Narrow Results

We introduce a class of analytically tractable jump processes with contagion effects by generalizing the classical Hawkes process. This model framework combines the characteristics of three popular point processes in the literature: (1) Cox process with CIR intensity; (2) Cox process with Poisson shot-noise intensity; (3) Hawkes process with exponentially decaying intensity. Hence, it can be considered as a self-exciting and externally-exciting point process with mean-reverting stochastic intensity. Essential probabilistic properties such as moments, the Laplace transform of intensity process, and the probability generating function of point process as well as some important asymptotics have been derived. Some special cases and a method for change of measure are discussed. This point process may be applicable to modeling contagious arrivals of events for various circumstances (such as jumps, transactions, losses, defaults, catastrophes) in finance, insurance and economics with both endogenous and exogenous risk factors within one framework. More specifically, these exogenous factors could contain relatively short-lived shocks and long-lasting risk drivers. We make a simple application to calculate the default probability for credit risk and to price defaultable zero-coupon bonds.

We propose an analytical pricing method for stop-loss reinsurance contracts and catastrophe insurance derivatives using a Cox process with jump diffusion Cox–Ingersoll–Ross (CIR) intensity. The expected payoff of these contracts is expressed by the Laplace transform of the integration of the jump diffusion CIR process and the first moment of the aggregate loss. To confirm that the proposed analytical formula provides stable and accurate insurance derivative prices, we simulate them using a full Monte Carlo method compared to those obtained from its theoretical expectation. It shows that it is much faster way to obtain them than the full Monte Carlo method. We also conduct sensitivity analysis by changing the relevant parameters in the loss intensity providing their figures.

Let $X$ be a locally compact Polish space and $\sigma$ a nonatomic reference measure on $X$ (typically $X={ℝ}^d$ and $\sigma$ is the Lebesgue measure). Let $X^2\ni (x,y)\mapsto 𝕂(x,y)\in {ℂ}^{2\times 2}$ be a $2\times 2$-matrix-valued kernel that satisfies ${𝕂}^T(x,y)=𝕂(y,x)$. We say that a point process $\mu$ in $X$ is hafnian with correlation kernel $𝕂(x,y)$ if, for each $n\in \mathbb{N}$, the $n$th correlation function of $\mu$ (with respect to ${\sigma }^{\otimes n}$) exists and is given by $k^{(n)}(x_1,\dots ,x_n)=haf{[𝕂(x_i,x_j)]}_{i,j=1,\dots ,n}$. Here $haf(C)$ denotes the hafnian of a symmetric matrix $C$. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process ${\mathrm{\Pi }}_R$ is a Poisson point process in $X$ with random intensity $R(x)$. Let $G(x)$ be a complex Gaussian field on $X$ satisfying ${\int }_{\mathrm{\Delta }}𝔼(|G(x){|}^2)\sigma (dx)<\infty$ for each compact $\mathrm{\Delta }\subset X$. Then the Cox process ${\mathrm{\Pi }}_R$ with $R(x)=|G(x){|}^2$ is a hafnian point process. The main result of the paper is that each such process ${\mathrm{\Pi }}_R$ is the joint spectral measure of a rigorously defined particle density of a representation of the canonical commutation relations (CCRs), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free.

This chapter extends the Margrabe formula such that it is suitable for accounting for any type jump of stocks. Despite the fact that prices of an exchange option are characterized by jumps, it seems no study has explored those price jumps of an exchange option. The jump in this chapter is illustrated by a Poisson process. Moreover, the Poisson process can be extended into Cox process in case there is more than one jump. The results illustrate that incompleteness in an exchange option leads to a premium which in turn increases an option value whilst hedging strategies reveal mixed-bag type of results.

This chapter extends the Margrabe formula such that it is suitable for accounting for any type jump of stocks. Despite the fact that prices of an exchange option are characterized by jumps, it seems no study has explored those price jumps of an exchange option. The jump in this chapter is illustrated by a Poisson process. Moreover, the Poisson process can be extended into Cox process in case there is more than one jump. The results illustrate that incompleteness in an exchange option leads to a premium which in turn increases an option value while hedging strategies reveal mixed-bag type of results.