MULTI-FACTOR LEVY MODELS FOR PRICING FINANCIAL AND … · MULTI-FACTOR LEVY MODELS 781 Brockhaus...

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 17, Number 4, Winter 2009

MULTI-FACTOR LEVY MODELS FOR PRICING

FINANCIAL AND ENERGY DERIVATIVES

ANATOLIY SWISHCHUK

ABSTRACT. We introduce one-factor and multi-factor α-

stable Levy-based models to price financial and energy deriva-tives. These models include, in particular, as one-factor mod-els, the Levy-based geometric motion model, the Ornstein-Uhlenbeck [75], the Vasicek [106], the Cox-Ingersoll-Ross [26],the continuous-time GARCH, the Ho-Lee [53], the Hull-White[56] and the Heath-Jarrrow-Morton [49] models, and as multi-factor models, various combinations of the previous models.For example, we introduce new multi-factor models such as theLevy-based Heston model, Levy-based SABR/LIBOR marketmodels, and Levy-based Schwartz-Smith and Schwartz mod-els. Using change of time method for SDEs driven by α-stableLevy processes we present the solutions of these equations insimple and compact forms (Propositions 1 and 2). We applythis method to price many financial and energy derivatives suchas variance swaps, options, forward and futures contracts.

1 Introduction In this section, we give a literature review on thechange of time method (CTM), and an overview on multi-factor Gaus-sian, LIBOR market and SABR models, swaps and energy derivatives.

1.1 Change of time method (CTM): short literature historyTo the best of the author’s knowledge, Bochner [12] was the first tointroduce the notion of change of time (namely, time-changed Brownianmotion). Clark [24] introduced Bochner’s change of time into financialeconomics. Feller [35] introduced subordinated process X(T (t)) for aMarkov process X(t) with T (t) a process having independent increments.(T (t) was called randomized operational time). Johnson [63] introduceda time-changed stochastic volatility model (SVM) in continuous time.

This paper is dedicated to the 30th Anniversary of CAIMS and supported byNSERC.

Keywords: α-stable Levy processes, multi-factor Levy models, change of timemethod, variance swaps, SABR and LIBOR models, forward and futures contracts.

Copyright c©Applied Mathematics Institute, University of Alberta.

777

778 ANATOLIY SWISHCHUK

Johnson and Shanno [64] studied pricing of options using a time-changedstochastic volatility (SV) model. Ikeda and Watanabe [57] introducedand studied the change of time for the solution of stochastic differentialequations (SDEs). Barndorff-Nielsen et al. [6] studied the relationshipbetween subordination and SVM using a change of time, (the T (t)-chronometer). Carr et al. [17] used subordinated processes to constructSV for Levy processes, (T (t) being business time).

The change of time method is closely associated with the embed-ding problem: to embed a process X(t) in Brownian motion is to finda Wiener process process W (t) and an increasing family of stoppingtimes T (t) such that W (T (t)) has the same joint distribution as X(t).Skorokhod [96] first treated the embedding problem, showing that thesum of any sequence of independent random variables with mean zeroand finite variation could be embedded in Brownian motion using stop-ping times. Dambis [27], and Dubins and Schwartz [29] independentlyshowed that every continuous martingale could be embedded in Brow-nian motion. Knight [68] discovered a multivariate extension of theDambis [27], and Dubins and Schwartz [29] result. Huff [55] showedthat every process of pathwise bounded variation could be embedded inBrownian motion. Monroe [73] proved that every right-continuous mar-tingale could be embedded in a Brownian motion. Monroe [74] provedthat a process can be embedded in Brownian motion if and only if thisprocess is a local semimartingale. Meyer [71] and Papangelou [76] in-dependently discovered Knight’s result [68] for point processes.

Rosinski and Woyczynski [87] considered time changes for integralsover a stable Levy processes. Kallenberg [66] considered time changerepresentations for stable integrals.

Levy processes can also be used as a time change for other Levyprocesses (subordinators). Madan and Seneta [70] introduced VarianceGamma (VG) process (Brownian motion with drift time changed by agamma process). Geman et al. [39] considered time changes (businesstimes) for Levy processes. Carr et al. [17] used a change of time tointroduce stochastic volatility into a Levy model to achieve leverageeffect and a long-term skew. Kallsen and Shiryaev [67] showed that theRosinski-Woyczynski-Kallenberg result can not be extended to any otherLevy processes other than the symmetric α-stable processes. Swishchuk[99, 102] applied a change of time method for options and swaps pricingfor Gaussian models.

The book, “Change of Time and Change of Measure,” by Barndorff-Nielsen and Shiryaev [5] states the main ideas and results of the stochas-tic theory of change of time and change of measure.

MULTI-FACTOR LEVY MODELS 779

1.2 Stochastic differential equations (SDEs) driven by Levyprocesses Girsanov [45] used the change of time method to constructa weak solution to a specific SDE driven by Brownian motion.

The existence and uniqueness of solutions for SDEs driven by Levyprocesses have been studied in [4]. The existence and uniqueness ofsolutions for SDEs driven by general semimartingale with jumps havebeen studied in [58, 80].

Janicki et al. [61] proved that there exists a unique solution of theSDE for continuous drift b and diffusion coefficient σ and α-stable Levyprocess Sα((t − s)1/α, β, δ), β ∈ [−1, +1].

Zanzotto [109] had also considered solutions of one-dimensional SDEsdriven by stable Levy motion.

Cartea and Howison [21] considered option pricing with Levy-stableprocesses generated by Levy-stable integrated variance.

1.3 Multi-factor Gaussian models: literature review Eydelandand Geman [34] proposed extending the Heston stochastic volatilitymodel [51] to gas or electricity prices by introducing mean-reversion inthe spot price and leaving the CIR model for the variance, resulting ina two-state variable model for commodity prices.

Geman [41] proposed a three-state variable for oil prices by intro-ducing mean-reversion in the spot price, geometric Brownian motionin the equilibrium (or mean-reverting) price and the CIR model in thevariance.

