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Subordinated Levy Processes and Applications to Crude Oil Options

Author(s):
Noureddine Krichene
Published Date:
September 2005
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I. Introduction

A suggested approach to oil markets analysis is to treat oil as an asset, besides its role as a commodity, and to recognize the influence of derivatives markets on oil price behavior. Restricting attention to fundamentals in modeling and forecasting short-term oil prices would be unrealistic and would omit a key aspect of oil markets, which is the asset role of crude oil and the preponderant presence of financial investors, other than the traditional operators, who are the producers and the consumers, in the markets. Sizable demand for futures contracts by institutional investors and speculative activity by hedge funds and commercial entities could exert pressure on prices and cause volatility of oil futures prices to rise to excessive levels. Similarly, in other commodity and currency markets, high volatility would stimulate speculation, which in turn contributes to higher volatility and to volatility clustering.

It is well known that asset markets can experience frequent jumps of different magnitudes and can overshoot or undershoot their equilibrium. Yet policymakers and central bankers would like to monitor regularly these markets, gauge the market sentiment, and forecast price distributions. They turn to derivatives markets, and particularly to options prices, for such forecasts as these markets account for both the consumption and investment aspects of an asset. The purpose of this paper is therefore to model asset prices’ uncertainty and option pricing in the context of Levy processes, which are capable of handling discontinuities and are known to behave properly under time aggregation.

Levy processes have gained considerable interest in financial modeling as they were found to overcome many of the shortcomings associated with the Black-Scholes’ model (1973) and to offer a more general tool for modeling uncertainty in asset prices. They are seen as a random walk in continuous time with jumps occurring at random times. They include the Brownian motion and the Poisson processes as particular cases. Their general form has a drift, a Brownian motion, and a compound Poisson process which distinguishes small size jumps and large jumps. The probability distributions associated with Levy processes are infinitely divisible and offer more flexibility for fitting financial data, particularly high-frequency data, and can have skewed shapes and slow decaying tails.

While the Black-Scholes model and diffusion processes constitute the main framework for derivatives pricing, they can, nonetheless, have inconsistencies with market data, typically in relation to the implied volatility and to the dynamics of the asset’s price. The model’s implied volatility tends to vary both in relation to the state and time, exhibiting a smile or a smirk. Volatility could even be seen as stochastic. The dynamics of the asset’s price may exhibit jumps of different sizes, with small jumps occurring more often than large jumps, leading both to asymmetries and fat tails in the asset’s returns, contradicting thus the Brownian motion assumption underlying the Black-Scholes model. These empirical features of assets’ returns were noted by Mandelbrot (1963) and Fama (1965). As the normal distribution did not fit the data, they proposed the use of stable distributions, similar to the Pareto distributions, which are capable of accommodating skewness and the slowly decaying tails of the empirical distribution, in contrast to the thin and rapidly decaying tails of the normal distribution. Clark (1973), however, noted that stable distributions have an infinite variance and heavy tails, whereas assets’ returns distributions have a finite variance and semi-heavy tails. He proposed the use of Bochner (1955) concept of subordinated processes. Namely, the price process could be modeled as a Brownian motion, time changed by a random and independent subordinator, which is an increasing positive stochastic process. While Clark proposed the use of volume as subordinator, the number of transactions could also be used for measuring assets’ returns. His finding was that, when measured in relation to volume, cotton futures tend to be normally distributed.

This paper addresses option pricing in the context of a Levy market model with an application consisting of deriving crude oil price density forecast from crude oil options. In view of the paramount relevance of Levy processes in financial modeling, Section II describes their key properties and particularly their characteristic function, a major tool for studying their distributional properties. Section III discusses a methodology for constructing Levy processes through subordination and describes examples of probability distributions that are obtained through subordination and that were found to fit adequately financial time series. As Levy processes may exhibit jumps, the markets become incomplete and there is an infinite number of martingale measures compatible with absence of arbitrage that can be used for pricing contingent claims. In this respect, Section IV presents Esscher transform as a procedure for selecting a martingale measure. It is known that probability distributions associated with Levy processes are not always available in closed form and may involve many special functions. The corresponding characteristic function may, however, be readily available. In Section V, the paper presents a methodology for pricing contingent claims based on characteristic functions. In Section VI, the paper discusses an application of the Levy pricing model to crude oil options and attempts to infer the density forecast of future oil prices at a given time horizon. Findings point to high volatility of oil futures prices and a right-skewed market expectations, implying greater probability mass on upward deviations from the mean. Consequently, extracting densities from options prices allows one to analyze the role of crude oil as an asset, besides its role as a commodity, and improves oil market modeling. Section VII concludes.

II. Modeling Uncertainty in ASSETS PRICES: Levy Processes

A. From Random Walks to Levy Processes

A Levy process (LP) is defined as a cadlag (continu à droite et limite à gauche, right continuous and left limit (RCLL)) stochastic process (Xt)t≥0 on a probability space (Ω,F,P) with values in R such that X0 = 0 and possesses the following properties: (i) independent increments: for every increasing sequence of times t0,…,tn, the random variables Xt0,Xt1Xt0,,XtnXtn1 are independent, (ii) stationary increments: the law of Xt+hXt does not depend on t, and (iii) stochastic continuity: ∀ ε > 0, limh0P(|Xt+hXt|ε)=0. i.e., discontinuity occurs at random times. Levy processes are limits of random walks and are infinitely divisible into independent and identically distributed (i.i.d.) random variables. For any Xt, t > 0, and n ≥ 2, an LP can be written as: Xt=k=1n(Xtk/nXt(k1)/n), where the successive increments are independent and stationary. An equivalent way to express the infinite divisibility is: Xt(n)=Y1++Yn, where the random variables Y1, Y2,…,Yn are i.i.d. The probability distribution of Xt is the same as that for Y1 + …+ Yn. The most common examples of infinitely divisible laws are: the Gaussian, the Gamma, α − stable, and the Poisson distributions.

