Additive normal tempered stable processes for equity derivatives and power law scaling
AAdditive normal tempered stable processes for equity derivatives and power law scaling
Michele Azzone ‡ & Roberto Baviera ‡ September 17, 2019 ( ‡ ) Politecnico di Milano, Department of Mathematics, 32 p.zza L. da Vinci, Milano Abstract
We introduce a simple model for equity index derivatives. The model generalizes well knownL´evy Normal Tempered Stable processes (e.g. NIG and VG) with time dependent parameters.It accurately fits Equity index implied volatility surfaces in the whole time range of quotedinstruments, including small time horizon (few days) and long time horizon options (years).We prove that the model is an Additive process that is constructed using an Additive subordi-nator. This allows us to use classical L´evy-type pricing techniques. We discuss the calibrationissues in detail and we show that, in terms of mean squared error, calibration is on averagetwo orders of magnitude better than both L´evy processes and Self-similar alternatives.We show that even if the model loses the classical stationarity property of L´evy processes, itpresents interesting scaling properties for the calibrated parameters.
Keywords : Additive process, volatility surface, calibration.
JEL Classification : C51, G13.
Address for correspondence:
Roberto BavieraDepartment of MathematicsPolitecnico di Milano32 p.zza Leonardo da VinciI-20133 Milano, ItalyTel. +39-02-2399 4630Fax. +39-02-2399 [email protected] 1 a r X i v : . [ q -f i n . M F ] S e p dditive normal tempered stable processesfor equity derivatives and power law scaling1 Introduction Following the seminal work of Madan and Seneta (1990), L´evy processes have become a powerfulmodeling solution that provides parsimonious models that are consistent with option prices andwith underlying asset prices. There are several advantages of this modeling approach: this modelclass admits a simple closed formula for the most liquid derivative contracts (Carr and Madan 1999,Lewis 2001) and it allows us to obtain a volatility surface that can reproduce in a parsimoniousway some of the key features observed in the market data. In particular, the class of L´evy normaltempered stable processes (LTS) appears to be rather flexible and it involves very few parameters.LTS are obtained via the well-established L´evy subordination technique (see e.g. Cont and Tankov2003, Schoutens 2003). Specifically, most of these applications involve two processes in the LTSfamily: Normal Inverse Gaussian (NIG) (Barndorff-Nielsen 1997) and Variance Gamma (VG)(Madan et al. σ , which controls the average level of the volatility surface; k , which is related to the convexity of the implied volatility surface; and η , which is linked to thevolatility skew (for a definition see, e.g. Gatheral 2011, Ch.3, p. 35).Unfortunately, the recent literature has shown that these models do not reproduce the impliedvolatilities that are observed in the market data at different time horizons with sufficient precision(see e.g. Cont and Tankov 2003, and references therein). Additive processes have been proposedto overcome this problem. Additive processes are an extension of L´evy processes that considerindependent but not stationary increments. Given an Additive process, for every fixed time t , it isalways possible to define a L´evy process that at time t has the same law of the Additive process.This feature allows us to maintain several properties (both analytical and numerical) of the L´evyprocesses.The probability description of Additive processes is well-established (Sato 1999) but the applica-tions in quantitative finance are relatively few. A first application of Additive processes to optionpricing is developed by Carr et al. (2007), who investigate Self-similar processes in derivativemodeling. Benth and Sgarra (2012) use Additive processes, which they call time-inhomogeneousLevy processes, in the electricity market. In their paper, the electricity spot price is characterizedby Ornstein-Uhlenbeck processes, which are driven by Additive processes. More recently, Li et al. (2016) have considered a larger class of Additive processes. Their paper studies Additive subor-dination, which (they show) is a useful technique for constructing time inhomogeneous Markovprocesses with an analytically tractable characteristic function. This technique is a natural gen-eralization of L´evy subordination.We introduce a new class of stochastic processes that extends the family of Additive processesproposed by Li et al. (2016). We prove that, under some condition on the time-dependent modelparameters, this is a class of Additive processes, which are named Additive tempered stableprocesses (ATS), and we obtain a closed formula expression for the ATS characteristic function.ATS processes cannot be obtained via a time-change as in the Additive subordination of Li et al. (2016). This case corresponds to a specific subclass of ATS process. The main advantage of thisnew class of models is the possibility to exactly calibrate the term structure of observed impliedvolatility surfaces, while maintaining the parsimony of LTS.2e calibrate the ATS process on the S&P 500 and EURO STOXX 50 implied volatility surfacesof the 30th May 2013. The ATS calibration is on average two orders of magnitude better thanthe corresponding LTS in terms of mean squared error.We show that the calibrated time-dependent parameters present an interesting and statisticallyrelevant self-similar behavior. Having observed this scaling behavior of two model parameters, weare able to prove the additivity of the calibrated process.The main contributions of this paper are threefold: • We introduce a new broad family of stochastic processes, which we call Additive temperedstable processes. We show that, under some hypotheses on the model time-dependent pa-rameters, ATS is a family of Additive processes. We introduce a subcase of ATS withself-similar time-dependent parameters. • We calibrate the ATS processes on S&P 500 and EURO STOXX 50 volatility surfaces. Weshow that ATS has better calibration features (in terms of both the Mean Squared Error andthe Average Percentage Error) than LTS and Self-similar processes (constructed extendingthe same LTS). • We consider a rescaled ATS process using as new time the implied volatility term-structure.We show that the calibrated parameters exhibit a self-similar behavior w.r.t. the new time.The statistical relevance of the scaling properties is determined.The rest of the paper is organized as follows. In Section , we introduce a new family of processesas a natural extension of the corresponding L´evy processes and we prove under which conditionsthese processes are Additive. In Section , we describe the dataset used in the calibration and thecalibration results for LTS, ATS and Self-similar processes. In Section we check the calibratedprocess additivity and we present an interesting scaling property in the calibrated parameters.Finally; Section concludes. L´evy normal tempered stable processes (LTS) are commonly used in the financial industry forderivative pricing. According to this class of models, the underlying forward with expiry T is anexponential L´evy; i.e. F t ( T ) := F ( T ) exp( f t ) , with f t a LTS f t = µ S t + σ W S t + ϕ t ∀ t ∈ [0 , T ] , (1)where µ, σ are two real parameters ( µ ∈ R , σ ∈ R + ), while the ϕ is obtained by imposing themartingale condition to F t ( T ). W t is a Brownian motion and S t is a L´evy tempered stablesubordinator independent from the Brownian motion, such as an Inverse Gaussian process forNIG or a Gamma process for VG. This theory is well known and can be found in many excellenttextbooks (see e.g. Cont and Tankov 2003, Schoutens 2003).In some applications, it is more suitable a different parametrization scheme where µ := − (cid:18) η + 12 (cid:19) σ . with η ∈ R . The parameter η controls the volatility skew; that is, the ATM-forward slope of theimplied volatility as a function of the moneyness x := ln F ( T ) /K . In particular, it can be proventhat for η = 0 the smile is symmetric, as