Gibson and Schwartz [44] note that the convenience yield has beenshown to be a key factor driving the relationship between spot andfutures prices, and they proposed the two-state variable model for oil-contingent claim pricing.

Continuing to the more complex level of two factor models, keepingthe same mean-reverting level Lt, the SDEs for spot price St and Lt canbe generalized by either allowing the long run mean Lt or the volatility σto be governed by an SDE. This leads to two distinct two factor models,with different dynamics. The first model assumes a stochastic long runmean and was introduced by Pilipovic [77] The author describes a two-factor mean-reverting model where spot prices revert to a long termequilibrium level which is itself a random variable. Pilipovich derives aclosed-form solution for forward prices to her model when the spot andlong term prices are uncorrelated, but does not discuss option pricing inher two-factor model.

Gibson and Schwartz [44], Schwartz [92], and Hilliard and Reis [52]all analyze versions of the same two-factor model that allows for a


stochastic convenience yield and permits a high level of analytical tractabil-ity. The first factor is the spot price process which is assumed to followthe geometric Brownian motion (GBM) and the second factor is theinstantaneous convenience yield of the spot energy and is assumed tofollow the mean reverting process.

Schwartz [92] extends his two-factor model to include stochastic in-terest rates. In this three-factor model the short term rate is assumedto follow the Vasicek mean-reverting process [106].

Fouque et al. [36] considered multi-factor stochastic volatility modelthat is a function of two processes in the form of geometric Brownianmotions.

Fouque and Han [37] found that two-factor SV models provide abetter fit to the term structure of implied volatility than one factor SVmodels by capturing the behavior at short and long maturities.

Chernov et al. [22] used two-factor SV family models to obtain com-parable empirical goodness-of-fit.

Molina et al. [72] found a strong evidence of two-factor SV modelswith well-separated time scales in foreign exchange data.

Carmona and Ludkovski [16] reviewed the literature of spot conve-nience yield models, and analyzed in detail two new extensions. First,they discussed a variant of the Gibson-Schwartz model with time-depend-ent parameters and second, they described a new three-factor affinemodel with stochastic convenience yield and stochastic market price ofrisk.

1.4 Pricing swaps overview Demeterfi et al. [28] explained theproperties and the theory of both variance and volatility swaps. Theyderived an analytical formula for the theoretical fair value in the pres-ence of realistic volatility skews, and pointed out that volatility swapscan be replicated by dynamically trading the more straightforward vari-ance swap.

Javaheri et al. [62] discussed the valuation and hedging of aGARCH(1,1) stochastic volatility model. They used a general and flex-ible PDE approach to determine the first two moments of the realizedvariance in a continuous or discrete context. Then they approximate theexpected realized volatility via a convexity adjustment.

Working paper by Theoret et al. [105] presented an analytical so-lution for pricing of volatility swaps, proposed by Javaheri et al. [62].They priced volatility swaps within framework of GARCH(1,1) stochas-tic volatility model and applied the analytical solution to price a swapon volatility of the S&P60 Canada Index (5-year historical period 1997–2002).


Brockhaus and Long [15] provided an analytical approximation forthe valuation of volatility swaps and analyzed other options with volatil-ity exposure.

In [99] we found the values of variance and volatility swaps for finan-cial markets with underlying asset and variance that follow the Hestonmodel [51]. We also studied covariance and correlation swaps for finan-cial markets. As an application, we provided a numerical example usingS&P60 Canada Index to price swap on the volatility.

Variance swaps for financial markets with underlying asset and one-factor and multi-factor stochastic volatilities with delay are modelledand priced in [100] and [101], respectively.

In [33], we found the value of variance swap for financial markets withMarkov stochastic volatility. In Swishchuk (2010) we found the valuesof variance and volatility swaps for semi-Markov volatility.

Swishchuk [102] contains applications of the change of time methodto Gaussian financial models such as geometric Brownian motion, theHeston model and a mean-reverting model to price options and differentkinds of swaps.

1.5 Libor market and SABR models: short literature reviewThe basic log-normal Forward Libor (also known as Libor Market, orBGM) model has proved to be an essential tool for pricing and risk-managing interest rate derivatives. First introduced in [13, 60], forwardLibor models are in the mainstream of interest rate modeling. For gen-eral information regarding the model, textbooks such as [14, 83] providea good starting point. Various extensions of forward Libor models thatattempt to incorporate volatility smiles of interest rates have been pro-posed. Local volatility type extensions were pioneered in [1]. A stochas-tic volatility extension is proposed in Andersen and Brotherton-Ratcliffe[3] and further extended in [2]. A different approach to stochastic volatil-ity forward Libor models is described in [83]. Jump-diffusion forwardLibor models are treated in [46, 47]. In [94], a stochastic volatility/jumpdiffusion forward Libor model is advocated.

The SABR and the Libor market models (LMM) have become indus-try standards for pricing plain-vanilla and complex interest rate prod-ucts, respectively. For a description of the SABR model; see, for ex-ample, [48]. Several stochastic-volatility extensions of the LMM existthat do provide a consistent dynamic description of the evolution of theforward rates (see, for example, [1, 65, 84, 85]), but these extensionsare not equivalent to the SABR model. Piterbarg [78, 79] presents anapproach based on displaced diffusion that is similar in spirit to a dy-


namical extension of the SABR model. Henry-Labordere [50] obtainssome interesting exact results, but his attempt to unify the BGM andSABR model using an application of hyperbolic geometry is very com-plex and the computational issues are daunting. Rebonato [82] proposedan extension of the LMM that recovers the SABR caplet prices almostexactly for all strikes and maturities. Many smiles and skews are usuallymanaged by using local volatility models by Dupire [30].

1.6 Energy derivatives’ overview Black’s model [11] and Schwartz’smodel [92] have become a standard approach to the problem of pricingoptions on commodities. These models have the advantage of mathe-matical convenience, in that they give rise to closed-form solutions forsome types of options (see [108]).

