A General Framework for Time-changed Markov ... - SSRN papers

A General Framework for Time-changed Markov Processes and Applications Zhenyu Cui

∗

J. Lars Kirkby

†

Duy Nguyen‡

August 20, 2018

Abstract In this paper, we propose a general approximation framework for the valuation of (pathdependent) options under time-changed Markov processes. The underlying background process is assumed to be a general Markov process, and we consider the case when the stochastic time change is constructed from either discrete or continuous additive functionals of another independent Markov process. We first approximate the underlying Markov process by a continuous time Markov chain (CTMC), and derive the functional equation characterizing the double transforms of the transition matrix of the resulting time-changed CTMC. Then we develop a two-layer approximation scheme by further approximating the driving process in constructing the time change using an independent CTMC. We obtain a single Laplace transform expression. Our framework incorporates existing time-changed Markov models in the literature as special cases, such as the time-changed diffusion process and the time-changed Lévy process. Numerical experiments illustrate the accuracy of our method. Keywords: Finance, Time change, Markov Process, Option Pricing, Continuous-time Markov Chains, Laplace transform, Subordination, Variance Swaps, Bermudan options AMS subject classifications: 91G80, 93E11, 93E20

∗

School of Business, Stevens Institute of Technology, Hoboken, NJ 07310. Email: [email protected] Corresponding author. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30318, Email: [email protected] ‡ Department of Mathematics, Marist College, 3399 North Road, Poughkeepsie, NY 12601, Email: [email protected] †

Electronic copy available at: https://ssrn.com/abstract=3235766

1

Introduction

Markov processes, as one of the most widely studied stochastic processes, are ubiquitous in economics and finance. They offer a rich theoretical structure which incorporates practical features needed in applications, such as jumps and leverage effects. As a seminal example, the geometric Brownian motion (GBM), which is the foundation of the Black and Scholes model, enjoys great success in financial applications1 mainly due to the availability of closed-form solutions for option pricing and portfolio choice. In recent years, there has been increased interest in developing alternative stochastic models that incorporate practical aspects exhibited by asset prices and financial environments, which include but are not limited to stochastic volatility, jump movements, regime switching, clustering of trading activities, leverage effects, self-exciting jumps (e.g. clustered defaults), etc. The recent experience of the 2008 financial crisis and the April 2010 Flash Crash indicate that volatility risk and jump risk are eminent in financial markets, and their modeling is of practical importance. Thus a more flexible framework that accounts for these features is desirable. On the other hand, analytical tractability is of utmost importance in developing stochastic models useful for finance, since closed-form explicit solutions of key quantities make transparent the dependence of financial quantities on model parameters and allows for better interpretation and calibration of the models from data. Representative analytically tractable models include Merton’s jump diffusion model (Merton (1976)), Kou’s double exponential jump diffusion model (Kou (2002), Kou and Wang (2004)), the mixed exponential jump diffusion model (Cai and Kou (2011)), and a recent model with general discrete jump sizes (Fu et al. (2017)). Another strand of popular analytically tractable stochastic models include the stochastic volatility models (see Heston (1993)), continuous-time affine stochastic processes (see Duffie et al. (2000)), discrete-time affine GARCH processes (Heston and Nandi (2000)), and the recently introduced “polynomial processes” (see Cuchiero et al. (2012), Filipović and Larsson (2016b) and Filipović et al. (2017)). For an overview of the literature on analytically tractable stochastic models in finance, please refer to Sundaresan (2000), Broadie and Detemple (2004), Kou (2007) and references therein. Based on existing analytically tractable stochastic processes, a natural way to introduce practical features, such as clustering of trading activities, is to subordinate2 the underlying stochastic process on a so-called stochastic clock. There are important recent studies on developing analytically tractable subordinated stochastic processes in modeling the equity and commodity markets. As noted in Li and Zhang (2016), “The pricing problem for these processes presents unique challenges, as existing numerical PIDE schemes fail to be efficient and the applicability of transform methods to many subordinate diffusions is unclear.” This motivates the search for alternative computational approaches to handle pricing problems associated with subordinated or time-changed Markov processes. Existing theoretical work along the direction of subordinate diffusions only focuses on the few cases where the explicit eigenvalues and eigenfunctions are known, and this may fail to capture essential market features and the scope of applications is limited. Above all, it motivates us to develop a novel continuous-time Markov chain (CTMC) approximation technique in options pricing for subordinated/time-changed Markov processes. Our approach is aligned with recent efforts in developing efficient valuation framework for (exotic) option pricing using CTMCs. In the case of barrier options, valuation methods based on CTMC have been developed in Mijatović and Pistorius (2013), Li and Zhang (2016, 2018b), and 1 Since Black and Scholes (1973) until now, majority of the finance literature are based on this model. See some recent literature in He (2009), DeMarzo et al. (2012), Lan et al. (2013), Drechsler (2014), etc. 2 Using subordinated stochastic processes to model asset prices dated back to Clark (1973). The stochastic clock naturally represents the cumulative number of trades, and can be understood as the transaction clock (see Ané and Geman (2000)).

1 Electronic copy available at: https://ssrn.com/abstract=3235766

for other exotic path-dependent options such as the Bermudan, Asian, realized volatility derivatives and cliquet options (or equity-linked insurance products), please refer to Cui et al. (2017b,c, 2018b); Kirkby et al. (2017). There are also direct links between the modeling of stochastic time change and exact simulation methodology for multivariate diffusion processes, such as stochastic volatility models Broadie and Kaya (2006); Cai et al. (2017) and Hawkes processes Dassios and Zhao (2013, 2017). Future research aims at investigating this relationship and the application of Markov chain approximation. In this paper, we focus on developing the general valuation methodology for a large class of timechanged Markov processes, and the emphasis is on the unified approach that can handle the full range3 of practically relevant model parameters. We demonstrate the weak convergence of the proposed CTMC approximation scheme to the exact values. The contribution of the paper is three-fold: 1. We provide a detailed analysis of the connection of our framework to the vast amount of existing literature on time-changed Markov processes, and present a unified theoretical and computational framework for continuously and discretely monitored options under general time-changed Markov processes. 2. We develop an accurate and efficient Markov chain approximation method for options pricing based on time-changed Markov processes. The method is general in that it covers most of the proposed time-changed Markov process models in the literature, and is also applicable to a broad range of parameters for practical use. The method yields explicit closed-form approximations for European options, and recursive semi-closed-form formulae for other exotic path-dependent options, while currently the literature on the valuation of exotic path-dependent options under time-changed Markov processes is scarce. 3. We propose a one-layer CTMC approximation when the explicit Laplace transform of the stochastic time change is available, and obtain the double transform of the transition density. Otherwise, when the Laplace transform of the stochastic time change is not available, we carry out a twolayer CTMC approximation and obtain a single Laplace transform expression for the transition density. The two-layer approximation method can handle general classes of time-changed Markov processes. The paper is organized as follows: Section 2 describes the general theoretical framework. Section 3 considers the modeling framework of the stochastic time change process and discusses specific examples under which closed-form examples are available. Section 4 considers the underlying background stochastic process and discusses some important special cases that have appeared in the literature. Section 5 contains numerical examples with detailed comparison of our proposed approach to the existing literature. Section 6 concludes the paper with discussions of future research directions. 3

Note that some existing methods work well in certain range of model parameters but not for others. For example, on page 865 of Linetsky (2004), it is stated: . . . the dimensionless time to expiration τ = (σ 2 T )/4 is the crucial parameter that controls the numerical convergence of the series . . . The larger the value of τ , the faster the convergence. For smaller values of this parameter, convergence significantly slows down . . ..


2

General Theoretical Framework

We start with a frictionless and no-arbitrage market, and assume that an appropriate market price of risk4 has been chosen, i.e. the equivalent martingale measure (EMM) Q is determined. We then have a filtered probability space (Ω, F, {Ft }t≥0 , Q), on which all stochastic processes are defined, and all subsequent expectations are taken with respect to the risk-neutral measure Q unless stated otherwise. Let {Yt }t≥0 denote a one-dimensional time-homogeneous Markov process, which serves as the underlying background stochastic process. We assume a model of the form St = S0 eµt exp(Yt ), where E[exp(Yt )] = 1. This model incorporates the case of equities by setting µ = r − q, and the case of futures by setting µ = 0, where r, q ∈ R are the interest rate and the dividend yield5 . A rich variety of models can be obtained by observing Yt on a stochastic clock, known as a random time change. Definition 1 A random time change τ is a family of stopping times {τt ; t ≥ 0} such that t → τt is increasing and right continuous. We assume that τ0 = 0 and for any fixed 0 < t < ∞, τt is finite a.s, i.e. Q(τt < ∞) = 1, and that the random variable τt is a stopping time adapted to the filtration Ft . In general, a time-changed Markov process is obtained by time changing a one dimensional Markov process Yt . Natural candidates for the stochastic time change6 are the discrete and continuous additive functionals: Z t n X Bn := h(Xti ), n ≥ 0, At := h(Xu )du, t ≥ 0, (1) 0

i=0

where {Xt }t≥0 is also a time-homogeneous Markov process which is independent 7 from Yt , and h(·) is any non-negative measurable function.8 In the discrete case, the time instant is defined as ti := iδ, where we assume equally spaced (δ > 0) discrete-time observations, and the last observation time instant is given by tn := t = nδ. The time-change functionals Bn and At in (1) are stochastically increasing with respect to n and t respectively, and they include many important examples as special cases, e.g. the affine process, the quadratic process, and the recently proposed polynomial processes (see Filipović and Larsson (2016a), Larsson and Pulido (2017)), etc. For a time-changed Markov process, there is limited analytical tractability except in special cases where the semigroup properties of the underlying stochastic process are well-studied. In what follows, we develop a theoretical treatment to accommodate general time changed Markov processes, which supports several CTMC approximation schemes. In particular, we consider the following approaches to forming tractable time-changed models, with greater flexibility than those offered in the existing literature: 0. Yet := Yτt , a standard time-changed process. n n n 1. Yet y := Yτt y , where Yt y is an ny -state CTMC that approximates the underlying Yt . 4

In a complete market, the market price of risk is uniquely defined through the risk-neutral probability. In an incomplete market, the market price of risk is not unique, and in this case, we assume that a suitable one is chosen. 5 Note that Yt can also be used to model the underlying asset price, or the log asset price process. In this case, we have E[e−µt Yt | Fs ] = e−µs Ys . 6 In the extant literature (see for example Madan et al. (1998), Carr et al. (2003)), it is sometimes assumed that E[τt ] = t, which has the intuitive interpretation as requiring that the expected readings from respectively the stochastic clock and the original clock are the same. In this paper, we consider general stochastic time changes, which are not subject to this constraint. 7 Borrowing terminology from the Lévy processes, applying an independent stochastic time change to an underlying stochastic process is usually called “subordination”. 8 In general, we assume that {Yt }t≥0 and {Xt }t≥0 have right-continuous-with-left-limit (c` adl` ag) sample paths.


2. Y¯t := Yτ¯t , where τ¯t is constructed from additive functionals of a CTMC and it approximates the stochastic time change τt . n n ,n 3. Ybt y x := Yτ¯t y , where both Y is approximated by a CTMC with ny states and τ is constructed from additive functionals of an approximating CTMC with nx states.

It is important to distinguish our work from that of Li and Linetsky (2014) and several other papers on subordinated diffusions, which apply a spectral representation of the density of the underlying diffusion. This requires an explicit closed-form expression for the spectral semigroups, which limits the applicability of their method. In addition, in the actual implementation, a truncation of the spectral expansion series is needed, and some special functions involved in the eigenfunctions may cause numerical instabilities for certain parameter ranges; see for example Linetsky (2004) for such a discussion in the pricing of Asian options. In comparison, our method is applicable to the general case when the underlying background stochastic process is a Markov process. Although our method involves approximating the underlying background Markov process by a CTMC, the trade-off is that it can handle a significantly wider range of underlying background processes, and is not subject to restrictions in the model parameters. Another operational advantage is that it yields closed form expressions, involving only simple matrices with no special functions involved, which results in very simple numerical implementations. Moreover, the method is computationally efficient, as illustrated by the numerical experiments to follow. To summarize, our proposed method has stable and consistent performance across a large number of underlying background Markov processes and across all parameter ranges, and offers a competitive set of tools for approximating Markov processes within a time-changed framework.

2.1

Underlying Driving Process

The first approach is to use a CTMC to approximate the underlying background Markov process Y , while keeping the stochastic time change process in its current general form (case 1 above). At this n current stage, assume that we have a CTMC, Yt y , which has a finite state space SY := {y1 , ..., yny }, n and a transition probability matrix PY (t) of the size ny × ny . Yt y will be used to approximate ny the underlying process Yt , and details of the construction of Yt for several important cases will be n n given in Section 4. The risk-neutral transition probabilities (PY )ij (t) = Q[Yt y = yj |Y0 y = yi ] for i, j ∈ {1, 2, ..., ny } are determined by the transition (rate) matrix Λ = (λij )i,j=1,...,ny , and satisfy the P k matrix exponential PY (t) = exp(Λt) = ∞ n=0 (Λt) /(k!). We recall some background knowledge for the CTMC below. ny

Lemma 1 Assume that the transition rate matrix of Yt Λ = QDQ−1 ,

(and hence PY ) is diagonalized9 as follows

PY (t) = QeDt Q−1 ,

(2)

where D := diag(α1 , α2 , . . . , αny ) is a diagonal matrix of the eigenvalues of Λ, Q = (qij )i,j=1,...,ny is a matrix whose columns are the corresponding eigenvectors, and we write Q−1 = (˜ qij )i,j=1,...,ny . The ny ny e transition density matrix of the time-changed process Yt := Yτt is represented as: n n (PYe )ij (t) = Q[Yet y = yj |Ye0 y = yi ] =

ny X

qik q˜kj E[eαk τt ].

k=1 9

This assumption is always satisfied in our construction, see Proposition 16.


(3)

Proof: Note that we have n n n (PYe )ij (t) = Q[Yet y = yj |Ye0 y = yi ] = E[1{Yτny =yj } |Y0 y = yi ] t " ny # X α k τt = E[(PY )ij (τt )] = E qik q˜kj e .

(4)

k=1

The result follows immediately from the above equality.

Remark 2 Similar to the observation of Li and Linetsky (2014), we note that the time variable enters the density expressions for the approximating CTMC only through the terms eαk τt , thus after the stochastic time change, the density has the same form as the original density expression, but with eαk t replaced by E[eαk τt ]. Here we have not specified the form of the general stochastic time change τt . In the case where τt is constructed by a Lévy subordinator, then we have E[eαk τt ] = eφ(αk )t , where φ(·) is the characteristic exponent of the subordinator. Another explicit example is when τt is constructed through an integrated CIR process, and a closed-form representation for E[eαk τt ] is also available, i.e. the zero coupon bond price is available in closed-form. More details about tractable examples are illustrated in Section 3.1. n

In general, if Yt y is a weakly convergent approximation to the true process Yt , the time-changed CTMC will also converge weakly to the true time-changed process. n Proposition 3 Suppose that Yt y =⇒ Yt (weak convergence) as ny → ∞, and let Yet := Yτt for an n n n independent time change τt . Then Yet y =⇒ Yet as ny → ∞, where Yet y := Yτt y is the time-changed CTMC approximation. n

n

Proof: From the weak convergence of Yt y to Yt for a fixed t, it holds that E[exp(iYt y ξ)] → E[exp(iYt ξ)] as ny → ∞, for each fixed t. It then follows that n

n

lim E[exp(iYet y ξ)] = lim Eζ [E[exp(iYζ y ξ)|τt = ζ]]

ny →∞

ny →∞

n

= Eζ [ lim E[exp(iYζ y ξ)|τt = ζ]] ny →∞

= Eζ [E[exp(iYζ ξ)|τt = ζ]] = E[exp(iYet ξ)], n

where passage of the limit is justified by the Portmanteau theorem by noticing that |E[exp(iYζ y ξ)|τt = ζ]| ≤ 1. This completes the proof. A closed-form pricing formula for European options follows immediately from Lemma 1. Recall that the price of an European call option on an underlying St = g(Yt ) is given by the risk-neutral expectation C = e−rt E[(g(Yt ) − K)+ |Y0 = y0 ],

(5)

where we have modeled the underlying as St = g(Yt ) = S0 eµt exp(Yt ), where Yt is the driving Markov process. We will denote the payoff generically as H(Yt ). It follows that C≈e

−rt

ny X j=1

H(yj )

ny X

qi0 ,k q˜k,j E[eαk τt ],

k=1

E[eαk τt ]

where is either known in closed form (see Section 3.1), or further approximated through independent CTMCs (see Section 3.3). Here the i0 -th state is chosen such that y0 = yi0 . In the following discussion, we will develop closed-form and semi-closed-form pricing algorithms to approximate the value of several more exotic payoffs. From Proposition 3, the pricing formula is consistent for a weakly convergent CTMC. 5 Electronic copy available at: https://ssrn.com/abstract=3235766

3

Modeling the Stochastic Time Change

n n Note that equation (4) is a key representation that links the probability density function of Yet y = Yτt y to a finite sum of weighted quantities of E [eαk τt ]. Theoretically the next step is to either determine an exact closed-form expression for it (see two explicit examples in Section 3.1.1 and Section 3.1.2) without further approximation or to determine a general exact theoretical characterization of it through functional equations (see Section 3.2). In the general case where we do not have an exact expression for E [eαk τt ], or the associated functional equation is hard to solve accurately, we develop a general and accurate approximation for the driving process behind the stochastic time change through CTMCs.

3.1

Important Cases for the Stochastic Time Change

Table 1 summarizes the stochastic time changes that have appeared in the literature and also our newly proposed stochastic time changes that are based on continuous/discrete additive functionals. Note that the stochastic times changes in the existing literature are special cases of our general setting of additive functionals. Stochastic Time Change

Representative Literature

Integral activity process

Carr and Wu (2004)

Lévy subordinator

Mendoza-Arriaga et al. (2010)

Lévy subordinator time-changed


by integral activity process Discrete Additive Functionals

This paper

Continuous Additive Functionals

This paper

Table 1: Classification of Time Change There are three main categories of stochastic time changes: Rt 1. Time integral of an activity rate process, i.e. τt := 0 h(Xu )du, where {Xt }t≥0 is usually assumed to be a time-homogeneous diffusion process representing the intensity process. In the existing literature, X is often taken to be the CIR process. 2. A Lévy subordinator, i.e. τt := Tt , where {Tt }t≥0 is a non-decreasing Lévy process with positive jumps and a nonnegative drift. Note that we can also construct the stochastic time change through multiple layers of Lévy subordinators, see Appendix A.2. 3. A Lévy subordinator time-changed by the time integral of an activity rate process. This is the situation of a “double-layer time change”, i.e. we time change the stochastic time change process. For example, we can construct a subordinator as τt := TR t h(Xu )du , where T· is a non-decreasing 0 Lévy process with positive jumps and a nonnegative drift. In the following, we shall illustrate some prominent examples of stochastic time changes that are commonly employed in the literature and yield closed-form expressions. The reader is invited to refer to Appendix A for additional examples.


3.1.1

Closed-form example: integral of CIR process

As a concrete example, consider the case when we specify the time change to be an integrated CIR process, i.e. a CIR stochastic clock, which is commonly used in the literature; see for example Carr and Wu (2004). The corresponding Laplace transform is available in closed-form due to the well-known zero coupon bond pricing formula in the CIR stochastic interest rate model. Recall that the dynamics of the CIR process is given by p dXt = κ(θ − Xt )dt + σv Xt dWt , X0 = x0 . (6) Rt With τt := 0 Xu du, we have the following Laplace transform (see Cairns (2004)) Lx0 (t, α) := Ex0 [e−ατt ] = exp (A(t, αt) − B(t, αt)x0 ) ,

(7)

where Ex0 [·] := EQ [·|X0 = x0 ], with 2κθ A(t, s) = 2 log σv

(γ(s)+κ)t

2γ(s)e 2 (γ(s) + κ)(eγ(s)t − 1) + 2γ(s)

2s(eγ(s)t − 1) B(t, s) = , t(γ(s) + κ)(eγ(s)t − 1) + 2γ(s)

! , r

γ(s) =

κ2 + 2

σv2 s . t

The transition density in (3) is then given as (PYe )ij (t) =

ny X

qik q˜kj Lx0 (t, −αk ).

