (cid:90) Tt e − r ( u − t ) exp (cid:26) ξ ( u − t ) (cid:27) (cid:18) σ t φ / ( u, T, α ) + σ t φ ( u, T, α ) (cid:19) χ ( u, T, α ) d u + Cρ − α ) (cid:90) Tt e − r ( u − t ) exp (cid:26) ξ ( u − t ) (cid:27) σ t φ / ( u, T, α ) χ ( u, T, α ) d u + 3 Cρξ (2 − α ) (cid:90) Tt e − r ( u − t ) exp (cid:26) ξ ( u − t ) (cid:27) (cid:18) σ t φ ( u, T, α ) + σ t φ / ( u, T, α ) (cid:19) χ ( u, T, α ) d u. Substituting φ ( u, T, α ) and using the upper-bound for χ ( u, T, α ) when α ≥ , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) ΛΓ BS ( u, X u , v u ) σ u d (cid:104) W, M (cid:105) u (cid:35) − ΛΓ BS ( t, X t , v t ) U t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cρξ − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:26) ξ ( u − t ) (cid:27) σ t ξ [exp { ξ ( T − u ) } −
1] + 2 σ t √ ξ [exp { (2 − α ) ξ ( T − u ) } − + σ t (cid:17)(cid:18) − (cid:8) ξ ( T − u ) (cid:9) + 4 exp (cid:26) ξ ( T − u ) (cid:27) + 5 (cid:19) d u + 2 Cρ − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:26) ξ ( u − t ) (cid:27)(cid:16) σ t ξ [exp { (2 − α ) ξ ( T − u ) } − + σ t ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:17)(cid:18) − (cid:8) ξ ( T − u ) (cid:9) + 4 exp (cid:26) ξ ( T − u ) (cid:27) + 5 (cid:19) d u + 6 Cρ − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:26) ξ ( u − t ) (cid:27) σ t ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:18) − (cid:8) ξ ( T − u ) (cid:9) + 4 exp (cid:26) ξ ( T − u ) (cid:27) + 5 (cid:19) d u + 6 Cρξ − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:26) ξ ( u − t ) (cid:27)(cid:32) σ t ξ [exp { (2 − α ) ξ ( T − u ) } −
1] + σ t √ ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:33)(cid:18) − (cid:8) ξ ( T − u ) (cid:9) + 4 exp (cid:26) ξ ( T − u ) (cid:27) + 5 (cid:19) d u. The above upper-bound error is difficult to interpret. In order to do this, we derive a Taylorexpansion for one of the terms. Then, the following error behaviour is retrieved:
C ρξ σ t T / α − (cid:32) √ ρ (cid:112) − ( α − ξ + 21 √ T (cid:33) . B.2 Upper-bound for term (II)
For matter of convenience, we define the function χ ( t, T , α ) := (cid:90) Tt exp (cid:8) (8 − α ) ξ ( s − t ) (cid:9) (cid:2) exp (cid:8) (2 − α ) ξ ( T − s ) (cid:9) − (cid:3) d s We can re-write R t as R t = σ t − α ) ξ χ ( t, T , α ) . It is easy to find that d R t = d σ t − α ) ξ χ ( t, T , α ) + σ t − α ) ξ χ (cid:48) ( t, T , α ) d t = 4 ξσ t d W t + 2(8 − α ) σ t d t − α ) ξ χ ( t, T , α ) + σ t − α ) ξ χ (cid:48) ( t, T , α ) d t. If α ≥ , we can find an upper-bound for χ ( t, T, α ) which is χ ( t, T, α ) ≤ (cid:90) Tt exp (cid:8) ξ ( s − t ) (cid:9) (cid:2) exp (cid:8) ξ ( T − s ) (cid:9) − (cid:3) d s ξ (cid:0) exp (cid:8) ξ ( T − t ) (cid:9) − (cid:1) (cid:0) exp (cid:8) ξ ( T − t ) (cid:9) + 3 (cid:1) . We can re-write the decomposition formula as E t (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M, M (cid:105) u (cid:35) − Γ BS ( t, X t , v t ) R t = 18 E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:0) ∂ x − ∂ x + 3 ∂ x − ∂ x (cid:1) Γ BS ( u, X u , v u ) R u d (cid:104) M, M (cid:105) u (cid:35) + ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:0) ∂ x − ∂ x + ∂ x (cid:1) Γ BS ( u, X u , v u ) R u σ u d (cid:104) W, M (cid:105) u (cid:35) + ρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:0) ∂ x − ∂ x (cid:1) Γ BS ( u, X u , v u ) σ u d (cid:104) W, R (cid:105) u (cid:35) + 12 E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:0) ∂ x − ∂ x + ∂ x (cid:1) Γ BS ( u, X u , v u ) d (cid:104) M, R (cid:105) u (cid:35) . Applying Lemma 3.1 and using the definition of a u , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E t (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M, M (cid:105) u (cid:35) − Γ BS ( t, X t , v t ) R t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 3 a u + 3 a u + 1 a u (cid:19) R u d (cid:104) M, M (cid:105) u (cid:35) + Cρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 2 a u + 1 a u (cid:19) R u σ u d (cid:104) W, M (cid:105) u (cid:35) + Cρ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 1 a u (cid:19) σ u d (cid:104) W, R (cid:105) u (cid:35) + C