A drawback of single-factor mean-reverting models lies in the case ofoptions pricing: the fact the long-term rate is fixed results in a model-implied volatility term structure that has the volatilities going to zeroas expiration time increases.

Using single-factor nonmean-reverting models has also a drawback: itwill impact valuation and hedging. The differences between the distri-butions are particularly obvious when pricing out-of -the-money options,where the tails of the distribution play a very important role. Thus, if alognormal model, for example, is used to price a far out-of-the-money op-tion, the price can be very different from a mean-reverting model’s price(see [40]). A popular model used for modeling energy and agriculturalcommodities and introduced by Schwartz [91] aims at resembling thegeometric Brownian motion while introducing mean-reversion to a long-term value in the drift term (see [92]). This mean-reverting model is aone-factor version of the two-factor model made popular in the contextof energy modelling by Pilipovic [77].

Villaplana [107] proposed the introduction of two sources of risk Xand Y representing, respectively, short-term and long-term shocks, anddescribes the spot price St. Geman and Roncoroni [40] introduced ajump-reversion model for electricity prices. The two-factor model foroil-contingent claim pricing was proposed by Gibson and Schwartz [44].Eydeland and Geman [34] proposed extending the Heston stochasticvolatility model [51] to gas or electricity prices by introducing mean-reversion in the spot price and proposing two-factor model. Geman [41]introduced three-factor model for commodity prices taking into accountstochastic equilibrium level and stochastic volatility. Bjork and Landen[10] investigated the term structure of forward and futures prices formodels where the price processes are allowed to be driven by a general


market point process as well as by a multidimensional Wiener process.Benth et al. [8] applied independent increments processes (see [69, 95])to model and price electricity, gas and temperature derivatives (for-wards, futures, swaps, options). Swishchuk [103] considers a risky assetin energy markets following mean-reverting stochastic process. An ex-plicit expression for a European option price based on this asset, using achange of time method, is derived. A numerical example for the AECONatural Gas Index (1 May 1998-30 April 1999) is presented.

1.7 Organization of the paper The paper organized as follows. Sec-tion 2 introduces one-factor and multi-factor Gaussian models. Thechange of time method for Gaussian models and solutions of the in-troduced Gaussian models are presented in Section 3. α-stable Levyprocesses and their properties are defined in Section 4. One-factor andmulti-factor Levy-based models are presented in Section 5. The changeof time method and solution of these equations using the change of timemethod are introduced in Section 6. Applications of Levy-based modelsin financial and energy markets are presented in Section 7. Conclusionsare stated in Section 8.

2 One-factor and multi-factor Gaussian models in financeIn this section, we introduce well-known one-factor and multi-factormodels (used in finance) described by SDEs driven by Brownian mo-tion (so-called Gaussian models).

2.1 One-factor Gaussian models For one-factor Gaussian modelswe define the following well-known processes.

1. The Geometric Brownian Motion:

dS(t) = µS(t) dt + σS(t) dW (t).

2. The Continuous-Time GARCH Process:

dS(t) = µ(b − S(t)) dt + σS(t) dW (t).

3. The Ornstein-Uhlenbeck Process [75]:

dS(t) = −µS(t) dt + σ dW (t).

4. The Vasicek Process [106]:

dS(t) = µ(b − S(t)) dt + σ dW (t).


5. The Cox-Ingersoll-Ross Process [26]:

dS(t) = k(θ − S(t)) dt + γ√

S(t) dW (t).

6. The Ho and Lee Process [53]:

dS(t) = θ(t) dt + σ dW (t).

7. The Hull and White Process [56]:

dS(t) = (a(t) − b(t)S(t)) dt + σ(t) dW (t).

8. The Heath, Jarrow and Morton Process [49]: Define the forwardinterest rate f(t, s) for t ≤ s, characterized by the following equalityP (t, u) = exp[−

∫ u

tf(t, s) ds] for any maturity u. f(t, s) represents

the instantaneous interest rate at time s as anticipated by the marketat time t. It is natural to set f(t, t) = r(t). The process f(t, u)0≤t≤u

satisfies an equation

f(t, u) = f(0, u) +

∫ t

0

a(v, u) dv +

∫ t

0

b(f(v, u)) dW (v),

where the processes a and b are continuous. We note that the lastSDE may be written in the following form:

df(t, u) = b(f(t, u))

( ∫ u

t

b(f(t, s))) ds + b(f(t, u)

)dW (t),

where

W (t) = W (t) −∫ t

0

q(s) ds and q(t) =

∫ u

t

b(f(t, s)) ds − a(t, u)

b(f(t, u)).

2.2 Multi-factor Gaussian models For the multi-factor Gaussianmodel, we give one example of a two-factor continuous-time GARCHmodel:

dS(t) = µ(b(t) − S(t)) dt + σS(t) dW 1(t),

db(t) = ξb(t) dt + ηb(t) dW 2(t),

where W 1 and W 2 may be correlated and µ, ξ ∈ R, σ, η > 0.Other multi-factor models driven by Brownian motions can be ob-

tained using various combinations of the above-mentioned processes; seeSection 2.1.


3 Change of time method (CTM) for SDEs driven by Brow-nian motion

Definition 1. A time change is a right-continuous increasing [0, +∞]-valued process (Tt)t∈R+

such that Tt is a stopping time for any t ∈ R+.

By Ft := FTt we define the time-changed filtration (Ft)t∈R+. The inverse

time change (Tt)t∈R+is defined as Tt := infs ∈ R+ : Ts > t. (See [57].)

We consider the following SDE driven by a Brownian motion:

dX(t) = a(t, X(t)) dW (t),

where W (t) is a Brownian motion and a(t, X) is a continuous and mea-surable by t and X function on [0, +∞) × R.

Theorem ([57, Chapter IV, Theorem 4.3]). Let W (t) be an one-

dimensional Ft-Wiener process with W (0) = 0, given on a probabilityspace (Ω,F , (Ft)t≥0, P ) and let X(0) be an F0-adapted random variable.