shown in the next Lemma.3 emma 2.1. If η = 0 , then the implied volatility surface of a LTS, as a function of the moneyness x := ln F ( T ) /K , is symmetric.Proof . See Appendix A.It has been observed that LTS processes do not properly describe short and long maturity atthe same time, while they allow an excellent calibration for a fixed maturity (see, e.g. Cont andTankov 2003, Ch.14, and references therein).L´evy normal tempered stable processes are pure jump models with independent and stationaryincrements. The key question is as follows: is it reasonable to consider stationary increments whenmodeling implied volatility?Jump stationarity is a feature that significantly simplifies the model’s characteristics but it israther difficult to justify a priori from a financial point of view.For example, a market maker in the option market does not consider the consequences of a jumpto be equivalent on options with different maturities. He cares about the amount of trading inthe underlying required to replicate the option after a jump arrival. Gamma is the Greek measurethat quantifies the amount of this hedging and, generally, it decreases with time-to-maturity. Theimpact of such a jump on the hedging policy is inhomogeneous with option maturity. Althoughit can have a significant impact for short maturities, for options with long maturities, the delta-hedging replication changes slightly, even in presence of a large jump. Hence, a priori , it is notprobable that stationary increments can adequately model implied volatilities.For this reason, we would like to select a model that allows independent but non-stationaryincrements. The simplest way to obtain this modeling feature is to consider model (1) but withtime-dependent parameters. We would desire to model forward exponential with f t = − (cid:18) η t + 12 (cid:19) σ t S t + σ t W S t + ϕ t t (2)where S t is an Additive subordinator independent from the Brownian motion (i.e. a naturalextension of a L´evy subordinator, see, e.g. Sato 1999), σ t and η t deterministic functions of timewith σ t > σ t t an increasing function of t .Unfortunately, this process cannot be obtained as a Brownian motion subordinated with an Ad-ditive subordinator, as in Li et al. (2016). This requires us to carefully build the Additive process.To preserve the Additive property, we need a set of model’s conditions that can be statisticallytested. Once this construction has been realized, we can select the forward price F t ( T ) as amartingale process, in a similar way to the LTS case. The deterministic function of time ϕ t canbe chosen s.t. the process F t ( T ) satisfies this property, as shown at the end of this Section (cf. Theorem 2.13 ).As we will underline in Section , this approach has powerful implications in model calibration,allowing us i) to cut the volatility surface into slices, each one containing options with the samematurity and ii) to calibrate each slice separately.Let us mention two interesting characteristics in model (2): i) for a fixed t the marginal distributionof the forward price is exactly the same of the corresponding L´evy process and ii) it is possible toreproduce exactly the term structure of volatility observed in the market place. We show that thevolatility term structure can be quite general: model (2) is well posed for any bounded σ t s.t. σ t t is an increasing function of time. The former characteristic allows us to price European optionswith a formula as simple as in the L´evy case. The latter is a crucial degree of freedom for marketmakers because they desire to adapt the volatility term structure to the set of events—which caninfluence the underlying —that are known in advance at value date. These events are typically4ither macro events (e.g. main political elections, change in central bank monetary policy) orrelated to the underlying of interest (e.g. dividend payments).In the rest of this section, we prove that a large class of processes (2) are Additive and we callthem Additive normal tempered stable processes (or ATS). In Section we show that once theterm structure has been taken into account, the remaining parameters η t and κ t present a powerlaw scaling and only two free parameters are left for modeling the whole implied volatility surface. In this subsection, we recall the basic definitions and key properties of Additive processes. Thenotation follows closely the one in Sato (1999).
Definition 2.2. Additive process (see, e.g. Cont and Tankov 2003, Def.14.1 p.455) . A c´adl´ag stochastic process on R { X t } t ≥ is an Additive process if and only if it satisfies thefollowing conditions:1. X =0 almost surely;2. Independent increments: for every positive real increasing sequence t , ......, t n the randomvariables X t − X t , ....., X t n − X t n − are independent;3. Stochastic continuity: ∀ (cid:15) > , lim h → P [ | X t + h − X t | > (cid:15) ] = 0 . We call ( A t , ν t , γ t ) the generating triplet that characterizes the Additive process { X t } t ≥ .Notice that a L´evy process is an Additive process by definition; that is, a L´evy process is a processwith stationary increments that satisfies the three conditions of Definition 2.2 . Theorem 2.3. Main Additive properties.
Let { g t } t ≥ be a system of infinitely divisible probability measures on R with generating triplet ( A t , ν t , γ t ) satisfying the following conditions 1, 2 and 3. Then, there exists, uniquely up to identityin law, an Additive process { X t } t ≥ on R s.t. X t has law g t for t ≥ .1. A = 0 , ν = 0 , γ = 0 ;2. Given t, s s.t. ≤ s ≤ t then A s ≤ A t and ν s ( B ) ≤ ν t ( B ) , B ∈ B ( R ) ;3. Given t, s s.t. ≤ s ≤ t then as s → t , A s → A t , ν s ( B ) → ν t ( B ) and γ s → γ t , where B ∈ B ( R ) and B ⊂ { x : | x | > (cid:15) > } .Conversely, given { g t } t ≥ , the law of an Additive process is a system of infinitely divisible proba-bility measure on R with generating triplet ( A t , ν t , γ t ) satisfying conditions 1, 2 and 3.Proof. See Sato (1999), Th.9.8 p.52.
Theorem 2.3 provides a powerful link between process marginal characteristic functions andprocess additivity. In the rest of this section, we use this result to prove the additivity of ATSprocesses.We introduce an Additive subordinator imposing some conditions on an Additive process charac-teristic function. ATS processes are constructed using an Additive subordinator.5 efinition 2.4. Additive subordinator.
An Additive subordinator is an Additive process with infinitely divisible distribution for everyfixed time t satisfying A t = 0, b t := γ t − (cid:82) ≤ x ≤ x ν t ( dx ) ≥ ν t s.t. i) (cid:82) R ( | x | ∧ ν t ( dx ) < ∞ ,ii) ν t (( −∞ , ν t an integral measuredefined on R that can be identified with the real function ν t ( x ) s.t. (cid:82) B ν t ( x ) dx = ν t ( B ) ∀ B ∈ B ( R )and B ⊂ { x : | x | > (cid:15) > } . Hereinafter, when we define a measure through ν t ( x ), we refer to theintegral measure characterized by this real function. Proposition 2.5. Additive subordinator properties.
A subordinator { S t } t ≥ is almost surely positive and almost surely non-decreasing and the followingholds: ln E (cid:2) e iuS t (cid:3) = ib t u + (cid:90) x> (cid:0) e iux − (cid:1) ν t ( x ) dx (3) Proof.
See Appendix A.The following Theorem introduces three different transformations under which process additivityis preserved. We will use these results in the proof of ATS processes additivity (cf.
Theorem2.11 ); for example, the second transformation is the Additive subordination of Li et al. (2016),which is a key ingredient in ATS construction.
Theorem 2.6. Building new Additive processes from known ones
To construct new Additive processes, three are the basic types of transformations under whichAdditive process class is invariant:1. The sum of two independent Additive processes is an Additive process;2. Given { X t } t ≥ a L´evy process and { S t } t ≥ an Additive subordinator then { X S t } t ≥ is anAdditive process;3. Given { X t } t ≥ an Additive process and r t a real continuous increasing function of time s.t. r = 0 then { X r t } t ≥ is an Additive process.Proof. See Appendix A.