B. The Characteristic Function of the Levy Process: The Levy-Khintchine Formula

Suppose ϕ(u) is the characteristic function (CF) of a distribution. If for every positive integer n, ϕ(u) is the n th power of a CF, then the distribution is infinitely divisible. For every such distribution there can be defined a stochastic process, X = {Xt,t ≥ 0}, which starts at zero and has independent and stationary increments. If (Xt)t≥0 is an LP, then for any t > 0, the distribution of Xt is infinitely divisible and has a CF Φ(u)=E[eiuX1]=etψ(u), uR, t ≥ 0. The cumulant characteristic function ψ(u) = log ϕ(u) is often called the characteristic exponent and satisfies the Levy-Khintchine formula:23ψ(u)=iγuσ22u2+R\{0}[eiux1iux1|x|<1(x)]v(dx), where γR is the drift parameter, σ2 ≥ 0 is the volatility parameter, and v is a Levy measure on R\{0} with inf(1,x2)v(dx)<. The sums of all jumps smaller than some ε > 0 does not converge. However, the sum of the jumps compensated by their mean does converge. This peculiarity leads to the necessity of the compensator term iux1(|x|<1). If the Levy measure is of the form ν(dx) = f (x)dx, then f (x) is called the Levy density. In the same way the instant volatility describes the local uncertainty of a diffusion, the Levy density describes the local uncertainty of a pure jump process.

The Levy-Khintchine formula allows to study the distributional properties of an LP. Another key concept, the Levy-Ito decomposition theorem, allows one to describe the structure of an LP’s sample paths. Let X = {Xt,t ≥ 0} be an LP, then the distribution of X1 has as parameters (γ,σ2, ν) and X decomposes as: Xt= γt + σWt+ Jt+ Mt, where Wt is a Brownian motion. The instantaneous jump ΔXt = XtXt follows an independent Poisson point process with intensity measureν, Jt=stΔXs1{|ΔXs|>1}, and M is a martingale with jumps ΔMt=ΔXt1{|ΔXs|1}. It is interesting to observe that there is a one-to-one correspondence between an LP and its CF. More precisely, starting with the Levy-Khintchine formula, three processes, X(1), X(2), and X(3), can be built as follows: denoting Xt(1)=γt+σWt, where Wt is a standard P − Brownian motion, the CF of X(1) is straightforward and equal to:ϕ1(u)=iγu+12σ2u2. Now, consider the process Xt(2)=Jt=j=1NtYj, where N is a Poisson process whose intensity λ is defined by λ=|x|>1v(dx) and Y1,Y2, …, Yn are independent random variables, independent of the process N and with common distribution 1|x|>1v(dx). Xt(2) is a compound Poisson process whose CF is eϕ2(u), where Φ2(u)=(eiux1)1|x|>1v(dx). The last term in the Levy-Khintchine formula is the CF of an LP X(3) obtained as the limit of compound Poisson processes (different from Xt(2)). Hence: X=X(1) + X(2) + X(3). Each of X(1), X(2), X(3) is a semi-martingale, so is X .4

An important implication of the Levy-Ito decomposition is that every LP is a combination of a Brownian motion with drift and a possibly infinite sum of independent compound Poisson processes. This also means that every LP can be approximated with arbitrary precision by a jump-diffusion process. In particular, the Levy measure ν describes the arrival rates for jumps of every possible size for each component of Xt. Jumps of sizes in the set A occur according to a Poisson process with intensity parameter Av(dx), where A is an arbitrary interval bounded away from zero. The Levy measure of the process X may also be defined as v(A)=E{0s11A(ΔXs)}. Integration of the Levy density over a particular spatial domain provides the arrival rates of jumps sized in this domain. Levy process X has infinite activity if the integral of the measure ν on the real line is infinite. This case characterizes a high rate of arrival of jumps of different sizes.

C. Finite-Activity Versus Infinite Activity Levy Jumps

A pure jump LP can display either finite or infinite activity. In the former case, the aggregate jump arrival rate is finite, while in the latter case, an infinite number of jumps can occur in any finite time interval. A pure jump LP exhibits finite activity if the following integral is finite: R\{0}v(dx)=λ<. The classical example of a finite-activity jump process is the compound Poisson jump process of Merton (1976). For such process, the integral R\{0}v(dx)=λ<, where λ is the Poisson intensity. Conditional on one jump occurring, the Merton model assumes that the jump magnitude is normally distributed with mean α and variance σ2. The Levy measure of the Merton process is given by: v(dx)=λ12πσ2exp((xα)22σ2)dx. Obviously, one can choose any distribution, F(x), for the jump size under the compound Poisson framework and obtain the following Levy measure; ν(dx) = λdF(x). Kou (2002) assumes a double-exponential conditional distribution for the jump size. The Levy measure in this case is given by v(dx)=λdF(x)=λ12ηexp(|xα|η)dx. Eraker et al. (2003) incorporate compound Poisson jumps into the stochastic volatility process; jump size is controlled by one-sided exponential density. The Levy measure in this case is given by v(dx)=λdF(x)=λ1ηexp(xη)dx, x>0. Based on the Levy-Khintchine formula, the characteristic exponent corresponding to these compound Poisson jump components is given by ψ(u)=R\{0}(1eiux)λdF(x)=λ(1Φ(u)), where ϕ(u) denotes the CF of the jump size distribution F(x), ϕ(u)=R\{0}eiuxdF(x). An infinite activity jump process can generate an infinite number of jumps within any finite time interval. The integral of the Levy measure R\{0}v(dx)= is no longer finite. Examples in this class include the normal inverse Gaussian (NIG) model of Barndorff-Nielsen (1998), the generalized hyperbolic class of Eberlein et al. (1998), the variance-gamma (VG) model of Madan and Milne (1991), and the CGMY model of Carr et al. (2002).

III. Subordination and Time Changed-Levy Processes

Three key pieces of evidence on financial securities: jumps, stochastic volatility, and leverage effect are easily addressed when uncertainty in the economy is governed by a time-changed LP. A stochastic time change to the LP amounts to stochastically altering the clock on which the LP is run. Intuitively, one can regard the original clock as a calendar time and the new random clock as a business time. A more active business day implies a faster business clock. Randomness in business activity generates randomness in volatility. Furthermore, if innovations in the LP are correlated with innovations in the random clock on which it is run, this correlation will capture the leverage effect.