Define a continuous process V = V (t) by the equality

V (t) = X(0) + W (t).

Let Tt be the change of time process:

Tt =

∫ t

0

a−2(Ts, X(0) + W (s)) ds.

IfX(t) := V (Tt) = X(0) + W (Tt)

and Ft := FbTt, then there exists Ft-adapted Wiener process W = W (t)

such that (X(t), W (t)) is a solution of dX(t) = a(t, X(t))dW (t) on the

probability space (Ω,F , Ft, P ), where Tt is the inverse time change Tt.

3.1 Solutions to one-factor and multi-factor Gaussian modelsusing the CTM

3.1.1 Solution of one-factor Gaussian SIRMs using the CTM We usethe change of time method (see [57]) to get the solutions to the fol-lowing equations (see [102]). W (t) below is an standard Brownian mo-

tion, and W (t) is a (Tt)t∈R+-adapted standard Brownian motion on

(Ω,F , (Ft)t∈R+, P ).


1. The Geometric Brownian Motion:

dS(t) = µS(t) dt + σS(t) dW (t).

Solution:S(t) = eµt[S(0) + W (Tt)],

where Tt = σ2∫ t

0[S(0) + W (Ts)]

2 ds.

2. The Continuous-Time GARCH Process:

dS(t) = µ(b − S(t)) dt + σS(t) dW (t).

Solution:S(t) = e−µt(S(0) − b + W (Tt)) + b,

where Tt = σ2∫ t

0 [S(0) − b + W (Ts) + eµsb)2 ds.

3. The Ornstein-Uhlenbeck Process:

dS(t) = −µS(t) dt + σ dW (t)

Solution:S(t) = e−µt[S(0) + W (Tt)],

where Tt = σ2∫ t

0(eµs[S(0) + W (Ts)])

2 ds.

4. The Vasicek Process:

dS(t) = µ(b − S(t)) dt + σ dW (t)

Solution:S(t) = e−µt[S(0) − b + W (Tt)],

where Tt = σ2∫ t

0(eµs[S(0) − b + W (Ts)] + b)2 ds.

5. The Cox-Ingersoll-Ross Process:

dS2(t) = k(θ − S2(t)) dt + γS(t) dW (t).

Solution:S2(t) = e−kt[S2

0 − θ2 + W (Tt)] + θ2,

where Tt = γ−2∫ t

0 [ekTs(S20 − θ2 + W (s)) + θ2e2kTs ]−1 ds.


6. The Ho and Lee Process:

dS(t) = θ(t) dt + σ dW (t).

Solution:

S(t) = S(0) + W (σ2t) +

∫ t

0

θ(s) ds.

7. The Hull and White Process:

dS(t) = (a(t) − b(t)S(t)) dt + σ(t) dW (t).

Solution:

S(t) = exp[−∫ t

0

b(s)ds][S(0) − a(s)

b(s)+ W (Tt)],

where Tt =∫ t

0 σ2(s)[S(0) − a(s)b(s) + W (Ts) + exp[

∫ s

0 b(u)du]a(s)b(s) ]

2 ds.

8. The Heath, Jarrow and Morton Process:

f(t, u) = f(0, u) +

∫ t

0

a(v, u) dv +

∫ t

0

b(f(v, u)) dW (v).

Solution:

f(t, u) = f(0, u) + W (Tt) +

∫ t

0

a(v, u)dv,

where Tt =∫ t

0b2(f(0, u) + W (Ts) +

∫ s

0a(v, u)dv) ds.

3.1.2 Solution of multi-factor Gaussian models using CTM Solutionsof multi-factor models driven by Brownian motions can be obtainedusing various combinations of solutions of the above-mentioned pro-cesses; see Section 3.1, and CTM. We give one example of a two-factorcontinuous-time GARCH model driven by Brownian motions:

dS(t) = r(t)S(t)) dt + σS(t) dW 1(t),

dr(t) = a(m − r(t)) dt + σ2r(t) dW 2(t),

where W 1 and W 2 may be correlated, m ∈ R, σ, a > 0.The solution, using the CTM for the first and the second equations

is

S(t) = eR

t0

rsds[S0 + W 1(T 1t )]

= eR t0

e−as[r0−m+cW 2( bT 2s )]ds[S0 + W 1(T 1

t )],

where T i and W i are defined in 1. and 2., Section 3.1.1.


4 α-stable Levy processes and their properties

4.1 Levy processes

Definition 2. By Levy process we mean a stochastically continuousprocess with stationary and independent increments (see [4, 90]).

Examples of Levy processes L(t) include: a linear deterministic func-tion L(t) = γt; Brownian motion with drift; Poisson process, compoundPoisson process; jump-diffusion process; variance-gamma (VG), inverseGaussian (IG), normal inverse Gaussian (NIG), generalized hyperbolicand α-stable processes (see [89]).

4.2 Levy-Khintchine formula and Levy-Ito decomposition forfor Levy processes L(t) The characteristic function of the Levy pro-cess follows the following formula (so-called Levy-Khintchine formula).

E(ei(u,L(t))) = exp

t

[i(u, γ) − 1

2(u, Au)

+

∫

Rd−0

[ei(u,y) − 1 − i(u, y)1B1(0)]ν(dy)

],

where (γ, A, ν) is the Levy-Khintchine triplet.If L is a Levy process, then there exists γ ∈ Rd, a Brownian mo-

tion BA with covariance matrix A and an independent Poisson randommeasure N on R+ × (Rd − 0) such that, for each t ≥ 0, L(t) has thefollowing decomposition (Levy-Ito decomposition)

L(t) = γt + BA(t) +

∫

|x|<1

xN(t, dx) +

∫

|x|≥1

xN(t, dx),

where N is a Poisson counting measure and N is a compensated Poissonmeasure (see [4]).

