### Citations

1032 | An Equilibrium Characterization of the Term Structure
- Vasicek
- 1977
(Show Context)
Citation Context ...asset models, namely a geometric Brownian motion with deterministic interest rate (constant short rate r), and a geometric Brownian motion with stochastic interest rates given by a Vasicek model (see =-=[46]-=-). In the first case, we have the classical Black-Scholes (BS) setup, so the asset process under the risk-neutral measureQ evolves according to the SDE: dAt = rAt dt++AAt dWt , A0 = P(1+ x0), where r ... |

612 |
Financial modelling with Jump Processes
- CONT, TANKOV
- 2003
(Show Context)
Citation Context ...Ys)ds } Vt0(“alive”) ∣∣∣∣Ft]︸ ︷︷ ︸ =:F(t,Yt ) +EQ [ exp { − ∫ t0 t rs ds }( 1− exp { − ∫ t0 t !(x+ s,Ys)ds }) Vt0(“dead”) ∣∣∣∣Ft] . Applying Itô’s formula for Lévy processes (see e.g. Prop. 8.18 in =-=[18]-=-), we obtain dF(t,Yt) = k(t,Yt−,F(t,Yt−))dt+dMt , with drift term k(t,Yt−,F(t,Yt−)) and local martingale part Mt . Since, by construction,( exp ( − ∫ t 0 rs+!(x+ s,Ys)ds ) F(t,Yt) ) t∈["−1,t0 ] is aQ-... |

557 |
A general version of the fundamental theorem of asset pricing
- Delbaen, Schachermayer
- 1994
(Show Context)
Citation Context ...al Martingale Measure (see [44]). This result can be extended to incomplete market settings when choosing QF to be the Minimal Martingale Measure for the financial market (see e.g. [42]). However, in =-=[22]-=-, Delbaen and Schachermayer quote “the use of mortality tables in insurance” as “an example that this technique [change of measure] in fact has a long history” in actuarial sciences, indicating that t... |

511 | Valuing american options by simulation: a simple least-squares approach
- Longstaff, Schwartz
- 2001
(Show Context)
Citation Context ... high number of P(I)DEs needing to be solved may slow down the algorithm considerably. 93.3 A least-squares Monte Carlo approach The least-squaresMonte Carlo (LSM) approach by Longstaff and Schwartz (=-=[35]-=-) was originally presented for pricing American options but has recently also been applied to the valuation of insurance contracts (see e.g. [5] and [41]). We present the algorithm for life insurance ... |

353 |
On Cox processes and credit risky securities
- Lando
- 1998
(Show Context)
Citation Context ...t = F Ft ∨GMt , and Q = QF ⊗PM is the product measure of independent financial and biometric events. We further let F = (Ft)t∈[0,T ], where Ft =F Ft ∨FMt . A slight extension of the results by Lando (=-=[34]-=-, Prop. 3.1) now yields that for anFt-measurable payment Ct , we have for u≤ t3 BuEQ [ B−1t Ct 1{Tx>t} ∣∣Gu] = 1{Tx>u}BuE Q [ B−1t Ct exp { − ∫ t u !(x+ s,Ys)ds }∣∣∣∣Fu] , which can be readily applied... |

262 |
Processes of normal inverse Gaussian type
- Barndorff-Nielsen
- 1998
(Show Context)
Citation Context ... ( 2+(x−m)2 , where K1 denotes the modified Bessel function of the third kind with index 1, and an NIG process is defined as a Lévy process (Xt)t∈[0,T ] at zero with Xt ∼ NIG(, ,0 ,( · t,m · t) (see =-=[7]-=- or [43] for more details). As in the classical BS model, we assume a constant short rate r and define our exponential Lévy (NIG) model by At = A0eXt , where Xt ∼ NIG(, ,0 ,( · t,m · t) under “a” ris... |

153 |
Dynamic programming and optimal control, vol
- Bertsekas
- 2007
(Show Context)
Citation Context ... value in what follows, we need to rely on so-called “nested simulations”. We do not allow for surrenders at inception of the contract, so we define C(0,y0,d0) := 0. By the Bellman equation (see e.g. =-=[11]-=- for an introduction to dynamic programming and optimal control) the contract value at time " , " ∈ {0, . . . ,T −1}, is the maximum of the exercise value and the continuation value. The latter is the... |

