A family of density expansions for Lévy-type processes

A family of density expansions for Lévy-type processes

Matthew Lorig ORFE Department, Princeton University, Princeton, USA. Work partially supported by NSF grant DMS-0739195    Stefano Pagliarani Dipartimento di Matematica, Università di Padova, Padova, Italy    Andrea Pascucci Dipartimento di Matematica, Università di Bologna, Bologna, Italy
August 19, 2019
Abstract

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Lévy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Lévy measure. Generalizing and extending the novel adjoint expansion technique of Pagliarani, Pascucci, and Riga (2013), we derive a family of asymptotic expansions for the transition density of the underlying as well as for European-style option prices and defaultable bond prices. For the density expansion, we also provide error bounds for the truncated asymptotic series. Our method is numerically efficient; approximate transition densities and European option prices are computed via Fourier transforms; approximate bond prices are computed as finite series. Additionally, as in Pagliarani et al. (2013), for models with Gaussian-type jumps, approximate option prices can be computed in closed form. Sample Mathematica code is provided.

To the memory of our dear friend and esteemed colleague Peter Laurence.

Keywords: Local volatility; Lévy-type process; Asymptotic expansion; Pseudo-differential calculus; Defaultable asset

1 Introduction and literature review

A local volatility model is a model in which the volatility of an asset is a function of time and the present level of . That is, . Among local volatility models, perhaps the most well-known is the constant elasticity of variance (CEV) model of Cox (1975). One advantage of local volatility models is that transition densities of the underlying – as well as European option prices – are often available in closed-form as infinite series of special functions (see Linetsky (2007) and references therein). Another advantage of local volatility models is that, for models whose transition density is not available in closed form, accurate density and option price approximations are readily available (see, Pagliarani and Pascucci (2011), for example). Finally, Dupire (1994) shows that one can always find a local volatility function that fits the market’s implied volatility surface exactly. Thus, local volatility models are quite flexible.

Despite the above advantages, local volatility models do suffer some shortcomings. Most notably, local volatility models do not allow for the underlying to experience jumps, the need for which is well-documented in literature (see Eraker (2004) and references therein). Recently, there has been much interest in combining local volatility models and models with jumps. Andersen and Andreasen (2000), for example, discuss extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps (i.e., jumps with finite activity). And Benhamou, Gobet, and Miri (2009) derive analytically tractable option pricing approximations for models that include local volatility and a Poisson jump process. Their approach relies on asymptotic expansions around small diffusion and small jump frequency/size limits. More recently, Pagliarani, Pascucci, and Riga (2013) consider general local volatility models with independent Lévy jumps (possibly infinite activity). Unlike, Benhamou et al. (2009), Pagliarani et al. (2013) make no small jump intensity/size assumption. Rather the authors construct an approximated solution by expanding the local volatility function as a power series. While all of the methods described in this paragraph allow for local volatility and independent jumps, none of these methods allow for state-dependent jumps.

Stochastic jump-intensity was recently identified as an important feature of equity models (see Christoffersen, Jacobs, and Ornthanalai (2009)). A locally dependent Lévy measure allows for this possibility. Recently, two different approaches have been taken to modeling assets with locally-dependent jump measures. Mendoza-Arriaga, Carr, and Linetsky (2010) time-change a local volatility model with a Lévy subordinator. In addition to admitting exact option-pricing formulas, the subordination technique results in a locally-dependent Lévy measure. Jacquier and Lorig (2013) considers another class of models that allow for state-dependent jumps. The author builds a Lévy-type processes with local volatility, local default intensity, and a local Lévy measure by considering state-dependent perturbations around a constant coefficient Lévy process. In addition to pricing formula, the author provides an exact expansion for the induced implied volatility surface.

In this paper, we consider scalar Lévy-type processes with regular coefficients, which naturally include all the models mentioned above. Generalizing and extending the methods of Pagliarani et al. (2013), we derive a family of asymptotic expansions for the transition densities of these processes, as well as for European-style derivative prices and defaultable bond prices. The key contributions of this manuscript are as follows: {itemize*}

We allow for a locally-dependent Lévy measure and local default intensity, whereas Pagliarani et al. (2013) consider a locally independent Lévy measure and do not allow for the possibility of default. A state-dependent Lévy measure is an important feature because it allows for incorporating local dependence into infinite activity Lévy models that have no diffusion component, such as Variance Gamma (Madan, Carr, and Chang (1998)) and CGMY/Kobol (Boyarchenko and Levendorskii (2002); Carr, Geman, Madan, and Yor (2002)).

