Lévy Process as Combination of a Brownian Motion with Drift and Infinite Sum of Independent Compound Poisson Processes: Introduction to Martingales. Part 4.

Every piecewise constant Lévy process Xt0 can be represented in the form for some Poisson random measure with intensity measure of the form ν(dx)dt where ν is a finite measure, defined by

ν(A) = E[#{t ∈ [0,1] : ∆Xt0 ≠ 0, ∆Xt0 ∈ A}], A ∈ B(Rd) —– (1)

Given a Brownian motion with drift γt + Wt, independent from X0, the sum Xt = Xt0 + γt + Wt defines another Lévy process, which can be decomposed as:

Xt = γt + Wt + ∑s∈[0,t] ΔXs = γt + Wt + ∫[0,t]xRd xJX (ds x dx) —– (2)

where JX is a Poisson random measure on [0,∞[×Rd with intensity ν(dx)dt.

Can every Lévy process be represented in this form? Given a Lévy process Xt, we can still define its Lévy measure ν as above. ν(A) is still finite for any compact set A such that 0 ∉ A: if this were not true, the process would have an infinite number of jumps of finite size on [0, T], which contradicts the cadlag property. So ν defines a Radon measure on Rd \ {0}. But ν is not necessarily a finite measure: the above restriction still allows it to blow up at zero and X may have an infinite number of small jumps on [0, T]. In this case the sum of the jumps becomes an infinite series and its convergence imposes some conditions on the measure ν, under which we obtain a decomposition of X.

Let (Xt)t≥0 be a Lévy process on Rd and ν its Lévy measure.

ν is a Radon measure on Rd \ {0} and verifies:

|x|≤1 |x|2 v(dx) < ∞

The jump measure of X, denoted by JX, is a Poisson random measure on [0,∞[×Rd with intensity measure ν(dx)dt.

∃ a vector γ and a d-dimensional Brownian motion (Bt)t≥0 with covariance matrix A such that

Xt = γt + Bt + Xtl + limε↓0 X’εt —– (3)

where

Xtl = ∫|x|≥1,s∈[0,t] xJX (ds x dx)

X’εt = ∫ε≤|x|<1,s∈[0,t] x{JX (ds x dx) – ν(dx)ds}

≡ ∫ε≤|x|<1,s∈[0,t] xJ’X (ds x dx)

The terms in (3) are independent and the convergence in the last term is almost sure and uniform in t on [0,T].

The Lévy-Itô decomposition entails that for every Lévy process ∃ a vector γ, a positive definite matrix A and a positive measure ν that uniquely determine its distribution. The triplet (A,ν,γ) is called characteristic tripletor Lévy triplet of the process Xt. γt + Bt is a continuous Gaussian Lévy process and every Gaussian Lévy process is continuous and can be written in this form and can be described by two parameters: the drift γ and the covariance matrix of Brownian motion, denoted by A. The other two terms are discontinuous processes incorporating the jumps of Xt and are described by the Lévy measure ν. The condition ∫|y|≥1 ν(dy) < ∞ means that X has a finite number of jumps with absolute value larger than 1. So the sum

Xtl = ∑|∆Xs|≥10≤s≤t ∆Xs

contains almost surely a finite number of terms and Xtl is a compound Poisson process. There is nothing special about the threshold ∆X = 1: for any ε > 0, the sum of jumps with amplitude between ε and 1:

Xεt = ∑1>|∆Xs|≥ε0≤s≤t ∆Xs = ∫ε≤|x|≤1,s∈[0,t] xJX(ds x dx) —– (4)

is again a well-defined compound Poisson process. However, contrarily to the compound Poisson case, ν can have a singularity at zero: there can be infinitely many small jumps and their sum does not necessarily converge. This prevents us from making ε go to 0 directly in (4). In order to obtain convergence we have to center the remainder term, i.e., replace the jump integral by its compensated version,

X’εt = ∫ε≤|x|≤1,s∈[0,t] xJ’X (ds x dx) —– (5)

which, is a martingale. While Xε can be interpreted as an infinite superposition of independent Poisson processes, X’εshould be seen as an infinite superposition of independent compensated, i.e., centered Poisson processes to which a central-limit type argument can be applied to show convergence. An important implication of the Lévy-Itô decomposition is that every Lévy process is a combination of a Brownian motion with drift and a possibly infinite sum of independent compound Poisson processes. This also means that every Lévy process can be approximated with arbitrary precision by a jump-diffusion process, that is by the sum of Brownian motion with drift and a compound Poisson process.

