Jump models in financial modelling

Introduction

This is an outline of a seminar's contents.

The main reference for the seminar is Rama Cont and Tankov. Purely mathematical texts on the same subject are Protter and Jacod and Shiryaev. Texts being more devoted to finance are Shreve and Shiryaev.

Home reading

Rama Cont and Tankov: Chapter 1 and the introduction of each Chapter 2--15.

Protter: Chapter I, in particular section 4.

Prior knowledge

• measures, measurable functions - Radon Nikodym theorem - random variables - characteristic functions (Fourier transforms), how to find moments from characteristic functions - moment generating functions (how to find moments from characteristic functions)

• special distributions (exponential - gamma - Gaussian)

• convergence of random variables (almost sure, in probability, in distribution)

• stochastic process - cadlag and caglad - filtrations and histories - non-anticipating (adapted) - stopping time - martingale - optional sampling (stopping) theorem

• Levy process - characterization of continuous Levy processes -Poisson process - compensated Poisson process - counting process - compound Poisson process - characteristic function of a compound Poisson process

Examples of Levy processes

Concepts and facts

• point processes - marked point processes - characterization of Poisson and compound Poisson processes (without proof)
• jump diffusion - Levy measure of jump diffusion - Fourier transform of jump diffusion
• infinitely divisible distribution - convolution semigroups - Levy processes have infinitely divisible distributions - examples (Gaussian, Poisson, compound Poisson, Gamma, Cauchy)
• for every infinitely divisible distribution there is a Levy process (without proof)
• Fourier transform of Levy processes ${\varphi }_t(u$) - zeros of ${\varphi }_t(u)$ - dependence of $t$ - examples (Gaussian, Poisson, compound Poisson, Gamma, Cauchy)
• Gamma process as limit of compound Poisson processes (via Fourier transforms) - limit behaviour of the Levy measures - Gamma process is FV-process - Levy measure of Gamma process - order of singularity at zero
• Cauchy process as limit of compound Poisson processes (via Fourier transforms) - limit behaviour of the Levy measures - Levy measure of Cauchy process - order of singularity at zero

Review questions

1. Explain the notions of a point process and a marked point process.
2. Which Levy processes can be characterized by path properties ?
3. Which path properties characterize special Levy processes ?
4. What are jump diffusions ? Give the Fourier transform of jump diffusions.
5. Explain the notion of infinitely divisible distributions. What is the relation between infinitely divisble distributions and Levy processes ?
6. Explain the Gamma process and its Fourier transform.
7. Describe, how a Gamma process can be approximated by jump diffusions. What does it tell us about small and large jumps ?
8. Explain the Cauchy process and its Fourier transform.
9. Describe, how a Cauchy process can be approximated by jump diffusions. What does it tell us about small and large jumps ?
10. Which compound Poisson processes can be written as a linear combination of independent Poisson processes ?

\end{enumerate}

Problems

1. Analyze the Levy processes with the following Fourier transforms (expectation, variance, path properties, decomposition as a jump diffusion, Levy measure, jump intensity, jump height distribution, martingale property (yes/no), compensator).
   1. $\log {\varphi }_t(u)=t(-iu-5u^2+e^{2iu}+e^{-iu/2}-2)$
   2. $\log {\varphi }_t(u)=t(iu+3/2e^{2iu}+1/2e^{-iu/2}-2)$
   3. $\log {\varphi }_t(u)=t(iu-u^2)$
   4. $\log {\varphi }_t(u)=t(e^{3iu}-1-3iu)$
2. Find the Fourier transform of a jump diffusion with variance 2, jumping with intensity 3, having jump heights +1 and -1 with equal probability.
3. Find the Fourier transform of a jump diffusion with variance 1, jumping with intensity 1, having jump heights uniformly distributed on $[-2,0]$.
4. Find the Levy measure of a sum of five independent Poisson processes with intensities $1,2,\dots ,5$.
5. Find the Levy measure of a linear combination of five independent Poisson processes with intensity 1 and weights $1,2,\dots ,5$.