A. Construction of Levy Processes by Subordination

Clark (1973) proposed Bochner’s (1955) concept of subordinated stochastic process as a model to account for non-normality of returns. He showed that finite-variance distributions subordinate to the normal distribution fit cotton futures better than stable distributions. Writing the return process X (t) as a subordinated process X (t) = Z (T (t)), where the subordinator T (t) is an increasing Levy process with independent and stationary increments, and using historical data on returns (represented by X (t)) and volume (represented by T (t)), he was able to show that the distribution of Z computed in relation to T did satisfy classical normality tests. He also showed that the kurtosis of the increments of Z (T (t)) is an increasing function of the variance of the increments of T (t).5

Monroe (1978) proved that every semi-martingale Xt can be written as a time-changed Brownian motion, where the random time Tt is a positive and increasing semi-martingale. By such result, there exists a Brownian motion (W(u),u ≥ 0) and a random time change T(t) where T(t) is an increasing stochastic process such that: X(t) = W(T(t)). As an implication, every semi-martingale can also be written as a time-changed LP Xt = ZT((t)). The distribution of increments, ΔZ(T(t)), is said to be subordinate to the distribution of increments, ΔZ(t); T(t) is a clock measuring the speed of the evolution. Furthermore, every semi-martingale Xt starting at zero (X0 = 0), can be uniquely represented in the form: Xt=αt+Xtc+0t|x|>1xdμ+0t|x|1xd(μν), where Xtc is the continuous component, µ is the counting measure of the semi-martingale, and ν is its compensator.

An important result from Monroe’s theorem relates to modeling the return distribution as a mixture of normals with a view to account for the observed fat tails of the return. Choosing for simplicity a discrete fT for T, the following holds: P(X(t)dx)=ζP(Z(t)dx|T(t)=ζ)fT(ζ). Now let Z(t) = W(t), and assuming the independence of the processes W and T, this yields: P(X(t)dx)=ζP(W(ζ)dx)fT(ζ). Hence the distribution of X appears as a mixture of normal distributions, where the mixing factor is the density of the time change, which itself accounts for the market activity measured by volume or number of trades.

A simple example of a subordinated LP is a compound Poisson process with a finite arrival rate, i.e., a random walk time changed by a Poisson process expressed as follows: X(t)=i=1N(t)Yi, where N(t) is a Poisson process with arrival rate λt , and the sequence of Yi is i.i.d. with density f(y)=2exp(y22σ2)σπ for y > 0. The CF of Xt is Φx(u)=exp(2λt(exp(σ2u22)1)).

Another example of a time changed LP is to subordinate a Brownian motion (Wt)t≥0 with drift µ by an independent positive process (Tt)t≥0, yielding a new LP: Xt = σW(Tt) + µTt. This process is a Brownian motion if it is observed on a new time scale, which is the stochastic time scale given by Tt. It is worth noticing that the constant volatility in the arithmetic Brownian motion W is going to give rise to stochastic volatility for the stochastic price X when W is compounded with a stochastic time T. Time changes appear a natural tool to handle stochastic volatility. The interpretation of an LP as a subordinated Brownian motion is easier to understand than general Levy models.

B. The Characteristic Function of a Subordinated Process

The CF of a subordinated process X is obtained by composition of the Laplace exponent6 of T with the characteristic exponent of Z. Since the time-changed process Xt=ZTt is a stochastic process evaluated at a stochastic time, its CF involves expectation over two sources of randomness, ΦXt(u)=E[eiuZTt]=E[E[eiuZξ|Tt=ξ]], where the inside expectation is taken on ZTt, conditional on a fixed value of Tt = ξ, and the outside expectation is on all possible values of Tt. If the random time Tt is independent of Zt, the randomness due to the LP can be integrated out using equation: ΦZt(u)=E[eiuZt]=etψz(u). Simple computation yields: ΦXt(u)=E[eiuZTt]=E[E[eiuZξ|Tt=ξ]]=EeTtψz(u)=LTt(ψz(u)). Under independence, the CF of Xt=ZTt is just the Laplace transform of Tt evaluated at the characteristic component of Zt.7 We can introduce asymmetry to the distribution of Xt=ZTt by introducing correlation between the time change and the Levy innovations in Z.

C. Examples of Subordinated LP

A simple approach to defining a parametric LP is to obtain an LP by subordinating a Brownian motion with an independent increasing LP. Here the CF of the resulting process can be obtained immediately, but there may not be always an explicit formula for the Levy measure. Due to the conditionally Gaussian structure of the process, simulation and computation can be considerably simplified.

The Inverse Gaussian Process

Let T(a,b) be the first time a Brownian motion with drift b > 0, i.e., (Ws+ bs, s ≥ 0), reaches the positive level a > 0. It is well known that this random time follows the so-called Inverse Gaussian law, IG(a, b), and has a CF: ΦIG(u;a,b)=exp(a(2iu+b2b)). The IG distribution is infinitely divisible and the IG process X(IG)={Xt(IG),t0}, with parameters a,b > 0, is defined as the process which starts at zero and has independent and stationary increments over the interval [s,s + t], s,t ≥ 0 such that: E(exp(iuXt(IG)))=ΦIG(u;at,b)=exp(at(2iu+b2b)). The density function of the IG(a,b) law is explicitly given by: fIG(x;a,b)=a2πexp(ab)x3/2exp(12(a2x1+b2x)), x > 0. Its Levy measure is given by νIG(dx)=(2π)1/2ax3/2exp(12b2x)1(x>0)dx.

The Generalized Inverse Gaussian Process

The Inverse Gaussian IG(a,b) law can be generalized to the Generalized Inverse Gaussian distribution GIG(λ,a,b). This distribution is given by fGIG(x;λ,a,b)=(b/a)λ2Kλ(ab)xλ1exp(12(a2x1+b2x)), x>0.

The CF is given by ΦGIG(u;λ,a,b)=1Kλ(ab)(12iu/b2)λ/2Kλ(ab12iub2), where Kλ(x) denotes the modified Bessel function of the third kind with index λ. The GIG(λ,a,b) distribution is infinitely divisible. The GIG process is defined as the LP where the increment over the interval [s,s + t], s,t ≥ 0 has the CF (ϕGIG(u;λ,a,b))t. The Levy measure has a density: ν(x)=x1exp(12b2x)(a20exp(xz)g(z)dz+max[0,λ]), where g(z)=(π2a2z(J|λ|2(a2z)+N|λ|2(a2z)))1 and J and N are Bessel functions.