(8)

k=1

3.1.2

Closed-form example: L´ evy Subordinators

Another important and tractable case for the time-change process is the Lévy subordinator, τt , which comprises the class of pure jump Lévy process {Xt }t≥0 with drift parameter γ ≥ 0, and Lévy measure R ν, which satisfies the integrability condition (0,∞) (s ∧ 1)ν(ds) < ∞. We require that {Xt }t≥0 is spectrally positive, and intuitively this refers to Lévy processes which exhibit only positive jumps. Formally, from Kuznetsov et al. (2012), we say that a Lévy process is spectrally negative if the jump measure is carried by (−∞, 0), i.e. ν(0, ∞) = 0. Thus we say that a Lévy process X is spectrally positive when −X is spectrally negative. The Laplace transform of τt is characterized by Z −λτt −φ(λ)t E[e ]=e , φ(λ) := γλ + (1 − e−λs ν(ds)). (9) (0,∞)

For example, a common class of Lévy subordinators are given by ( γλ − CΓ(−α)((λ + η)α − η α ), φ(λ) = γλ + C ln(1 + λ/η),

α 6= 0 α = 0,

where C > 0, α < 1, and η ≥ 0. This contains the tempered stable subordinators when α ∈ (0, 1), the Inverse Gaussian processes as the special case of α = 1/2, and the Gamma process when α = 0. n Hence, from (4) the transition probabilities of Yet y satisfy (PYe )ij (t) =

ny X

qik q˜kj e−φ(−αk )t ,

k=1

where we note that αk ≥ 0. 7 Electronic copy available at: https://ssrn.com/abstract=3235766

(10)

3.1.3

Closed-form example: Markovian Hawkes process

We next consider an example of a pure jump point process, the Hawkes process, which has become popular in queuing theory and modeling high frequency trade arrivals10 . Consider τt = Nt , where Nt is a Markovian Hawkes process with kernel h(t) = ae−bt , that is, Nt is a simple point process with intensity ν + Zt− , where dZt = −bZt dt + adNt , Z0 = z0 , (11) and the Zt process has the following infinitesimal generator: Af (z) = −bz

∂f + (ν + z)[f (z + a) − f (z)]. ∂z

(12)

In this case, Zt is Markovian and so is the pair (Nt , Zt ). By its affine structure, see e.g. Errais et al. (2010), one can compute β Ez0 [e−βNt ] = e(C(t)+ a )z+D(t) , (13) where C(t), D(t) are the solutions of the following ordinary differential equation (ODE): C 0 (t) = −bC(t) + eC(t)a − 1 −

βb , a

D0 (t) = ν(eD(t)a − 1),

(14) (15)

with the initial condition C(0) = −β/a and D(0) = 0. The above ODEs can be numerically solved. Remark 4 This model has been subsequently generalized in (Dassios and Zhao, 2011, Theorem 3.1), and we believe that their results can be applied in our case. It would be interesting to extend and adapt their results within our framework, which we leave for future research.

3.2

Time changes constructed from additive functionals: General theory

Previous examples showcase special situations where the Laplace transform of the stochastic time change has a closed-form expression. However, this restricts the modeling to a few stylized cases that may not capture the empirical features of the financial data. Thus in this section we shall present a general exact theoretical characterization of the “double transform” of the stochastic time change through functional equations. The central objects of study are the following general double transforms: l(z; x) :=

∞ X

z n Ex [e−θBn ],

n=0 Z ∞

m(µ; x) :=

e−µt Ex [e−θAt ]dt,

|z| < 1, Re(µ) > 0,

(16) (17)

0

where Bn and At are defined in (1), Re(θ) ≥ 0, Ex [·] = EQ [·|X0 = x], and nδ = t. In subsequent discussions, in order to emphasize the dependence of the double transform on θ, we shall also equivalently use l(z, θ; x) and l(z; x) interchangeably. Similarly for m(µ, θ; x) and m(µ; x). Note that (16) corresponds to the “Z-Laplace transform”, where the first step is a Z-transform, and (17) corresponds to the “double Laplace transform”. In the following, we call them “double transforms” in general. The next result characterizes the double transforms of the above objects of interest through functional equations. 10

The Hawkes process is first introduced in Hawkes (1971). It is a representative point process that exhibit both self-exciting and mutual exciting cluster behavior, which makes it suitable for modeling seismology and the order arrivals in the high frequency trading setting. For some related applications of Hawkes process to finance, please refer to Zhu (2015), A¨ıt-Sahalia et al. (2015), Bacry et al. (2015), Dassios and Zhao (2013, 2017), and references therein.


Theorem 5 Let h(·) be defined as in (1). The following hold: (i) Given δ > 0, and for any z such that |z| < 1, if there exists a function f (·) := f (x) that solves the functional equation exp (θh(x)) · f (x) − zEx [f (Xδ )] = 1, and is bounded, i.e.,

(18)

| f (x) |≤ C < ∞ for some constant C > 0, then we must have f (x) =

sup x∈[0,∞)

l(z; x), which is defined in (16). (ii) Assume that {Xt }t≥0 has right-continuous-with-left-limit (c` adl` ag) sample paths and the infinitesimal generator G. For any µ such that Re(µ) > 0, if there exists a function f (x) that solves the functional equation (θh(x) + µ − G)f (x) = 1,

(19)

and is bounded, then we must have f (x) = m(µ; x), where m(µ; x) is defined in (17). Proof: (i) The first part can be proved following similar arguments as in Cai et al. (2015). Suppose that f (·) is any bounded solution to the functional equation (18) if it exists, then we introduce the more general discrete process (Mn )n=0,1,... as follows Mn := f (Xtn ) · exp (θh(Xtn )) · z n · exp (−θBn ) +

n−1 X

z k · exp (−θBk ) .

(20)

k=0

First note that n−1 X n |Mn | ≤ f (Xtn ) exp (θh(Xtn )) · |z exp (−θBn ) | + |z k exp (−θBk ) | k=0

= |1 + zPδ f (Xtn )| · |z n exp (−θBn ) | +

n−1 X

|z k exp (−θBk ) |

k=0

1 < ∞, ≤1+C + 1 − |z|

(21)

where Pδ f (x) := Ex [f (Xδ )]. Moreover, we have that Ex [Mn |F (n−1) ] − Mn−1 = z n exp (−θBn−1 ) Pδ f (Xtn−1 ) + z n−1 exp (−θBn−1 ) − f (Xtn−1 ) exp θh(Xtn−1 ) · z n−1 exp (−θBn−1 ) = z n−1 exp (−θBn−1 ) zPδ f (Xtn−1 ) + 1 − f (Xtn−1 ) exp θh(Xtn−1 ) = 0.

(22)

It follows that {Mn , n ≥ 0} is a martingale with respect to {F (n) , n ≥ 0}, where F (n) := Fnδ , i.e. P k −θBk almost surely, then we have Ex [Mn ] = M0 = f (x). Note that lim Mn = ∞ k=0 z e n→∞

f (x) = lim Ex [Mn ] = Ex n→∞

h

i

lim Mn = Ex

n→∞

"∞ X

# z k e−θBk

k=0

=: l(z; x),

(23)

and the second equality above is due to the condition (21) and the application of the Lebesgue dominated convergence theorem. This completes the proof of part (i).


(ii) Let us define u(t, x) := Ex [e−θAt ], and from the extended Feynman-Kac formula for Markov processes (Rogers and Williams, 2000, cf. III.19), we have ∂u(t, x) = Gu(t, x) − h(x)u(t, x), ∂t

u(0, x) = 1.

(24)

R∞ Recall that m(µ; x) = 0 e−µt u(t, x)dt is the Laplace transform of u(t, x) with respect to t. By taking Laplace transform with respect to t on both sides of the equation (24), we get µ · m(µ; x) − u(0, x) = µ · m(µ; x) − 1 = (G − θh(x))m(µ; x), which implies that m(µ; x) satisfies the equation (θh(x) + µ − G)m(µ; x) = 1. This completes the proof of part (ii). ny e Combining the representation of the density of Yt in (4) and the double transform result in Theorem 5, we obtain a functional equation characterization for the Laplace transform of the probability density function of the (approximating) time-changed CTMC. n Proposition 6 Consider the time-changed CTMC, Yet y , subordinated by either the discrete or continuous additive functionals τn := Bn and τt := At . We define its Z−transform and the Laplace transform respectively as below

U (z) :=

∞ X

n

z (PYe )ij (nδ) =

n=0 ∞

Z V (µ) := 0

ny X

qik q˜kj l(z, −αk ; x),

k=1

e−µt (PYe )ij (t)dt =

ny X

qik q˜kj m(µ, −αk ; x),

k=1

where l(z, −αk ; x) :=

∞ X

z n E [eαk τn ] ,

n=0 Z ∞

m(µ, −αk ; x) :=

e−µt E [eαk τt ] dt.

(25)

0

Then we have that l(·, αk ; ·) and m(·, αk ; ·) satisfy respectively the following functional equations: exp (−αk h(x)) · f (x) − zEx [f (Xδ )] = 1,

and

(−αk h(x) + µ − G)f (x) = 1.

(26)

Proof: The proof follows directly from the application of Theorem 5. The above result provides the theoretical characterization when the stochastic time change is assumed to be constructed from additive functionals of an independent general Markov process. It yields closed-form solutions when the corresponding functional equations have closed-form solutions. For example, the closed-form examples demonstrated in Section 3.1 all belong to this category.


3.3

Time change by additive functionals: CTMC approximation

While Theorem 5 provides a general characterization, if the associated functional equation does not admit closed-form solutions, then numerical approaches for solving the functional equation have to be employed, which require a numerical inversion of the double transform. This direct approach leads to multiple sources of discretization errors, and it motivates us to develop a general approximation method based on CTMCs. The intuitive idea behind our approach is to use a CTMC to approximate the driving Markov process behind the construction of the stochastic time change, and then solve the associated functional equation associated with the CTMC, which turns out to be a matrix equation. Thus in the following, we investigate the case when the stochastic time change is constructed based on additive functionals of a CTMC. Consider a non-negative CTMC {Xtnx , t ≥ 0} with finite state space SX := {x1 , . . . , xnx }, which is a finite state Markov chain approximation to Xt . The transition nx probability matrix of Xtnx is denoted as PX (t) with elements pX = xj | X0nx = xi ), ij (t) = Q(Xt and its transition rate matrix is denoted GX . Define x = (x1 , . . . , xnx )T , and by convention, I is the identity matrix and 1 is an nx × 1 column vector with all entries equal to 1. Define a diagonal matrix DX = (dij )nx ×nx with djj = h(xj ), j = 1, . . . , nx . Similar to Proposition 6 of Cui et al. (2018a), we have the following results. Proposition 7 (Single transform in the CTMC setting) Define the functionals Bnnx and Ant x of the CTMC Xtnx by Bnnx

:=

n X

h(Xtnix ),

n ≥ 0,

Ant x

i=0

Z :=

t

h(Xunx )du, t ≥ 0.

(27)

0

i h nx (i) Let gd (n; x) := Ex e−θBn . Then, with δ := t/n, gd (n; x) = (e−θDX PX (δ))n e−θDX 1.

(28)

h i nx (ii) Let gc (t; x) := Ex e−θAt . Then, gc (t; x) = e(GX −θDX )t 1.