Unlike Benhamou et al. (2009), we make no small diffusion or small jump size/intensity assumption. Our formulae are valid for any Lévy type process with smooth and bounded coefficients, independent of the relative size of the coefficients.

Whereas Pagliarani et al. (2013) expand the local volatility and drift functions as a Taylor series about an arbitrary point, i.e. , in order to achieve their approximation result, we expand the local volatility, drift, killing rate and Lévy measure in an arbitrary basis, i.e. . This is advantageous because the Taylor series typically converges only locally, whereas other choices of the basis functions may provide global convergence in suitable functional spaces.

Using techniques from pseudo-differential calculus, we provide explicit formulae for the Fourier transform of every term in the transition density and option-pricing expansions. In the case of state dependent Gaussian jumps the respective inverse Fourier transforms can be explicitly computed, thus providing closed form approximations for densities and prices. In the general case, the density and pricing approximations can be computed quickly and easily as inverse Fourier transforms. Additionally, when considering defaultable bonds, approximate prices are computed as a finite sum; no numerical integration is required even in the general case.

For models with Gaussian-type jumps, we provide pointwise error estimates for transition densities. Thus, we extend the previous results of Pagliarani et al. (2013) where only the purely diffusive case is considered. Additionally, our error estimates allow for jumps with locally dependent mean, variance and intensity. Thus, for models with Gaussian-type jumps, our results also extend the results of Benhamou et al. (2009), where only the case of a constant Lévy measure is considered.

The rest of this paper proceeds as follows. In Section, 2 we introduce a general class of exponential Lévy-type models with locally-dependent volatility, default intensity and Lévy measure. We also describe our modeling assumptions. Next, in Section 3, we introduce the European option-pricing problem and derive a partial integro-differential equation (PIDE) for the price of an option. In Section 4 we derive a formal asymptotic expansion (in fact, a family of asymptotic expansions) for the function that solves the option pricing PIDE (Theorem 1). Next, in Section 5, we provide rigorous error estimates for our asymptotic expansion for models with Gaussian-type jumps (Theorem 2). Lastly, in Section 6, we provide numerical examples that illustrate the effectiveness and versatility of our methods. Technical proofs are provided in the Appendix. Some concluding remarks are given in Section 7.

We mention specifically that the arguments needed to provide rigorous error estimates for our asymptotic expansions are quite extensive. As such, in this manuscript, we provide only an outline of the proof of Theorem 2. The full proof of Theorem 2, as well as further numerical examples, can be found in a companion paper Lorig, Pagliarani, and Pascucci (2013).

2 General Lévy-type exponential martingales

For simplicity, we assume a frictionless market, no arbitrage, zero interest rates and no dividends. Our results can easily be extended to include locally dependent interest rates and dividends. We take, as given, an equivalent martingale measure , chosen by the market on a complete filtered probability space satisfying the usual hypothesis of completeness and right continuity. The filtration represents the history of the market. All stochastic processes defined below live on this probability space and all expectations are taken with respect to . We consider a defaultable asset whose risk-neutral dynamics are given by

(1)

Here, is a Lévy-type process with local drift function , local volatility function and state-dependent Lévy measure . We shall denote by the filtration generated by . The random variable has an exponential distribution and is independent of . Note that , which represents the default time of , is constructed here trough the so-called canonical construction (see Bielecki and Rutkowski (2001)), and is the first arrival time of a doubly stochastic Poisson process with local intensity function . This way of modeling default is also considered in a local volatility setting in Carr and Linetsky (2006); Linetsky (2006), and for exponential Lévy models in Capponi et al. (2013).

We assume that the coefficients are measurable in and suitably smooth in to ensure the existence of a solution to (1) (see Oksendal and Sulem (2005), Theorem 1.19). We also assume the following boundedness condition which is rather standard in the financial applications: there exists a Lévy measure

(2)

such that

(3)

Since is not -measurable we introduce the filtration in order to keep track of the event . The filtration of a market observer, then, is . In the absence of arbitrage, must be an -martingale. Thus, the drift is fixed by , and in order to satisfy the martingale condition111 We provide a derivation of the martingale condition in Section 3 Remark 1 below.

(4)

We remark that the existence of the density of is not strictly necessary in our analysis. Indeed, since our formulae are carried out in Fourier space, we provide approximations of the characteristic function of and all of our computations are still formally correct even when dealing with distributions that are not absolutely continuous with respect to the Lebesgue measure.