The Variance Gamma Process

A Variance Gamma (VG) process is defined as a Brownian motion with drift θ and volatility σ time-changed by a Gamma process. More precisely, let G = {Gt,t ≥ 0} be a Gamma process with parameters a = 1/υ > 0 and b = 1/υ > 0.8 Let W = {Wt, t ≥ 0} denote a Brownian motion, and let σ > 0 and θR; then the VG process X(VG)={Xt(VG),t0}, with parameters σ > 0, υ > 0, and θ, can be defined as Xt(VG)=θGt+σWGt. The CF is given by ΦVG(u;σ,ν,θ)=(1iuθυ+12σ2υu2)1υ. The distribution is infinitively divisible and the process X(VG)={Xt(VG),t0} is defined as the process which starts at zero, has independent and stationary increments and for which Xs+t(VG)Xs(VG) follows a VG(σt,υ/t,tθ) law over the interval [s,t + s]. The CF is: E[exp(iuXt(VG))]=ΦVG(u;σt,υ/t,tθ)=ΦVG(u;σ,υ,θ)t=(1iuθυ+12σ2υu2)tυ.

The two additional parameters in the VG distribution, which are the drift of the Brownian motion, θ, and the volatility of the time change,υ, provide control over skewness and kurtosis, respectively. Namely, when θ < 0, the distribution is negatively skewed, and vice versa. Moreover, larger values of υ indicate frequent jumps and contribute to fatter tails. The Levy measure has infinite mass, and hence a VG process has infinitely many jumps in any finite time interval. The moments of VG(σ,υ,θ) are: the mean = θ; the variance = σ2 + υθ2; skewness =θυ(3σ2+2υθ2)(σ2+υθ2)3/2; and kurtosis = 3(1 + 2υυσ4 (σ2 + υθ2)− 2). Clearly, skewness is influenced by θ, and kurtosis by υ.

The Normal Inverse Gaussian Process

The Normal Inverse Gaussian (NIG) process can be related to a Brownian motion time-changed by an Inverse Gaussian process. Let W = {Wt, t ≥ 0} be a Brownian motion and let IG = {IGt,t ≥ 0} be an IG process with parameters a = 1 and b=δα2β2,with α > 0,−α < β < α and δ > 0; then the process: Xt = βδ2IGt + δWt is an NIG process with parameters α,β,δ.9NIG(α,β,δ) has a CF given by ΦNIG(u;α,β,δ)=exp(δ(α2(β+iu)2α2β2)). This is an infinitely divisible CF. Therefore, the NIG X(NIG)={Xt(NIG),t0}, with X0(NIG)=0 has stationary and independent increments. Xt(NIG) has a NIG(α,β,t,δ) law with a CF: ΦNIG(u;α,β,tδ)=exp[tδ(α2(β+iu)2α2β2)]. The Levy measure is given by vNIG(dx)=δαπexp(βx)K1(α|x|)|x|dx, where Kλ (x) denotes the modified Bessel function of the third kind with index λ. The integral of νNIG over the real line is infinite; hence, the NIG process has infinite activity. The density of the NIG(α,β,δ) distribution is given by fNIG(x;α,β,δ)=αδπexp(δα2β2+βx)K1(αδ2+x2)δ2+x2. The moments are: mean =δβα2β2; variance α2δ(α2β2)−3/2; skewness 3βα−1δ−1/2 (α2β2)−1/4; and kurtosis =3(1+α2+4β2δα2α2β2). The NIG has semi-heavy tails, in particular fNIG(x;α,β,δ)~|x|−3/2 exp((±α + β)x) as x→±∞ up to a multiplicative constant.

The Generalized Hyperbolic Process

The Generalized Hyperbolic (GH) distribution can be represented as a normal variance- mean mixture. Let T be a GIG random variable and W be an independent standard normal variable. Then the law of TW+μT, where µ is a constant, is called normal variance-mean mixture with mixing distribution GIGfGH(x;α,β,δ,υ,μ)=0fNormal(x;μ+βw,w)fGIG(w;υ,δ,α2β2)dw. The GH distribution, GH (α,β,δ,υ), is defined in Barndorff-Nielsen (1998) through its CF ΦGH(u;α,β,δ,υ,μ)=eiμu(α2β2α2(β+iu)2)υ/2Kυ(δα2(β+iu)2)Kυ(δα2β2), where Kυ is the modified Bessel function of the third kind. This is an infinitely divisible CF. The GH Levy process X(GH)={Xt(GH),t0}, with X0(GH)=0, stationary and independently distributed increments, has the CF: E[exp(iuXt(GH))]=(ΦGH(u;α,β,δ,υ,μ))t. The density of GH (α,β,δ,υ,µ) distribution is given by fGH(x;α,β,δ,υ,μ)=a(α,β,δ,υ)(δ2+(xμ)2)(υ1/2)/2K(υ1/2)(αδ2+(xμ)2)exp(β(xμ)), a(α,β,δ,υ)=(α2β2)υ/22παυ12δυKυ(δα2β2). The GH distribution has semi-heavy tails: in particular fGH(x;α,β,δ,υ)~|x|υ−1 exp ((±α + β)x) as x→±∞ up to a multiplicative constant.

Some of the above processes are special cases of the GH process. The Variance-Gamma process can be obtained from the GH process by taking: υ = σ2/v, α(2/v)+(θ2/σ2), β = θ/σ2 and δ → 0. The Hyperbolic process (HYP) is obtained when υ = 1. In this case, X1(HYP) follows HYP(α,β,δ) with CF:ΦHYP(u;α,β,δ)=(α2β2α2(β+iu)2)1/2K1(δα2(β+iu)2)K1(δα2β2). The density reduces to fHYP(x;α,β,δ)=(α2β2)1/22πδK1(δα2β2))exp(αδ2+x2+βx), The Normal Inverse Gaussian process is obtained when υ = −1/2, thus: GH (α,β,δ, −1/2) = NIG(α,β,δ).