Remark (Levy Processes in Finance). The most commonly usedLevy processes in finance include Brownian motion with drift (the onlycontinuous Levy process), the Merton process=Brownian motion+ drift+Gaussian jumps, the Kou process=Brownian motion +drift + expo-nential jumps and variance gamma (VG), inverse Gaussian (IG), normalinverse Gaussian (NIG), generalized hyperbolic (GH) and α-stable Levyprocesses.


4.3 α-stable distributions and Levy processes In this section, weintroduce α-stable distributions and Levy processes, and describe theirproperties.

4.3.1 Symmetric α-stable (SαS) distribution The characteristic func-tion of the SαS distribution is defined as follows:

φ(u) = e(iδu−σ|u|α),

where α is the characteristic exponent (0 < α ≤ 2), δ ∈ (−∞, +∞) isthe location parameter, and σ > 0 is the dispersion.

For values of α ∈ (1, 2] the location parameter δ corresponds to themean of the α-stable distribution, while for 0 < α ≤ 1, δ correspondsto its median. The dispersion parameter σ corresponds to the spreadof the distribution around its location parameter δ. The characteristicexponent α determines the shape of the distribution.

A stable distribution is called standard if δ = 0 and σ = 1. If a randomvariable L is stable with parameters α, δ and σ, then (L − δ)/σ1/α issdandard with characteristic exponent α. By letting α take the values1/2, 1 and 2, we get three important special cases: the Levy (α = 1/2),Cauchy (α = 1) and the Gaussian (α = 2) distributions:

f1/2(γ, δ; x) =

(t

2√

π

)x−3/2 e−t2/(4x),

f1(γ, δ; x) =1

π

γ

γ2 + (x − δ)2,

f2(γ, δ; x) =1√4πγ

exp

[− (x − δ)2

4γ

].

Unfortunately, no closed form expression exists for general α-stabledistributions other than the Levy, the Cauchy and the Gaussian. How-ever, power series expansions can be derived for the density fα(δ, σ; x).Its tails (algebraic tails) decay at a lower rate than the Gaussian densitytails (exponential tails).

The smaller the characteristic exponent α is, the heavier the tails ofthe α-stable density.

This implies that random variables following α-stable distributionwith small characteristic exponent are highly impulsive, and it is thisheavy-tail characteristic that makes this density appropriate for model-ing noise which is impulsive in nature, for example, electricity prices or


volatility. Only moments of order less than α exist for the non-Gaussianfamily of α-stable distribution. The fractional lower order moments withzero location parameter and dispersion σ are given by

E|X |p = D(p, α)σp/α for 0 < p < α,

D(p, α) =2pΓ(p+1

2 )Γ(1 − pα )

α√

πΓ(1 − p2 )

,

where Γ(·) is the Gamma function ([89]).Since the SαS r.v. has infinite variance, the covariation of two jointly

SαS real r.v. with dispersions γx and γy defined by

[X, Y ]α =E[X |Y |p−2Y ]

E[|Y |p] γy

has often been used instead of the covariance (and correlation), whereγy = [Y, Y ]α is the dispersion of r.v. Y.

4.3.2 α-stable Levy processes

Definition 3. Let α ∈ (0, 2]. An α-stable Levy process L such thatL1 (or equivalently any Lt) has a strictly α-stable distribution (i.e.,L1 ≡ Sα(σ, β, δ)) for some α ∈ (0, 2] \ 1, σ ∈ R+, β ∈ [−1, 1], δ = 0or α = 1, σ ∈ R+, β = 0, δ ∈ R). We call L a symmetric α-stable

Levy process if the distribution of L1 is even symmetric α-stable (i.e.,L1 ≡ Sα(σ, 0, 0) for some α ∈ (0, 2], σ ∈ R+.) A process L is called(Tt)t∈R+

-adapted if L is constant on [Tt−, Tt] for any t ∈ R+ (see [89]).

4.3.3 Properties of α-stable Levy processes The α-stable Levy processes

are the only self-similar Levy processes such that L(at)Law= a1/αL(t), a ≥

0. They are either Brownian motion or pure jump. They have character-istic exponent and Levy-Khintchine triplet known in closed form. Theyalso have only 4 parameters, but infinite variance (except for Brownianmotion). The α-stable Levy Processes are semimartingales (in this way,∫ t

0fsdLs can be defined) and α-stable Levy Processes are pure discon-

tinuous Markov processes with generator

Af(x) =

∫

R−0

[f(x + y) − f(x) − yf ′(y)1|y|<1(y)]Kα

|y|1+αdy.

E|L(t)|p is finite or infinite according as 0 < p < α or p > α, respec-tively. In particular, for an α-stable process, EL(t) = δt (1 < α < 2)([89]).


5 Stochastic differential equations driven by α-stable Levyprocesses Consider the following SDE driven by an α-stable Levyprocess L(t) :

(1) dZt = b(t, Zt−) dt + σ(t, Zt−) dL(t).

Janicki et al. [61] proved that this equation has a weak solution forcontinuous coefficients a and b.

We consider below one-factor and multi-factor models described bySDEs driven by α-stable Levy process L(t).

5.1 One-factor α-stable Levy models L(t) below is a symmetric α-stable Levy process. We define below various processes via SDE drivenby α-stable Levy process.

1. The Geometric α-stable Levy motion:

dS(t) = µS(t−) dt + σS(t−) dL(t).

2. The Ornstein-Uhlenbeck Process Driven by α-stable Levy motion:

dS(t) = −µS(t−) dt + σ dL(t).

3. The Vasicek Process Driven by α-stable Levy motion:

dS(t) = µ(b − S(t−)) dt + σ dL(t).

4. The Continuous-Time GARCH Process Driven by α-stable Levy mo-tion:

dS(t) = µ(b − S(t−)) dt + σS(t−) dL(t).

5. The Cox-Ingersoll-Ross Process Driven by α-stable Levy motion:

dS(t) = k(θ − S(t−)) dt + γ√

S(t−) dL(t).