3 Option pricing

We consider a European derivative expiring at time with payoff and we denote by its no-arbitrage price. For convenience, we introduce

(5)
Proposition 1.

The price is given by

(6)

The proof can be found in Section 2.2 of Linetsky (2006). Because our notation differs from that of Linetsky (2006), and because a short proof is possible by using the results of Jeanblanc, Yor, and Chesney (2009), for the reader’s convenience, we provide a derivation of Proposition 1 here.

Proof.

Using risk-neutral pricing, the value of the derivative at time is given by the conditional expectation of the option payoff

(7)
(8)
(9)
(10)
(11)

where we have used Corollary 7.3.4.2 from Jeanblanc, Yor, and Chesney (2009) to write

(12)

Remark 1.

By Proposition 1 with and , we have that the martingale condition is equivalent to

(13)

Therefore, we see that is a martingale if and only if the process is a martingale. The drift condition (4) follows by applying the Itô’s formula to the process and setting the drift term to zero.

From (6) one sees that, in order to compute the price of an option, we must evaluate functions of the form222Note: we can accommodate stochastic interest rates and dividends of the form and by simply making the change: and .

(14)

By a direct application of the Feynman-Kac representation theorem, see for instance (Pascucci, 2011, Theorem 14.50), the classical solution of the following Cauchy problem,

(15)

when it exists, is equal to the function in (14), where

(16)
(17)

is the characteristic operator of the SDE (1). In order to shorten the notation, in the sequel we will suppress the explicit dependence on in by referring to it just as .

Sufficient conditions for the existence and uniqueness of solutions of second order elliptic integro-differential equations are given in Theorem II.3.1 of Garroni and Menaldi (1992). We denote by the fundamental solution of the operator , which is defined as the solution of with . Note that represents also the transition density of 333Here with we denote the process .

(18)

Note also that is not a probability density since (due to the possibility that ) we have

(19)

Given the existence of the fundamental solution of , we have that for any that is integrable with respect to the density , the Cauchy problem (15) has a classical solution that can be represented as

(20)
Remark 2.

If is the generator of a scalar Markov process and contains , the Schwartz space of rapidly decaying functions on , then must have the following form:

(21)

where , , is a Lévy measure for every and (see *hoh1998pseudo, Proposition 2.10). If one enforces on the drift and integrability conditions (3) and (4), which are needed to ensure that is a martingale, and allow setting , then the operators (17) and (21) coincide (in the time-homogeneous case). Thus, the class of models we consider in this paper encompasses all non-negative scalar Markov martingales that satisfy the regularity and boundedness conditions of Section 2.

Remark 3.

In what follows we shall systematically make use of the language of pseudo-differential calculus. More precisely, let us denote by

(22)

the so-called oscillating exponential function. Then can be characterized by its action on oscillating exponential functions. Indeed, we have

(23)

where

(24)
(25)

is called the symbol of . Noting that

(26)

for any analytic function , we have

(27)

Then can be represented as

(28)

since by (24) and (27)

(29)
(30)

If coefficients are independent of , then we have the usual characterization of as a multiplication by operator in the Fourier space:

where and denote the (direct) Fourier and inverse Fourier transform operators respectively:

(31)

Moreover, if the coefficients are independent of both and , then is the generator of a Lévy process and is the characteristic exponent of :

(32)

4 Density and option price expansions (a formal description)

Our goal is to construct an approximate solution of Cauchy problem (15). We assume that the symbol of admits an expansion of the form

(33)

where is of the form

(34)
(35)

and is some expansion basis with being an analytic function for each , and (see Examples 1, 2 and 3 below). Note that is the symbol of an operator

(36)

so that

(37)

Thus, formally the generator can be written as follows

(38)

Note that is the generator of a time-dependent Lévy-type process . In the time-independent case is a Lévy process and is its characteristic exponent.

Example 1 (Taylor series expansion).

Pagliarani, Pascucci, and Riga (2013) approximate the drift and diffusion coefficients of as a power series about an arbitrary point . In our more general setting, this corresponds to setting and expanding the diffusion and killing coefficients and , as well as the Lévy measure as follows:

(39)

In this case, (33) and (38) become (respectively)

(40)

where, for all , the symbol is given by (35) with coefficients given by (39). The choice of is somewhat arbitrary. However, a convenient choice that seems to work well in most applications is to choose near , the current level of . Hereafter, to simplify notation, when discussing implementation of the Taylor-series expansion, we suppress the -dependence: , and .