The normal inverse Gaussian, the Variance-Gamma, and the hyperbolic motion are Levy processes which share the property of being pure jump and infinite activity models. Their empirical performance in modeling skewness, leptokurtosis, and the implied volatility smile in option prices made them more appealing than the classical diffusions or jump-diffusion models. Their representation as time-changed Brown motions allows to model the time change which itself reflects the intensity of economic activity through news arrival and trades. The tractability of their CF allows to recover option prices through fast Fourier transform (FFT). Eberlein et al. (1998) showed that the hyperbolic distribution allows an almost perfect fit to financial data, both in spot and derivatives markets. The knowledge of the CF enables to recover the probability distribution through numerical inversion as F(x)=12+12π0eiuxΦ(u)eiuxΦ(u)iudu.

The Appendix describes the estimation of the parameters of an LP using the CF.

IV. Market Incompleteness and Esscher Transform

When uncertainty is modeled by an LP, except when X is a Brownian motion or a Poisson process, the Levy process is an incomplete model. A perfect hedge cannot be obtained and there is always a residual risk which cannot be hedged. In a Levy market, there are many different equivalent martingale measures under which the discounted asset price process is a martingale. The existence of a martingale measure is related to the absence of arbitrage, while the uniqueness of the equivalent martingale measure is related to market completeness, i.e., perfect hedging. A contingent claim can be perfectly hedged if there exists a predictable strategy which can replicate the claim in the sense that there is a dynamic portfolio, investing in a riskless bond and the asset, such that at every time point the value of the portfolio matches the value of the claim. The portfolio must be self-financing. A market is called complete if for every integral contingent claim there exists an admissible self-financing strategy replicating the claim.10

One approach for finding an equivalent martingale measure is the Esscher transform proposed by Gerber and Shiu (1994). An Esscher transform of a stock-price process induces an equivalent probability measure on the process. The Esscher parameter is determined so that the discounted price of a security is a martingale under the new probability measure. Let S(t) = S(0)eX(t),11 where {X(t}t≥0 is an LP defined on a probability space (Ω,F,P), with stationary and independent increments and X(0) = 0. For each t, the random variable X(t), seen as the continuous compounded rate of return over the t periods, has an infinitely divisible distribution with a probability density under P given by f(x,t), t > 0. The moment generating function (MGF) is assumed to exist and is defined as M(u,t)=E[euX(t)]=euxf(x,t)dx. By assuming that M(u,t) is continuous at t = 0, it follows from the infinite divisibility that M(u,t) = [M(u,1)]t. Let h be a real number such that M(h)=ehxf(x)dx exists. The Esscher transform (with parameter h) of the original distribution is defined as f(x;h)=ehxf(x)M(h). The Esscher transform (with parameter h) of the process {X (t)}t≥0 is defined as an LP process with stationary and independent increments, where the new probability density of X(t), t > 0, is f(x,t;h)=ehxf(x,t)ehyf(y,t)dy=ehxf(x,t)M(h,t).

The corresponding MGF is M(u,t;h)=euxf(x,t;h)dx=M(u+h,t)M(h,t) and M(u,t;h) = [M(u,1;h)]t. The parameter h is determined so that the modified probability measure, denoted by Q, is an equivalent martingale measure. The idea is to find h = h*, so that the discounted stock price process {ertS(t)}t≥0 is a martingale with respect to the probability measure corresponding to h*. The martingale condition is: S(0) = EQ[e–rtS(t)] = e–rtEQ[S(t)]. The parameter h* is a solution to S(0)=EQ[ertS(t)]=ertEQ[S(0)eX(t)]=ertS(0)Ep[e(h+1)X(t)]Ep[ehX(t)]=ertS(0)M(u+h,t)M(h,t). This condition is equivalent to the following equation: 1 = ertEQ[eX(t)], or ert = M(1,t;h*). The solution does not depend on t. Therefore, setting t = 1, yields er= M(1,1;h*), or in logarithm form: r = log[M(1,1;h*)] = log[M(1 + h*,1)] − log[M(h*,1)]. The risk-neutral probability measure obtained as an Esscher equivalent measure is given by: dQdP|Ft=ehXtE(ehXt)=exp(hXttlog(M(h))). The parameter h is the solution to r = log M(h + 1) − log M(h).

Let ϕ(u) = E{exp(uiX1)} denote the CF of X1. Since the MGF is, up to a change of variable u ↔−iu, a CF, yielding M(u) =ϕ(−iu) or equivalently M(iu) = ϕ(u), the CF ϕ(h) of the Esscher transform measure is given by ϕ(h)(u)=Φ(uih)Φ(ih) and remains infinitely divisible. The condition on MGF yields an identical condition on the CF, namely for the discounted price to be a martingale, the following has to hold: er=Φ(i(h+1))Φ(ih) or r = logϕ(−i(h + 1)) − logϕ(−ih). Moreover, if the characteristic triplet of the process Xt under P are (σ2,ν(dx),b), then they become, under Q, (σ(h)2,ν(h) (dx),b(h)), with σ(h)2=σ2,v(h)(dx)=vh*x(dx), and b(h)=b+h*σ2+{|x|1}x(eh*x1)v(dx). Miyahara (2004) showed that Esscher transform could be identified with the minimum entropy martingale measure.

An alternative approach for computing a risk-neutral measure, similar to the Esscher transform, can also be proposed (Carr et al., 2003). Let (Xt)t≥0 be a real-valued process with independent increments. Then (eiuXtE[eiuXt])t0 is a martingale ∀uR. For example, if the asset price St is modified as St=So exp[rt + Xt] where Xt is an LP. The resulting risk-neutral process for the log price is: log S(t) = (log S(0) + rt −log E[exp(X(t)])+ X(t)). The CF of the log price is: E[exp(iu log(S(t)))] = exp(iu((log S(0) + rt − log E[exp(X(t)])E[exp(iuX(t))].