6. The Ho and Lee Process Driven by α-stable Levy motion:

dS(t) = θ(t−) dt + σ dL(t).

7. The Hull and White Process Driven by α-stable Levy motion:

dS(t) = (a(t−) − b(t−)S(t−)) dt + σ(t) dL(t).


8. The Heath, Jarrow and Morton Process Driven by α-stable Levymotion: Define the forward interest rate f(t, s), for t ≤ s, that repre-sents the instantaneous interest rate at time s as anticipated by themarket at time t. The process f(t, u)0≤t≤u satisfies an equation

f(t, u) = f(0, u) +

∫ t

0

a(v, u) dv +

∫ t

0

b(f(v, u)) dL(v),

where the processes a and b are continuous.

We note that Eberlein and Raible [32] considered Levy-based term struc-ture models.

5.2 Multi-factor α-stable Levy models Multi-factor models drivenby α-stable Levy motions can be obtained using various combinationsof the above-mentioned processes. We give one example of a two-factorcontinuous-time GARCH model driven by α-stable Levy motions.

dS(t) = r(t−)S(t−)) dt + σS(t−) dL1(t),

dr(t) = a(m − r(t−)) dt + σ2r(t−) dL2(t),

where L1 and L2 may be correlated, m ∈ R, σi, a > 0, i = 1, 2.Also, we can consider various combinations of models, presented above,

i.e., mixed models containing Brownian and Levy motions. For example,

dS(t) = µ(b(t−) − S(t−)) dt + σS(t−) dL(t),

db(t) = ξb(t) dt + ηb(t) dW (t),

where the Brownian motion W (t) and Levy process L(t) may be corre-lated.

6 Change of time method (CTM) for SDEs driven by Levyprocesses We denote by Lα

a.s. the family of all real measurable Ft-adapted processes a on Ω × [0, +∞) such that for every T > 0,

∫ T

0

|a(t, ω)|αdt < +∞ a.s.

We consider the following SDE driven by a Levy motion:

dX(t) = a(t, X(t−)) dL(t),

where L(t) is an α-stable Levy process.


Theorem ([87, Theorem 3.1, p. 277]). Let a ∈ Lαa.s. be such that

T (u) :=∫ u

0 |a|αdt → +∞ a.s. as u → +∞. If T (t) := infu : T (u) >

t and Ft = FbT (t), then the time-changed ctochastic integral L(t) =∫ bT (t)

0adL(t) is an Ft − α-stable Levy process, where L(t) is Ft-adapted

and Ft-α-stable Levy process. Consequently, a.s. for each t > 0∫ t

0adL =

L(T (t)), i.e., the stochastic integral with respect to a α-stable Levy pro-cess is nothing but another α-stable Levy process with randomly changedtime scale.

6.1 Solutions of one-factor Levy models using the CTM Belowwe give the solutions to the one-factor Levy models decribed by SDEsdriven by α-stable Levy process introduced in Section 5.1.

Proposition 1. Let L(t) be a symmetric α-stable Levy process, and L is

a (Tt)t∈R+-adapted symmetric α-stable Levy process on (Ω,F , (Ft)t∈R+

,P )). Then, we have the following solutions for the above-mentioned one-factor Levy models 1–8 (Section 5.1):

1. The Geometric α-stable Levy Motion:

dS(t) = µS(t−) dt + σS(t−) dL(t).

Solution:S(t) = eµt[S(0) + L(Tt)],

where Tt = σα∫ t

0[S(0) + L(Ts)]

α ds.

2. The Ornstein-Uhlenbeck Process Driven by α-stable Levy Motion:

dS(t) = −µS(t−) dt + σ dL(t).

Solution:S(t) = e−µt[S(0) + L(Tt)],


0 (eµs[S(0) + L(Ts)])α ds.

3. The Vasicek Process Driven by α-stable Levy Motion:

dS(t) = µ(b − S(t−)) dt + σ dL(t).

Solution:S(t) = e−µt[S(0) − b + L(Tt)],


0 (eµs[S(0) − b + L(Ts)] + b)α ds.


4. The Continuous-Time GARCH Process Driven by α-stable Levy pro-cess:

dS(t) = µ(b − S(t−)) dt + σS(t−) dL(t).

Solution:S(t) = e−µt(S(0) − b + L(Tt)) + b,

where Tt=σα∫ t

0 [S(0) − b + L(Ts) + eµsb]α ds.

5. The Cox-Ingersoll-Ross Process Driven by α-stable Levy Motion:

dS(t) = k(θ2 − S(t−)) dt + γ√

S(t−) dL(t).

Solution:S2(t) = e−kt[S2

0 − θ2 + L(Tt)] + θ2,

where Tt = γα∫ t

0[ek bTs(S2

0 − θ2 + L(Ts)) + θ2e2k bTs ]α/2 ds.

6. The Ho and Lee Process Driven by α-stable Levy Motion:

dS(t) = θ(t−) dt + σ dL(t).

Solution:

S(t) = S(0) + L(σαt) +

∫ t

0

θ(s) ds.

7. The Hull and White Process Driven by α-stable Levy Motion:

dS(t) = (a(t−) − b(t−)S(t−)) dt + σ(t−) dL(t).

Solution:

S(t) = exp[−∫ t

0

b(s) ds][S(0) − a(s)

b(s)+ L(Tt)],

where Tt =∫ t

0σα(s)[S(0) − a(s)

b(s) + L(Ts) + exp[∫ s

0b(u) du]a(s)

b(s) ]α ds.

8. The Heath, Jarrow and Morton Process Driven by α-stable Levy Mo-tion:

f(t, u) = f(0, u) +

∫ t

0

a(v, u) dv +

∫ t

0

b(f(v, u)) dL(v).

Solution:

f(t, u) = f(0, u) + L(Tt) +

∫ t

0

a(v, u) dv,

where Tt =∫ t

0 bα(f(0, u) + L(Ts) +∫ s

0 a(v, u) dv) ds.