Remark . Theorem 2.6 can be useful from a practical point of view. Unfortunately, the first twotransformations cannot be easily relaxed.On the one hand, the sum of two generic Additive process is not necessarily Additive. As acounterexample, consider the process { W t + W S t } t ≥ , where { S t } t ≥ is an Additive subordinator:the independence of increments does not hold. In fact, let r , s and t be three real positive constantss.t. 0 < r < s < t , the increment ( W t + W S t ) − ( W s + W S s )is not independent from W r + W S r if S r is larger than s with a positive probability.On the other hand, the subordination of an Additive process with an Additive subordinator isnot necessarily Additive. As a counterexample, consider the Additive process { X t = t } and thesubordinator { S t } t ≥ : { X S t } t ≥ is not Additive. The independence of increments does not hold,because X S t − X S s = S t − S s = ( S t − S s ) ( S t + S s )and, while S t − S s is independent from S s , S t + S s is dependent from S s (and then from S s ).The third transformation of Theorem 2.6 is very useful because it tells us that any deterministictime-change transforms an Additive process in an Additive process. This property plays a keyrole in Section . 6 .3 The model is Additive In this subsection, we prove the main theoretical results of this paper. We demonstrate under whichconditions the process (2) is Additive trough a constructive proof: we first introduce an Additivesubordinator (called TSS) via its triplet, then we show the role it plays in model construction.Finally we consider i) a subcase of (2) that can be statistically tested on market data and ii)the additional conditions that should be imposed in order to obtain a martingale process for theforwards.We define a stochastic process through its marginal characteristic functions, we then prove thatthis is an Additive subordinator. The selected characteristic function is the one of a L´evy temperedstable subordinator (see e.g. Cont and Tankov 2003) but with a time-dependent parameter k t . Definition 2.7. The process S t . The process { S t } t ≥ is characterized by the triplet (0 , V t , Γ t ) V t ( x ) := t Γ(1 − α ) (cid:18) − αk t (cid:19) − α (cid:18) e − (1 − α ) x/k t x α (cid:19) I x> Γ t := (cid:90) x V t ( x ) dx , (4)where t ∈ R + , α ∈ [0 , − α ) is the gamma function in 1 − α , and k t a positive non-decreasingcontinuous function of time s.t. tk − αt is o (1) for small t and non-decreasing. Proposition 2.8. S t is an Additive subordinator. The process { S t } t ≥ in Definition 2.7 is an Additive subordinator with b t = 0 . k t t is the varianceof the subordinator.Proof. See Appendix A.We call the process S t of Definition 2.7 an Additive tempered stable subordinator (TSS).
Corollary 2.9. σ t S t is an Additive subordinator. Let { S t } t ≥ be a TSS and σ t a positive continuous function of time s.t.1. t σ t is o (1) for small t ;2. tk − αt σ αt is o (1) for small t and non-decreasing;3. σ t k t is non-decreasing;then { σ t S t } t ≥ is an Additive subordinator with b t = 0 .Proof. See Appendix A.
Proposition 2.10. Properties of a TSS.
Let { S t } t ≥ be a TSS, then properties , and hold true.1. E [ S t ] = t ;2. V ar [ S t ] = k t t ; . ln L ( u ) := ln E (cid:2) e − uS t (cid:3) = tk t − αα (cid:26) − (cid:18) u k t − α (cid:19) α (cid:27) if < α < − tk t ln (1 + u k t ) if α = 0 . Proof.
The proof is analogous to the one in Cont and Tankov (2003) for the L´evy case. Theexpected values and the Laplace transform expressions of TSS can be easily obtained for a fixed t , because ATS marginal distributions are equal to marginal distributions of a L´evy TSS.Two are the TSS commonly used Inverse Gaussian (cid:18) α = 12 (cid:19) : ln L ( u ) = tk t (cid:110) − (cid:112) uk t (cid:111) Gamma ( α = 0) : ln L ( u ) = − tk t ln (1 + uk t ) . We are now able to prove the main results of this section: under certain conditions on the time-dependent parameters, the ATS family is a family of Additive processes. We also derive ATScharacteristic function through the TSS Laplace exponent (cf. property 3 of
Proposition 2.10 ). Theorem 2.11. The process f t is Additive The process { f t } t ≥ in (2) with { S t } t ≥ a TSS (4) is Additive when the following conditions holdtrue:1. η t has the same sign ∀ t > ;2. | η t | σ t k t is non-decreasing;3. t | η t | α σ αt k − αt is non-decreasing and o (1) for small t .Moreover, f t has characteristic function φ c ( u ) := E (cid:2) e iuf t (cid:3) = L (cid:18) iu (cid:18)
12 + η t (cid:19) σ t + u σ t (cid:19) e iuϕ t t . (5) Proof.
See Appendix A.It can be noticed that φ c ( u ) is analytic in a strip that includes the points u = 0 , − i . Theproof is similar to the LTS case (see, e.g. Cont and Tankov 2003). Theorem 2.11 characterizescompletely ATS processes; unfortunately, in general, the three conditions are rather difficult toverify statistically.We can introduce a subcase of ATS determined by self-similar functions of time. In Section , weshow that this family of processes describes accurately market implied volatility surfaces. Powerscaling functions of time allow to rewrite Theorem 2.11 conditions as simple inequality on thescaling parameters.
Corollary 2.12.
Consider the process { f t } t ≥ in (2) with { S t } t ≥ a TSS (4) where k t = ¯ k t β and η t = ¯ η t δ and σ t = ¯ σ with α ∈ [0 , , ¯ σ, ¯ k ∈ R + and ¯ η, β, δ ∈ R . The process is an ATS if1. ≤ β < − α , . δ ≥ − β and δ > β (1 − α ) − α for α ∈ (0 , , where the second condition reduces to δ ≥ − β for α = 0 .Proof. By direct verification of the conditions in
Theorem 2.11 .It is interesting to observe that the LTS case falls in the subcase described by this
Corollary .This corresponds to the case with both k t and η t time independent; that is, β and δ equal to zero.With Theorem 2.11 we have fully characterized the ATS processes of interest. Finally, to modelthe forward, the process { F t ( T ) } t ≥ should be a martingale. In the next theorem we prove that,imposing a condition on ϕ t , the ATS process is a martingale w.r.t. the proper filtration. Theorem 2.13. Martingale process.
The forward { F t ( T ) } t ≥ , modeled via an exponential Additive process characterized by a process { f t } t ≥ satisfying the conditions of Theorem 2.11 is a martingale w.r.t. the filtration {F t } t ≥ generated by the Additive subordinator and the Brownian motion, if and only if ϕ t t = − ln L (cid:0) σ t η t (cid:1) . (6) Proof.
See Appendix A.
