Jump models in financial modelling

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## Introduction

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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].

### Home reading

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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

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* 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

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### Concepts and facts

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* 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  ϕ t ( u {\displaystyle \phi _{t}(u} ) - zeros of  ϕ t ( u ) {\displaystyle \phi _{t}(u)} \- dependence of  t {\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

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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

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

### Proofs

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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  ν | B ( R ) {\displaystyle \nu |{\mathcal {B}}(\mathbb {R} )} there is a compound Poisson process with Levy measure  ν {\displaystyle \nu } .
5. Show that the characteristic function of a Levy process satisfies  ϕ t ( u ) = exp ⁡ ( t ψ ( u ) ) {\displaystyle \phi _{t}(u)=\exp(t\psi (u))} .
6. Let  ( X t ) {\displaystyle (X_{t})} be a Levy process. Show that  Z t = e i u X t / E ( e i u X t ) {\displaystyle Z_{t}=e^{iuX_{t}}/E(e^{iuX_{t}})} is a martingale.

## Jump measures and decomposition of Levy processes

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### Concepts and facts

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* 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 ) {\displaystyle N_{t}(B)} of a cadlag process - jump heights in sets  B {\displaystyle B} bounded away from zero - finiteness of the jump measure of a cadlag process
* properties of  B ↦ N t ( B ) {\displaystyle B\mapsto N_{t}(B)} \- properties of  t ↦ N t ( B ) {\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  ( | x | ≥ ϵ ) {\displaystyle (|x|\geq \epsilon )} , \epsilon>0
* Z t = ∫ B f ( x ) N t ( d x ) {\displaystyle Z_{t}=\int _{B}f(x)\,N_{t}(dx)} is a compound Poisson process for  B ¯ ⊆ R ∖ { 0 } {\displaystyle {\overline {B}}\subseteq \mathbb {R} \setminus \\{0\\}} \- expectation and variance of  ( Z t ) {\displaystyle (Z_{t})} \- Fourier transform of  ( Z t ) {\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  ( a , σ 2 , ν ) {\displaystyle (a,\sigma ^{2},\nu )} (w.r.t. a particular centering function  h ( x ) {\displaystyle h(x)} )
* characterization of FV-Levy processes (without proof)

### Review questions

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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

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1. Let  ( X t ) {\displaystyle (X_{t})} be a Levy process with Levy measure
1. ν = 2 ϵ 1 + ϵ − 1 {\displaystyle \nu =2\epsilon _{1}+\epsilon _{-1}} ,
2. ν ( d x ) = 1 [ − 1 , 1 ] ( x ) d x {\displaystyle \nu (dx)=1_{[-1,1]}(x)\,dx} .
2. Find expectation and variance of
1. ∑ s ≤ t Δ X s 1 ( Δ X s > 1 / 2 ) {\displaystyle \sum _{s\leq t}\Delta X_{s}\,1_{(\Delta X_{s}>1/2)}}
2. ∑ s ≤ t ( Δ X s ) 2 1 ( Δ X s < − 1 / 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  h ( x ) = x 1 ( | x | ≤ 1 ) {\displaystyle h(x)=x\,1_{(|x|\leq 1)}} ):
1. ( − 1 , 0 , e − x 1 ( x > 0 ) d x ) {\displaystyle (-1,0,e^{-x}1_{(x>0)}dx)}
2. ( 0 , 1 , x − 1 / 2 1 ( x > 0 ) d x ) {\displaystyle (0,1,x^{-1/2}1_{(x>0)}dx)}
3. ( 0 , 0 , x 2 e − x 2 d x ) {\displaystyle (0,0,x^{2}e^{-x^{2}}dx)}
4. ( 0 , 0 , 1 / x 2 e − x 2 d x ) {\displaystyle (0,0,1/x^{2}e^{-x^{2}}dx)}

### Proofs

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1. Every Levy measure  ν {\displaystyle \nu } satisfies  ν ( | x | ≥ 1 ) < ∞ {\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  ( N t ( B ) ) {\displaystyle (N_{t}(B))} be a Poisson jump measure and let  ν ( B ) {\displaystyle \nu (B)} be its Levy measure. Prove the formulas:
1. E ( ∫ f ( x ) N t ( d x ) ) = t ∫ f ( x ) ν ( d x ) {\displaystyle E{\Big (}\int f(x)\,N_{t}(dx){\Big )}=t\int f(x)\,\nu (dx)}
2. V ( ∫ f ( x ) N t ( d x ) ) = t ∫ f 2 ( x ) ν ( d x ) {\displaystyle V{\Big (}\int f(x)\,N_{t}(dx){\Big )}=t\int f^{2}(x)\,\nu (dx)}
3. E ( exp ⁡ ( ∫ f ( x ) N t ( d x ) ) ) ) = exp ⁡ ( t ∫ ( e f ( x ) − 1 ) ν ( d x ) ) {\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  ν {\displaystyle \nu } satisfies  ∫ | x | ≤ 1 x 2 ν ( d x ) < ∞ {\displaystyle \int _{|x|\leq 1}x^{2}\,\nu (dx)<\infty } .

## Stochastic analysis for processes with jumps

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### Concepts and facts

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* general Ito-formula - proof by induction
* Ito-formula for processes with isolated jumps - direct proof
* solving  d S t = S t − d X t {\displaystyle dS_{t}=S_{t-}\,dX_{t}} when  ( X t ) {\displaystyle (X_{t})} is a semimartingale with isolated jumps
* Poisson process  N t {\displaystyle N_{t}} : - solving  d S t = S t − d N t {\displaystyle dS_{t}=S_{t-}\,dN_{t}} \- solving  d S t = α S t − d t + σ S t − d N t {\displaystyle dS_{t}=\alpha S_{t-}\,dt+\sigma S_{t-}\,dN_{t}} \- martingale solutions

### Review questions

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

### Problems

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

## Financial models with jumps, pricing and hedging

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### Concepts and facts

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* 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

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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

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1. Let  ( N t ) {\displaystyle (N_{t})} be a Poisson process under  P {\displaystyle P} . Show that for every  λ > 0 {\displaystyle \lambda >0} one may find an equivalent probability measure  Q {\displaystyle Q} such that  ( N t ) {\displaystyle (N_{t})} has intensity  λ {\displaystyle \lambda } .

## References

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1. ↑ 1.0 1.1 Rama Cont; Peter Tankov (26 October 2012). Financial Modelling with Jump Processes, Second Edition. CRC PressINC. ISBN 978-1-4200-8219-7. http://books.google.com/books?id=-fZtKgAACAAJ. Retrieved 24 January 2013.
2. ↑ 2.0 2.1 Philip Protter (24 May 2005). Stochastic Integration and Differential Equations: Version 2.1. Springer. ISBN 978-3-540-00313-7. http://books.google.com/books?id=mJkFuqwr5xgC. 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. http://books.google.com/books?id=sUgXKpUIdHwC. 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. http://books.google.com/books?id=O8kD1NwQBsQC. 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. http://books.google.com/books?id=oiIG5FmJxWgC. Retrieved 24 January 2013.