Proof. The approach is to eliminate drift, reduce obtained SDE to theabove-mentioned form dX(t) = a(t, X(t−))dL(t) and then to use theabove-mentioned Rosinski-Woyczynski result [87].


6.2 Solution of multi-factor Levy models using CTM Solutionof multi-factor models driven by α-stable Levy motions (see Section 6.1)can be obtained using various combinations of solutions of the above-mentioned processes and the CTM. We give one example of two-factorcontinuous-time GARCH model driven by α-stable Levy motions.

Proposition 2. Let we have the following two-factor Levy-based model:

dS(t) = r(t−)S(t−)) dt + σ1S(t−) dL1(t),

dr(t) = a(m − r(t−)) dt + σ2r(t−) dL2(t),

where L1 and L2 may be correlated, m ∈ R, σi, a > 0, i = 1, 2.

Then the solution of the two-factor Levy model using the CTM is(applying CTM for the first and the second equations, respectively)

S(t) = eR t0

rsds[S0 + L1(T 1t )]

= eR t0

e−as[r0−m+bL2( bT 2s )]ds[S0 + L1(T 1

t )],

where T i are defined in 1. and 4., respectively, Section 6.1.

Proof. The approach is to eliminate drifts in both equations, reduce theobtained SDEs to the above-mentioned form dX(t) = a(t, X(t−)) dL(t),and then to use the above-mentioned Rosinski-Woyczynski result [87].

Important Remark. Kallsen and Shiryaev [67] showed that the Rosin-ski and Woyczynski [87] statement cannot be extended to any otherLevy process but α-stable processes. If one considers only nonnegativeintegrands a in dX(t) = a(t, X(t−)) dL(t), then we can extend theirstatement to asymmetric α-stable Levy processes.

7 Applications in financial and energy markets In this sec-tion, we consider various applications of the change of time method forLevy-based SDEs arising in financial and energy markets: swap andoption pricing, interest derivatives pricing and forward and futures con-tracts pricing.


7.1 Variance swaps for Levy-based Heston model Assume thatin the risk-neutral world the underlying asset St and the variance followthe following model:

dSt

St= rt dt + σt dwt,

dσ2t = k(θ2 − σ2

t ) dt + γσt dLt,

where rt is the deterministic interest rate, σ0 and θ are the short andlong volatilities, k > 0 is a reversion speed, γ > 0 is a volatility (ofvolatility) parameter, wt and Lt are independent standard Wiener andα-stable Levy processes (α ∈ (0, 2]).

The solution for the second equation has the following form:

σ2(t) = e−kt[σ20 − θ2 + L(Tt)] + θ2,

where

Tt = γα

∫ t

0

[ek bTs(σ20 − θ2 + L(Ts)) + θ2e2k bTs ]α/2 ds.

A variance swap is a forward contract on annualized variance, thesquare of the realized volatility. Its payoff at expiration is equal to

N(σ2R(S) − Kvar),

where σ2R(S) is the realized stock variance (quoted in annual terms) over

the life of the contract,

σ2R(S) :=

1

T

∫ T

0

σ2(s) ds,

Kvar is the delivery price for variance, and N is the notional amount.Valuing a variance forward contract or swap is no different from valu-

ing any other derivative security. The value of a forward contract P onfuture realized variance with strike price Kvar is the expected presentvalue of the future payoff in the risk-neutral world:

Pvar = Ee−rT (σ2R(S) − Kvar),

where r is the risk-free discount rate corresponding to the expirationdate T, and E denotes the expectation.


The realized variance in our case is

σ2R(S) :=

1

T

∫ T

0

σ2(s) ds =1

T

∫ T

0

e−ks[σ20 − θ2 + L(Ts)] + θ2 ds.

The value of the variance swap then is

Pvar = Ee−rT (σ2R(S) − Kvar)

= E

e−rT

(1

T

∫ T

0

e−ks[σ20 − θ2 + L(Ts)] + θ2 ds − Kvar

).

Thus, for calculating variance swaps we need to know only Eσ2R(S),

namely, the mean value of the underlying variance, or EL(Ts).Only moments of order less than α exist for the non-Gaussian family

of α-stable distributions. We suppose that 1 < α < 2 to find EL(Ts).The value of a variance swap for the Levy-based Heston model is

Pvar = e−rT

[1 − e−kT

kT(σ2

0 − θ2) + θ2 +δT

2− Kvar

],

where δ is a location parameter.If δ = 0, then the value of a variance swap for the Levy-based Heston

model is

Pvar = e−rT

[1 − e−kT

kT(σ2

0 − θ2) + θ2 − Kvar

],

which coincides with the well-known result by Brockhaus and Long [15]and Swishchuk [99].

7.2 Volatility swaps for Levy-based Heston model? A stock volatil-ity swap is a forward contract on the annualized volatility. Its payoff atexpiration is equal to

N(σR(S) − Kvol),

where σR(S) is the realized stock volatility (quoted in annual terms)over the life of contract,

σR(S) :=

√1

T

∫ T

0

σ2s ds ,

σt is a stochastic stock volatility, Kvol is the annualized volatility deliveryprice, and N is the notional amount.


To calculate volatility swaps we need more. From Brockhaus-Long

approximation [15] (which used the second order Taylor expansion forthe function

√x) we have

E√

σ2R(S)

≈

√EV − V arV

8EV 3/2,

where V := σ2R(S), and V arV

8EV 3/2 is the convexity adjustment.

Thus, to calculate the value of volatility swaps

Pvol = e−rT (EσR(S) − Kvol),

we need both EV and V arV .For SαS processes only the moments of order p < α exist, α ∈ (0, 2].Since the SαS r.v. has infinite variance, the covariation of two jointly

SαS real r.v. with dispersions γx and γy defined by

[X, Y ]α =E[X |Y |p−2Y ]

E[|Y |p] γy,

where γy = [Y, Y ]α is the dispersion of r.v. Y, has often been usedinstead of the covariance (and correlation).

