# 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[1]. Purely mathematical texts on the same subject are Protter[2] and Jacod and Shiryaev[3]. Texts being more devoted to finance are Shreve[4] and Shiryaev[5].

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

Protter[2]: Chapter I, in particular section 4.

### Prior knowledge

• 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 ${\displaystyle \phi _{t}(u}$) - zeros of ${\displaystyle \phi _{t}(u)}$ - dependence of ${\displaystyle t}$ - 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. ${\displaystyle \log \phi _{t}(u)=t(-iu-5u^{2}+e^{2iu}+e^{-iu/2}-2)}$
2. ${\displaystyle \log \phi _{t}(u)=t(iu+3/2e^{2iu}+1/2e^{-iu/2}-2)}$
3. ${\displaystyle \log \phi _{t}(u)=t(iu-u^{2})}$
4. ${\displaystyle \log \phi _{t}(u)=t(e^{3iu}-1-3iu)}$
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 ${\displaystyle [-2,0]}$.
4. Find the Levy measure of a sum of five independent Poisson processes with intensities ${\displaystyle 1,2,\ldots ,5}$.
5. Find the Levy measure of a linear combination of five independent Poisson processes with intensity 1 and weights ${\displaystyle 1,2,\ldots ,5}$.

### 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 ${\displaystyle \nu |{\mathcal {B}}(\mathbb {R} )}$ there is a compound Poisson process with Levy measure ${\displaystyle \nu }$.
5. Show that the characteristic function of a Levy process satisfies ${\displaystyle \phi _{t}(u)=\exp(t\psi (u))}$.
6. Let ${\displaystyle (X_{t})}$ be a Levy process. Show that ${\displaystyle Z_{t}=e^{iuX_{t}}/E(e^{iuX_{t}})}$ 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 ${\displaystyle N_{t}(B)}$ of a cadlag process - jump heights in sets ${\displaystyle B}$ bounded away from zero - finiteness of the jump measure of a cadlag process
• properties of ${\displaystyle B\mapsto N_{t}(B)}$ - properties of ${\displaystyle t\mapsto N_{t}(B)}$
• Poissonian jump measure (two properties) - Levy processes have Poissonian jump measures
• Levy measure of a Levy process - Levy measures are bounded on ${\displaystyle (|x|\geq \epsilon )}$, \epsilon>0
• ${\displaystyle Z_{t}=\int _{B}f(x)\,N_{t}(dx)}$ is a compound Poisson process for ${\displaystyle {\overline {B}}\subseteq \mathbb {R} \setminus \{0\}}$ - expectation and variance of ${\displaystyle (Z_{t})}$ - Fourier transform of ${\displaystyle (Z_{t})}$
• 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 ${\displaystyle (a,\sigma ^{2},\nu )}$ (w.r.t. a particular centering function ${\displaystyle h(x)}$)
• 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 ${\displaystyle (X_{t})}$ be a Levy process with Levy measure
1. ${\displaystyle \nu =2\epsilon _{1}+\epsilon _{-1}}$,
2. ${\displaystyle \nu (dx)=1_{[-1,1]}(x)\,dx}$.
2. Find expectation and variance of
1. ${\displaystyle \sum _{s\leq t}\Delta X_{s}\,1_{(\Delta X_{s}>1/2)}}$
2. ${\displaystyle \sum _{s\leq t}(\Delta X_{s})^{2}\,1_{(\Delta X_{s}<-1/2)}}$
3. Find the moments and the Fourier transform of a Levy process with characteristics (let ${\displaystyle h(x)=x\,1_{(|x|\leq 1)}}$):
1. ${\displaystyle (-1,0,e^{-x}1_{(x>0)}dx)}$
2. ${\displaystyle (0,1,x^{-1/2}1_{(x>0)}dx)}$
3. ${\displaystyle (0,0,x^{2}e^{-x^{2}}dx)}$
4. ${\displaystyle (0,0,1/x^{2}e^{-x^{2}}dx)}$