(29)

Proof: The detailed proof is contained in the Appendix. The next result demonstrates the weak convergence of the stochastic time change approximation. Proposition 8 Suppose that Xt is the stochastic process behind the construction of the stochastic time change process, and let Xtnx =⇒ Xt , where Xtnx is the CTMC approximation. Then we have the weak convergence Bnnx =⇒ Bn and Ant x =⇒ At as nx → ∞. Assuming further that Yt has continuous paths, it holds that Yet =⇒ Yt as nx → ∞ when Yt is subordinated by τ¯t = Ant x or τ¯n = Bnnx . Proof: From the weak convergence11 of Xtnx to Xt for a fixed t due to the construction of the CTMC, it holds that E[exp(iξXtnx )] → E[exp(iξXt )] as nx → ∞, for each fixed t. For the case of discrete additive functional, application of Theorem 9 in Song et al. (2013) implies that Bnnx =⇒ Bn as nx → ∞ n P by noticing the resemblance of the payoff h(Xti ) to that of a discrete arithmetic Asian option. For i=0

the case of continuous additive functional, the continuous mapping theorem implies that Ant x =⇒ At as nx → ∞. Finally, we have limnx →∞ E[exp(iξYτ¯t )] = E[limnx →∞ E[exp(iξYτ¯t (ω))]] = E[exp(iξYτt )], 11

We assume that the grid for X nx is increased with nx to progressively cover the domain of Xt .


which follows from the continuous paths of Yt for any fixed ω ∈ Ω. This completes the proof.

Above all, we have the following result on an explicit matrix approximation to the density function of a general time-changed Markov process, whose underlying driving process is a Markov process, and whose stochastic time-change is constructed based on additive functionals of another independent Markov process. The main idea is to use two independent CTMCs, Y ny and X nx on SY × SX to approximate respectively the two independent Markov processes Y and X. Then we construct the n ,n n n n ,n approximate time-changed CTMC process as Ybt y x := Yτ¯t y or Ybn y x := Yτ¯ny , where τ¯t and τ¯n are the stochastic time changes constructed in terms of additive functionals of the approximating CTMC X nx . n ,n Corollary 9 After approximating the time-changed Markov process Yτt by Ybt y x through a two-layer CTMC, we have the following explicit expressions for the approximate density functions:  ny X    qik q˜kj (eαk DX PX (δ))n eαk DX 1, (Discrete case)   n ,n n ,n Q(Ybt y x = yj | Yb0 y x = yi , X0nx = x) = k=1 ny X    qik q˜kj e(GX +αk DX )t 1, (Continuous case)   k=1

(30) Proof: The proof is straightforward following the results of Proposition 7 and Lemma 1 and replacing θ by −αk . Next we establish the convergence of the two-layer CTMC approximation. Without loss of genRt erality, we consider the case of a continuous additive functional subordinator, i.e. τt := 0 h(Xu )du. The case of discrete additive functionals follows similarly. n Proposition 10 Suppose that Yt y =⇒ Yt , and let Yet := Yτt for an independent time change τt := Rt n ,n n At = 0 h(Xu )du. Denote Ybt y x := Yτ¯t y as the double-layer CTMC approximation, where τ¯t := Rt nx nx 0 h(Xu )du. Suppose further that Xt =⇒ Xt , and that Yt has continuous sample paths. Then we n ,n have Ybt y x =⇒ Yet as ny → ∞ and nx → ∞.

Proof: Due to the independence of the two CTMCs, we can first condition on the stochastic time change process, then we have lim

ny ,nx

lim E[exp(iYbt

nx →∞ ny →∞

ξ)] = lim

n

lim Eζ [E[exp(iYζ y ξ)|¯ τt = ζ]]

nx →∞ ny →∞

n

= lim Eζ [ lim E[exp(iYζ y ξ)|¯ τt = ζ]] nx →∞

ny →∞

= lim Eζ [E[exp(iYζ ξ)|¯ τt = ζ]] nx →∞

= lim E[exp(iYτ¯t ξ)] = E[exp(iYτt ξ)] = E[exp(iYet ξ)], nx →∞

(31)

where the second last equality follows from the continuous sample paths of Yt . This completes the proof. A closed-form approximation for the prices of European options follows immediately from (30). In particular, the price of a contract with payoff H(Yt ) at time t > 0 is given by −rt

C≈e

ny X j=1

H(yj )

ny X

(GX +αk DX )t qi0 ,k q˜k,j · e> 1, i0 e

k=1


for the continuous case, and similarly for the discrete case where the corresponding sum is replaced by that of (30). 3.3.1

Barrier and Bermudan Options

Note that Corollary 9 provides the necessary framework for pricing discretely monitored exotic options using the double-layer approximation methodology. Let ∆ := T /M denote the discrete monitoring time step, with M monitoring points on [0, T ], where T > 0 is the contract maturity. Given the Rt CTMC time change approximation τ¯t := 0 h(Xsnx )ds, define the nx × nx transition matrix Γ(k) by Γ(k) = e(GX +αk DX )∆ ,

k = 1, . . . , ny ,

(32)

n

y being the eigenvalues of Λ. An analogous representation holds in the discrete functional with {αk }k=1 case. For a discretely monitored exotic contract, at each observation point tm := ∆m we define (1) (n ) (l) V m = [V m , . . . , V m x ] to be the ny × nx value matrix with column vectors V m of dimension ny × 1. Let ym ≡ y := [y1 , . . . , yny ]> denote the grid on SY at an arbitrary time step m = 0, . . . , M . In (l) particular, V m denotes the value vector along ym conditioned on Xtnmx = xl . We have the following corollary, which summarizes the price recursion for Bermudan and discrete barrier options in the Markov chain model.

n ,n

Corollary 11 Consider the Markov chain (Yt y x , Xtnx ) on SY × SX = y × x with initial value (y0 , x0 ), and let ∆ := T /M denote the monitoring step size. Let H(sm ) denote the intrinsic payoff,12 where sm := S0 eµ·m∆ exp(y) denotes the underlying values along SY at time tm . The initial price of n ,n a Bermudan option on (Yt y x , Xtnx ) satisfies the recursion:  (p)   V M = H(sM )   P x (p) (p) (33) M (l,p) V m+1 , m = M − 1, . . . , 0 C m = e−r∆ np=1     V (p) = max{C (p) , H(s )}, m m m where M (l,p) is defined in (36) below. Now let 1B (s) denote the knock-out indicator, where 1B (si ) = 0 n ,n if si ∈ B for a fixed knock-out region B. The initial price of the barrier contract on (Yt y x , Xtnx ) satisfies the recursion   V (p) = H(sM ) ◦ 1B (sM ) M P (34) nx −r∆ (l,p) V (p)  V (p) M ◦ 1 (s ), m = M − 1, . . . , 0 m =e m B m+1 p=1 In either case, the initial value can be recovered from V l00 (yj0 ), where xl0 = x0 and yj0 = y0 . The recursive valuation approach for Bermudan and discrete barrier options is collected in Algorithm 1. Proof: First note that for l, p = 1, . . . , nx , the joint transition density of the Markov process n ,n (Yb y x , X nx ) satisfies t

t

n ,n n ,n nx P∆ (yj , xp |yi , xl ) := Q(Yb∆ y x = yj , X∆ = xp | Yb0 y x = yi , X0nx = xl )

=

ny X

(k)

qik q˜kj Γl,p .

(35)

k=1 12

For example H(sm ) = (K − sm )+ for a put.


(l)

We solve the pricing problem recursively in terms of the value functions V m (yi ) = V m (yi , xl ), m = M, M − 1, . . . 0, using the principal of dynamic programming. The value function is initialized in the final stage according to the final payoff, for example a call payoff V M (yi , xl ) := (S0 eµT exp(yi ) − K)+ ,

(yi , xl ) ∈ SY × SX , (l)

which depends only on yi at expiration. More generally, we have that V M (yi , xl ) = H(yi ). Let C m denote the continuation value for the Bermudan contract, when Xtnmx = xl . In vector form, it follows from (35) that −r∆ C (l) m =e

= e−r∆

nx X p=1 nx X

(p)

P∆ (ym+1 , xp |ym , xl )V m+1 (p)

(n )

(1)

QD (l,p) Q−1 V m+1 ,

D (l,p) := diag(Γl,p , . . . , Γl,py )

p=1

= e−r∆

nx X

(p)

M (l,p) V m+1 ,

M (l,p) := QD (l,p) Q−1

(ny × ny ),

(36)

p=1

and (33) follows. The proof for discrete barrier contract proceeds analogously. This completes the proof.

4

Modeling the Underlying Driving Process

In this section, we shall summarize cases for the underlying driving Markov process to which the previous time change processes are applied, and provide corresponding theoretical analysis. Table 2 summarizes the underlying background stochastic processes that have appeared in some representative literature, which are all special cases of our general setting. In the following, we shall consider two main categories: the time-changed Lévy processes, and the time-changed diffusion processes (with the Ornstein-Uhlenbeck and CIR as two popular choices), and they are summarizes in Table 2. Underlying Driving Processes

Previous Literature

Lévy processes

Carr et al. (2003), Carr and Wu (2004)

General Diffusion Processes

Li et al. (2015)

Ornstein-Uhlenbeck Processes

Li and Mendoza-Arriaga (2013)

CIR Processes


Table 2: Classification of Underlying Driving Stochastic Processes

4.1

Time-changed L´ evy Process

We first provide a theoretical characterization of the characteristic function of the general time-changed Lévy process, where the stochastic time change is constructed based on general additive functionals of an independent Markov process X. Explicit matrix expressions for the characteristic function of ¯ This the time-changed Lévy process are provided when X is approximated through the CTMC X. framework includes two special cases considered in the literature, where the stochastic time change is either constructed based on the Lévy subordinator or the integral of a CIR processes. 14 Electronic copy available at: https://ssrn.com/abstract=3235766

Denote L as a standard Lévy process, and (τt )t≥0 is a positive increasing process serving as the stochastic time change. In the following, we assume that the underlying driver L and the stochastic time change τt are independent, which is consistent with Carr et al. (2003). For the case where there is a non-zero leverage effect introduced into the underlying driver and the time change, please refer to Carr and Wu (2004) and Huang and Wu (2004). Due to the independence between the underlying background Lévy process and the stochastic time change, we can also refer to it as a subordinated Lévy process. The Lévy process is determined by its characteristic exponent, ψL (·), from which the characteristic function is given by φL (θ; t) := E[exp(iθLt )] = exp(ψL (θ)t). More specifically, we model the underlying stock price as St = S0 eµt exp(Lτt ), t ≥ 0, where we normalize the process by setting E[exp(Lt )] = 1 for all t ≥ 0, which moreover holds for any stopping time.13 In Carr et al. (2003), they considered three particular time-changed Lévy processes: the normal inverse Gaussian (NIG) model (see Barndorff-Nielsen (1997)), the variance gamma (VG) model (see Madan et al. (1998)), and the CGMY model. A common stochastic time change chosen in that paper is the time integral of the CIR or the square root process. For a comprehensive review of the relevant literature, please refer to Wu (2007) and references therein. Proposition 12 Consider the time-changed Lévy processes Lτt and Lτn , where {Lt }t≥0 is a standard Rt P Lévy process with characteristic exponent14 ψL (·), τt := 0 h(Xs )ds, and τn := nj=1 h(Xj ). Assume that {Lt }t≥0 has infinitesimal generator given by G, and that τt is independent of {Lt }t≥0 . Define the following transforms of the characteristic functions of the time-changed Lévy processes: UL (z) :=

∞ X

i h z n E eiθLτn

Z VL (µ) :=

∞

i h e−µt E eiθLτt dt,

(37)

0

n=0

then UL (·) and VL (·) satisfy respectively the following functional equations: exp (ψL (θ)h(x)) · f (x) − zEx [f (X∆ )] = 1,

and

(ψL (θ)h(x) + µ − G)f (x) = 1. Proof: From the Lévy-Khintchine theorem, we have h i h i h i i h E eiθLτn = E eψL (θ)τn and E eiθLτt = E eψL (θ)τt .