Cadlag Stochasticities: Lévy Processes. Part 1.

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A compound Poisson process with a Gaussian distribution of jump sizes, and a jump diffusion of a Lévy process with Gaussian component and finite jump intensity.

A cadlag stochastic process (Xt)t≥0 on (Ω,F,P) with values in Rd such that X0 = 0 is called a Lévy process if it possesses the following properties:

1. Independent increments: for every increasing sequence of times t0 . . . tn, the random variables Xt0, Xt1 − Xt0 , . . . , Xtn − Xtn−1 are independent.

2. Stationary increments: the law of Xt+h − Xt does not depend on t.

3. Stochastic continuity: ∀ε > 0, limh→0 P(|Xt+h − Xt| ≥ ε) = 0.

A sample function x on a well-ordered set T is cadlag if it is continuous from the right and limited from the left at every point. That is, for every t0 ∈ T, t ↓ t0 implies x(t) → x(t0), and for t ↑ t0, limt↑t0 x(t)exists, but need not be x(t0). A stochastic process X is cadlag if almost all its sample paths are cadlag.

The third condition does not imply in any way that the sample paths are continuous, and is verified by the Poisson process. It serves to exclude processes with jumps at fixed (nonrandom) times, which can be regarded as “calendar effects” and means that for given time t, the probability of seeing a jump at t is zero: discontinuities occur at random times.

If we sample a Lévy process at regular time intervals 0, ∆, 2∆, . . ., we obtain a random walk: defining Sn(∆) ≡ Xn∆, we can write Sn(∆) = ∑k=0n−1 Yk where Yk = X(k+1)∆ − Xk∆ are independent and identically dependent random variables whose distribution is the same as the distribution of X. Since this can be done for any sampling interval ∆ we see that by specifying a Lévy process one can specify a whole family of random walks Sn(∆).

Choosing n∆ = t, we see that for any t > 0 and any n ≥ 1, Xt = Sn(∆) can be represented as a sum of n independent and identically distributed random variables whose distribution is that of Xt/n: Xt can be “divided” into n independent and identically distributed parts. A distribution having this property is said to be infinitely divisible.

A probability distribution F on Rd is said to be infinitely divisible if for any integer n ≥ 2, ∃ n independent and identically distributed random variables Y1, …Yn such that Y1 + … + Yn has distribution F.

Since the distribution of independent and identically distributed sums is given by convolution of the distribution of the summands, denoting by μ the distribution of Yk-s, F = μ ∗ μ ∗ ··· ∗ μ is the nth convolution of μ. So an infinitely divisible distribution can also be defined as a distribution F for which the nth convolution root is still a probability distribution, for any n ≥ 2.

fig1

Thus, if X is a Lévy process, for any t > 0 the distribution of Xt is infinitely divisible. This puts a constraint on the possible choices of distributions for Xt: whereas the increments of a discrete-time random walk can have arbitrary distribution, the distribution of increments of a Lévy process has to be infinitely divisible.

The most common examples of infinitely divisible laws are: the Gaussian distribution, the gamma distribution, α-stable distributions and the Poisson distribution: a random variable having any of these distributions can be decomposed into a sum of n independent and identically distributed parts having the same distribution but with modified parameters. Conversely, given an infinitely divisible distribution F, it is easy to see that for any n ≥ 1 by chopping it into n independent and identically distributed components we can construct a random walk model on a time grid with step size 1/n such that the law of the position at t = 1 is given by F. In the limit, this procedure can be used to construct a continuous time Lévy process (Xt)t≥0 such that the law of X1 if given by F. Let (Xt)t≥0 be a Lévy process. Then for every t, Xt has an infinitely divisible distribution. Conversely, if F is an infinitely divisible distribution then ∃ a Lévy process (Xt) such that the distribution of X1 is given by F.

Forward Pricing in Commodity Markets. Note Quote.