### Proofs

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

## 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 ${\displaystyle dS_{t}=S_{t-}\,dX_{t}}$ when ${\displaystyle (X_{t})}$ is a semimartingale with isolated jumps
• Poisson process ${\displaystyle N_{t}}$: - solving ${\displaystyle dS_{t}=S_{t-}\,dN_{t}}$ - solving ${\displaystyle dS_{t}=\alpha S_{t-}\,dt+\sigma S_{t-}\,dN_{t}}$ - martingale solutions

### Review questions

1. Explain the solution of ${\displaystyle dS_{t}=S_{t-}\,dX_{t}}$ when ${\displaystyle (X_{t})}$ has isolated jumps.
2. What is the stochastic exponential of a compound Poisson process ?

### Problems

1. Find the solution of ${\displaystyle dS_{t}=-S_{t-}dt+2S_{t-}dN_{t}}$ where ${\displaystyle (N_{t})}$ is a Poisson process with intensity ${\displaystyle \lambda }$.
2. In the preceding problem choose ${\displaystyle \lambda }$ such that ${\displaystyle (S_{t})}$ is a martingale.
3. Find the solution of ${\displaystyle dS_{t}=S_{t-}dt-S_{t-}/2dN_{t}}$ where ${\displaystyle (N_{t})}$ is a Poisson process with intensity ${\displaystyle \lambda }$.
4. In the preceding problem choose ${\displaystyle \lambda }$ such that ${\displaystyle (S_{t})}$ is a martingale.
5. Find the solution of ${\displaystyle dS_{t}=-S_{t-}dt+S_{t-}dW_{t}+2S_{t-}dN_{t}}$ where ${\displaystyle (W_{t})}$ is a Wiener process and ${\displaystyle (N_{t})}$ is an independent Poisson process with intensity ${\displaystyle \lambda }$.
6. In the preceding problem choose ${\displaystyle \lambda }$ such that ${\displaystyle (S_{t})}$ is a martingale.
7. Let ${\displaystyle S_{t}=W_{t}+N_{t}}$ where ${\displaystyle (W_{t})}$ is a Wiener process and ${\displaystyle (N_{t})}$ is an independent Poisson process with intensity ${\displaystyle \lambda }$. Expand ${\displaystyle e^{t-S_{t}}}$ by Ito's formula.
8. Let ${\displaystyle (S_{t})}$ be the solution of ${\displaystyle dS_{t}=-S_{t-}dt+2S_{t-}dN_{t}}$ where ${\displaystyle (N_{t})}$ is a Poisson process with intensity ${\displaystyle \lambda }$. Expand ${\displaystyle e^{S_{t}+2t}}$ 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 ${\displaystyle (N_{t})}$ be a Poisson process under ${\displaystyle P}$. Show that for every ${\displaystyle \lambda >0}$ one may find an equivalent probability measure ${\displaystyle Q}$ such that ${\displaystyle (N_{t})}$ has intensity ${\displaystyle \lambda }$.

## References

1. Rama Cont; Peter Tankov (26 October 2012). Financial Modelling with Jump Processes, Second Edition. CRC PressINC. ISBN 978-1-4200-8219-7. Retrieved 24 January 2013.
2. Philip Protter (24 May 2005). Stochastic Integration and Differential Equations: Version 2.1. Springer. ISBN 978-3-540-00313-7. Retrieved 24 January 2013.
3. Jean Jacod; Albert N. Shiryaev (31 December 1987). Limit theorems for stochastic processes. Springer-Verlag. ISBN 978-3-540-17882-8. Retrieved 24 January 2013.
4. Steven E. Shreve (3 June 2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer. ISBN 978-0-387-40101-0. Retrieved 24 January 2013.
5. Albert N. Shiryaev (1 February 1999). Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific. ISBN 978-981-02-3605-2. Retrieved 24 January 2013.