E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) a u + 2 a u + 1 a u (cid:19) d (cid:104) M, R (cid:105) u (cid:35) . Noting that a u = σ u φ / ( u, T, α ) , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E t (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M, M (cid:105) u (cid:35) − Γ BS ( t, X t , v t ) R t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C − α ) E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) σ u φ / ( u, T, α ) + 3 σ u φ ( u, T, α ) + 3 σ u φ / ( u, T, α ) + σ u (cid:19) χ ( u, T, α ) d u (cid:35) + Cρ − α ) ξ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) σ u φ ( u, T, α ) + 2 σ u φ / ( u, T, α ) + σ u φ ( u, T, α ) (cid:19) χ ( u, T, α ) d u (cid:35) + 2 Cρ (2 − α ) ξ E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) σ u φ ( u, T, α ) + σ u φ / ( u, T, α ) (cid:19) χ ( u, T, α ) d u (cid:35) + 2 C (2 − α ) E t (cid:34)(cid:90) Tt e − r ( u − t ) (cid:18) σ u φ / ( u, T, α ) + 2 σ u φ ( u, T, α ) + σ u φ / ( u, T, α ) (cid:19) χ ( u, T, α ) d u (cid:35) . Being σ u the only stochastic component, we can get the expectation inside. Each power of σ u hasa different forward value, in this case, we can bound all terms by exp (cid:8) ξ ( u − t ) (cid:9) . We have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E t (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M, M (cid:105) u (cid:35) − Γ BS ( t, X t , v t ) R t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:8) ξ ( u − t ) (cid:9)(cid:18) σ t φ / ( u, T, α ) + 3 σ t φ ( u, T, α ) + 3 σ t φ / ( u, T, α ) + σ u (cid:19) χ ( u, T, α ) d u + Cρ − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:8) ξ ( u − t ) (cid:9)(cid:18) σ t φ ( u, T, α ) + 2 σ t φ / ( u, T, α ) + σ t φ ( u, T, α ) (cid:19) χ ( u, T, α ) d u + 2 Cρ (2 − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:8) ξ ( u − t ) (cid:9)(cid:18) σ t φ ( u, T, α ) + σ t φ / ( u, T, α ) (cid:19) χ ( u, T, α ) d u + 2 C (2 − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:8) ξ ( u − t ) (cid:9)(cid:18) σ t φ / ( u, T, α ) + 2 σ t φ ( u, T, α ) + σ t φ / ( u, T, α ) (cid:19) χ ( u, T, α ) d u. Substituting φ ( u, T, α ) and using the upper-bound for χ ( u, T, α ) when α ≥ , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E t (cid:34)(cid:90) Tt e − r ( u − t ) Γ BS ( u, X u , v u ) d (cid:104) M, M (cid:105) u (cid:35) − Γ BS ( t, X t , v t ) R t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:8) ξ ( u − t ) (cid:9)(cid:16) σ t (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − +3 σ t (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } −
1] + 3 σ t (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − + σ u (cid:17)(cid:0) exp (cid:8) ξ ( T − u ) (cid:9) − (cid:1) (cid:0) exp (cid:8) ξ ( T − u ) (cid:9) + 3 (cid:1) d u + Cρ − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:8) ξ ( u − t ) (cid:9)(cid:16) σ t (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − + 2 σ t (cid:32) (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:33) + σ t (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:17)(cid:0) exp (cid:8) ξ ( T − u ) (cid:9) − (cid:1) (cid:0) exp (cid:8) ξ ( T − u ) (cid:9) + 3 (cid:1) d u + Cρ − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:8) ξ ( u − t ) (cid:9)(cid:16) σ t (cid:32) (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:33) + σ t (cid:32) (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:33)(cid:17)(cid:0) exp (cid:8) ξ ( T − u ) (cid:9) − (cid:1) (cid:0) exp (cid:8) ξ ( T − u ) (cid:9) + 3 (cid:1) d u + C − α ) ξ (cid:90) Tt e − r ( u − t ) exp (cid:8) ξ ( u − t ) (cid:9)(cid:16) σ t (cid:32) (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:33) + 2 σ t (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − σ t (cid:32) (2 − α ) ξ [exp { (2 − α ) ξ ( T − u ) } − (cid:33)(cid:17)(cid:0) exp (cid:8) ξ ( T − u ) (cid:9) − (cid:1) (cid:0) exp (cid:8) ξ ( T − u ) (cid:9) + 3 (cid:1) d u. The above upper-bound error is difficult to interpret. In order to this, we do a Taylor analysis ofone term. Then, the following error behavior is retrieved
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