**9. References**


measure explicitly the process risk effects.

*Wolters Kluwer Financial Services / FRSGlobal, Switzerland* 

July 5. Available from: http://www.ceiops.org.

Int. Congress of Actuaries. 1988; vol. 3: 1-17.

transitions.

annuities).

**Author details** 

Werner Hürlimann

**9. References** 

419.

viewed as a sub-model of the model with Poisson-Gamma time dependent biometric

Moreover, a detailed analysis for a single cohort of identical endowment contracts with waiver of premium by disability has been undertaken in Section 7. Besides a complete Markov chain specification, which seems to be missing in the literature, the numerical illustration has shown, as expected, that the cohort size is a main driving factor of process risk. Due to the statistical law of large numbers, the larger the cohort size the less solvency capital is actually required. In contrast to the life annuity "longevity risk" study in [1], the stochastic approach penalizes almost all insurers (except the very large ones) because the current standard approach prescribes almost negligible solvency capital ratios and does not

The interested actuary might challenge the proposed approach with alternatives from other regulatory environments than Solvency II. Moreover, it is important to point out that a lot of technical issues remain to be settled properly. They are not only regulatory specific but also related to the complex mathematics of related software products and go beyond the Markov chain model. Today's life insurance contracts include many embedded options and are henceforth even more complex. A challenging issue is the definition of capital requirements for unit-linked contracts without and with guarantee and variable annuities with guaranteed minimum benefits (so-called variable GMXB

[1] Hürlimann, W. Biometric solvency risk for portfolios of general life contracts (I) The single life multiple decrement case. North Amer. Actuarial J. 2010; 14(4): 400-

[2] QIS5 (2010). Technical Specifications QIS5 – CEIOPS Quantitative Impact Study 5. 2010

[3] Amsler, M.-H. Sur la modélisation des risques vie par les chaînes de Markov.Trans. 23rd

[4] Hoem, J.M. Markov chain models in life insurance. Blätter DGVFM. 1969; 9: 91-107. [5] Hoem, J.M. The versatility of the Markov chain as a tool in the mathematics of life

[6] Koller, M. Stochastische Modelle in der Lebensversicherung. Berlin: Springer; 2000.

Insurance. Trans. 23rd Int. Congress of Actuaries. 1988; vol. 3: 171-202.

	- [28] Christiansen, M. Biometric worst-case scenarios for multi-state life insurance policies. Insurance: Math. and Economics. 2010; 47(2): 190-197.

**Chapter 0**

**Chapter 2**

**Boundary-Value Problems for Second Order PDEs**

**Arising in Risk Management and Cellular Neural**

This work deals with the Dirichlet problem for some PDEs of second order with non-negative characteristic form. One main motivation is to study some boundary-value problems for PDEs of Black-Scholes type arising in the pricing problem for financial options of barrier type. Barrier options on stocks have been traded since the end of the Sixties and the market for these options has been dramatically expanding, making barrier options the most popular ones among the exotic. The class of standard barrier options includes 'in' barriers and 'out' barriers, which are activated (knocked in) and, respectively, extinguished (knocked out) if the underlying asset price crosses the barrier before expiration. Moreover, each class includes 'down' or 'up' options, depending on whether the barrier is below or above the current asset price and thus can be breached from above or below. Therefore there are eight types of standard barrier options, depending on their 'in' or 'out', 'down' or 'up', and 'call' or 'put' attributes. It is possible to include a cash rebate, which is paid out at option expiration if an 'in' ('out') option has not been knocked in (has been knocked out, respectively) during its lifetime. One can consider barrier options with rebates of several types, terminal payoffs of different forms (e.g. power options), more than one underlying assets and/or barriers, and allow for time-dependent barriers, thus enriching this class still further. On the other hand, a large variety of new exotic barriers have been designed to accommodate investors' preferences. Another motivation for the study of such options is related to credit risk theory. Several credit-risk models build on the barrier option formalism, since the default event can be modeled throughout a signalling variable hitting a pre-specified boundary value (See [3],[8] among others). As a consequence, a substantial body of academic literature provides pricing methods for valuating barrier options, starting from the seminal work of [18], where an exact formula is offered for a down-and-out European call with zero rebate. Further extensions

> ©2012 Slavova et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 Slavova et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Networks Approach**

http://dx.doi.org/10.5772/49936

**1. Introduction**

Rossella Agliardi, Petar Popivanov and Angela Slavova

Additional information is available at the end of the chapter

cited.