V. Option Pricing Using Characteristic Functions

Characteristic functions were useful in simplifying the complexity of option pricing. Under martingale pricing, the value of an option is a convolution of a discounted pay-off function with the state price density. Using the Feynman-Kac formula, which stipulates that if f(S,t) satisfies a partial integro-differential equation (PIDE), with final condition: f (S,T) = g(S) for all S, then the solution is given by:12

The conditional expectation is computed with respect to a risk-neutral transition probability density p(ST | St). From Breeden and Litzenberger (1978), the state price density is: p(ST|St)exp(tt+τr(ς)dς)=2K2|K=SC(t;K,T), where C(t;K,T) is the value of a call option.13 However, for many stochastic processes, particularly Levy processes, the transition densities are often complicated and may not be readily available in closed form. The CF of the underlying stochastic process may be readily available in closed form. Using the CF, it turns out to be much easier to compute option prices as an integral in the Fourier space and interpret this integral as a Parseval identity. Let ϕ(z;t;τ) be the conditional CF of the state price density ϕ(z,t)=0eizSTexp{tTr(ς)dς}p(ST|St)dST, Lewis (2001) has shown that the option value can be expressed as a convolution in the Fourier space: C(St,τ;K)=erτ2πiωiω+eizXΦ(z)g^(z)dz,14 where ln St=Xt+rτ,g^(z)=exp(izs)g(s)ds, z=u+iw, and st= ln St. Being expressed as a complex-valued integral, the option value can thus be computed using residue calculus.

Carr and Madan (1999) proposed the use of fast Fourier transform (FFT) for pricing options. Let k = ln(K), where K is the strike price, CT(k) =value of a T – maturity call strike K, and cT (k) ≡ exp(α k)CT(k) for α > 0, the damped option price for which a Fourier transform exists. This transform is expressed as: ψT(z)=eizkcT(k)dk. Carr and Madan (1999) showed that transforms are related as follows: ψT(z)=erTΦT(z(α+1)i)α2+αz2+i(2α+1)z. Knowledge of the CF of ϕT(z − (α +1)i), which is the CF of the log of the asset price under the risk neutral-measure, implies knowledge of the Fourier transform of the value of the option. The option price can therefore be computed via Fourier inversion as: CT(k)=exp(αk)2πeizkψ(z)dz=exp(αk)π0eizkψ(z)dz.

The Fourier inversion can be approximated discretely via an N -point sum with a grid spacing of Δ in the Fourier domain. The inversion integral can be approximated using an integration rule, such as Simpson’s or the trapezoidal rule, as 0eixzψ(z)dzj=0N1ei(2πN)xzjψ˜jΔ.

The points zj are equidistant with grid spacing Δ, zj=jΔ. The value of Δ should be sufficiently small to approximate the integral well enough, while the value of NΔ should be large enough to assume the CF is equal to zero for z>z¯=NΔ. In general, the values ψ˜j are set equal to ψ˜j=ψ(zj)wj, where wj are the weights of the integration rule. Appropriate values of the coefficient α are chosen to ensure the boundedness of the truncation error.

VI. Application to Crude Oil Options: The Inverse Problem

An application of the above analysis to crude oil options is undertaken in this section with the objective of estimating, from observed options’ market values, density forecast for crude oil prices at a given maturity date. The estimation of the risk-neutral distribution is known as the inverse problem in option pricing models. While the pricing problem is concerned with computing values of options given model’s parameters, the inverse problem consists of backing out the parameters describing risk-neutral dynamics from observed prices. The inverse problem is also known as model calibration, whereby parameters are extracted from observed market values for the options. In accordance with the above analysis, crude oil prices are assumed to follow an exponential Levy process with triplet (σ(Φ),ν(Φ),γ(Φ)) under the risk-neutral measure and Φ denotes the parameters of distribution. The calibration procedure is based on minimizing the quadratic pricing error: Φ^=arg minΦ1Ni=1N(CΦ(T,Ki)Ci(T,Ki))2, under the put-call parity constraint: St+ PuttCallt = Ker(Tt), where CΦ denotes the call option computed for the exponential Levy model with triplet (σ(Φ),ν(Φ),γ(Φ)), Ci denotes the observed prices of call options for maturity T and strikes Ki. The put option values are the observed market values.15

The calibration exercise is applied to the Variance-Gamma model. The data set consists of observed crude oil options on July 5, 2005 for maturity end-September 2005; the risk-free interest was the three-month U.S. treasury bill rate; and the crude futures price was US$ 59.81 per barrel. The estimation yielded the following parameters: σ^2=0.32,υ^=0.20 θ^=0.10. The estimated σ^2 indicates high volatility characterizing the oil market; υ^ shows fat tails, implying higher probability than the normal distribution for important deviations of prices from the futures level; finally, θ^ indicates positive skewness, meaning that the market was according higher probability for upward deviations from the expected mean (Figure 1). The robustness of the results is confirmed by the use of the following linear model: A = D.q, where A is a vector of call and put options prices, D is a pay-off matrix, and q is a vector of Arrow-Debreu prices. Owing to the high volatility, market expectations can change dramatically during intra-day trading or from one day to the other and thus can change dramatically the density forecast for a given maturity time. For this reason, the calibration results need to be interpreted with caution.

Figure 1.Crude oil prices density forecast on July, 5, 2005 for end-September 2005.

These findings support the asset’s view to crude oil markets. Traders in derivatives markets are hedgers, arbitrageurs, and speculators. Many types of investors participate in the crude oil futures market, including speculative and non speculative traders. The latter group includes institutional investors (e.g., pension funds) who seek to diversify their portfolios with less correlated assets, whereas the former group includes hedge funds and commercial entities registered with the Commodity Futures Trading Commission (CFTC). The aggregate of all large-traders’ position reported to the CFTC usually represents 70-90 percent of total open interests in any given derivatives market.16 Commercial traders occasionally take speculative short-term positions during periods of large price swings. High volatility and volatility clustering increase the speculative activity and add pressure on futures prices. Furthermore, very low interest rates reduce considerably the cost of shorting bonds as well as the cost of margin requirements and increase the volume of activity in the futures market. For instance, increased demand for long contracts would exert an upward pressure on futures prices. A significant portion in crude oil price increase could be attributed to derivatives markets and to the role of crude oil as an asset rather than as a commodity.

Subsequently, the forecast of oil prices relying only on the role of crude oil as a commodity would certainly omit the significant impact of derivatives markets. The use of Levy processes and their corresponding inverse problem would allow one to study the role of asset markets in the behavior of crude oil prices. The commodity aspect is also important. To the extent that demand is acting against a short-term fixed crude oil supply and bottlenecks in refining and distribution capacity, it causes frequent and large jumps in crude oil prices and suggests the use of Levy processes for modeling these jumps.