Example 2 (Two-point Taylor series expansion).

Suppose is an analytic function with domain and . Then the two-point Taylor series of is given by

(41)

where

(42)

For the derivation of this result we refer the reader to Estes and Lancaster (1972); Lopez and Temme (2002). Note truncating the two-point Taylor series expansion (41) at results in an expansion which of which is of order ).

The advantage of using a two-point Taylor series is that, by considering the first derivatives of a function at two points and , one can achieve a more accurate approximation of over a wider range of values than if one were to approximate using derivatives at a single point (i.e., the usual Taylor series approximation).

If we associate expansion (41) with an expansion of the form then , which is affine in . Thus, the terms in the two-point Taylor series expansion would not be a suitable basis in (33) since . However, one can always introduce a constant and define a function

so that (43)

Then, one can express as

(44)

where the are as given in (42) with . If we associate expansion (44) with an expansion of the form , then we see that and one can choose . Thus, as written in (44), the terms of the two-point Taylor series can be used as a suitable basis in (33).

Consider the following case: suppose , and are of the form

(45)

so that with

(46)
(47)

It is certainly plausible that the symbol of would have such a form since, from a modeling perspective, it makes sense that default intensity, volatility and jump-intensity would be proportional. Indeed, the Jump-to-default CEV model (JDCEV) of Carr and Linetsky (2006); Carr and Madan (2010) has a similar restriction on the form of the drift, volatility and killing coefficients.

Now, under the dynamics of (45), observe that and can be written as in (33) and (38) respectively with and

(48)

As above (capital “C”) are given by (42) with and

(49)

As in example 1, the choice of , and is somewhat arbitrary. But, a choice that seems to work well is to set and where is a constant and . It is also a good idea to check that, for a given choice of and , the two-point Taylor series expansion provides a good approximation of in the region of interest.

Note we assumed the form (45) only for sake of simplicity. Indeed, the general case can be accommodated by suitably extending expansion (33) to the more general form

(50)

where for are related to the diffusion, jump and default symbols respectively. For brevity, however, we omit the details of the general case.

Example 3 (Non-local approximation in weighted -spaces).

Suppose is a fixed orthonormal basis in some (possibly weighted) space and that for all . Then we can represent in the form (33) where now the are given by

(51)

A typical example would be to choose Hermite polynomials centered at as basis functions, which (as normalized below) are orthonormal under a Gaussian weighting

(52)

In this case, we have

(53)

Once again, the choice of is arbitrary. But, it is logical to choose near , the present level of the underlying . Note that, in the case of an orthonormal basis, differentiability of the coefficients is not required. This is a significant advantage over the Taylor and two-point Taylor basis functions considered in Examples 1 and 2, which do require differentiability of the coefficients.

Now, returning to Cauchy problem (15), we suppose that can be written as follows

(54)

Following Pagliarani et al. (2013), we insert expansions (38) and (54) into Cauchy problem (15) and find

(55)
(56)

We are now in a position to find the explicit expression for , the Fourier transform of in (55)-(56).

Theorem 1.

Suppose and let denote its Fourier transform. Suppose further that and its Fourier transform exist, and that both the left and right hand side of (55)-(56) belong to . Then is given by

(57)
(58)

Note that the operator acts on everything to the right of it.

Proof.

See Appendix A. ∎

Remark 4.

To compute survival probabilities over the interval , one assumes a payoff function . Note that the Fourier transform of a constant is simply a Dirac delta function: . Thus, when computing survival probabilities, (possibly defaultable) bond prices and credit spreads, no numerical integration is required. Rather, one simply uses the following identity

(59)

to compute inverse Fourier transforms.

Remark 5.

Assuming , one recovers using

(60)

As previously mentioned, to obtain the FK transition densities one simply sets . In this case, becomes .

When the coefficients are time-homogeneous, then the results of Theorem 1 simplify considerably, as we show in the following corollary.

Corollary 1 (Time-homogeneous case).

Suppose that has time-homogeneous dynamics with the local variance, default intensity and Lévy measure given by , and respectively. Then the symbol is independent of . Define

(61)

Then, for we have

(62)

where

(63)
(64)
Proof.

The proof is an algebraic computation. For brevity, we omit the details. ∎

Example 4.

Consider the Taylor density expansion of Example 1. That is, . Then, in the time-homogeneous case, we find that and are given explicitly by

(65)
(66)
(67)
(68)