Proofs

1. Is every driftless (i.e. centered) process a martingale ? (Give a counter example for the general case. Prove it for processes with independent increments.)
2. Any finite sum of independent Poisson processes is a Poisson process. Find the Levy measure.
3. Any linear combination of independent Poisson processes is a compound Poisson process. Find the Levy measure.
4. For every finite measure $\nu |ℬ(ℝ)$ there is a compound Poisson process with Levy measure $\nu$.
5. Show that the characteristic function of a Levy process satisfies ${\varphi }_t(u)=\exp (t\psi (u))$.
6. Let $(X_t)$ be a Levy process. Show that $Z_t=e^{iu{X}_t}/E(e^{iu{X}_t})$ is a martingale.

Jump measures and decomposition of Levy processes

Concepts and facts

• Levy processes with uniformly bounded jumps have moments of all orders (without proof)
• counting measure of a finite set - representation of sums as integrals
• jump measure $N_t(B)$ of a cadlag process - jump heights in sets $B$ bounded away from zero - finiteness of the jump measure of a cadlag process
• properties of $B\mapsto N_t(B)$ - properties of $t\mapsto N_t(B)$
• Poissonian jump measure (two properties) - Levy processes have Poissonian jump measures
• Levy measure of a Levy process - Levy measures are bounded on $(|x|\ge ϵ)$, \epsilon>0
• $Z_t=\underset{B}{\int }f(x)N_t(dx)$ is a compound Poisson process for $\bar{B}\subseteq ℝ\setminus \{0\}$ - expectation and variance of $(Z_t)$ - Fourier transform of $(Z_t)$
• elimination of big jumps from Levy processes - Levy processes are semimartingales
• moments of Levy processes and Levy measures
• relation between quadratic variation and the singularity of Levy measures
• decomposition of Levy processes (big jumps - continuous part - compensated small jumps) - relation to the Fourier transform - Levy-Khintchine formula - uniquenesss (without proof) - predictable characteristics $(a,{\sigma }^2,\nu )$ (w.r.t. a particular centering function $h(x)$)
• characterization of FV-Levy processes (without proof)

Review questions

1. How many jumps can occur on a single path of a stochastic process ?
2. Explain, why the jump measure of a Levy process leads to Poisson processes.
3. What is the Levy measure of a Levy process ?
4. Explain the integral representation of sums of jump expressions.
5. What is a Poissonian jump measure ? Why are the jump measures of Levy processes of Poissonian type ?
6. How to extract big jumps from a stochastic process ?
7. Refer the moment properties of integrals w.r.t. Poissonian jump measures.
8. Discuss the properties of the singularity of a Levy measure.
9. Describe the basic steps of the decomposition of a Levy process. What are the predictable characteristics ? What about uniqueness ?

Problems

1. Let $(X_t)$ be a Levy process with Levy measure
   1. $\nu =2{ϵ}_1+{ϵ}_{-1}$,
   2. $\nu (dx)=1_{[-1,1]}(x)dx$.
2. Find expectation and variance of
   1. $\sum _{s\le t}\mathrm{\Delta }{X}_s{1}_{(\mathrm{\Delta }{X}_s>1/2)}$
   2. $\sum _{s\le t}(\mathrm{\Delta }{X}_s{)}^2{1}_{(\mathrm{\Delta }{X}_s<-1/2)}$
3. Find the moments and the Fourier transform of a Levy process with characteristics (let $h(x)=x{1}_{(|x|\le 1)}$):
   1. $(-1,0,e^{-x}{1}_{(x>0)}dx)$
   2. $(0,1,x^{-1/2}{1}_{(x>0)}dx)$
   3. $(0,0,x^2{e}^{-x^2}dx)$
   4. $(0,0,1/x^2{e}^{-x^2}dx)$