VII. Conclusions

The paper has addressed option pricing models from the perspective of Levy processes, which offer better tools for analyzing skewness, fat tails, and stochastic volatility in high-frequency financial data than the classical diffusions or jump-diffusion models. The concept of subordination plays an important role in building an LP and amounts to measuring returns in relation to the level of activity and news, instead of calendar time. High level of activity or important news may cause higher volatility in returns. The Normal Inverse Gaussian, the Variance-Gamma, and the General Hyperbolic motions are Levy processes which are time-changed Brown motions and share the property of being pure jump and infinite activity models. Their empirical performance in modeling skewness, leptokurtosis, and the implied volatility smile in option prices was deemed consistent with data. Levy processes, however, lead to incomplete markets and an infinite number of martingale measures that are compatible with no arbitrage. The Esscher measure constitutes a procedure, among many others, for obtaining a martingale measure. The role of characteristic functions in option pricing has become prominent, particularly in the context of processes that do not have easily available distribution functions. Fourier transforms offer an efficient tool for option pricing when CFs are available in closed forms.

The paper has addressed the inverse problem and attempted to extract a risk-neutral distribution from crude oil options. The results indicate that market expectations were positively skewed, namely the market put a higher probability mass on crude oil prices remaining above the futures’ level. This outcome is in conformity with the sustained pressure on oil prices in the recent past. The Levy market model described in this paper is highly relevant to the work of the Fund. It provides an adequate tool for analyzing high-frequency data, gauge market sentiment, and design appropriate policy responses. The findings of the calibration could be relevant for policymaking. This may require assessing factors causing pressure on crude oil demand, including low interest rates and depreciating currencies, and seeking greater energy efficiency and inter-energy substitution. They also point to the importance of derivatives markets in influencing crude oil prices. A high speculative activity associated with high volatility in futures prices would lead to volatility clustering and hence greater uncertainty in crude oil futures prices. Energy modeling would need therefore to look at the role of crude oil as an asset besides its role as a commodity. The latter aspect remains important. To the extent that demand pressure is acting against a short-term fixed crude oil supply and bottlenecks in the refining and distribution capacity, frequent and sizable jumps in crude oil prices will take place. Levy processes and their corresponding inverse problems would provide a framework for assessing both the asset and commodity aspects in crude oil prices behavior.

APPENDIX

Empirical Characteristic Function and Estimation in the Frequency Domain

The lack of a tractable form of the probability density function makes estimation via maximum likelihood of the parameters of the distribution extremely difficult. Moreover, the likelihood function can be unbounded over the parameter space. Consequently, alternative methods, based on the characteristic function, were proposed, (e.g., Parzen (1962), and Feuerverger and McDunnough (1981a and 1981b)), to deal with inference problems involving such distribution. Being a Fourier transform of the probability density function the characteristic function (CF) is always bounded; it can have a closed form expression; and it retains all the information in the sample. The basic idea of the estimation in the frequency domain, called also the empirical characteristic function (ECF) procedure, is to match the CF derived from the model and the ECF obtained from the data. Because the minimization of the distance between the ECF and CF over a grid of points in the Fourier domain is equivalent to matching a finite number of moments, the ECF method is in essence equivalent to the Generalized Method of Moments (GMM). Feuerverger (1990) proves that, under some regularity conditions, the resulting estimate can be made to have arbitrarily high asymptotic efficiency provided that the sample of observations is sufficiently large and the grid of points is sufficiently fine and extended.

Suppose X1,…, Xn are i.i.d. realization of the same variable X with density f (x;θ) and distribution function Fθ (x). The parameter θRl is the parameter of interest with true value θ0. We wish to estimate θ from a realization {X1, X2,…, Xn}. Define the theoretical CF as: Φθ(u)=eiuxdFθ(x) and its empirical counterpart (ECF) as Φn(u)=eiuxdFn(x)=1n1nexp(iuXj)=1nj=1ncos(uXj)+i1nj=1nsin(uXj). The focus here is on moment conditions of the form: h(u,Xj;θ) = exp(iuXj)−ϕ(u,θ). Obviously, E(h(u, Xj;θ0)) = 0, ∀u. Suppose q discrete points u1,u2,…,uq in the Fourier space are used and define m(Xj;θ) = (Re[h(u1,Xj;θ)],…,Re[h(uq,Xj;θ)],Im[h(u1,Xj;θ)],…,Im[h(uq,Xj;θ)]). By construction E(m(Xj;θ0)) = 0. This forms 2q (usually larger than l) moment conditions. Evaluating the ECF and the CF at the grid points yields: Vn = (Re[ϕn(u1)],…,Re[ϕn(uq)], Im[ϕn(u1)],…,Im[ϕn(uq)])′ and Vθ = (Re[ϕ(u1;θ)],…,Re[ϕ(uq;θ)],Im[ϕ(u1;θ)],…,Im[ϕ(uq;θ)])′. Obviously: 1nj=1nm(Xj;θ)=VnVθ. From Feuerverger and McDunnough (1981a, Theorem 2.1) we have that T(VTVθ) converges in distribution to a 2q − dimensional normal distribution with zero mean and covariance matrix: Ω=(ΩRRΩRIΩIRΩII) where the elements in the partitions associated with uj and uk are given by:

The ECF method estimates θ by finding the θ that minimizes (VTVθ)Ω^1(VTVθ), where Ω^ is a consistent estimator of Ω. This procedure can be thought of as the second stage GMM estimation or the non-linear regression of Vn on Vθ and hence yields GMM efficient estimators. The asymptotic properties of the ECF estimator θ^ have been examined by Feuerverger and McDunnough (1981a). The basic result is that θ^ is strongly consistent and asymptotically normal with covariance 1n[(Vθθ)Ω^1(Vθθ)]1. The asymptotic efficiency of the ECF procedure depends essentially on the choice of {uj}. Feuerverger and McDunnough (1981a) argued that in some cases one can obtain full asymptotic efficiency of the procedure (in terms of achieving the Cramer-Rao lower bound) by selecting the grid of points {uj} to be sufficiently fine and extended.