In this subsection we describe the dataset (and the filtering techniques) considered in modelcalibration.We analyze all quoted S&P 500 and EURO STOXX 50 option prices observed at 11:00 am NewYork time of the 30th May 2013. The dataset is composed of real market quotes (no smoothingor interpolation). Let us recall that the options on these two indices are the most liquid optionsin the equity market at world level. For both indices, options expire on the third Friday of themonths of March, June, September and December in the front year and June and December in thenext year. In the EURO STOXX 50 case also December contracts for the following three yearsare available. The dataset also includes the risk-free interest rate curves bootstrapped from (USDand EUR) OIS curves. Financial data are provided by Bloomberg.The dataset contains call and also put bid and ask prices in a regular grid of strikes for eachavailable maturity. We filter out the options that do not satisfy the two liquidity criteria discussedin Azzone and Baviera (2019, section 2).As forward price, we use the synthetic forward because this allows a perfect synchronization withoption prices and, for several maturities, it identifies the most liquid forward in the market. Thesynthetic forward is obtained The synthetic forward price F ( T ) is obtained following the sameprocedure of Azzone and Baviera (2019, section 4). In Figure 1 we show, for a given underlyingand a given maturity, the values considered in the forward price construction and the value selectedby the algorithm. 9igure 1: An example of the construction of the synthetic forward price: bid, ask and mid forward pricesof EURO STOXX 50 for the JUN14 maturity. Only prices not discarded by the two liquidity criteria,described in the text, are considered. Prices are obtained using the put-call parity relationship on quotedoptions. According to the algorithm described in text, the values related to the lowest strike (K= 1700)are also discarded from the forward price computation. We observe that the discarded forward price isrelated to a DOTM put and lies on the left of already discarded values due to the two liquidity criteria.We show in red the corresponding forward bid-ask prices and with a diamond the forward price F ( T )relative to this expiry. In Figure 2 we plot the bid, ask and mid synthetic forward prices for the different maturitiesavailable for the S&P 500 and the EURO STOXX 50.Figure 2: Term structure of the synthetic forward prices: we report also observed bidand ask prices for every maturity. On the left hand we plot the S&P 500 index case andon the right hand the EURO STOXX 50 index case.10 .2 Calibration
In this subsection, we describe the model calibration procedure. We also compare the performanceof ATS processes with the performance of LTS processes and of Self-similar processes in Carr et al. (2007). Hereinafter, we focus on α = 1 / α = 0 (VG), which are the two (ATS andSelf-similar) generalizations of the two most frequently used LTS processes.As already underlined in Section , for every fixed maturity T , the marginal distribution of anAdditive tempered stable process is equal to the marginal distribution of a L´evy tempered stableprocess. A different L´evy NIG and VG is calibrated for every different maturity and the threetime-dependent parameters k T , η T , σ T are obtained.Beneath the ATS processes, we consider the calibration of the standard L´evy processes and ofthe (four parameters) Self-similar processes proposed by Carr et al. (2007). Option prices arecomputed using the Lewis (2001) formula C ( K, T ) = B T F ( T ) (cid:26) − e − x/ (cid:90) ∞−∞ dz π e − iz x φ c (cid:18) − z − i (cid:19) z + (cid:27) (7)where φ c ( u ) is analytical in the strip 0 ≤ (cid:61) ( u ) ≤ x is the moneyness . The calibrationis performed minimizing the Euclidean distance between model and market prices. The simplexmethod is used to calibrate every maturity of the ATS process. For L´evy processes and Self-similar processes, because standard routines for global minimum algorithms are not satisfactory,we consider a differential evolution algorithm together with a multi-start simplex method.The calibration performance is reported in Table 1 in terms of Mean Squared Error (MSE) andMean Absolute Percentage Error (MAPE). It is possible to observe that Self-similar processesslightly improve L´evy performance, as reported in the literature (see e.g Carr et al. α ∈ [0 , .
56 1 . . .
13% 1 . . %S&P 500 VG 8 .
49 2 . . .
31% 1 . . %Euro Stoxx 50 NIG 22 .
15 9 . . .
75% 0 . . %Euro Stoxx 50 VG 55 .
81 9 . . .
85% 0 . . %Table 1: Calibration performance for the S&P 500 and EURO STOXX 50 in terms of MSE and MAPE.In the NIG and VG cases we consider the standard L´evy process, the Self-similar process and the corre-sponding ATS process. Self-similar processes perform better than L´evy processes but ATS improvementis far more significant: two orders of magnitude for MSE and one order for magnitude of MAPE.
Figure 3 shows the differences of MSE w.r.t. different maturities for S&P 500 volatility surfacecalibrated with a NIG process. Self-similar and L´evy LTS have a MSE of the same order ofmagnitude, while the improvement of ATS is of two orders of magnitude and particularly significantfor the short time. The short time improvement in implied volatility calibration is particularlyevident, as shown in Figure 4. 11igure 3:
MSE w.r.t. different maturities (in years) for S&P 500 volatility surface calibrated with a NIGprocess. Self-similar (circles) and L´evy (triangles) have a MSE of the same order of magnitude, while theimprovement of ATS (squares) is of two orders of magnitude and particularly significant for the shorttime.
In Figure 4, we plot the market implied volatility and the volatility replicated via ATS, LTS andSelf-similar processes at 1 and 8 month maturities. We observe that the ATS implied volatility isthe closest to the market implied volatility in any case and it significantly improves both LTS andSelf-similar processes, particularly for small maturities. Similar results hold for all other ATS.Figure 4:
Implied volatility smile for S&P 500 at a given maturity: 1 month (on the left) and 8 months(on the right). The ATS process, self-similar LTS process and LTS process implied volatility are plottedtogether with the market implied volatility. ATS reproduces the smile significantly better then thealternatives, the improvement is particularly evident for small maturities.
In Figure 5 we plot the market and the ATS implied volatility skews (for a definition see, e.g.Gatheral 2011, Ch.3, p.35) for EUROSTOXX 50 w.r.t. the maturities. We observe that thecalibrated ATS replicate accurately the market implied volatility skews.12igure 5:
The market and the ATS implied volatility skews for EUROSTOXX 50 w.r.t. the maturities.ATS replicate market implied volatility skews behaviour.
In this subsection, the calibrated process is rescaled through a deterministic time-change θ := T σ T .We statistically test whether the rescaled process is Additive w.r.t. θ , showing that it verifies theconditions of Corollary 2.12 . This fact implies, thanks to
Theorem 2.6 , that the forwardexponential (2) is also an ATS process.Given a generic ATS process { f T } T ≥ , it is always possible to define ˆ k θ := k T σ T and ˆ η θ := η T andto construct a new process, as follows:ˆ f θ := − (cid:18) ˆ η θ + 12 (cid:19) S θ + W ( S θ ) + ˆ ϕ θ θ , (8)where ˆ ϕ θ θ = − ln ( L S θ (ˆ η θ )) and S θ is a tempered stable subordinator with variance ˆ k θ θ . { ˆ f θ } θ ≥ is an Additive tempered stable process w.r.t. θ if the conditions of Corollary 2.12 hold.We calibrate the ATS process and analyze the rescaled parameters, in both S&P 500 and EUROSTOXX 50 cases. We observe a self-similar behaviour of ˆ k θ and ˆ η θ ; that is,ˆ k θ = ¯ kθ β ˆ η θ = ¯ ηθ δ , (9)where ¯ k is a positive constant and ¯ η , β and δ are real constant parameters.To investigate this behavior and to infer the value of the scaling parameters we consider equations(9) in log-log scale.In Figures 6 and 7 we plot the weighted regression lines and the observed time dependent pa-rameters ln ˆ k θ and ln ˆ η θ with their confidence intervals for S&P 500 and EURO STOXX 50. Asconfidence intervals, we have considered two standard deviations, respectively, of ln ˆ k θ and ofln ˆ η θ . In Appendix B, we discuss the estimation of the standard deviations via a confidenceinterval propagation technique and the selection of the weights.13igure 6: Weighted regression line and the observed time dependent parameters ln ˆ k θ and ln ˆ η θ w.r.t.ln θ for the NIG calibrated model for S&P 500. We plot a confidence interval equal to two times thecorresponding standard deviation. Figure 7:
Weighted regression line and the observed time dependent parameters ln ˆ k θ and ln ˆ η θ w.r.t.ln θ for the VG model calibrated on EURO STOXX 50. We plot a confidence interval equal to two timesthe corresponding standard deviation. The fitted regression lines provides us with an estimation of β and δ . The scaling parametersappear qualitatively compatible to β = 1 and δ = − in all observed cases; it is interesting toobserve that, with such scaling parameters, the process (8) is Additive according to Corollary2.12 . We can test whether there is statistical evidence that our hypothesis is consistent withmarket data. We observe that, in both volatility surfaces and for both models (VG and NIG), thescaling parameters are consistent with our hypothesis β = 1 and δ = − . The estimated scalingparameters together with the p-value of statistical tests are reported in Table 2. In all cases weaccept the null hypotheses with a 5% threshold. Notice that all p-values, except the S&P 500 VG β , are above 15%. 14urface Model Parameter Parameter’s Value p-valueS&P 500 NIG β .
