A decoupled time-changed (DTC) L\'evy  process is a generalized time-changed L\'evy process whose continuous and discontinuous parts are allowed to follow separate random time scalings . Disentangling the stochastic time change in two components not only provides a powerful unitary framework for the models already present in the literature, but in principle opens up new possibilities for the choice of an asset's log-price dynamics. Following this new idea, we devise a general martingale structure for the price process, and obtain an inverse-Fourier pricing equation for claims paying at maturity a function of the asset value and its realized volatility. Thus with a single formula we are able to capture prices for derivatives depending on either an asset or its volatility , but also joint payoffs are allowed , like the target volatility option (TVO) , which shall be discussed. Numerical computations validating our techniques are provided. Notably, the DTC theory allows to incorporate valuation formulae from various asset models in a single software implementation .
In this paper we propose a general derivative pricing framework which employs decoupled time-changed (DTC) L\'evy processes to model the underlying asset of contingent claims. A DTC L\'evy process is a generalized time-changed L\'evy process whose continuous and pure jump parts are allowed to follow separate random time scalings ; we devise the martingale structure for a DTC L\'evy-driven asset and revisit many popular models which fall under this framework. Postulating different time changes for the underlying L\'evy decomposition allows to introduce asset price models consistent with the assumption of a correlated pair of continuous and jump market activities; we study one illustrative DTC model having this property by assuming that the instantaneous activity rates follow the the so-called Wishart process. The theory developed is applied to the problem of pricing claims depending not only on the price or the volatility of an underlying asset , but also to more sophisticated derivatives that pay-off on the joint performance of these two financial variables , like the target volatility option (TVO) . We solve the pricing problem through a Fourier-inversion method; numerical computations validating our technique are provided .
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In this paper we propose a general derivative pricing framework which employs
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processes to model the underlying asset of contingent claims. A DTC L\'evy
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discontinuous
pure jump
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. Disentangling the stochastic time change in two components not only provides a powerful unitary framework for the models already present in the literature, but in principle opens up new possibilities for the choice of an asset's log-price dynamics. Following this new idea, we devise a general
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the price process, and obtain an inverse-Fourier pricing equation for claims paying at maturity a function of the asset value and its realized volatility. Thus with
R
single formula we are able to capture prices for derivatives depending on either an asset or its volatility
DTC L\'evy-driven asset and revisit many popular models which fall under this framework. Postulating different time changes for the underlying L\'evy decomposition allows to introduce asset price models consistent with the assumption of a correlated pair of continuous and jump market activities; we study one illustrative DTC model having this property by assuming that the instantaneous activity rates follow the the so-called Wishart process. The theory developed is applied to the problem of pricing claims depending not only on the price or the volatility of an underlying asset
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joint payoffs are allowed
to more sophisticated derivatives that pay-off on the joint performance of these two financial variables
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, which shall be discussed. Numerical
. We solve the pricing problem through a Fourier-inversion method; numerical
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techniques are provided. Notably, the DTC theory allows to incorporate valuation formulae from various asset models in a single software implementation
technique are provided