One of possible ways to get volatility swaps for the Levy-based Hestonmodel is to use covariation.

7.3 Gaussian- and Levy-based SABR/LIBOR market modelsSABR model (see [48]) and the Libor Market Model (LMM) (BGM[13], [78]) have become industry standards for pricing plain-vanilla andcomplex interest rate products, respectively.

The Gaussian-based SABR model [48] is a stochastic volatility modelin which the forward value satisfies the following SDE:

dFt = σtFβt dW 1

t ,

dσt = νσt dW 2t .

In a similar way, we introduce the Levy-based SABR model, a stochas-tic volatility model in which the forward value satisfies the followingSDE:

dFt = σtFβt dWt,

dσt = νσt dLt,


where L(t) is an α-stable Levy process.The solution of Levy-based SABR model using a change of time

method has the following expression:

Ft = F0 + W (T 1t ),

T 1t =

∫ t

0

σ−2T 1

s(F0 + W (s))−2β ds,

σt = σ0 + L(T 2t ),

T 2t = ν−α

∫ t

0

(σ0 + L(s))−α ds.

The expressions for Ft and σt give the possibility to calculate manyfinancial derivatives.

7.4 Energy forwards and futures Random variables following α-stable distribution with small characteristic exponent are highly impul-sive, and it is this heavy-tail characteristic that makes this density ap-propriate for modeling noise which is impulsive in nature, for example,energy prices such as electricity. Here, we introduce two Levy-basedmodels in energy market: two-factor Levy-based Schwartz-Smith andthree-factor Schwartz models. We show how solve them using the changeof time method.

7.4.1 Levy-based Schwartz-Smith model We introduce the Levy-basedSchwartz-Smith model

ln(St) = κt + ξt,

dκt = (−kκt − λκ) dt + σκ dLκ,

dξt = (µξ − λξ) dt + σξ dWξ ,

where St is the current spot price, κt is the short-term deviation inprices, and ξt is the equilibrium price level.

Let Ft,T denote the market price for a futures contract with maturityT, then

ln(Ft,T ) = e−k(T−t)κt + ξt + A(T − t),

where A(T − t) is a deterministic function with explicit expression. Wenote that κt, using change of time for α-stable processes, can be pre-


sented in the following form:

κt = e−kt

[κ0 +

λκ

k+ Lκ(Tt)

],

Tt = σακ

∫ t

0

(e−ks

[κ0 +

λκ

k+ Lκ(Ts)

]− λκ

k

)α

ds.

In this way, the market price for a futures contract with maturity Thas the following form:

ln(Ft,T ) = e−kT

[κ0 +

λκ

k+ Lκ(Tt)

]

+ ξ0 + (µξ − λξ)t + σξWξ + A(T − t),

where the Levy process Lκ and Wiener process Wξ may be correlated.If α ∈ (1, 2], then we can calculate the value of Levy-based futures

contracts.

7.4.2 Levy-based Schwartz model We also introduce a Levy-Based Schwartzmodel

d ln(St) = (rt − δt)St dt + Stσ1 dW1,

dδt = k(a − δt) dt + σ2 dL,

drt = a(m − rt) dt + σ3 dW2,

where the Wiener processes W1, W2 and α-stable Levy process L maybe correlated. δt and rt are the instantaneous convenience yield andinterest rate, respectively.

We note that

δt = ekt(δ0 − a + L(Tt)),

Tt = σα2

∫ t

0

(eks[δ0 − a + L(Ts)] + a)α ds

and

rt = eat(r0 − m + W2(Tt)),

Tt = σ23

∫ t

0

(eas[r0 − m + W2(Ts)] + m)2 ds.


The solution for ln[St] is

ln[St] = eR t0[eas(r0−m+cW2( bT 2

s ))−eks(δ0−a+bL( bTs))]ds[ln S0 + W1(T1t )].

In this way, the futures contract has the following form:

ln(Ft,T ) =1 − e−k(T−t)

kδt +

1 − e−a(T−t)

art + ln(St) + C(T − t)

=1 − e−k(T−t)

k[ekt(δ0 − a + L(Tt))]

+1 − e−a(T−t)

aeat(r0 − m + W2(T

2t ))

+ exp∫ t

0

(eas(r0 − m + W2(T2s ))

− eks(δ0 − a + L(Ts))) ds[ln(S0) + W1(T1t )] + C(T − t),

where C(T − t) is a deterministic explicit function. If α > 1, then wecan calculate the value of a futures contract.

8 Conclusion and future work We introduced one-factor andmulti-factor α-stable Levy-based models to price financial and energyderivatives. These models include, in particular, as one-factor-Levy-based geometric Brownin motion model, the Ornshtein-Uhlenbeck [75],the Vasicek [106], the Cox-Ingersoll-Ross [26], the continuous-timeGARCH, the Ho-Lee [53], the Hull-White [56] and the Heath-Jarrrow-Morton [49] models, and multi-factor-various combinations of these mod-els. For example, we introduce such new multi-factor models as Levy-based Heston model, Levy-based SABR/LIBOR models, Levy-basedSchwartz-Smith and Schwartz models. Using the change of time methodfor Levy-based SDEs we present the solutions of these equations in verysimple and compact form. We apply this method to the pricing of manyfinancial and energy derivatives such as variance swaps, options, forwardand futures contracts. Future research will be associated with imple-menting numerical methods for calculating these Levy-based models.

Acknowledgements The research of this paper is dedicated to the30th Anniversary of CAIMS and supported by NSERC. The authorwould like to thank very much to the organizers Tony Ware and Adam


Metzler of the mathematical finance session of 2009 Summer CAIMSmeeting, UWO, London, ON, June 10–14, for the kind invitation tosubmit this paper. The author also thanks to the anonymous referee formany valuable comments that definitely improved the present paper.The remaining errors are mine.

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Department of Mathematics and Statistics, University of Calgary,2500 University Drive NW, Calgary, AB, Canada T2N 1N4.E-mail address: [email protected]

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