10 0 . δ − .
47 0 . β .
08 0 . δ − .
50 0 . β .
02 0 . δ − .
44 0 . β .
01 0 . δ − .
48 0 . Scaling parameters calibrated from S&P 500 and EURO STOXX 50 volatility surfaces, param-eter estimates are provided together with the p-values of the statistical tests that verify whether it ispossible to accept the null hypothesis β = 1 and δ = − . This result has two major implications.First, from an “experimental” point of view, we have observed what seems to be a stylized fact ofthis model class: both ˆ η θ and ˆ k θ scale as power law. This property should be tested on a largerdatabase.Second, from a “theoretical” point of view, we can state that it is Additive the original process inreal time { f T } T ≥ in (2). This fact is a consequence of the properties of volatility term structure σ T (it is always observed on real data that σ T T is non-decreasing) and of property 3 of Theorem2.6 . This theorem states that if (cid:110) ˆ f θ (cid:111) θ ≥ is an Additive process then (cid:110) ˆ f T σ T (cid:111) T ≥ is an Additiveprocess w.r.t. T ; moreover, for every T the processes f T and ˆ f T σ T have the same marginal law. In this section, we compare ATS with the two classes of Additive processes already present in thefinancial literature, the Self-similar processes (see, e.g. Carr et al. et al. η t parameter and another to the skewness and to the excess kurtosis of the calibratedexponential forward. We consider two statistical tests to show whether these two alternativeAdditive classes can properly describe some stylized facts observed in the market data.A first test is build to verify the adequacy of Self-similar Processes. Given a model for underlyingdynamics (e.g. chosen α in the Normal Tempered Stable model), it is possible to compute skewnessand kurtosis. For example, a Self-similar Process has skewness and kurtosis constant over time,as it can be deduced by definition (see, e.g. Carr et al. √ t : we reject the null hypothesis of no slope in all of the cases that weanalyzed (both indices and both tempered stable models; that is, NIG and VG) with p-values ofthe order of 10 − . Similar results hold in all ATS cases.It is interesting to observe that ATS short time skewness and kurtosis are asymptotic to thesquared root of time (available upon request); thus, the process { f t } t ≥ has skewness and kurtosisconsistent with the behaviour observed in the market.15igure 8: Observed time dependent skewness (kurtosis) w.r.t. √ t for the NIG calibrated model on S&P500 volatility surfaces. We plot a confidence interval equal to two times the standard deviation. Thebehaviour is not consistent with a Self-similar process. The other statistical test aims to verify the adequacy of Additive processes obtained throughAdditive subordination (Li et al. η θ is equal to a constant ¯ η (and for a generic term structure σ T ), falls within this class.In Figure 6 and 7 we have already shown the time scaling ˆ η θ . We can statistically test the nullhypothesis of constant ˆ η θ . For both volatility surfaces and for both tested tempered stable models(NIG and VG) we reject the null hypothesis of a constant ˆ η θ with p-values below 10 − . As alreadyobserved, ATS processes are characterized by a power law scaling in ˆ η θ , such as the one observedin market data.In this section we have shown that some power law scalings are observed in market data. Thesestylized facts are extremely relevant. On the one hand, they enable us to verify that an Additivemodel cannot be rejected when analyzing volatility surfaces. On the other hand, they allow us todiscard other Additive models already present in the financial literature. In this paper we introduce a new broad family of stochastic processes that we call Additive normaltempered stable processes (ATS) and we prove that, under some hypotheses on the model time-dependent parameters, ATS is a family of Additive processes. An interesting subcase of ATSpresents a power-law scaling of the time-dependent parameters.We have considered all quoted options on S&P500 and EURO STOXX 50 at 11:00 am New Yorktime of the 30th May 2013;. The dataset considers options on a time horizons starting from twoweeks and up to several years. We calibrate the ATS processes on the options of both indices,showing that ATS present better calibration features than LTS and Self-similar processes. Theobserved improvement of ATS is even of two orders of magnitude in terms of MSE, as presentedin Table 1.The quality of ATS calibration results looks quite incredible. In Sections and , we have shownthat once the volatility term structure has been taken into account, the whole implied volatilitysurface is calibrated accurately with only two free parameters.16e also construct a rescaled ATS process using as new time the implied volatility term-structure.We show that the rescaled process calibrated parameters exhibit a power-law behavior. Statisticalrelevance of the scaling properties is discussed in detail.Finally, we have compared some model consequences with the two alternative Additive processespresent in the financial literature. These two classes fail to reproduce some stylized facts observedin market data, which are adequately described by ATS processes. Acknowledgements
We thank F. Benth, P. Carr and J. Gatheral for enlightening discussions on this topic. We alsothank G. Guatteri, M.P. Gregoratti, P. Spreij and all partecipants to WSMF 2019 in Lunterenand to VCMF 2019 in Vienna. R.B. feels indebted to P. Laurence for several helpful and wisesuggestions on the subject.
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Symbol Description A t diffusion term of the Additive process { X t } t ≥ b t drift term of an Additive subordinator { S t } t ≥ B T discount factor between value date and T B ( R ) Borel set on R C ( T, K ) Call option price at value date with maturity T and strike K { f t } t ≥ ATS process that models the forward exponent (cid:110) ˆ f θ (cid:111) θ ≥ rescaled ATS process F t ( T ) price at time t of a Forward contract with maturity T { g t } t ≥ system of ATS marginal infinitely divisible distributions k t ATS subordinator variance parameterˆ k θ rescaled ATS subordinator variance parameter¯ k constant part of the rescaled ATS subordinator variance parameter ˆ k θ L Laplace transform of the subordinator S t N n ( µ, Σ) n-dimensional Gaussian distribution with mean µ and variance Σ P ( T, K ) Put option price with maturity T and strike K { S t } t ≥ Subordinator (either LTS or ATS) V t ATS subordinator jump measure term W t Brownian motion x option moneyness α tempered stable subordinator distribution parameter β scaling parameter of k t Γ( ∗ ) Gamma function evaluated in *Γ t tempered stable subordinator drift term δ scaling parameter of η t ϕ LTS process drift term ϕ t ATS process deterministic drift termˆ ϕ θ rescaled ATS deterministic drift term φ c characteristic function of the forward exponent f t η LTS skew parameter η t ATS skew parameterˆ η θ rescaled ATS skew parameter¯ η constant part of the rescaled ATS skew parameter ˆ η θ µ drift term of the LTS process equivalent to − ( η + 1 / σ ν t jump measure of the Additive process { X t } t ≥ σ LTS diffusion parameter σ t ATS diffusion parameter¯ σ ATS constant diffusion parameter σ B ( x ) implied (Black) volatility w.r.t. moneyness, for a given maturity θ rescaled maturity, defined as σ T T horthands ATS Additive normal tempered stable processB&S Black and Scholescf. confrontLTS L´evy normal tempered stable processMAPE Mean Absolute Percentage ErrorMSE Mean Squared ErrorNIG Normal Inverse Gaussian processp.d.f. probability density functionr.v. random variables.t. such thatTSS tempered stable subordinatorVG Variance Gamma processw.r.t. with respect to20 ppendix A Proofs
Proof of Lemma 2.1
It is enough to prove the proposition in the call case. If η = 0 we can write the call price accordingto the Lewis formula (7) C ( T, K ) = B T F ( T ) (cid:26) − e − x/ (cid:90) + ∞−∞ dω π e − ixω L [ σ T ( ω + 1 / / ω + 1 / (cid:27) , where B T is the discount factor between the value date and the maturity and L is the Laplacetransform of the subordinator. Let us notice that the function of x that multiplies e x/ is symmetricin x , whatever the Laplace transform L .The Black call price with same strike and maturity, according to Lewis formula, is C B ( T, K ) = B T F ( T ) (cid:26) − e − x/ (cid:90) + ∞−∞ dω π e − ixω exp [ − σ B ( x ) T ( ω + 1 / / ω + 1 / (cid:27) , where the implied volatility σ B ( x ) is obtained imposing the equality of the two prices, or equiva-lently (cid:90) + ∞−∞ dω π e − ixω L [ σ T ( ω + 1 / / ω + 1 / (cid:90) + ∞−∞ dω π e − ixω exp [ − σ B ( x ) T ( ω + 1 / / ω + 1 / . Due to the symmetry in x of the left-hand part of the equation, the above equality is only satisfiedif even the right-hand part has the same symmetry and then the (positive) implied volatility issymmetric. Proof of Proposition 2.5
This proof extends to the Additive case the one in Cont and Tankov (2003) for L´evy subordination(Cor.3.1 and Prop.3.10, pp.84-85). Define L ( x ) := I | x |≤ xν t ( x ) and M ( x ) := ( e iux − ν t ( x ). Wehave thatln E (cid:2) e iuS t (cid:3) = iγ t u + (cid:90) R (cid:0) e iux − − I | x |≤ iux (cid:1) ν t ( x ) dx = iγ t u + (cid:90) R ( iuL ( x ) + M ( x )) dx . The first equality is due to the definition of an Additive process characteristic function withno diffusion. L ( x ) is integrable w.r.t. x thanks to the conditions on ν t in Definition 2.4 .The sum of iuL ( x ) and M ( x ) is integrable, because E (cid:2) e iuS t (cid:3) is a well defined characteristicfunction, thus M ( x ) is integrable too. We can split the integral and check the thesis defining b t := γ t − (cid:82) ≤ x ≤ xν t ( dx ). This proves equation (3).By Definition 2.4 , b t is positive (i.e. positive drift) and there is no possibility of negative jumps;hence, the process is almost surely non-decreasing. By Definition 2.2 , X = 0 almost surely andbecause the process is almost surely non-decreasing, the process is almost surely positive. Proof of Theorem 2.6
We separately prove the three points in the theorem. point 1.
Let { L t } t ≥ and { G t } t ≥ be independent Additive processes with triplets ( α t , β t , γ t ) and( a t , b t , g t ). Define { Y t } t ≥ := { L t + G t } t ≥ . The following holds ∀ t > E [ e iuY t ] = E [ e iuL t ] E [ e iuG t ]= e − u ( α t + a t )2 + i ( γ t + g t ) u + (cid:90) R (cid:0) e iux − − I | x |≤ iux (cid:1) ( β t + b t )( dx ) . Theorem 2.3 { Y t } t ≥ is an Additive process with triplet ( α t + a t , γ t + g t , β t + b t ). point 2. We prove the thesis verifying the three conditions of an Additive process in
Definition2.2 . The proof extends the one in Cont and Tankov (2003, Th.4.2, p.120) on L´evy subordination.1. Condition 1 holds by
Definition 2.2 . For the processes { S t } t ≥ and { X t } t ≥ , S = X = 0almost surely. Thus, X S = 0 almost surely.2. We prove the independence of increments. Let F S be the sigma-algebra generated by theprocess { S t } t ≥ ; for any increasing time sequence t , t , . . . , t N , let us write the characteristicfunction of the vector of increments: E (cid:20) e i (cid:80) Ni =1 u j (cid:16) X St ( j ) − X St ( j − (cid:17) (cid:21) = E (cid:34) E (cid:34) N (cid:89) j =1 e iu j (cid:16) X St ( j ) − X St ( j − (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F s (cid:35)(cid:35) = E (cid:34) N (cid:89) j =1 E (cid:20) e iu j (cid:16) X St ( j ) − X St ( j − (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) F s (cid:21)(cid:35) = E (cid:34) N (cid:89) j =1 e ( S t ( j ) − S t ( j − ) ψ ( u j ) (cid:35) (10)= N (cid:89) j =1 E (cid:104) e ( S t ( j ) − S t ( j − ) ψ ( u j ) (cid:105) = N (cid:89) j =1 E (cid:20) e iu j (cid:16) X St ( j ) − X St ( j − (cid:17) (cid:21) , (11)where equality (10) is due to the independence of { X t } t ≥ increments and to the characteristicfunction of the L´evy process; equality (11) to the independence of { S t } t ≥ increments.3. Stochastic continuity w.r.t. time follows from stochastic continuity of the two processes. point 3. We prove the thesis using the definition of Additive process, similarly to the previouspoint.1. By hypothesis r = 0 and by Definition 2.2 X = 0 almost surely. Thus, X r = 0 almostsurely.2. Independence of increments follows from the monotonicity of r t .3. Stochastic continuity w.r.t. time follows from stochastic continuity of the Additive processand continuity of the function r t . Proof of Proposition 2.8
We check whether the conditions of
Definition 2.4 on the generating triplet of an Additivesubordinator are satisfied by { S t } t ≥ . Let us observe that there is no diffusion term and (cid:90) ∞ ( | x | ∧ V t ( x ) dx ≤ (cid:90) ∞ x V t ( x ) dx = t (cid:90) ∞ − α ) (cid:18) − αk t (cid:19) − α e − (1 − α ) xkt x α dx = t , (12)where the last equality is due to the definition of Γ(1 − α ). Moreover V t ( −∞ ,
0) = 0 and b t is nullby direct substitution of Γ t in the formula of Definition 2.4 .We show that { S t } t ≥ is an Additive process using Theorem 2.3 ; that is, we check whether thetriplet introduced in
Definition 2.7 satisfies the Theorem conditions.1. The triplet has no diffusion term. 22. V t is not decreasing in t because t/k − αt and k t are non-decreasing functions of t (see Defi-nition 2.7 ).3. For t > V t ( B ), where B ∈ B ( R + ) and B ⊂ { x : | x | > (cid:15) > } , is due tothe composition of continuous functions. For t = 0 we can extend V t ( B ) and Γ t to 0 sinceboth converge to 0 as t →
0. The convergence of Γ t to 0 is due to Γ t positiveness and to thecondition Γ t ≤ t (see equation (12)). The convergence of V t ( B ) to 0 is due to the dominatedconvergence Theorem. We observe that, ∀ x ∈ R + s.t. | x | > (cid:15) > V t ( x ) is finite and adecreasing function in t .An Additive process that satisfies the conditions on the triplet of Definition 2.4 is a subordinator.The following technical Lemma is used in the proof of
Corollary 2.9 and
Theorem 2.11 : Lemma A.1.
Let { S t } t ≥ be an Additive subordinator, with generating triplet (0 , V t , Γ t ) and b t = 0 ,and let η t be a finite and continuous function of t in (0 , ∞ ) then, if η t (cid:54) = 0 , η t S t characteristicfunction is: ln E [ e iuS t η t ] = (cid:90) R (cid:0) e ium − (cid:1) V t (cid:16) mη t (cid:17) | η t | dm Proof. If η t is different from zero, then the characteristic function of S t η t is:ln E [ e iuS t η t ] = (cid:90) R ( e iuxη t − V t ( x ) dx (13)defining m := η t x and changing the integration variable, the thesis follows. Proof of Corollary 2.9
Notice that if σ s (cid:54) = 0 for a given s , then σ t (cid:54) = 0 ∀ t > s to satisfy conditions 2 or 3. Thus, thereare two cases of interest.First, if σ t = 0 ∀ t then S t σ t = 0 which is an Additive subordinator.Second, if σ t (cid:54) = 0 ∀ t , we obtain the marginal characteristic function expression by Lemma A.1 . S t σ t has the characteristic function of a process which family of infinitely divisible distributionhas no diffusion term and is characterized by the jump measure ˆ V t and the drift term ˆΓ t : ˆ V t ( m ) := V t (cid:16) mσ t (cid:17) σ t = tσ αt k − αt − α ) (1 − α ) − α e − (1 − α ) mktσ t m α I m ≥ ˆΓ t := (cid:90) m ˆ V t ( m ) dm . Then, it is possible to show that { σ t S t } t ≥ is an Additive subordinator, following the same stepsin the proof of Proposition 2.8 .We will now prove a technical result that is useful in the proof of
Theorem 2.11 . Theorem A.2.
Let { X S t ,t } t ≥ { Y S t ,t } t ≥ be two Additive process satisfying the conditions below and { S t } t ≥ be anAdditive subordinator w.r.t. t , then { R t } t ≥ := { X S t ,t + Y S t ,t } t ≥ is an Additive process.1. { X s,t } s,t ≥ and { Y s,t } s,t ≥ are two independent families of random variables also independentfrom { S t } t ≥ . .a E (cid:2) e iu ( X s,t − X r,p ) (cid:3) = e A ( u )( B ( s ) C ( t ) − B ( r ) C ( p )) , where A and C are real functions and B is a realand invertible function ∀ s > r, t > p ∈ R + .2.b E (cid:2) e iu ( Y s,t − Y r,p ) (cid:3) = e a ( u )( b ( s ) c ( t ) − b ( r ) c ( p )) , where a and c are real functions and b is a real andinvertible function ∀ s > r, t > p ∈ R + .3.a { B ( S t ) C ( t ) } t ≥ is an Additive process.3.b { b ( S t ) c ( t ) } t ≥ is an Additive process.Proof. We prove that { R t } t ≥ is an Additive process showing that it satisfies the three conditionsof Definition 2.2 .1. R = 0 almost surely since by Definition 2.2 X S , = Y S , = 0 almost surely.2. ∀ t > s > r ≥ R t − R s = ( X S t ,t − X S s ,s + Y S t ,t − Y S s ,s ) is independentfrom R s − R r = ( X S s ,s − X S r ,r + Y S s ,s − Y S r ,r ). For the properties of Additive process( X S t ,t − X S s ,s ) ⊥⊥ ( X S s ,s − X S r ,r ) and ( Y S t ,t − Y S s ,s ) ⊥⊥ ( Y S s ,s − Y S r ,r ).It remains to prove ( X S t ,t − X S s ,s ) ⊥⊥ ( Y S s ,s − Y S r ,r ) and ( Y S t ,t − Y S s ,s ) ⊥⊥ ( X S s ,s − X S r ,r ). Wedenote with S the sigma algebra generated by the process { S t } t ≥ . We denote M the smallestsigma algebra generated by the random vector { S s , S r } . Notice that M is equivalent to thesigma algebra generated by the vector { B ( S s ) C ( s ) − B ( S r ) C ( r ) , B ( S r ) C ( r ) } , because B isan invertible function and C a real deterministic function of time, moreover { B ( S t ) C ( t ) } t ≥ is an Additive process. Hence, B ( S t ) C ( t ) − B ( S s ) C ( s ) ⊥⊥ M .Let us prove that ( X S t ,t − X S s ,s ) ⊥⊥ ( Y S s ,s − Y S r ,r ): E (cid:104) e iu ( X St,t − X Ss,s ) + iu ( Y Ss,s − Y Sr,r ) (cid:105) = E (cid:104) E (cid:104) e iu ( X St,t − X Ss,s ) + iu ( Y Ss,s − Y Sr,r ) (cid:12)(cid:12)(cid:12) S (cid:105)(cid:105) = E (cid:2) e A ( u )( B ( S t ) C ( t ) − B ( S s ) C ( s ))+ a ( u )( b ( S s ) c ( s ) − b ( S r ) c ( r )) (cid:3) (14)= E (cid:2) E (cid:2) e A ( u )( B ( S t ) C ( t ) − B ( S s ) C ( s ))+ a ( u )( b ( S s ) c ( s ) − b ( S r ) c ( r )) (cid:12)(cid:12) M (cid:3)(cid:3) = E (cid:2) E (cid:2) e A ( u )( B ( S t ) C ( t ) − B ( S s ) C ( s )) (cid:3) e a ( u )( b ( S s ) c ( s ) − b ( S r ) c ( r )) (cid:3) (15)= E (cid:2) e A ( u )( B ( S t ) C ( t ) − B ( S s ) C ( s )) (cid:3) E (cid:2) e a ( u )( b ( S s ) c ( s ) − b ( S r ) c ( r )) (cid:3) = E (cid:104) e iu ( X St,t − X Ss,s ) (cid:105) E (cid:104) e iu ( Y Ss,s − Y Sr,r ) (cid:105) , (16)because equality (14) is due to conditions 2.a and 2.b; equality (15) is due to the Additiveproperty of the process { B ( S t ) C ( t ) } t ≥ (i.e. condition 3.a) and to the fact that S r and S s are M -measurable; equality (16) is due to conditions 2.a and 2.b.With a similar procedure it is straightforward to prove that also ( Y S t ,t − Y S s ,s ) ⊥⊥ ( X S s ,s − X S r ,r ).This prove the independence { R t } t ≥ increments.3. Stochastic continuity follows from stochastic continuity of { X S t ,t } t ≥ and { Y S t ,t } t ≥ . Proof of Theorem 2.11
By the properties of a Gaussian r.v. σ t W S t has the same law of W σ t S t . In this proof, we will usesecond formulation. We prove that { η t σ t S t } t ≥ is an Additive process.Notice that if η t σ t = 0 ∀ t , then S t σ t η t = 0, which is an Additive process.If η t σ t is different from zero, then we obtain the characteristic function expression by Lemma A.1 .24 t σ t η t has the characteristic function of a process which family of infinitely divisible distributionhas no diffusion term and is characterized by the jump measure ˆ V t and the drift term φ t :ˆ V t ( m ) := V t (cid:16) mσ t η t (cid:17) σ t | η t | = t σ αt η αt k − αt sign ( η t )Γ(1 − α ) (cid:18) − αk (cid:19) − α e − (1 − α ) mσ t ηtkt m α I mηt ≥ ˆΓ t := (cid:90) − m ˆ V t ( m ) dm . If η t is positive, then ˆ V t ( m ) is defined on [0 , ∞ ). If η t is negative, then the jump measure is definedon ( −∞ ,
0] hence the sign of η t match the negative sign of m i.e. we can collect the always positiveterms (cid:0) η t m (cid:1) α and (cid:16) sign ( η t ) m (cid:17) . Thanks to the hypotheses on k t , η t and σ t the triplet (0 , ˆ V t , ˆΓ t ) satisfythe conditions of Theorem 2.3 thus it exists an Additive process equal in law to { η t σ t S t } t ≥ .1. ˆ V t is increasing in t . The jump measure is a positive product of an exponential and a functionincreasing in t .2. Both ˆ V t and ˆΓ t can be extended to 0 thanks to dominated convergence theorem. Theseresults hold both for negative and positive η t .3. Continuity is given by the composition of continuous functions.We prove that the process { f t } t ≥ is Additive via Theorem A.2 , using as Additive subordinator (cid:110) ˆ S t (cid:111) t ≥ the process { σ t S t } t ≥ .1. (cid:0) + η t (cid:1) σ t S t is Additive by Theorem A.2 . We define X s,t := s , Y s,t := η t s it is straightfor-ward to check that { X s,t } s ≥ ,t ≥ , { Y s,t } s ≥ ,t ≥ and (cid:110) ˆ S t (cid:111) t ≥ verify the conditions of Theo-rem A.2 .2. − (cid:0) + η t (cid:1) σ t S t + W ( S t σ t ) is Additive by Theorem A.2 . We define X s,t := − (cid:0) + η t (cid:1) s and Y s,t := W s , it is straightforward to check that { X s,t } s ≥ ,t ≥ , { Y s,t } s ≥ ,t ≥ and (cid:110) ˆ S t (cid:111) t ≥ verify the conditions of Theorem A.2 .3. ϕ t t is a continuous function of time, null in t = 0. Thus, it is Additive.4. Let us observe that tϕ t + (cid:0)(cid:0) + η t (cid:1) σ t S t + σ t W ( S t ) (cid:1) is the sum of two independent Additiveprocesses. For Theorem 2.6 this sum is an Additive process.For all t > S t E [ e iuf t ] = E (cid:20) E (cid:20) e iuϕ t t − iu ( + η t ) σ t S t + iuW σ t St (cid:12)(cid:12)(cid:12)(cid:12) S t (cid:21)(cid:21) == E (cid:20) e iuϕ t t − iu ( + η t ) σ t S t − u σ t S t (cid:21) = L (cid:18) iu (cid:18)
12 + η t (cid:19) σ t + u σ t (cid:19) e iuϕ t t , where L is the Laplace transform of S t . 25 roof of Theorem 2.13 A forward contract, valued in t with delivery in T , is F t ( T ) = F ( T ) e f t .We prove the sufficient condition. If the forward is martingale E [ F t ( T ) |F ] = F ( T ) . This is equivalent to impose that E (cid:2) e f t (cid:12)(cid:12) F (cid:3) = 1 , (17)that is, the characteristic function of f t computed in − i is equal to one. From equation (5) E [ e f t |F ] = L (cid:18)(cid:18) η t + 12 (cid:19) σ t − σ t (cid:19) e ϕ t t = L (cid:0) σ t η t (cid:1) e ϕ t t . (18)Imposing the condition (17), we get ϕ t .For the necessary condition, we follow two steps. First, given ϕ t by equation (6) we prove that E [ e f t |F ] = 1 , ∀ t >
0. This fact is a consequence of equation (18).Second, we check the martingale condition; that is, ∀ s, t s.t 0 < s < tE [ F t ( T ) |F s ] = F ( T ) E (cid:2) e f t − f s + f s (cid:12)(cid:12) F s (cid:3) = e f s F ( T ) E (cid:2) e f t − f s (cid:3) = F s ( T ) E (cid:2) e f t − f s (cid:3) . The theorem is proven once we prove that E (cid:2) e f t − f s (cid:3) = 1.This equality holds because f t is Additive; that is, process increments are independent E (cid:2) e f t |F (cid:3) = E (cid:2) e f t − f s |F (cid:3) E (cid:2) e f s |F (cid:3) , then E (cid:2) e f t − f s (cid:3) = E (cid:2) e f t − f s |F (cid:3) = E (cid:2) e f t |F (cid:3) E [ e f s |F ] = 1 . Appendix B Parameter estimation
In physics and engineering, all measurements are subject to some uncertainties or “errors”. Erroranalysis is a vital part of any quantitative study (see, e.g. Taylor 1997). In this appendix, weestimate pricing errors and “propagate” them to model parameters. This is a crucial passage toverify the quality of the proposed model.First, we estimate pricing errors. In finance, the idea of considering the bid ask spread in marketprices as a sort of measurement error of “true” prices is well known and goes back to the seminalpaper of Roll (1984). He considers the price y = y ∗ + q ( y ask − y bid ) /
2, where y is the observedprice, y ∗ the unobserved true price and q a binomial r.v. that takes value in {− , } with equalprobability, where − y is Σ y = ( y ask − y bid ) /
2. More recently, George et al. (1991) propose an extended formulationof the price y = y ∗ + πq ( y ask − y bid ) /
2, where π is the unobserved proportion of the spread dueto the so-called order processing cost; π is estimated from market data as a value 0.8 and in allcases analyzed in George et al. (1991) is observed a value greater than 0.5. Conservatively, π canbe chosen as 0.5, obtaining the relation Σ y = ( y ask − y bid ) / y as a Gaussian random variable with a mean equal to the mid-market price ( y ask + y bid ) / y ask − y bid (cid:39) × .
96 Σ y ).For this reason, in this paper we consider the measurement error in prices as Gaussian and relatedto the bid-ask spread via Σ y = ( y ask − y bid ) /
4. With this choice the relation between pricesstandard deviation and bid-ask spread is equal to the one obtained by George et al. (1991).Second, we “propagate” to model parameters this measurement error in prices. In applied statis-tics, the propagation of uncertainties is a standard technique (see, e.g. Taylor 1997, Ryan 2008).We briefly recall some of the main results.Consider the linear model y = Zg + (cid:15) , where y ∈ R n is the response vector, Z ∈ R n × ( r +1) is the explanatory variables matrix, (cid:15) d = N n (0 , Σ), Σ ∈ R n × n is the diagonal response vector variance-covariance matrix, g ∈ R r +1 is theunobserved coefficient vector. We perform a weighted linear regression with weights W ∈ R n × n , adiagonal matrix. The least square solution isˆ g = ( Z (cid:48) W Z ) − Z (cid:48) W Y , where Y ∈ R n is the observed response vector (see e.g. Ryan 2008, ch.3, pp.115-116). Thus, ˆ g isthe Gaussian linear combination of Gaussian random variables:ˆ g d = N r +1 (cid:16) g, ( Z (cid:48) W Z ) − Z (cid:48) W Σ W Z (cid:48) ( Z (cid:48) W Z ) − (cid:17) . (19)In the weighted non-linear regression case, it is possible to obtain a similar result (see e.g. Seberand Wild 1989, ch.2, pp.21-24). Consider the model y i = f ( g, z i ) + (cid:15) i where y i is the i th component of the response vector y ∈ R n , (cid:15) i is the i th component of the errorvector (cid:15) d = N n (0 , Σ), z i ∈ R r +1 is the i th row of the explanatory variables matrix. Similarly, thecoefficients of a non-linear regression are:ˆ g d = N r +1 (cid:16) g, ( F (cid:48) W F ) − F (cid:48) W Σ W (cid:48) F ( F (cid:48) W F ) − (cid:17) , (20)where F ∈ R n × ( r +1) is s.t. its ( i, j ) element is F i,j = ∂f∂g j | g,z i and g j is the j th component of g .In the literature, the case that takes into account Gaussian correlated errors on both the responsevector and the explanatory variables is available for the fitting of a straight line (see e.g. York1968). Consider the model y i = a + bz i , y i and z i subjected to Gaussian errors with variance Σ z i and Σ y i and correlation r i . Theestimated slope ˆ b can be obtained through a fast iterative procedure. Its first order approximationis ˆ b d = N ( b, Σ b ) , (21)where the expression of Σ b is reported in York (1968, 1 st equation in p.324).In this paper, the calibration procedure is divided into two steps.First, for a given maturity T , we deal with the non-linear problem and we calibrate from marketdata the three time-dependent parameters k T , σ T and η T on options with different strikes. Thedistribution of the estimated parameters can be obtained using equation (20). We construct Σthrough all observed bid and ask prices at the given maturity: the diagonal value is equal to( y ask − y bid ) /