(38)

(39)

Combining the above two expressions with the characterization in Theorem 5, we have the desired result. This completes the proof. Remark 13 Note that we have provided a theoretical characterization of the single transform of the characteristic function of the time-changed Lévy process, which to the best of authors’ knowledge is a new result. Proposition 14 We approximate the Markov process X by the nx -state CTMC X nx , and construct Rt P the approximate stochastic time changes as τ¯n := nj=1 h(Xjnx ) and τ¯t := 0 h(Xsnx )ds. Then we have the following explicit matrix expressions for the characteristic functions: h i x iθLτ¯n ΦL (θ) := E e = (eψL (θ)∆·DX PX (∆))n eψL (θ)∆·DX 1, (40) τ¯n 13

In particular, for any t ≥ 0, starting with an non-normalized Lévy process L0t , note that E[exp(L0t − tψL0 (−i))] = 1. Hence, assume that we are working with the normalized characteristic exponent ψL (θ) := ψL0 (θ) − iθ · ψL0 (−i). 14 See Table 1 on page 127 of Carr and Wu (2004) for a list of common Lévy processes with explicit expressions for their characteristic exponents.


nx where [ΦL τ¯n (θ)]j := E [exp(iθLτ¯n )|X0 = xj ], j = 1, . . . , nx , and where DX = (dij )nx ×nx is a diagonal matrix with djj = h(xj ), j = 1, . . . , nx . Similarly i h x iθLτ¯t = e(GX +ψL (θ)DX )t 1, (41) ΦL (θ) := E e τ¯t nx where [ΦL τ¯t (θ)]j := E [exp(iθLτ¯t )|X0 = xj ], j = 1, . . . , nx .

Proof: The proof directly follows from combining the representations in (39) and Proposition 7. There are vast applications of the time-changed Lévy processes to options pricing, including European options (Carr et al. (2003)), barrier options and Bermudan options (Zeng and Kwok (2014)), average options (Yamazaki (2014)), discretely monitored path-dependent options (Umezawa and Yamazaki (2015)), discretely monitored arithmetic Asian options (Zeng and Kwok (2016)), and variance swaps Lorig et al. (2016). In this paper, we examine the case of European options as well as discretely sampled or realized variance swaps. Given the nature of the approximation framework, we also note the potential for future applications to timer options Bernard and Cui (2011); Cui et al. (2017a) as well as occupation time derivatives Cui et al. (2018a); Kirkby (2017a), which depend crucially on additive functionals of the underlying process (or the latent volatility process). Remark 15 We can combine the results in Proposition 14 with, for example, the PROJ method for European options Kirkby (2015, 2017b), and price different types of exotic options under the timechanged Lévy process with general stochastic time changes. The only required input is the characteristic function of the log return process eiθµt ·ΦL τ¯t (θ). This approach is illustrated in the numerical experiments section. 4.1.1

Realized Variance Swaps

Consider the asset price St = S0 eµt exp(Lτt ) on a finite trading horizon [0, T ]. The discrete realized variance across M monitoring dates 0 = t0 < t1 < . . . < tM = T on [0, T ] is defined by VM

M 1 X Stm 2 := log . T Stm−1

(42)

m=1

A discrete variance swap is a forward contract that exchanges the discrete realized variance with a fixed strike. The fair strike KV ar of the variance swap is given by KV ar = E[VM ], which ensures a price of zero at initialization. We first note that Stm log = (r − q)∆ + Lτm − Lτm−1 := Ztm . Stm−1 Conditioned on the value of Xtm−1 , we have Lτm − Lτm−1 ∼ Lτ∆ , from which Ztm |(Xtm−1 = x) ∼ Z∆ |(X0 = x) = µ∆ + Lτ∆ |(X0 = x). To illustrate the methodology, consider the continuous functional approximation τ¯t := and let Z¯t denote the approximation of Zt corresponding to Lτ¯tm , and similarly VM =

M M 1 X 1 X ¯ 2 (Ztm )2 ≈ V¯M := Ztm . T T m=1

m=1


Rt 0

h(Xsnx )ds,

(43)

The CTMC structure yields # " M nx M X h i X X 2 nx ¯tm )2 |X nx = xj , X nx = xj ¯ pX (t ) · E ( Z E Ztm X0 = xj0 = m−1 j0 ,j 0 tm−1 0 m=1 j=1

m=1

=

nx M X X

[PXm−1 (∆)]j0 ,j · E (Z¯∆ )2 |X0nx = xj

m=1 j=1

=

M X

m−1 e> (∆)E = e> j0 PX j0

m=1

M X

! PXm−1 (∆) E,

m=1

where E = [E1 , . . . , Enx ]> with Ej := E (Z¯∆ )2 |X0nx = xj . We can then calculate Ej from the closed-form characteristic function ΦL τ¯t (θ) to obtain h i ¯∆ nx iθZ (GX +ψL (θ)DX )∆ ϕj (θ) := E e 1. (44) X0 = xj = eiθµ∆ · e> j e Finally, we use the relationship of the second moment with ϕj (θ) to estimate 2 ϕj (ε) − 2 + ϕj (−ε) ∂ Ej = − ϕj (θ) ≈− 2 ∂θ ε2 θ=0 for a small ε > 0,15 where we note that ϕj (0) = 1. This approach is highly accurate, as illustrated in Section 5.2.

4.2

Time-changed Diffusion Processes

We next discuss an important special case of the time-changed Markov process: the time-changed (subordinated) diffusion process. A detailed summary of related literature is given in Li et al. (2015). Consider a time-homogeneous diffusion {Yt }t≥0 , which represents the underlying stochastic process: dYt = µ(Yt )dt + σ(Yt )dWt ,

(45)

where {Wt }t≥0 is a standard Brownian motion.16 We can construct the time-changed diffusion process by subordinating this diffusion by an independent stochastic time change, τt , and the resulting subordinated process is denoted by Yet := Yτt . If µ(·) = 0 and σ(·) = 1, then Yτt becomes a timechanged Brownian motion, and (45) encompasses many well-known special cases (see e.g. Carr et al. (2003)). For example, a Normal Inverse Gaussian process (see Barndorff-Nielsen (1997)) with parameters (α, β, δ) can be represented as Yet = βδ 2 τt + δWτt , p where τt is an Inverse Gaussian process, IG(a, b), with parameters a = 1 and b = δ α2 − β 2 . Similarly, a Variance Gamma process (see Madan et al. (1998)) can be expressed in terms of a Gamma time-change. In these cases, the model for the asset price process is generally taken to be St = S0 e(r−q)t exp(Yet )/ω(t), where ω(t) = E[exp(Yet )] to ensure that the discounted process is a martingale More recently, Li and Mendoza-Arriaga (2013) proposed a subordinated Ornstein-Uhlenbeck (OU) model, where the underlying background process is the mean-reverting OU process: dYt = κ(θ − Yt )dt + σdWt , 15 16

A value of ε = 5e−3 works well in practice. We assume that µ(y) and σ(y) are continuous, and σ(y) > 0.


with a Lévy subordinator τt ≥ 0. Given their ability to model mean reversion, subordinated OU processes enjoy wide applicability in the context of stochastic modeling in commodity markets (see Li and Linetsky (2014)) as well as electricity markets. In general, we can also consider other underlying background processes, and construct a time-changed CIR model or a time-changed geometric Brownian motion, for example. The general class of time-changed processes Yet := Yτt resulting from (45) can be approximated in terms of time-changed CTMCs. We will first approximate the underlying background diffusion in (45) using a CTMC, while preserving the same stochastic clock, to arrive at the final approximation. ny The idea is to construct a rate matrix Λ = (λij )i,j=1,...,ny over a set of states SY := {yi }i=1 such ny that the resulting CTMC, Yt , is a locally consistent approximation to Yt (see Kushner (1990)). In Piccioni (1987) (and later extended in Chourdakis (2002)), a Markov chain approximation for Yt over ny a uniform grid {yi }i=1 is proposed. A more accurate approximation may be found on a non-uniformly spaced grid; refer to Kirkby et al. (2017) for implementation details and nonuniform grid selection17 . Given a set of (non-uniform) grid spacings ki := yi − yi−1 , Lo and Skindilias (2014) propose the locally consistent rate matrix  − µ (yi ) σ 2 (yi ) − (ki−1 µ− (yi ) + ki µ+ (yi ))   + , j = i − 1,    ki−1 (ki−1 + ki )  k+i−1 2 − + µ (yi ) σ (yi ) − (ki−1 µ (yi ) + ki µ (yi )) (46) λij = + , j = i + 1,   k k (k + k ) i i i−1 i     −λ − λ , j = i, i,i−1 i,i+1 n −1

y where λj,k = 0 for |j − k| > 1. Note that if k = {ki }i=1 is chosen such that

0
Λ2 > . . . > Λny . n Hence, the representation in (2) of Lemma 1 is valid for Yt y . Proof: The proof is contained in the Appendix C.2. 17

In particular, the moments of Yet can be used to determine the state-space grid width. For example, Var(Yτt ) = σ E[τt ] + θ2 Var(τt ). In the case where the Laplace transform of τt is known, it can be used to obtain E[τt ] and Var(τt ) using numerical derivatives. 18 In all experiments we use standard packages to compute the eigen-decomposition. However, as shown in Li and Zhang (2016), matrix structure can be used to reduce the complexity to O(n2y ) using the MR3 algorithm, after applying a transformation to Λ. 2


In addition to the previous result, which ensures that the approximation scheme produces a diagonalizable transition density, the CTMC will also converge weakly to the underlying process, as summarized next. In particular, all three approximation schemes are weakly convergent. Corollary 17 Suppose that Yt evolves as in (45), and let Yet := Yτt for an independent time change n τt . Let Yt y be defined on SY by its generator in (46). n 1. Then Yet y =⇒ Yet as ny → ∞.