Proofs

1. Every Levy measure $\nu$ satisfies $\nu (|x|\ge 1)<\mathrm{\infty }$.
2. The jump measure of any Levy process is a Poisson jump measure.
3. Every Levy process is a semimartingale.
4. Let $(N_t(B))$ be a Poisson jump measure and let $\nu (B)$ be its Levy measure. Prove the formulas:
   1. $E(\int f(x)N_t(dx))=t\int f(x)\nu (dx)$
   2. $V(\int f(x)N_t(dx))=t\int f^2(x)\nu (dx)$
   3. $E(\exp (\int f(x)N_t(dx))))=\exp (t\int (e^{f(x)}-1)\nu (dx))$
5. Every Levy measure $\nu$ satisfies $\underset{|x|\le 1}{\int }{x}^2\nu (dx)<\mathrm{\infty }$.

Stochastic analysis for processes with jumps

Concepts and facts

• general Ito-formula - proof by induction
• Ito-formula for processes with isolated jumps - direct proof
• solving $d{S}_t=S_{t-}d{X}_t$ when $(X_t)$ is a semimartingale with isolated jumps
• Poisson process $N_t$: - solving $d{S}_t=S_{t-}d{N}_t$ - solving $d{S}_t=\alpha S_{t-}dt+\sigma S_{t-}d{N}_t$ - martingale solutions

Review questions

1. Explain the solution of $d{S}_t=S_{t-}d{X}_t$ when $(X_t)$ has isolated jumps.
2. What is the stochastic exponential of a compound Poisson process ?

Problems

1. Find the solution of $d{S}_t=-S_{t-}dt+2S_{t-}d{N}_t$ where $(N_t)$ is a Poisson process with intensity $\lambda$.
2. In the preceding problem choose $\lambda$ such that $(S_t)$ is a martingale.
3. Find the solution of $d{S}_t=S_{t-}dt-S_{t-}/2d{N}_t$ where $(N_t)$ is a Poisson process with intensity $\lambda$.
4. In the preceding problem choose $\lambda$ such that $(S_t)$ is a martingale.
5. Find the solution of $d{S}_t=-S_{t-}dt+S_{t-}d{W}_t+2S_{t-}d{N}_t$ where $(W_t)$ is a Wiener process and $(N_t)$ is an independent Poisson process with intensity $\lambda$.
6. In the preceding problem choose $\lambda$ such that $(S_t)$ is a martingale.
7. Let $S_t=W_t+N_t$ where $(W_t)$ is a Wiener process and $(N_t)$ is an independent Poisson process with intensity $\lambda$. Expand $e^{t-S_t}$ by Ito's formula.
8. Let $(S_t)$ be the solution of $d{S}_t=-S_{t-}dt+2S_{t-}d{N}_t$ where $(N_t)$ is a Poisson process with intensity $\lambda$. Expand $e^{S_t+2t}$ by Ito's formula.

Financial models with jumps, pricing and hedging

Concepts and facts

• equivalent change of measure for Poisson processes (Escher transform) - existence of transforms for arbitrary intensities
• Poissonian stock models - risk neutral models - criterion for NA property - completeness - hedging
• Poisson-diffusion stock models - risk neutral models - criterion for NA property - incompleteness - hedging

Review questions

1. Describe Poissonian stock models. Which of them are risk neutral mdoels ?
2. Discuss the NA property for Poissonian stock models.
3. Discuss completeness of Poissonian stock models.
4. Describe Poisson-diffusion stock models. Which of them are risk neutral mdoels ?
5. Discuss the NA property for Poisson-diffusion stock models.
6. Discuss completeness of Poisson-diffusion stock models.

Proofs

1. Let $(N_t)$ be a Poisson process under $P$. Show that for every $\lambda >0$ one may find an equivalent probability measure $Q$ such that $(N_t)$ has intensity $\lambda$.