Besides being a GMM method, the ECF procedure is also equivalent to the maximum likelihood method. The likelihood equation is logfθ(x)θdFn(x)=0, or alternatively logfθ(x)θd[Fn(x)Fθ(x)]=0. This last equation may be transformed. Using Parseval identity, we obtain a Fourier domain version of the likelihood equation: wθ(u)[Φn(u)Φθ(u)]du=0, where wθ(u)=12πexp(iux)logfθ(x)θ, the Fourier transform of the score function logfθ(x)θ, is the optimal weight. Obviously, when the likelihood function has no closed form expression, the optimal weight is unknown.

References

The author gratefully acknowledges the help of Abdoulaye Bio-Tchané, Abbas Mirakhor, Kexue Liu, and Saeed Mahyoub.

The Levy-Khintchine formula is also written as; ψ(u)=iγu+σ22u2+|x|1(1eiux)v(dx)+|x|<1(1eiux+iux)v(dx)

Recalling the definition of the cumulant generating function (CGF) of a random variable, it follows that ψ is the CGF of X1:ψ=ψX1, the CGF of Xt varies linearly in t, ψXt=tψX1=tψ. The law of Xt is therefore determined by the knowledge of the law of X1: the only degree of freedom in specifying a LP is to specify the distribution of Xt for a single time (say, t = 1).

The transition operator for a Markov process is defined as follows Pt(x) = E[f(x + Xt)], verifying a semi-group property PtPs = Pt+s. A semi-group Pt can be described by the means of its infinitesimal generator L, which is a linear operator defined by: Lf=limt0(Ptff)t. Let (Xt)t≥0 be an LP on R, then the infinitesimal generator of Xt is defined for any fC02(R) as Lf(x)=12σ22fx2(x)+γfx(x)+R(f(x+y)f(x)yfx(x)1|y|1)v(dy). Computation of expectations of various functionals of Levy processes can be transformed into partial integro-differential equations involving the infinitesimal generator. Due to this fact, infinitesimal generators are important tools in option pricing.

In Clark (1973), if T (t) is a lognormal with independent increments distributed as N(μ,σ12) and Z is a normal process with independent increments distributed as N(0,σ22), then X (t) = Z(T (t)) has the following lognormal-normal increments: fLNN(y)=12πσ12σ220υ3/2exp((logυμ)22σ12)exp(y22υσ22)dυ.

Let (Tt)t≥0 be a subordinator, that is an LP whose trajectories are increasing. Since Tt is a positive random variable for all t, it is described by its Laplace transform rather than the Fourier transform. Let the characteristic triplet of T be (0, ρ, a). Then the moment generating function of Tt is: E[euTt]=etl(u) u0, where L(u)=au+0(eux1)ρ(dx), and L(u) is the Laplace exponent of T.

Let (Zt)t≥0 be an LP on R with charateristic exponent ψ (u) and triplet (σ,v,γ) and let (Tt)t≥0 be a subordinatior with Laplace exponent L(u) and triplet (0, ρ,a). Then X(t) = Z(T(t)) is an LP with CF E[eiuXt]=etL(ψ(u)), and triplet (σX,vXX) is given by: σ = ασ, vx(B)=αv(B)+0psZ(B)ρ(ds), ∀BB(R), γX=αγ+0ρ(ds)|z|1zpsz(dz), where ptZ is the probality density of Zt.

The probability density of the Gamma process with mean rate t and variance υt is well known: f(u)=u1υ1euυ/υtυΓ(tυ). Its Laplace transform is E[exp(λGtυ)]=(1+λυ)tυ. It results that the VG process has a simple CFΦVG(u)=(1iθυu+σ2υ2u2)tυ.

An equivalent parameterization of the NIG process is a Brownian motion with drift θ and volatility σ computed at a random time given by an IG(1,υ) process: XNIG(t;σ,υ,θ)=θIGtυ+σW(IGtυ). The CF is E[eiuXNIG(t)]=E[exp(iθuσ2u22)IGtυ].

The question of completeness is linked with the uniqueness of the martingale measure, which is in turn linked with the mathematical predictable representation property (PRP) of a martingale. In probability theory a martingale M is said to have the PRP if, for any square-integrable random variable H, we have H=E[H]+0TasdMs for some predictable process a = {as, 0 ≤ sT}. If such a representation exists, the predictable process as will give the necessary self-financing admissible strategy.

The asset price can alternatively be specified as a stochastic differential equation: dSt= µStdt + σStdXt, where St is left limit and Xt is an LP. The solution to this equation is the well-known Doleans-Dade or stochastic exponential given by: St=S0exp(μt+σXt)st(1+σΔXs)eσΔXs, with ΔXs = XsXs– and (1 + σΔXs)> 0.

When the asset price St is modeled as an exponential LP, Ito’s formula applied to f(St) shows that under the risk neutral measure St has the following infinitesimal generator: Ltf(x)=12c2σt2x2f(x)+rtxf(x)+R[f(x+σtxy)f(x)σtxyf(x)]v˜t(dy), where [σt,v˜(dx),rt] are the triplet under the risk-neutral measure. Let f(t S) be the solution of the following Cauchy problem: ft+Ltfrtf=0,f(T,S)=g(S). Then f(t,S) admits the representation f(t,S)=EQ[exp{tTrςdς}g(ST)|Ft].

The risk-neutral probabilities, discounted at the risk-free rate of interest, are interpreted economically as the prices of Arrow-Debreu (AD) securities, or the state prices. An AD security is a primitive security associated with a particular future state of the world; it pays $1 if that state occurs, and nothing otherwise. All contingent claims and derivatives can be expressed in terms of a portfolio of AD securities and priced accordingly. Given a vector of state prices, the price of any contingent claim may be determined by multiplying the claim’s payoff in each state by the corresponding state price, and then summing over all states.

This formula is based on Parseval’s identity: g(ST)p(ST)dST=12πg^(z)Φ*(z)dz, where g^(z) is the Fourier transform of g(ST) and ϕ*(z) is the conjugate of ϕ(z).

Cont and Tankov (2004) argued that the calibration problem could be an ill-posed problem and proposed the use of relative entropy, which is the Kullback-Leibler distance for measuring the proximity of two equivalent probability measures, as a regularization method with the prior distribution estimated from the statistical data via the maximum likelihood method. This regularization will enable to find a unique martingale measure.

Data for February 1, 2005 indicated that commercial traders held 67.1 percent of the open long positions and 69.2 percent of the short positions in crude oil futures on the New York Mercantile Exchange (NYMEX).

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