arXiv subjects
Publications and source records attributed to Oriol Zamora Font.
The main purpose of this work is the derivation of a functional partial differential equation (FPDE) for the calculations of equity-linked insurance policies, where the payment stream may depend on the whole past history of the financial asset. To this end, we employ variational techniques from the theory of functional Itô calculus.
We derive a semi-analytical pricing formula for European VIX call options under the Heston-Hawkes stochastic volatility model introduced in arXiv:2210.15343. This arbitrage-free model incorporates the volatility clustering feature by adding an independent compound Hawkes process to the Heston volatility. Using the Markov property of the exponential Hawkes an explicit expression of $\text{VIX}^2$ is derived as a linear combination of the variance and the Hawkes intensity. We apply qualitative ODE theory to study the existence of some generalized Riccati ODEs. Thereafter, we compute the joint characteristic function of the variance and the Hawkes intensity exploiting the exponential affine structure of the model. Finally, the pricing formula is obtained by applying standard Fourier techniques.
The main purpose of the paper is to derive Thiele's differential equation for unit-linked policies in the Heston-Hawkes stochastic volatility model introduced in arXiv:2210.15343. This model is an extension of the well-known Heston model that incorporates the volatility clustering feature by adding a compound Hawkes process in the volatility. Since the model is arbitrage-free, pricing unit-linked policies via the equivalence principle under a risk neutral probability measure is possible. Studying the moments of the variance and certain stochastic exponentials, a suitable family of risk neutral probability measures is found. The established and practical method to compute reserves in life insurance is by solving Thiele's equation, which is crucial to guarantee the solvency of the insurance company.
We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed.