Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes 1footnote 11footnote 1A MATHEMATICA® online supplement to this paper containing numerical reuslts is available at the website: http://lorenzotorricelli.it/Code/DTC_TVO_Implementation.nb.

Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes 111A Mathematica® online supplement to this paper containing numerical reuslts is available at the website: http://lorenzotorricelli.it/Code/DTC_TVO_Implementation.nb.

Lorenzo Torricelli222Department of Mathematics, University College London, lorenzo.torricelli.11@ucl.ac.uk
Abstract

In this paper we propose a general derivative pricing framework that employs decoupled time-changed (DTC) Lévy processes to model the underlying assets of contingent claims. A DTC Lévy process is a generalized time-changed Lévy process whose continuous and pure jump parts are allowed to follow separate random time scalings; we devise the martingale structure for a DTC Lévy-driven asset and revisit many popular models which fall under this framework. Postulating different time changes for the underlying Lévy decomposition allows the introduction of asset price models consistent with the assumption of a correlated pair of continuous and jump market activity rates; we study one illustrative DTC model of this kind based on the so-called Wishart process. The theory we develop is applied to the problem of pricing not only claims that depend on the price or the volatility of an underlying asset, but also more sophisticated derivatives whose payoffs rely on the joint performance of these two financial variables, such as the target volatility option (TVO). We solve the pricing problem through a Fourier-inversion method. Numerical analyses validating our techniques are provided. In particular, we present some evidence that correlating the activity rates could be beneficial for modeling the volatility skew dynamics.

Keywords: Derivative pricing; time changes; Lévy processes; joint asset and volatility derivatives; target volatility option; Wishart process

MSC: 91G20, 60G46

1 Introduction

The use of Lévy models in finance dates back to to the classic work of Merton (1976), who proposed that the log-price dynamics of a stock return should follow an exponential Brownian diffusion punctuated by a Poisson arrival process of normally distributed jumps. In that work, two of the main shortcomings of the Black-Scholes model, the continuity of the sample paths and the normality of returns, were addressed for the first time. Over the years, Lévy processes have proved to be a flexible and yet mathematically tractable instrument for asset price modeling and sampling. One of the easiest ways of producing a Lévy process is to use the principle of subordination of a Brownian motion . If is an increasing Lévy process independent of , then the subordinated process will still be of Lévy type. Subordination is the simplest example of a time change, that is, the operation whereby one considers the time evolution of a stochastic process as occurring at a random time.

Return models depending on time-changed Brownian motions have been conjectured since Clark (1973); further theoretical support to the financial use of time-changed models is given by Monroe’s (1978) theorem, asserting that any semimartingale can be viewed as a time change of a Brownian motion. Consequently, any semimartingale representing the log-price process of an asset can be considered as a re-scaled Wiener process. Empirical studies (Ane and Geman, 2000) confirmed that normality of returns can be recovered in a new price density based on the quantity and arrival times of orders, which justifies the interpretation of as “business time” or “stochastic clock”; the instantaneous variation of is hence the “activity rate” at which the market reacts to the arrival of information. Further advances were made by Carr and Wu (2004), who demonstrated that much more general time changes are potential candidates for asset price modeling, and effectively recovered many models from the standard literature by using a time-changed representation.

However, not all the possibilities in time change modeling have been exhausted by the current research. For example, the stochastic volatility model with jumps (SVJ) treated among the others by Bates (1996), and the stochastic volatility model with jumps and stochastic jump rate (SVJSJ) studied by Fang (2000), although retaining a time re-scaled structure, are not time-changed Lévy processes as they are understood in Carr and Wu (2004). Indeed, in these two classes of models the jump component does not follow the same time scaling as the continuous Brownian part: in the SVJ model the discontinuities have stationary increments, whereas in the SVJSJ model the jump rate is allowed to follow a stochastic process of its own. In other words, price models for which the “stochastic clock” runs at different paces for the “small” and “big” market movements have already been proposed and tested. The statistical analyses of Bates (1996) and Fang (2000) confirm that these models are capable of an excellent data fitting, in particular the SVJSJ model. As pointed out by Fang (2000), there are various other reasons for conjecturing a stochastic jump rate. If activity rates are to be interpreted as the frequencies of arrival of new market information, it seems unlikely that such rates could be taken as constant, as this would imply a constant information flow. Moreover, a constant jump rate implies stationary jump risk premia, which also seems unreasonable. Another stylized fact potentially captured by a model with a stochastic jump rate is the slow convergence of returns to the normal distribution, which is not a feature of stationary jump models. Despite all these considerations, the idea of a stochastic jump rate has never really caught on.

On the other hand, if we want to exogenously model the market activity, the hypothesis of independence between the jump and the continuous instantaneous rates as assumed by Fang (2000) seems to be overly simplistic, as in reality the two corresponding information flows may very well influence each other. For example, a crash or soaring of the market certainly impacts the day-to-day volume of trading in the days following such an event. Conversely, a sustained high activity trend over a long period, typically associated to falling prices, may eventually lead to a sudden, panic-driven plunge in the shares’ value. These and similar scenarios provide heuristic arguments for the assumption of a correlated pair of activity rates; nevertheless, to the best of this author’s knowledge, asset price models capturing this feature are not yet present in the literature.

Motivated by these arguments, the natural question that arises is whether it is possible to manufacture consistent general time-changed price processes in which the continuous and discontinuous parts of the underlying Lévy model follow two different, possibly correlated, stochastic time changes. We shall show that the answer is affirmative. The family of stochastic processes we investigate is that obtained by time-modifying the continuous and jump parts of a given Lévy process by two, in principle dependent, stochastic time scalings and satisfying a certain regularity condition (definition 3.1). We call such processes decoupled time changes. In a formula:

(1.1)

where and represent respectively the Brownian and jump components of .

The decoupled time-changed (DTC) approach suggested allows to embed in a unifying mathematical framework many previously-known models or classes of models, so that the DTC theory offers a natural generalization of some of the extant asset modeling research. In addition, the assumption of a pair of dependent activity rates can be captured by making use of decoupled time changes. To our knowledge, this last feature is new to the asset modeling literature. In section 7 we shall illustrate a practical example of a model having this property by considering an explicit asset evolution based on a multivariate version of the square-root process known as the Wishart process (e.g. Bru 1991; Gourieroux 2003; da Fonseca et al. 2007), which we use to model the instantaneous activity rates. In section 8.2 we provide some descriptive analysis showing that this model retains an increased flexibility for the purpose of modeling the volatility skew, compared to some popular existing jump asset price models.

A prior study supporting the financial use of DTC Lévy is given by the work of Huang and Wu (2004). The authors conduct a specification analysis of an SDE whose solution is equivalent to a DTC Lévy-based asset evolution as defined in this paper, give an overview of the “nesting” of models allowed by this setup, and then discuss the impact of various time change specifications in parameter estimation. However, their work does not provide any theoretical justification for the martingale property of the general asset price equation used. Furthermore, they do not explore the issue of dependence between the activity rates as related to analytical tractability, a natural ground of analysis provided by the model. Indeed, to ensure the existence of a semi-closed pricing equation for the “SV4” model in section E, the authors have to revert to a model with independent activities. By providing a general theoretical framework for DTC-based Lévy models, and devising an analytical DTC specification with true dependence between the stochastic volatility and the jump rate, this paper addresses both of these shortcomings.


From the perspective of the valuation of financial derivatives, the aim of this work is to gain some understanding of the impact on derivative pricing of the interactions between the volatility and the price of the underlying. To give an example, a recent market innovation is that of derivatives and investment strategies based on volatility-modified versions of plain vanilla products. Such contracts are able to replicate classic European payoffs under a perfect volatility foresight; at the same time, the component of the price that is due to a vega excess may be reduced by using the realized volatility as a normalizing factor. One example of such a product is the target volatility option. A target volatility (call) option (TVO) pays at maturity the amount:

(1.2)

for a strike price and a target volatility level , a constant that is written in the contract. Intuitively, the closer the realized volatility is to , the more this claim will behave like a call option; however, the presence of in the denominator decreases the sensitivity of to a change in volatility. It can be shown (Di Graziano and Torricelli, 2012) that the price of an at-the-money TVO is approximately that of an at-the-money Black-Scholes call having implied volatility ; such a constant thus represents the subjective volatility view of an investor, which may very well differ from the spot volatilities implied by the market.

In view of this increasing interaction between volatility and stock in the financial assets available in the market, being able to efficiently price derivatives like the TVO and other similar products is gaining relevance. The pricing problem of hybrid volatility/asset derivatives, with special emphasis on the target volatility option, has already been addressed by Di Graziano and Torricelli (2012) for a zero-correlation stochastic volatility model, and by Torricelli (2013), for a general stochastic volatility model. However, to our knowledge, a comprehensive pricing framework comparable to those available for plain vanilla derivatives (e.g Carr and Madan 1999; Lewis 2000; Lewis 2001; Carr and Wu 2004) has not yet been developed: this is one limitation we intend to overcome with this paper. The pricing technique we use is a well-known approach yielding a semi-closed analytical formula for the derivative price through an inverse Fourier integral. It should be apparent that in all the models we shall investigate there is no particular reason not to consider mixed price and volatility payoffs as the default input of pricing models e.g. for numerical implementation, as the introduction of the realized volatility does not cause the Fourier-inversion technique to break down. Clearly, pricing both vanilla and pure volatility derivatives is still possible within this framework, since the corresponding payoff types can be regarded as particular cases of our more general setting.


The remainder of the paper is organized as follows. In section 2 we lay out the assumptions; in section 3 we derive martingale properties for a decoupled time-changed Lévy model. Section 4 shows the fundamental relation linking the characteristic function of the log-price and its quadratic variation and the joint Laplace transform of the time changes as computed in an appropriate measure. Section 5 is dedicated to the derivation of a pricing formula for products whose payoffs depend jointly on and . We devote section 6 to characterizing the DTC structure of a number of known models and computing the joint characteristic function discussed in section 4 for each such model. In section 7 we introduce an exemplifying model of DTC type featuring correlation between the time changes/activity rates. In section 8 we implement our formulae to valuate different asset and volatility derivatives under various market conditions and asset price models. In this numerical section we also perform a sensitivity analysis of the model introduced in section 7 with respect to a correlation parameter. Finally, in section 9 we briefly summarize our work. The more technical proofs have been placed in the appendix.

2 Assumptions and notation

As customary, our market is represented by a filtered probability space satisfying the usual conditions. Throughout the paper we will assume that there exists a money market account process paying a constant interest rate .

Let be a non dividend-paying market asset. will denote its time-zero discounted value . The total realized variance on of is by definition the quadratic variation of the natural logarithm of , that is:

(2.1)

The limit runs over the supremum norm of all the possible partitions of . The total realized volatility is . The period realized variance and volatility (or realized variance/volatility tout court) are given respectively by and . If is a semimartingale, by taking the limit in (2.1) it is easy to check that:

(2.2)

The algebra of the square matrices of order with real entries is indicated by and the sub-algebra of the symmetric matrices by . Matrix product is denoted by juxtaposition; the scalar product between vectors is either indicated by multiplying on the left with the transposed vector or by the usual dot notation. The symbol stands for the trace operator.

If is an absolutely continuous random variable, we denote by its probability density function and by its characteristic function

(2.3)

For a Fourier-integrable function its Fourier transform will be denoted . For a complex-valued function or a complex plane subset, indicates the complex conjugate function or set.

When we say that a process is a martingale we mean a martingale with respect to its natural filtration. The notation for the conditional expectation of a stochastic process at time with respect to is . When the distribution of a process depends on other state variables (as in the case of a Markov process) the latter are implicitly understood to be given at time by . If is a process admitting conditional laws, the space of the integrable functions in the -conditional distribution of at time is indicated by . The notation for the bilateral Laplace transform of the distribution of conditional on is:

(2.4)

where for brevity we drop the dependence on and on the left hand side. The stochastic process of the left limits of is indicated . The symbol stands for the difference or for some prior time . Equalities are always understood to hold modulo almost sure equivalence.

If is an -dimensional Lévy process, the characteristic exponent of is the complex-valued function such that:

(2.5)

where lies in the subset of and where the left-hand side is finite.

For a given choice of truncation function (that is, a bounded function which is O around 0) the characteristic exponent has the unique Lévy-Khintchine representation:

(2.6)

where , is a non-negative definite matrix with real-valued entries, and is a Radon measure on having a density function that is integrable at and O around 0. We shall make the standard choice and drop the dependence of on . The triplet is then called the characteristic triplet or the Lévy characteristics of .

A stochastic time change is an -adapted càdlàg stochastic process, increasing and almost surely finite, such that is an -adapted stopping time for each . The time change of an -dimensional Lévy process according to is the -adapted process .

3 Definition, martingale relations and asset price dynamics

In this first section we introduce the notion of DTC Lévy process and devise an exponential martingale structure naturally associated to it. This construct serves a twofold purpose. In first place it allows to formulate a DTC-based asset price evolution whose discounted value enjoys the martingale property. According to general theory, this in turn enables to postulate the existence of a risk-neutral measure that correctly prices the market securities. Secondly, it defines a class of complex-valued martingales pivotal for the computations of the next section.

Let be the space of the -dimensional -supported Brownian motions with drift starting at , and be the space of the -supported pure jump Lévy processes starting at 0, that is, the class of the càdlàg -adapted processes with stationary and independent increments orthogonal333Two processes and are said to be orthogonal if for all . to all the elements of .

Every Lévy process such that can be decomposed as the orthogonal sum

(3.1)

with and . We shall refer to and respectively as the continuous and discontinuous parts of .

Time changes are fairly general mathematical objects, so we have to introduce some additional requirements in order for our discussion to proceed. One property we shall assume throughout is the so-called continuity with respect to the time change.

Definition 3.1.

Let be a time change on a filtration . An -adapted process is said to be -continuous444Jacod (1979) uses -adapted, and -synchronized is sometimes found; however, -continuous is also common in the literature, and in our view less ambiguous. if it is almost-surely constant on all the sets .

Obviously, a sufficient condition for -continuity is the almost sure continuity of . Hence, of particular relevance is the class of the absolutely continuous time changes, with respect to which every stochastic process is continuous. Given a pair of instantaneous rate of activity processes, that is, two exogenously-given càdlàg positive stochastic processes , valid time changes are given by the pathwise integrals:

(3.2)
(3.3)

The processes and describe the instantaneous impact of market trading and information arrival on the price, and formalize the concept of “business activity” over time.

A decoupled time change of a Lévy process is the sum of the (ordinary) time changes of its continuous and discontinuous part.

Definition 3.2.

Let be an -dimensional Lévy process and , two time changes such that is almost surely continuous and is -continuous. Then:

(3.4)

is the decoupled time change of according to and .

By (Jacod, 1979), corollaire 10.12, a first important property of is that it is an semimartingale.To avoid degenerate cases, in all that follows we always assume and to be such that and are Markov processes555In general, time changes of Markov processes are not Markovian; by using Dambis, Dubins and Schwarz’s theorem (Karatzas and Shreve 2000, theorem 4.6) one can manufacture a large class of counterexamples by starting from any continuous martingale that is not a Markov process..

We now define the class of exponential martingales canonically associated with when the time changes are absolutely continuous. The following proposition represents the main theoretical tool of this paper:

Proposition 3.3.

Let be an -dimensional Brownian motion with drift and a pure jump Lévy process in . Let and be two absolutely continuous time changes, set and ; define and denote by the domain of definition of . The process:

(3.5)

is a local martingale, and it is a martingale if and only if , where:

(3.6)

When , the exponential reduces to an ordinary time change of the type discussed by Carr an Wu (2004). Even in this simple case proposition 3.3 is not a consequence of applying Doob’s optional sampling theorem to the martingale , because the latter is not necessarily uniformly integrable. Indeed, time-transforming a process always preserves the semimartingale property, but the martingale property is only guaranteed to be maintained for uniformly integrable martingales; an actual example of an asset model of the form that is a strict supermartingale was given by Sin (1998). Hence, the set may very well trivialize to the empty set. This demonstrates that some choices of time changes are inherently unsuitable for time-changed asset price modeling. In the case of being a one-dimensional Brownian integral, sufficient requirements for (3.6) to be satisfied are the well-known Novikov and Kazamaki conditions (Karatzas and Shreve 2000, chapter 3), under which the set contains the whole of . The set is sometimes called the natural parameter set.


Having obtained martingale relations for a stochastic exponential involving , the risk-neutral dynamics for a DTC Lévy-driven asset are defined in the usual fashion. We have the following immediate corollary to proposition 3.3:

Corollary 3.4.

Let be a scalar Lévy process of characteristic triplet and a pair of absolutely continuous time changes. For a spot price value let, for :

(3.7)

with being such that (3.7) is a real number. The discounted process is a martingale, and therefore is a price process consistent with the no-arbitrage condition.

The stochastic process in (3.7) is the fundamental asset model we shall use throughout the rest of the paper.

4 Characteristic functions and the leverage-neutral measure

Characteristic functions of state variables are the essential component of the Fourier-inverse pricing methodology, because state price densities are analytically available only for a small number of models; in contrast, characteristic functions are computable in closed form in many instances (e.g. exponential Lévy models, Ito diffusions). This effectively means that in order to compute expectations (prices), the standard approach is not to integrate a payoff against a density function, but rather the payoff’s Fourier transform against the characteristic functions of the price transition densities. Famous examples include the FFT paper by Carr and Madan (1999), Lewis’s book (2000) and subsequent paper (2001).

The transform we are interested in is one associated with the price process (3.7). Compared to the usual inverse Fourier/Laplace framework the characteristic function we shall consider is not that of the discounted log-price alone, but one that incorporates also the quadratic variation of the log-process. Indeed, just as the characteristic function of the log-price allows for the derivation of pricing formulae for contingent claims , the joint characteristic function of and permits the valuation of payoffs of the form . This has been envisaged before by Carr and Sun (2007).

In the present section we compute this transform. There are normally two ways of computing characteristic functions/Laplace transforms of log-price densities. One is the analytical approach, which is popular for example in affine models, when the problem is ultimately reduced to solving a certain system of ODEs. The other is the probabilistic approach, in which the characteristic function of the log-price is linked with the Laplace transform of the integrated driving factors (where available) and then a change of measure is performed to keep track of correlations. As Carr and Wu (2004) show this technique is intimately connected with time-changed asset modeling; in what follows we extend it to the case of the underlying being modeled through a full DTC Lévy process.

First of all we must verify that the quadratic variation operator respects the additivity and time-changed structure of . We have the following “linearity/commutativity property”, of independent interest:

Proposition 4.1.

A DTC Lévy process is such that and are orthogonal. Furthermore, its quadratic variation satisfies:

(4.1)

That is, the quadratic variation of is the sum of the time changes of the quadratic variations of its continuous and discontinuous part.

Crucially, the processes and are orthogonal but not independent. Without the and -continuity assumption, this proposition would be false: a counterexample is provided in the appendix. Proposition 4.1 ensures that, in presence of time continuity of the Lévy continuous and jump parts with respect to the corresponding time changes, the quadratic variation of a DTC Lévy process is itself of DTC-type.

Now, for as in (3.7) define:

(4.2)

For each for which the right hand side is finite, is the Fourier transform is the joint transition function from time to time of and . The characteristic function can be completely characterized in terms of the Lévy triplet of and the joint -distribution of and by virtue of the following proposition.

Proposition 4.2.

Let be an asset evolution as in corollary 3.4, and define the family of absolutely-continuous measures having Radon-Nikodym derivative:

(4.3)

where , and is given by (3.5). For all such that , the characteristic function in (4.2) is given by:

(4.4)

with the notation indicating the bilateral Laplace transform of the conditional joint distribution of and taken under the measure , and

(4.5)
(4.6)

Notice that unlike the density processes used for standard numéraire changes, the new distributions implied by (4.3) also accounts for the quadratic variation as a factor. If we assume and to be pathwise integrals of the form (3.2) and (3.3), it is possible to interpret the Laplace transform (4.4) as being the analogue of a bivariate bond pricing formula, where the short rates are replaced by the instantaneous activity rates, and the pricing measure is not given once and for all, but varies as an effect of the correlation of with the underlying Lévy process. The financial insight of (4.4) is that it is possible to formulate a valuation theory by just modeling the joint term structure of the activity rates and and their correlation with the stock.

Also of interest is the interpretation of the measure . Let us consider the special case of being independent of and . In such a case it is straightforward to prove, by using the laws of the conditional expectation, that one obtains (4.4) with . Therefore, whenever there is no dependence between the time changes and the underlying Lévy process, no change of measure is needed in order to extract the characteristic function . In contrast, in the presence of correlation between and the time changes, the family gives a measurement of the impact of leverage on the price densities. Furthermore, in some well-behaved cases this change of measure can be absorbed in the -dynamics of the asset through a suitable parameter alteration of the distributions of and . In accordance with Carr and Wu (2004), we call the leverage-neutral measure and the leverage-neutral characteristic function. Just as prices in a risky market can be equivalently computed in a risk-neutral environment according to a different price distribution, valuations in the presence of leverage can be performed in a different economy with no leverage by means of an appropriate distributional modification.

5 Pricing and price sensitivities

The characteristic function found in section 4 is needed to obtain analytical formulae for the valuation of European-type derivatives with a sufficiently regular payoff . In the present section we find a semi-analytical formula based on an inversion integral that extends the standard Fourier-inversion machinery to our multivariate context.

Recall that since all the involved processes are Markovian, it makes sense to treat like a Gauss-Green integral kernel depending only on some given initial states at time . The following proposition extends both theorem 1 of Lewis (2000) and proposition 3.1 of Torricelli (2013):

Proposition 5.1.

Let , with given in corollary 3.4. Let for all , be a positive payoff function having analytical Fourier transform in a multi-strip

(5.1)

Suppose further that is analytical in

(5.2)

and that . If , then for every multi-line:

(5.3)

we have that the time- value of the contingent claim maturing at time is given by:

(5.4)

It is clear that modifying the asset dynamics specifications only acts on , whereas changing the claim to be priced only influences . Also, by setting either variable to 0, we are able to extract from (5.4) the prices of both plain vanilla and pure volatility derivatives. For example, the pricing integrals by Lewis (2000, 2001) are special cases of the above equation when does not depend on the realized volatility and is either obtained from a diffusion or a Lévy process. Moreover, equation (3.10) of Torricelli (2013) is recovered when is assumed to follow a stochastic volatility model.

In addition, this representation is useful if we are interested in the sensitivities of the claim value with respect to the underlying state variables. Let us consider for instance the Delta (sensitivity with respect to the change in the value of the underlying) and Gamma (sensitivity with respect to the rate of change in the value of the underlying) of valuations performed through formula (5.4). Call the integrand on the right hand side of (5.4); by differentiating (if possible) under the integral sign and noting that has no dependence on we see that:

(5.5)

and

(5.6)

Mutatis mutandis we can repeat this argument if we want to determine the price sensitivity with respect to the quadratic variation . Finally, as could also depend on other variables (e.g. an instantaneous rate of activity ) known at time , by calling one such variable we have:

(5.7)

This is especially well-suited to the case in which is exponentially-affine in , i.e.

(5.8)

for some functions and , when we have:

(5.9)

In section 6 we shall explicitly calculate for a number of decoupled time-changed models.

6 Specific model analysis

We now determine the DTC Lévy structure (3.7) of various popular asset price processes, and find for each of them the corresponding leverage-neutral characteristic function . Such a derivation allows for the full implementation of equation (5.4) for the pricing of joint asset and volatility derivatives in all the cases we deal with. What the discussion below should make apparent is that decoupled time changes offer a natural unifying framework for a priori different strains of financial asset models (e.g. continuous/jump diffusions, jump diffusions with stochastic volatility, Lévy processes). By classifying models through their DTC structure it is possible to recognize a “nesting” pattern linking different models, in which some can be considered particular cases of some others. This is of use for numerical purposes: as we shall see in section 8, one single implementation of equation (5.4) can produce values for several models, each one obtained by using a different instantiation of the code. Four categories of asset models are discussed: standard Lévy processes, stochastic volatility models, DTC jump diffusions and general exponentially-affine asset models. Throughout this section we assume in (3.7), so that and all of the involved processes are real-valued. The domain where the price processes are martingales is the whole complex plane, provided that the stochastic time changes and the underlying Lévy components are sufficiently well-behaved, in the sense of the usual theory (e.g. Novikov condition for the stochastic variance, decay of the jump distributions, integrability conditions on the Lévy measure etc.)

6.1 Lévy processes

In case of the Lévy process the DTC structure coincides with the underlying Lévy process. To determine no change of measure is necessary, so this function represents the joint conditional characteristic function of the log-price and its quadratic variation as given in the risk-neutral measure. Below, are provided the calculations for some popular models.

6.1.1 Black-Scholes model

The classic SDE with constant parameters driven by a Brownian motion :

(6.1)

can be trivially recovered from (3.7) by setting the triplet for the underlying Lévy process to be and letting , , so that . From (4.4), we immediately have:

(6.2)

6.1.2 Jump diffusion models

In their classic works, Merton and Kou (1976, 2002) proposed modeling the log-price dynamics as a finite-activity jump diffusion. The risk-neutral asset dynamics are given by:

(6.3)

where is a standard Brownian motion, is a Poisson counter of intensity , and is the jump size distribution. and are assumed to be independent, and the compensator equals For the discounted price to be a true martingale, conditions on the asymptotic behavior of must be imposed (see e.g. Cont and Tankov, 2003). In the Merton model is normally distributed , whereas Kou assumed for it an asymmetrically skewed double-exponential distribution, that is, the density function as given by:

(6.4)

for and .

In these models no time change is involved, so coincides with the underlying Lévy process having characteristic triplet . To completely characterize , observe that is just a bivariate compound Poisson process of joint jump density and intensity , whence:

(6.5)

where is the joint characteristic function of and . We conclude from (4.4) that has the exponential structure:

(6.6)

Now for the Merton model we have

(6.7)

and the integral converges for Im. For the Kou model we can write:

(6.8)

the characteristic function of the positive and negative parts are:

(6.9)
(6.10)

which both converge for Im.

6.1.3 Tempered stable Lévy and CGMY

Another way of obtaining Lévy distributions for the asset price is to directly specify an infinite activity Lévy measure . In such a case we have , with being a pure jump Lévy process of Lévy measure . The two instances we analyze here are the tempered stable Lévy process (e.g. Cont and Tankov 2003), and the CGMY (Carr et al. 2002) models. Both of these are obtained as an exponential smoothing of stable distributions; the latter can be viewed as a generalization of the former allowing for an asymmetrical skew between the distribution of positive and negative jumps. The Lévy density for a CGMY process is:

(6.11)

which is well defined for all , . When one has the tempered stable process. For simplicity in what follows we assume ; for such values the involved characteristic functions still exist, but lead to particular cases. Since

(6.12)

to fully characterize we only need to determine and . Letting , the exponent is given by the standard theory (Cont and Tankov 2003, proposition 4.2) as:

(6.13)

Set ; the positive part of is then seen to be:

(6.14)

Here is the Euler Gamma function and the confluent hypergeometric function. The multi-strip of convergence of (6.14) is the set . The determination has a similar expression.

6.2 Stochastic volatility and the Heston model

In a stochastic volatility model the asset process is given, in a risk neutral-measure, by the SDE

(6.15)

where is some continuous stochastic variance process. By the Dubins and Schwarz’s theorem any continuous martingale can be written as for a certain Brownian motion , which implies that the DTC structure of a stochastic volatility model corresponds to a standard Brownian motion time-changed by as in (3.2). In order to explicitly express the characteristic function we must make a specific choice for the dynamics in (6.15). For instance, we can make the popular choice of selecting a square-root (CIR) equation for the instantaneous variance:

(6.16)

for positive constants and a Brownian motion linearly correlated with through a correlation coefficient . For to be well-defined, the parameters , and need to satisfy the Feller condition . The system of SDEs (6.15)-(6.16) is the model by Heston (1993). As we change to the measure , the application of the complex-plane version of Girsanov’s theorem and a simple algebraic manipulation reveals that the leverage-neutral dynamics of are of the same form as (6.16), but with parameters:

(6.17)
(6.18)

(see also Carr and Wu 2004). Using equation (4.4), we determine as follows666Torricelli (2013) has independently found for the Heston model by augmenting the SDE system (6.15)-(6.16) with the equation , and solved the associated Fourier-transformed parabolic equation via the usual Feynman-Kac argument. As has to be the case, the two approaches coincide.:

(6.19)

where indicates the transform with respect to which is well-known analytically (e.g. Dufresne, 2001). The case reverts back to the Black-Scholes model, when (6.19) collapses to (6.2) with .

Other choices for are clearly possible, yielding different stochastic volatility models (the 3/2 model, GARCH, etc.). It is clear from the arguments above that, for an analytical expression for to exist it suffices that the Laplace transform of is known in closed form777See e.g. Lewis 2000, chapter 2, for the Laplace transform of the cited models. and that belongs to a class of models that are stable under the Girsanov transformation.

6.3 DTC jump diffusions

When the underlying Lévy process is represented by a finite activity jump diffusion, operating a decoupled time change amounts to either introducing a stochastic volatility coefficient in the continuous Brownian part, or making the intensity of the compound Poisson process stochastic, or both. Models carrying this structure have been prominently discussed by D.S. Bates (1996) and H. Fang (2000).

6.3.1 Stochastic volatility with jumps

The stochastic volatility model with jumps (SVJ) provides us with a first instance of a decoupled time change not otherwise obtainable as an ordinary time change. The SVJ model is in fact a Lévy decoupled time change with a time-changed continuous part and a time-homogeneous jump part. The dynamics for the asset price are given by the exponential jump diffusion:

(6.20)

for some Brownian motion , stochastic variance process , Poisson process and jump size having compensator . The underlying DTC structure of the Bates model is given by with the characteristic triplet for being and taking the form (3.2). By assuming as a jump distribution a normal random variable, and as a variance process the square-root equation:

(6.21)

we have the model by Bates (1996). For the discounted asset value to be a martingale, the parameters of the driving stochastic volatility and jump process must be subject to the requirements of both subsection 6.2 and subsection 6.1.2. It is straightforward to see that decomposes into:

(6.22)

where and are given respectively by (6.19) and (6.6)-(6.7). Therefore:

(6.23)

So far, we have encountered either exponential Lévy models, or exponentially-affine functions arising as solutions of a PDE problem. Here we have a mixture of the two: a time-homogeneous jump factor, modeled as a compound Poisson process, and a continuous diffusion factor, whose characteristic function solves a diffusion problem. The degenerate case , yields a Merton jump diffusion with diffusion coefficient .