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Part C Lévy Processes and Finance

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Part C Lévy Processes and Finance Part C Levy´ Processes and Finance Matthias Winkel1 University of Oxford HT 2007 1Departmental lecturer (Institute of Actuaries and Aon Lecturer in Statistics) at the Department of Statistics, University of Oxford MS3 Levy´ Processes and Finance Matthias Winkel – 16 lectures HT 2007 Prerequisites Part A Probability is a prerequisite. BS3a/OBS3a Applied Probability or B10 Martin- gales and Financial Mathematics would be useful, but are by no means essential; some material from these courses will be reviewed without proof. Aims L´evy processes form a central class of stochastic processes, contain both Brownian motion and the Poisson process, and are prototypes of Markov processes and semimartingales. Like Brownian motion, they are used in a multitude of applications ranging from biology and physics to insurance and finance. Like the Poisson process, they allow to model abrupt moves by jumps, which is an important feature for many applications. In the last ten years L´evy processes have seen a hugely increased attention as is reflected on the academic side by a number of excellent graduate texts and on the industrial side realising that they provide versatile stochastic models of financial markets. This continues to stimulate further research in both theoretical and applied directions. This course will give a solid introduction to some of the theory of L´evy processes as needed for financial and other applications. Synopsis Review of (compound) Poisson processes, Brownian motion (informal), Markov property. Connection with random walks, [Donsker’s theorem], Poisson limit theorem. Spatial Poisson processes, construction of L´evy processes. Special cases of increasing L´evy processes (subordinators) and processes with only positive jumps. Subordination. Examples and applications. Financial models driven by L´evy processes. Stochastic volatility. Level passage problems. Applications: option pricing, insurance ruin, dams. Simulation: via increments, via simulation of jumps, via subordination. Applications: option pricing, branching processes. Reading • J.F.C. Kingman: Poisson processes. Oxford University Press (1993), Ch.1-5, 8 • A.E. Kyprianou: Introductory lectures on fluctuations of L´evy processes with Ap- plications. Springer (2006), Ch. 1-3, 8-9 • W. Schoutens: L´evy processes in finance: pricing financial derivatives. Wiley (2003) Further reading • J. Bertoin: L´evy processes. Cambridge University Press (1996), Sect. 0.1-0.6, I.1, III.1-2, VII.1 • K. Sato: L´evy processes and infinite divisibility.Cambridge University Press (1999), Ch. 1-2, 4, 6, 9 Lecture 1 Introduction Reading: Kyprianou Chapter 1 Further reading: Sato Chapter 1, Schoutens Sections 5.1 and 5.3 In this lecture we give the general definition of a L´evy process, study some examples of L´evy processes and indicate some of their applications. By doing so, we will review some results from BS3a Applied Probability and B10 Martingales and Financial Mathematics. 1.1 Definition of L´evy processes Stochastic processes are collections of random variables Xt, t ≥ 0 (meaning t ∈ [0, ∞) as opposed to n ≥ 0 by which means n ∈ N = {0, 1, 2,...}). For us, all Xt, t ≥ 0, take values in a common state space, which we will choose specifically as R (or [0, ∞) or Rd for some d ≥ 2). We can think of Xt as the position of a particle at time t, changing as t varies. It is natural to suppose that the particle moves continuously in the sense that t 7→ Xt is continuous (with probability 1), or that it has jumps for some t ≥ 0: ∆Xt = Xt+ − Xt = lim Xt+ε − lim Xt ε. − ε 0 ε 0 − ↓ ↓ We will usually suppose that these limits exist for all t ≥ 0 and that in fact Xt+ = Xt, i.e. that t 7→ Xt is right-continuous with left limits Xt for all t ≥ 0 almost surely. The − path t 7→ Xt can then be viewed as a random right-continuous function. Definition 1 (L´evy process) A real-valued (or Rd-valued) stochastic process X = (Xt)t 0 is called a L´evy process if ≥ − − ≥ (i) the random variables Xt0 ,Xt1 Xt0 ,...,Xtn Xtn−1 are independent for all n 1 and 0 ≤ t0 < t1 <...<tn(independent increments), (ii) Xt+s − Xt has the same distribution as Xs for all s, t ≥ 0 (stationary increments), (iii) the paths t 7→ Xt are right-continuous with left limits (with probability 1). It is implicit in (ii) that P(X0 = 0) = 1 (choose s = 0). 1 2 Lecture Notes – MS3b L´evy Processes and Finance – Oxford HT 2008 Figure 1.1: Variance Gamma process and a L´evy process with no positive jumps Here the independence of n random variables is understood in the following sense: ( ) Definition 2 (Independence) Let Y j be an Rdj -valued random variable for j = 1,...,n. The random variables Y (1),...,Y (n) are called independent if, for all (Borel ( ) measurable) C j ⊂ Rdj P(Y (1) ∈ C(1),...,Y (n) ∈ C(n))= P(Y (1) ∈ C(1)) . P(Y (n) ∈ C(n)). (1) (j) (j1) (jn) An infinite collection (Y )j J is called independent if Y ,...,Y are independent for ∈ (1) (n) every finite subcollection. Infinite-dimensional random variables (Yi )i I1 ,..., (Yi )i In (1) (n) ∈ ⊂ ∈ are called independent if (Yi )i F1 ,..., (Yi )i Fn are independent for all finite Fj Ij. ∈ ∈ (j) (j) (j) × × (j) (j) It is sufficient to check (1) for rectangles of the form C =(a1 , b1 ] . (adj , bdj ]. 1.2 First main example: Poisson process Poisson processes are L´evy processes. We recall the definition as follows. An N(⊂ R)- valued stochastic process X =(Xt)t 0 is called a Poisson process with rate λ ∈ (0, ∞) if ≥ X satisfies (i)-(iii) and Poi P (λt)k λt ≥ ≥ (iv) (Xt = k)= k! e− , k 0, t 0 (Poisson distribution). The Poisson process is a continuous-time Markov chain. We will see that all L´evy pro- cesses have a Markov property. Also recall that Poisson processes have jumps of size 1 (spaced by independent exponential random variables Zn = Tn+1−Tn, n ≥ 0, with param- λs eter λ, i.e. with density λe− , s ≥ 0). In particular, {t ≥ 0 : ∆Xt =6 0} = {Tn, n ≥ 1} and ∆XTn = 1 almost surely (short a.s., i.e. with probability 1). We can define more general L´evy processes by putting Xt Ct = Yk, t ≥ 0, =1 Xk for a Poisson process (Xt)t 0 and independent identically distributed Yk, k ≥ 1. Such ≥ processes are called compound Poisson processes. The term “compound” stems from the ◦ representation Ct = S Xt = SXt for the random walk Sn = Y1 + . + Yn. You may think of Xt as the number of claims up to time t and of Yk as the size of the kth claim. Recall (from BS3a) that its moment generating function, if it exists, is given by γY1 E(exp{γCt}) = exp{−λt(E(e − 1)}. This will be an important building block of a general L´evy process. Lecture 1: Introduction 3 Figure 1.2: Poisson process and Brownian motion 1.3 Second main example: Brownian motion Brownian motion is a L´evy process. We recall (from B10b) the definition as follows. An R-valued stochastic process X =(Xt)t 0 is called Brownian motion if X satisfies (i)-(ii) ≥ and BM (iii) the paths t 7→ Xt are continuous almost surely, 2 (iv)BM P(X ≤ x)= x 1 exp − y dy, x ∈ R, t> 0. (Normal distribution). t √2πt 2t −∞ The paths of BrownianR motion aren continuous,o but turn out to be nowhere differentiable (we will not prove this). They exhibit erratic movements at all scales. This makes Brownian√ motion an appealing model for stock prices. Brownian motion has the scaling property ( cXt/c)t 0 ∼ X where “∼” means “has the same distribution as”. ≥ Brownian motion will be the other important building block of a general L´evy process. The canonical space for Brownian paths is the space C([0, ∞), R) of continuous real- valued functions f : [0, ∞) → R which can be equipped with the topology of locally uniform convergence, induced by the metric k d(f,g)= 2− min{dk(f,g), 1}, where dk(f,g) = sup |f(x) − g(x)|. [0 ] k 1 x ,k X≥ ∈ This metric topology is complete (Cauchy sequences converge) and separable (has a countable dense subset), two attributes important for the existence and properties of limits. The bigger space D([0, ∞), R) of right-continuous real-valued functions with left limits can also be equipped with the topology of locally uniform convergence. The space is still complete, but not separable. There is a weaker metric topology, called Skorohod’s topology, that is complete and separable. In the present course we will not develop this and only occasionally use the familiar uniform convergence for (right-continuous) functions f, fn : [0,k] → R, n ≥ 1: sup |fn(x) − f(x)|→ 0, as n →∞, x [0,k] ∈ which for stochastic processes X,X(n), n ≥ 1, with time range t ∈ [0, T ] takes the form | (n) − |→ →∞ sup Xt Xt 0, as n , t [0,T ] ∈ and will be as a convergence in probability or as almost sure convergence (from BS3a or 2 2 2 B10a) or as L -convergence, where Zn → Z in the L -sense means E(|Zn − Z| ) → 0. 4 Lecture Notes – MS3b L´evy Processes and Finance – Oxford HT 2008 1.4 Markov property The Markov property is a consequence of the independent increments property (and the stationary increments property): Proposition 3 (Markov property) Let X be a L´evy process and t ≥ 0 a fixed time, then the pre-t process (Xr)r t is independent of the post-t process (Xt+s − Xt)s 0, and the ≤ ≥ post-t process has the same distribution as X.
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