Pricing of Commodity Derivatives on Processes with Memory
Abstract
:1. Introduction
2. Generalised Langevin Equation
2.1. A Commodity Spot Market Model with Memory and Jumps
2.2. Analysis of the Langevin Equation
- Step 1.
- Computing the differential of in (10), we get
- Step 2.
- The differentiation under the stochastic integral in (11) follows by observingUsing the stochastic Fubini theorem (Veraar 2012, Theorem 2.2), we deduce that a sufficient condition for the exchange of integrals in (13) is given byObserveHence (14) follows from Assumption 1. To allow the exchange of the integrals in (12), we need to impose (see (Veraar 2012, Theorem 2.2))
3. Change of Measure
- 1.
- 2.
- 3.
- by the assumptions in this proposition.
- One can easily check that , for all , .
- Using a similar approach as in the proof of Lemma 1 and the integrability of M we obtain for in (5)Let us turn our attention to : one can easily show thatIndeed consider for the moment the functionThus, we see that and that the integrand is non-negative and less than 1 as is increasing for . It follows that for , or, . Letting , the desired inequality is reached. Hence, using again the uniform boundedness of ,
- The bound (29) and the uniform boundedness of , immediately provide
4. Pricing of Options and Forwards
4.1. Example I: Volterra Equation Driven by a Lévy Process
4.1.1. Spot Option Prices
4.1.2. Forward Prices
4.1.3. Options on Forwards
4.2. Example II: CARMA Processes
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Affine Processes
- , for .
- , are continuous in ,
- 1.
- ,
- 2.
Appendix B. Proof of Lemma 2
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1. | The proof of this extension requires a reasoning using the spectral representation applied to the function similarly as applied to the function in the proof of (Benth and Benth 2012, Proposition 5.1). |
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Benth, F.E.; Khedher, A.; Vanmaele, M. Pricing of Commodity Derivatives on Processes with Memory. Risks 2020, 8, 8. https://doi.org/10.3390/risks8010008
Benth FE, Khedher A, Vanmaele M. Pricing of Commodity Derivatives on Processes with Memory. Risks. 2020; 8(1):8. https://doi.org/10.3390/risks8010008
Chicago/Turabian StyleBenth, Fred Espen, Asma Khedher, and Michèle Vanmaele. 2020. "Pricing of Commodity Derivatives on Processes with Memory" Risks 8, no. 1: 8. https://doi.org/10.3390/risks8010008