If we assume further that Xtnx is constructed as in (46), and that Yt has continuous paths: Rt 2. It holds that Yet =⇒ Yt as nx → ∞ when Yt is subordinated by τ¯t = Ant x := 0 h(Xunx )du or n P τ¯n = Bnnx := h(Xtnix ). i=0

n ,n n ,n n 3. Finally we have Ybt y x =⇒ Yet as ny → ∞ and nx → ∞, where Ybt y x := Yτ¯t y is the double-layer CTMC approximation. n

Proof: Weak convergence19 of Yt y to Yt for a fixed t can be proved as in Cui et al. (2018a), and similarly for Xtnx . The three statements then follow respectively from Proposition 3, Proposition 8, and Proposition 10. By the weak convergence established in Proposition 17 for a fixed t > 0, it then follows that n E[H(Yet y )] → E[H(Yet )] for any bounded payoff H(·) which is continuous on C ⊂ [0, ∞) such that Q[Yet ∈ C] = 1 (and similarly for the other approximation schemes). Assuming that E[H(Yet )] < ∞, for any > 0 we can choose θ > 0 such that E[H(Yet ) − H(Yet )1{Yet ≤θ} ] < , and h i h i n E H(Yet y )1{Ye ny ≤θ} → E H(Yet )1{Yet ≤θ} . t

Hence, the Markov chain approximation will converge for any L1 payoffs, i.e. finitely valued European options. In addition, by Proposition 17 and the continuous mapping theorem, we have that n g(Yet y ) =⇒ g(Yt ), for any continuous function g. Theorem 9 of Song et al. (2013) implies that the value of certain path-dependent payoffs, (including discretely monitored barrier/Bermudan/Asian opn tion payoffs) written on g(Yet y ) will converge to those written on g(Yt ). The interested reader is invited to refer to Section 1.2 of Song et al. (2013) for detailed discussions. The connection between Markov chain approximation and PDEs is discussed in Section B, notably the work of Li and Zhang (2016), Li and Zhang (2018a,b) and Mijatović and Pistorius (2013). In particular, the quadratic rate of convergence of the Markov chain approximation is proved, assuming a sufficiently smooth payoff function. In Table 3 and Table 5 of Section 5, we provide numerical examples which verify this result. As expected, quadratic convergence is generally observed.

5

Numerical examples

This section illustrates the theoretical and computational framework with a series of numerical experiments. Several contract specifications are considered, including European, Bermudan, and Variance Swaps, along with a number of alternative models. Each of the three approximations frameworks is illustrated: time-changed Lévy (TC-L), time-changed diffusion (TC-D), and the time-changed doublelayer (TC-DL) approach. For performance comparisons, Cpu times in seconds (s) are reported, and all experiments are conducted in Matlab 8.5 on a personal computer with Intel(R) Core(TM) i7-6700 CPU @3.40GHz. 19

We assume that the grid for Y ny is increased with ny to progressively cover the domain of Yt .


5.1

European Options

We first consider the pricing of European options using the time-change diffusion (TC-D) approach. The time change τt is taken as a Lévy subordinator (section 3.1.2), namely the inverse Gaussian: r m2 v φ(λ) = 1+2 λ−1 , (48) v m where m = E[τ1 ] and v = Var[τ1 ], and the underlying follows the diffusion in (45). We then consider the following model specifications: 1. NIG model: St = S0 exp(˜ µt + Yet ), where µ ˜ := r − q + φ(−θ − σ 2 /2), which includes a martingale correction. The drift in diffusion components in (45) are respectively: µ(y) = θ, σ(y) = σ, with parameters θ = 0.1, σ = 0.3. 2. Subordinated CIR (SubCIR): St = S0 Yet , a commodity model on the level of the underlying (see √ Li et al. (2015) for details). In this case µ(y) = κ(θ − y) and σ(y) = σ y, with parameters κ = 0.3, θ = 0.8, σ = 0.3. 3. Subordinated OU (SubOU): St = S0 exp(Yet ), where µ(y) = κ(θ − y) and σ(y) = σ, with parameters κ = 0.5, θ = 1.0, σ = 0.3. This model is considered in for commodity prices (see Li and Linetsky (2014)). In each case we set m = 1, v = 1, r = 0.05, q = 0 in (48). Table 3 provides a comparison with the method of Li and Zhang (2016). Prices, errors, and convergence order for our method are reported in the columns under “TC-D”, along with their reported prices and errors in the columns under “LZ”, all using the benchmark prices provided in their paper. In the case of European options, their method coincides with our TC-D approach,20 with the exception of the Markov chain construction. Moreover, our use of a non-uniform grid (see also Li and Zhang (2018b)), results in more accurate approximations, as is evident from Table 3. We also provide an estimate of the convergence order, defined as − log2 (eny /eny /2 ), where eny denotes the pricing error with ny states. As expected from the theoretical analysis of Li and Zhang (2016, 2018b), quadratic convergence is generally observed, although smooth convergence is not guaranteed. This is seen in Figure 1, which compares convergence across strikes for the SubOU model. While the accuracy for TC-D is generally good, we find that the time-changed Lévy (TC-L) method is more accurate, but at a higher cpu cost as illustrated below.

5.1.1

Unifying Example: Heston Model

A prominent example that can be approximated in several ways is the Heston model, for which we know the exact analytic characteristic function. First, it can be viewed as a time-changed Lévy-process: 1 Yτt = − τt + Wτt , 2

St = S0 eµt exp(Yτt ),

Rt as discussed in Section 4.1. In particular, Yt is subordinated by τt := 0 Xs ds, where p (2) dXt = κ(θ − Xt )dt + σv Xt dWt , X0 = x0 , 20

(49)

(50)

For this reason, we have not included a cpu cost comparison, as the primary difference between the two approaches will be the choice of algorithm used to form the matrix exponential. We also note that Li and Zhang (2016) provide a specialized algorithm to reduce this cost from O(n3y ) to O(n2y ), which applies as well to our formulation. We expect that by adopting this approach, further computational gains can be made.


SubCIR

NIG

TC-D ny

Price

Error

128

11.08441

-2.68e-03

256

11.08641

-6.83e-04

512

11.08691

-1.83e-04

LZ Order

TC-D

Price

Error

Price

Error

11.05339

-3.37e-02

9.56108

-1.53e-03

1.97

11.07879

-8.29e-03

9.56231

-3.03e-04

1.90

11.08502

-2.06e-03

9.56257

-4.36e-05

LZ Order

Price

Error

9.52894

-3.37e-02

2.34

9.55409

-8.54e-03

2.80

9.56049

-2.15e-03

Table 3: European put options pricing with time-change diffusion (TC-D) method. Comparison with method of Li and Zhang (2016), denoted LZ. Parameters: K = S0 = 100, T = 1, r = 0.05, q = 0. SubCIR ref: 11.0871. NIG ref: 9.5626.

K = 95 K = 100 K = 105

-2.5

log10 (|err|)

-3

-3.5

-4

-4.5 50

100

150

200

250

300

ny

Figure 1: SubOU Model: European put convergence for TC-D across strikes; r = 0.05, T = 2, S0 = 100. (2)

and we assume that E[dWt dWt ] = 0. In this case, the Lévy exponent is given by ψY (ξ) = − 21 iξ − 12 ξ 2 , and we can obtain a closed-form approximation of the characteristic function in terms of the time change driver Xtnx . Equivalently, it can be viewed as a time-changed diffusion as in Section 4.2, where dYt = θdt+σdWt . From (49), θ = −0.5 and σ = 1, and the integrated CIR time change process τt which has a closed-form Laplace transform given in (7). In this case, the underlying process is approximated by Y ny , while τt is left in its original form. Finally, Heston’s model can be approximated with a double-layer CTMC, by combing the diffusion equations in (49) and (50). Heston’s model highlights the flexibility of the proposed framework, as a single model can be understood and formulated in various ways, the Markov chain approach allows us to target different components of the stochastic process for approximation, depending on what is known. It also suggests that in the case of several potential representations, one approach may be particularly well-suited for the model dynamics, as we illustrate next. 5.1.2

Method Comparison

The next set of experiments compare the three approximation approaches for European options. Figure 2 illustrates the option price convergence (relative pricing error) as a function of the number of states in the CTMC approximation. For a fair comparison, we set nx = ny , where nx and ny are n the number of states for the two CTMC approximations Yt y and Xtnx , respectively. The double-layer approximation approximates the process in both dimensions, and is thus less accurate than the other 21 Electronic copy available at: https://ssrn.com/abstract=3235766

case

η

σv

θ

v0

ρ

1

3.0

0.10

0.04

0.04

0

2

1.5

0.35

0.04

0.04

0

3

1.0

0.30

0.02

0.04

0

4

3.0

0.50

0.05

0.05

0

Table 4: Example test cases for the Heston model.

TC-Levy TC-Diffusion TC-DoubleLayer

TC-Levy TC-Diffusion TC-DoubleLayer

-2.5

-3

log10 (|rel.err|)

log10 (|rel.err|)

-2.5

-3.5

-3

-3.5

-4

-4

-4.5

-4.5 40

60

80

100

120

140

40

60

nx = ny

80

100

120

140

nx = ny

Figure 2: Heston Model: European call convergence (relative error): r = 0.05, T = 0.5, S0 = K = 100. Left: Case 3 from Table 4. Right: Case 4. two approaches. However, from Table 5, the quadratic rate of convergence is apparent for all three methods. Moreover, while the TC-L method is more accurate at each level of approximation, it is also the more expensive of the three approaches. Taking this into account, the TC-D method provides the more favorable tradeoff between accuracy and cpu cost, requiring just 5 milliseconds to compute the price to three correct decimals in this example. Despite the two layers of approximation, the TC-DL approach is also quite efficient, and extends readily to discretely monitored barrier and Bermudan options.

TC-L nx = ny

Price

Error

8

5.9357

4.68e-01

16

6.3910

1.29e-02

32

6.4005

64 128

TC-D Order

Cpu

Price

Error

0.054

6.1897

2.14e-01

5.18

0.105

6.3452

5.87e-02

3.38e-03

1.93

0.181

6.3927

6.4030

8.88e-04

1.93

0.451

6.4037

2.66e-04

1.74

2.035

TC-DL Order

Cpu

Price

Error

Order

Cpu

0.001

5.7603

6.44e-01

1.87

0.001

6.3327

7.12e-02

3.18

0.003

1.13e-02

2.38

0.001

6.3893

1.46e-02

2.28

0.011

6.4007

3.24e-03

1.80

0.002

6.3998

4.13e-03

1.83

0.045

6.4033

6.69e-04

2.28

0.005

6.4030

9.34e-04

2.14

0.311

0.003

Table 5: European call options under Heston model. Comparison of time-changed Lévy (TC-L), diffusion (TC-D), and double-layer (TC-DL) approaches, for Case 3 from Table 4. Ref price: 6.40393.

While the cost varies between the three approximation strategies, each of the approaches achieves practical accuracy at a modest Cpu cost. As previously mentioned, a more targeted implementation of 22 Electronic copy available at: https://ssrn.com/abstract=3235766

the eigen-decomposition and matrix exponential offer potential gains in computational performance. Table 6 provides a price comparison for the three methods across a range of S0 , with reference prices confirmed by the PROJ Fourier method Kirkby (2015). We note that further improvement can be made using the Richardson Extrapolation combined with payoff smoothing, as illustrated in Li and Zhang (2016). v0 = 0.04

v0 = 0.09

S0

PROJ

TC-L

TC-D

TC-DL

PROJ

TC-L

TC-D

TC-DL

90

0.8970

0.8970

0.8971

0.8971

1.9140

1.9140

1.9138

1.9138

95

2.2665

2.2665

2.2662

2.2662

3.6215

3.6215

3.6210

3.6210

100

4.6090

4.6090

4.6085

4.6085

6.0698

6.0698

6.0691

6.0691

105

7.9197

7.9197

7.9195

7.9195

9.2313

9.2313

9.2306

9.2306

110

11.9889

11.9889

11.9890

11.9890

12.9975

12.9975

12.9967

12.9967

Table 6: European call options under Heston model. Comparison of time-changed Lévy (TC-L), diffusion (TC-D), and double-layer (TC-DL) approaches, with nx = ny = 150. Parameters: K = 100, r = 0.05, η = 3, θ = 0.04, ρ = 0, σv = 0.1, T = 0.25.

5.2

Variance Swaps

We next illustrate the variance swap pricing methodology introduced in Section 4.1.1. We continue with the Heston example, which allows for a closed-form reference benchmark using the method of Bernard and Cui (2014). As demonstrated in Table 7 for three reference cases, exceptionally high accuracy is achieved by the method, which utilizes the time-changed Lévy (TC-L) framework. Moreover, for each level of monitoring dates M = {12, 52, 252}, prices are calculated in under 10 milliseconds, as reported in the column “Cpu”, which averages cpu times over the three cases.

Case 1

Case 2

Case 3

M

Cpu

Ref

Price

Error

Ref

Price

Error

Ref

Price

Error

12

0.005

0.04007605

0.04007605

1.5e-10

0.04009689

0.04009696

6.8e-08

0.03275451

0.03275456

5.1e-08

52

0.006

0.04001757

0.04001757

2.6e-09

0.04002260

0.04002261

7.4e-09

0.03266843

0.03266844

1.3e-08

252

0.008

0.04000363

0.04000363

2.7e-09

0.04000468

0.04000468

2.2e-09

0.03264779

0.03264779

3.7e-09

Table 7: Variance swaps in Heston model using TC-L, for three cases from Table 4; r = 0.05, T = 1, nx = 60, ε = 5e-03. Cpu (s) averaged over cases, with ref. prices from Bernard and Cui (2014).

5.3

Bermudan Options

As a final experiment, we illustrate the pricing of Bermudan options under the Heston model using the double-layer approximation. For comparison, alternative prices are also computed using the regimeswitching PROJ (RSP) method of Kirkby et al. (2017) and the least squares Monte Carlo (LSM) method of Longstaff and Schwartz (2001). Table 8 summarizes the results for several values of the


underlying, and two levels of initial volatility, v0 . There is close agreement between the results from the TC-DL method and the RSP method, which is further confirmed by the LSM results. v0 = 0.04

v0 = 0.09

S0

TC-DL

RSP

LSM

TC-DL

RSP

LSM

90

10.1744

10.1748

10.1730

11.0281

11.0288

11.0303

95

6.2675

6.2685

6.2646

7.5853

7.5865

7.5894

100

3.4702

3.4712

3.4725

4.9405

4.9420

4.9454

105

1.7168

1.7181

1.7148

3.0481

3.0494

3.0440

110

0.7610

0.7611

0.7617

1.7842

1.7857

1.7838

9.54

12.53

64.77

9.75

12.56

65.44

Cpu(s)

Table 8:

Bermudan put options under Heston model. Parameters: M = 50, K = 100, r = 0.05, η = 3, θ = 0.04, ρ = 0, σv = 0.1, T = 0.25, nx = ny = 90. RSP parameters: N = 210 , m0 = 90. LSM paths: 5,000,000.

6

Conclusion

We have developed a general framework for the valuation of options under time-changed Markov processes. The model proposed in this paper is a general one-dimensional Markov process subordinated by additive functionals of another independent Markov process. In particular, we have established an approximate closed-form density formula for a time-changed Markov process Yet := Yτt where τt is an additive functional of another independent Markov process Xt , t ≥ 0. Interesting special cases include time-changed Lévy processes as well as time-changed diffusions, such as the subordinated OU process (Li and Mendoza-Arriaga (2013)) and the subordinated CIR process (Mendoza-Arriaga et al. (2014)). Other prominent examples include the Normal Inverse Gaussian model and Heston’s model. More importantly, the modeling framework is applicable to background processes and time changes based on general additive functionals, which holds promise for many additional models beyond these wellstudied cases. This treatment unifies the recent development of time-changed Markov processes, and provides several approximation approaches to simplify a model into one for which closed-form pricing approximations can be applied. The flexibility of this framework allows us to harness alternative representations of a stochastic process to choose the appropriate approximation strategy. The choice of approximation method depends on what is known about the process, and can also account for each method’s relative performance for the specified model, as well as the contract type in the case of option pricing. Numerical examples for European and Bermudan options as well as variance swaps are provided to illustrate each of the three approximation strategies that we develop, including single and double layer CTMC approximation schemes. In the future, it would be interesting to extend our results to incorporate a general correlation between two Brownian motions in stochastic volatility models, which we leave as a project for future research.

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A A.1

Closed-form examples of stochastic time changes Integral of 3/2 process

As another example of an integrated activity rate process, consider the 3/2 process 3/2

dXt = κXt (θ − Xt )dt + σv Xt

X0 = x0 . Rt From Carr and Sun (2007), the closed-form Laplace transform of τt := 0 Xu du satisfies Ex0 [e−ατt ] =

Γ(γ − β) Γ(γ)

2 σv2 y(x0 )

dWt ,

β

M

β, γ,

−2 σv2 y(x0 )

,

with x0 κθt e −1 , κθ s 1 κ 1 κ 2 2α β=− + + + + 2, 2 σv2 2 σv2 σv κ γ =2 β+1+ 2 , σv y(x0 ) =

and M (·, ·, ·) is the confluent hypergeometric function defined as M (β, γ, z) =

∞ X (β)n z n n=0

(γ)n n!

,

with

(51)

(x)n = x(x + 1)(x + 2) . . . (x + n − 1).


(52)

A.2

Multiple Layers of L´ evy Subordinators

In this section, motivated from the discussions in Mendoza-Arriaga et al. (2010), we consider an interesting construction of the stochastic time change through multiple (compound) layers of subordinations based on a sequence of Lévy subordinators. Consider a sequence of Lévy subordinators (1) (2) (3) (n) denoted by Tt , Tt , Tt , . . . , Tt , with Lévy exponents given by φ(1) (·), φ(2) (·), . . . , φ(n) (·). We con(1) struct the “n-layer” stochastic time change as T (2) , and we can characterize its Laplace transform T···

(n) Tt

in closed-form as: E[e−λτt ] = e−φ

(n) (φ(n−1) (···φ(1) (λ)))t

.

(53)

The above formula utilizes the “exponential affine” structure of the Laplace transform of the time change, and thus we can nest all of them into one single formula as in (53).

A.3

Integral of CIR process with Hawkes jumps

Extending further, we may combine a diffusion process (e.g. CIR process) and a pureR point process t (e.g. Markovian Hawkes process) to construct the activity rate process. Consider τt = 0 Xs ds, where Xs is a CIR process with Hawkes jumps, which follows the dynamics: p dXt = θ(µ − Xt )dt + σ Xt dWt + adNt . (54) Here Nt is a simple point process with intensity ν + Xt− . Then, C(t), D(t) are the solutions of the following ODE: (see e.g. Zhu (2014)) 1 C 0 (t) = −θC(t) + σ 2 C(t)2 + (eaC(t) − 1) − β, 2 0 D (t) = θµC(t) + ν(eaC(t) − 1).

(55) (56)

with the initial conditions C(0) = 0 and D(0) = 0. The above ODEs can be numerically solved.

A.4

Converting stochastic volatility models to subordinated diffusions

As a final example, we consider a general class of stochastic volatility models:   dSt = µSt dt + m(vt )St dW (1) , t  dv = µ(v )dt + σ(v )dW (2) , t

(1)

t

t

(57)

t

(2)

where E[dWt dWt ] = 0, i.e. the driving Brownian motions are independent. We note that the stochastic volatility model considered in (57) encompasses many stochastic volatility models with zero correlation in the literature, including classic models such as Scott’s model (Scott (1987)), the HullWhite model (Hull and White (1987)), and Heston’s model (Heston (1993)), as well as recent modles such as the alpha-hypergeometric model (Da Fonseca and Martini (2016)), the 4/2 model (Grasselli (2017)), and the Jacobi model (Ackerer et al. (2018)). From Itô’s lemma, it is easy to rewrite the above dynamics as Z Z t 1 t 2 (1) St = S0 exp µt − m (Vs )ds + m(Vs )dWs . (58) 2 0 0 Rt

m2 (Vs )ds, from the Dambis-Dubins-Schwartz theRt (1) orem (see Ch.V, Theorem 1.6 and Theorem 1.7 of Revuz and Yor (1999)) we have 0 m(Vs )dWs = Bτt a.s., where B is a standard Brownian motion (usually called the Dambis-Dubins-Schwartz Brownian motion), whose detailed construction can be found in the proof of Theorem 2.1 in Cui and Ma (2016), With the stochastic time change denoted by τt :=

0


see also related applications in Bernard et al. (2017); Badescu et al. (2017). Thus we can rewrite the stock price as τt St = S0 exp µt − + Bτt , t ≥ 0, (59) 2 and we can treat it as a time-changed geometric Brownian motion. This is a special case of our general framework, i.e. St = S0 eµt eYτt , where Yt := Bt − t/2 is a drifted Brownian motion. Note that here S0 eµt is a constant for any fixed t > 0. In the numerical section, we illustrate using the well-known Heston model with zero correlation. Rt

2

Remark 1 When the Laplace transform E[e−λτt ] = E[e−λ 0 m (Vs )ds ] is known in closed-form, as is the case for Heston (recall section 3.1.1), the method developed in section 4.2 can be applied directly to Yt := Bt − t/2, where we note that E[exp(Yt )] = 1 for all t ≥ 0. When E[e−λτt ] is unknown, R t it can be estimated using Markov chain approximation of {Vs }, as discussed in section 4.2, where 0 m2 (Vs )ds is an additive functional of Vs and can be approximated by (27).

B

Connection with PDEs

In Li and Zhang (2016, 2018b), the connection is established between Markov chain approximation techniques and the numerical solution to classic PDEs (see also Mijatović and Pistorius (2013)). Consider the time-homogeneous diffusion dYt = µ(Yt )dt + σ(Yt )dWt , with natural boundaries21 −∞ ≤ l < r ≤ ∞. Given a well-behaved payoff H(Yt ) (for example, continuous on (l, r)), the expected value u(t, y) = Ey [H(Yt )] satisfies the PDE 1 ∂t u(t, y) = µ(y)∂y u(t, y) + σ 2 (y)∂yy u(t, y), 2 u(0, y) = H(y), y ∈ (l, r).

t > 0,

y ∈ (l, r),

(60)

n

Recall that the continuous process Yt is approximated by the CTMC Yt y with state space SY := ny {yi }i=1 . For ease of exposition, we assume (without loss of generality) a uniform step size k ≡ ki = yi − yi−1 . A semi-discrete approximation is made using a central difference in the space of y: 1 µ(y)∂y u(t, y) + σ 2 (y)∂yy u(t, y) 2 u(t, yi+1 ) − u(t, yi−1 ) 1 2 u(t, yi+1 ) − 2u(t, yi ) + u(t, yi−1 ) ≈ µ(yi ) + σ (yi ) , (61) 2k 2 k2 with appropriate boundary conditions (e.g. reflecting, killing, or absorbing). Let uk (t) be the approximate option price based on the discretization in (61). Then from the work of Mijatović and Pistorius (2013); Li and Zhang (2018b) we have that uk (t) satisfies the following (matrix-valued) ODE d uk (t) = Λk uk (t), dt

uk (0) = Hk ,

(62)

where uk and Hk = [H(y1 ), . . . , H(yny )]> are Rn column vectors, and Λk is the ny × ny tridiagonal generator matrix given (46) with constant step k. We can solve the ODE through matrix exponentials: uk (t) = eΛk t Hk .

(63)

Define πk g = (g(y1 ), g(y2 ), . . . , g(yny ))T , and kAk∞ = maxi,j |Ai,j |. Now consider the option written on the subordinated underlying process Yt , i.e. option written under Yτt . Let uτ (·), uτk (·) denote the n true value and the approximated option value written on Yτt and Yet y , respectively. Let qt (·) be the probability density function of τt . The following relation holds Z uτk (t) = uk (s)qt (ds). (64) [0,∞)

With this, we have the following result: 21 Note that we make the same assumptions (e.g. Assumption 2.1 to 2.3) as in Li and Zhang (2018b), to which we refer the reader for more details.


Proposition 18 (Li and Zhang (2018b)) Suppose that H is piecewise twice continuously differentiable (i.e., there are only a finite number of points in (l, r) where this is not true) and that at any nondifferentiable point y, there exists some δy > 0 such that H is Lipschitz continuous in (y − δy , y + δy ). Consider k ∈ (0, ε), where ε is sufficiently small such that ε ≤ δy for all non-differentiable points y. For any t > 0, there is some constant Ct > 0 independent of k such that || uτk (t) − πk uτ (t, ·) ||∞ ≤ Ct k 2 .

(65)

In Table 3 of Section 5, we provide numerical examples to demonstrate this result. As expected, quadratic convergence is generally observed.

C C.1

Proofs Proof of Proposition 7

(i) In the discrete case, the functional equation (18) has the following form eθDX − zPX (δ) f = 1.

(66)

It is easy to show that eθDX − zPX (δ) is strictly diagonally dominant, and by the Lévy-Desplanques theorem (Corollary 5.6.17 of Horn and Johnson (1985)), we have that eθDX − zPX (δ) is invertible, and we have −1 f = eθDX − zPX (δ) 1. (67) From equation (16), we observe that gd (n; x) can be treated as the coefficient of z n in the power series expansion of f = l(z; x) with respect to the transform variable z. This motivates us to expand the right hand side of (67) into an analytic power series. From Corollary 5.6.16 of Horn and Johnson (1985), if there is a matrix norm || · || (without loss of generality, we can norm, P take the maximum k . Thus we can (I − A) i.e. ||A|| = max{|aij |}) such that ||I − A|| < 1, then we have A−1 = ∞ k=0 1 specify our transform variable to satisfy |z| < min{1, e−rδ , ||(eθDX )−1 } to make sure that all the PX (δ)|| following series expansions with respect to z are well-defined. We have (eθDX − zPX (δ))−1 1 = (eθDX (I − z(eθDX )−1 PX (δ)))−1 1 = (I − z(eθDX )−1 PX (δ))−1 (eθDX )−1 1 = I + z(eθDX )−1 PX (δ) + z 2 ((eθDX )−1 PX (δ))2 + ... + z n ((eθDX )−1 PX (δ))n + ... (eθDX )−1 1 = (eθDX )−1 1 + z(eθDX )−1 PX (δ)(eθDX )−1 1 + ... + z n ((eθDX )−1 PX (δ))n (eθDX )−1 1 + ...

(68)

The coefficient of z n in (68) is ((eθDX )−1 PX (δ))n (eθDX )−1 1, and this is equal to the value of gd (n; x) defined in (28). This completes the proof of part (i). (ii) In the continuous case, the functional equation (19) has the following form (θDX + µI − GX )g = 1.

(69)

Similarly we can show that θDX + µI − GX is strictly diagonally dominant, and by the LévyDesplanques theorem (Corollary 5.6.17 of Horn and Johnson (1985)), we have that θDX + µI − GX is invertible, and we have g = (θDX + µI − GX )−1 1.


(70)

We assume |µ| > ||GX − θDX ||, so that the following series expansions with respect to µ are well-defined. Then we have −1 1 1 −1 (θDX + µI − GX ) 1 = µ I − − θDX + GX 1 µ µ 1 GX − θDX −1 = I− 1 µ µ ! GX − θDX GX − θDX 2 GX − θDX n 1 I+ + + ... + + ... 1 = µ µ µ µ =

1 (GX − θDX )1 (GX − θDX )2 1 (GX − θDX )n 1 + + + ... + + ... µ µ2 µ3 µn+1

(71)

By applying an inverse Laplace transform w.r.t. µ, we get n o −1 −1 (θD + µI − G ) 1 gc (t, x) = L−1 {m(µ, θ; x)} = L X X µ µ =

∞ X (GX − θDX )i i=0

i!

ti = e(GX −θDX )t 1.

(72)

This completes the proof of part (ii).

C.2

Proof of Proposition 16

The proof is scattered in Gantmakher and Krein (2002). We collect and adapt it into our setting here for the reader’s convenience. Define D0 (x) ≡ 1 and let Dk (x) be the characteristic polynomial of Λ when ny = k. By expanding along the last row, we have the following recurrence formula for Dk (x): Dk (x) = (λk,k − x)Dk−1 (x) − λk−1,k λk,k−1 Dk−2 (x),

k = 2, 3, . . .

(73)

Note that we have D1 (x) = λ1,1 − x. As a result of this, we can use (73) to calculate all Dk (x), k = 2, 3, . . . In the expression of Dk (x), the numbers λk−1,k and λk,k−1 enter only in the form of λk−1,k λk,k−1 which is positive since λk−1,k and λk,k−1 are positive for k = 2, 3, . . .. Let   p λ1,2 λ2,1  λ1,1  p  p  λ1,2 λ2,1  λ2,2 λ2,3 λ3,2     p .. ..  , Λ1 =  . . λ2,3 λ3,2    p .. ..   . . λny −1,ny λny ,ny −1     p λny −1,ny λny ,ny −1 λny ,ny then Λ1 is symmetric and has the same characteristic polynomial as that of Λ. Since all eigenvalue of Λ1 are real, so are those of Λ. Next, let u = (u1 , u2 , . . . , uny ) be an eigenvector corresponding to the eigenvalue λ of Dny (x) = 0. We have   (λ1,1 − λ)u1 − λ1,2 u2 = 0,       λ2,1 u1 + (λ2,2 − λ)u2 + λ2,3 u3 = 0, (74)   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .,      λ ny −1,ny uny −1 + (λny ,ny − λ)uny = 0. −1 −1 From (73), Dk−1 (λ) 6= 0 if Dk (λ) = 0. In addition, it is easy to see that uk = Cλ−1 1,2 λ2,3 . . . λk−1,k Dk−1 (λ) for k = 1, 2, . . . , ny and for some C 6= 0. Hence each eigenvalue λ corresponds to exactly one eigenvector (up to a scalar multiple C). It is known that, for real and symmetric matrices the multiplicity


of the eigenvalue is equal to that of the corresponding eigenvector (see for example Proposition 3, page 21 of Gantmakher and Krein (2002)); hence λ is a simple root of Dny (x). This implies that all eigenvalues of Dny (x) are simple. Finally, by choosing i such that |ui | = max{|uj | : j = 1, . . . , ny }, we have ny X

λij uj = λui − λij ui .

j6=i

This implies that |λ − λij | ≤

ny X j6=i

ny

X uj λij | | ≤ λij . ui j6=i

Hence λ is in the circle centered at λij with radius λ ≤ 0. This completes the proof.

Pny

j6=i λij .

Since λii ≤ 0 and

Pny

Algorithm 1 Recursive Algorithm: European, American, and Barrier Initialize state vectors y = [y1 , . . . , yny ]> , x = [x1 , . . . , xnx ]> n Approximate driving process Yt by Yt y , with rate matrix Λ = QDQ−1 Approximate time change driver Xt by Xtnx , with rate matrix GX Initialize diagonal matrix DX = (dij )nx ×nx with djj = h(xj ) Initialize Γ(k) in (32), k = 1, . . . , ny Initialize M (i,j) in (36), i, j = 1, . . . , ny Initialize Terminal Value, l = 1, . . . , nx : ( H(sM ) ◦ 1B (sM ), Barrier option (l) VM = H(sM ), Else for m = M − 1, . . . , 0 do for l = 1, . . . , nx do Pnx −r∆ (l,p) (p) V (l) V m+1 m =e p=1 M (r−q)m∆ sm = S0 e exp(y)  (l)  Barrier option V m ◦ 1B (sm ), (l) V m = max{V (l) , H(s )}, Bermudan option m m   (l) max{V m , H(sm )} ◦ 1B (sm ), Bermudan barrier option end for end for (l ) C = V 0 0 (yj0 )


j

λij = 0, we have