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We use the Hilbert space

Hα := {f ∈ AC(R+,C) : ∫0 |f′(x)|2 eαx dx < ∞}

where AC(R+,C) denotes the space of complex-valued absolutely continuous functions on R+. We endow Hα with the scalar product ⟨f,g⟩α := f(0) g(0) + ∫0 f′(x) g(x) eαx dx, and denote the associated norm by ∥ · ∥αFilipović shows that (Hα, ∥ · ∥α) is a separable Hilbert space. This space has been used in Filipović for term structure modelling of bonds and many mathematical properties have been derived therein. We will frequently refer to Hα as the Filipović space.

We next introduce our dynamics for the term structure of forward prices in a commodity market. Denote by f (t, x) the price at time t of a forward contract where time to delivery of the underlying commodity is x ≥ 0. We treat f as a stochastic process in time with values in the Filipović space Hα. More specifically, we assume that the process {f(t)}t≥0 follows the HJM-Musiela model which we formalize next. The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curve – instantaneous forward rate curve in particular (as opposed to simple forward rates). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the Gaussian Heath–Jarrow–Morton (HJM) model of forward rates. For direct modeling of simple forward rates the Brace–Gatarek–Musiela model represents an example.

On a complete filtered probability space (Ω,{Ft}t≥0,F,P), where the filtration is assumed to be complete and right continuous, we work with an Hα-valued Lévy process {L(t)}t≥0 for the construction of Hα-valued Lévy processes). In mathematical finance, Lévy processes are becoming extremely fashionable because they can describe the observed reality of financial markets in a more accurate way than models based on Brownian motion. In the ‘real’ world, we observe that asset price processes have jumps or spikes, and risk managers have to take them into consideration. Moreover, the empirical distribution of asset returns exhibits fat tails and skewness, behavior that deviates from normality. Hence, models that accurately fit return distributions are essential for the estimation of profit and loss (P&L) distributions. Similarly, in the ‘risk-neutral’ world, we observe that implied volatilities are constant neither across strike nor across maturities as stipulated by the Black and Scholes. Therefore, traders need models that can capture the behavior of the implied volatility smiles more accurately, in order to handle the risk of trades. Lévy processes provide us with the appropriate tools to adequately and consistently describe all these observations, both in the ‘real’ and in the ‘risk-neutral’ world. We assume that L has finite variance and mean equal to zero, and denote its covariance operator by Q. Let f0 ∈ Hα and f be the solution of the stochastic partial differential equation (SPDE)

df(t) = ∂xf(t)dt + β(t)dt + Ψ(t)dL(t), t≥0,f(0)=f

where β ∈ L ((Ω × R+, P, P ⊗ λ), Hα), P being the predictable σ-field, and

Ψ ∈ L2L(Hα) := ∪T>0 L2L,T (Hα)

where the latter space is defined as in Peszat and Zabczyk. For t ≥ 0, denote by Ut the shift semigroup on Hα defined by Utf = f(t + ·) for f ∈ Hα. It is shown in Filipović that {Ut}t≥0 is a C0-semigroup on Hα, with generator ∂x. Recall, that any C0-semigroup admits the bound ∥Utop ≤ Mewt for some w, M > 0 and any t ≥ 0. Here, ∥ · ∥op denotes the operator norm. Thus s → Ut−s β(s) is Bochner-integrable (The Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions). and s → Ut−s Ψ(s) is integrable with respect to L. The unique mild solution of SPDE is

f(t) = Utf0 + ∫t0 Ut−s β(s)ds+ ∫t0 Ut−s Ψ(s)dL(s)

If we model the forward price dynamics f in a risk-neutral setting, the drift coefficient β(t) will naturally be zero in order to ensure the (local) martingale property (In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters. In particular, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.) of the process t → f(t, τ − t), where τ ≥ t is the time of delivery of the forward. In this case, the probability P is to be interpreted as the equivalent martingale measure (also called the pricing measure). However, with a non-zero drift, the forward model is stated under the market probability and β can be related to the risk premium in the market. In energy markets like power and gas, the forward contracts deliver over a period, and forward prices can be expressed by integral operators on the Filipović space applied on f. The dynamics of f can also be considered as a model for the forward rate in fixed-income theory. This is indeed the traditional application area and point of analysis of the SPDE. Note, however, that the original no-arbitrage condition in the HJM approach for interest rate markets is different from the no-arbitrage condition. If f is understood as the forward rate modelled in the risk-neutral setting, there is a no-arbitrage relationship between the drift β, the volatility σ and the covariance of the driving noise L.