Jump models in financial modelling

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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  • 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  ) - zeros of   - dependence of   - 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.  
    2.  
    3.  
    4.  
  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  .
  4. Find the Levy measure of a sum of five independent Poisson processes with intensities  .
  5. Find the Levy measure of a linear combination of five independent Poisson processes with intensity 1 and weights  .

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   there is a compound Poisson process with Levy measure  .
  5. Show that the characteristic function of a Levy process satisfies  .
  6. Let   be a Levy process. Show that   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   of a cadlag process - jump heights in sets   bounded away from zero - finiteness of the jump measure of a cadlag process
  • properties of   - properties of  
  • Poissonian jump measure (two properties) - Levy processes have Poissonian jump measures
  • Levy measure of a Levy process - Levy measures are bounded on  , \epsilon>0
  •   is a compound Poisson process for   - expectation and variance of   - Fourier transform of  
  • 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   (w.r.t. a particular centering function  )
  • 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   be a Levy process with Levy measure
    1.  ,
    2.  .
  2. Find expectation and variance of
    1.  
    2.  
  3. Find the moments and the Fourier transform of a Levy process with characteristics (let  ):
    1.  
    2.  
    3.  
    4.  

Proofs

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  1. Every Levy measure   satisfies  .
  2. The jump measure of any Levy process is a Poisson jump measure.
  3. Every Levy process is a semimartingale.
  4. Let   be a Poisson jump measure and let   be its Levy measure. Prove the formulas:
    1.  
    2.  
    3.  
  5. Every Levy measure   satisfies  .

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   when   is a semimartingale with isolated jumps
  • Poisson process  : - solving   - solving   - martingale solutions


Review questions

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  1. Explain the solution of   when   has isolated jumps.
  2. What is the stochastic exponential of a compound Poisson process ?

Problems

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  1. Find the solution of   where   is a Poisson process with intensity  .
  2. In the preceding problem choose   such that   is a martingale.
  3. Find the solution of   where   is a Poisson process with intensity  .
  4. In the preceding problem choose   such that   is a martingale.
  5. Find the solution of   where   is a Wiener process and   is an independent Poisson process with intensity  .
  6. In the preceding problem choose   such that   is a martingale.
  7. Let   where   is a Wiener process and   is an independent Poisson process with intensity  . Expand   by Ito's formula.
  8. Let   be the solution of   where   is a Poisson process with intensity  . Expand   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   be a Poisson process under  . Show that for every   one may find an equivalent probability measure   such that   has intensity  .

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.