153 |
Lévy Processes in Finance: Pricing Financial Derivatives. Wiley Series in Probability and Statistics
- Schoutens
- 2003
(Show Context)
Citation Context ...r such analyses. We fix the parameters as indicated above (cf. Sec. 4.3), but as in the latter part of [47] choose an alternative value for the volatility parameter. For the S&P 500 index, Schoutens (=-=[43]-=-) finds an implied volatility of 18.12%, but since insurers’ asset portfolios contain a limited proportion of risky assets only,10 we choose + = 0.03624, which approximately corresponds to a portfolio... |

134 |
exercise and valuation of executive stock options
- Carpenter
- 1998
(Show Context)
Citation Context ...ic perspective, one could assume that policyholders will maximize their personal utility, which would lead to a non-trivial control problem similar as for the valuation of employee stock options (see =-=[16]-=-, [28], or references therein). However, the assumption of homogenous policyholders does not seem proximate. In particular, the implied assertion that options within contracts with the same characteri... |

105 |
The pricing of equity-linked life insurance policies with an asset value guarantee
- Brennan, Schwartz
- 1976
(Show Context)
Citation Context ...e denote the set of all possible values of (Yt ,Dt) by$t . Most models for the market-consistent valuation of life insurance contracts presented in literature fit into this framework. For example, in =-=[14]-=-, Brennan and Schwartz price equity-linked life insurance policies with an asset value guarantee. Here, the value of the contract at time t only depends on the value of the underlying asset which is m... |

99 | Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing
- Andersen, Andreasen
(Show Context)
Citation Context ...cess driving the financial market, PIDEs with non-local integral terms must be solved. Several numerical methods have been proposed for the solution, e.g. based on finite difference schemes (see e.g. =-=[4,19]-=-), based on wavelet methods ([37]), or Fourier transform based methods ([30,36]). While, in comparison to Monte Carlo simulations, the complexity does not increase exponentially in time, the high numb... |

90 |
On the minimal martingale measure and the Föllmer– Schweizer decomposition
- Schweizer
- 1995
(Show Context)
Citation Context ...eutrality of an insurer with respect to mortality risk (cf. [2]), Møller ([39]) points out that if PM denotes the physical measure, Q as defined above is the so-called Minimal Martingale Measure (see =-=[44]-=-). This result can be extended to incomplete market settings when choosing QF to be the Minimal Martingale Measure for the financial market (see e.g. [42]). However, in [22], Delbaen and Schachermayer... |

69 | Living with mortality: longevity bonds and other mortality-linked securities.”
- Blake, Cairns, et al.
- 2006
(Show Context)
Citation Context ...respect to mortality risk may not be adequate. Then, the measure choice depends on the availability of adequate mortality-linked securities traded in the market (see [21] for a particular example and =-=[13]-=- for a survey on mortality-linked securities) and/or the insurer’s 3 In what follows, we write !(x + t,Yt ) := !(x + t,YMt ) and r (t,Yt ) := r ( t,YFt ) , where (Yt)t∈[0,T ] :=( YFt ,Y M t ) t∈[0,T ]... |

66 | A finite difference scheme for option pricing in jump diffusion and exponential Levy models
- Cont, Voltchkova
- 2003
(Show Context)
Citation Context ...cess driving the financial market, PIDEs with non-local integral terms must be solved. Several numerical methods have been proposed for the solution, e.g. based on finite difference schemes (see e.g. =-=[4,19]-=-), based on wavelet methods ([37]), or Fourier transform based methods ([30,36]). While, in comparison to Monte Carlo simulations, the complexity does not increase exponentially in time, the high numb... |

66 |
Stochastic Mortality in Life Insurance: Market Reserves and MortalityLinked Insurance Contracts, Insurance:
- Dahl
- 2004
(Show Context)
Citation Context ...n by the short rate process. 3In order to include the mortality component, we fix another probability space ( !M,GM,PM ) and a homogenous population of x-year old individuals at inception. Similar to =-=[12,20]-=-, we assume that a q2-dimensional vector of locally bounded, adapted Lévy processes (YMt )t∈[0,T ] = (YM,(q1+1)t , . . . ,Y M,(q) t )t∈[0,T ], q = q1 + q2, on ( !M,GM,PM ) is given. Now let !(·, ·) :... |

59 | Fair Valuation of Life Insurance Liabilities: The Impact of Interest Rate
- Grosen, Jørgensen
- 2000
(Show Context)
Citation Context ...any. Furthermore, these contracts usually contain a surrender option, i.e. the policyholder is allowed to lapse the contract at time " ∈ {1, . . . ,T}. Such contracts are, for instance, considered in =-=[15,25,38]-=-. All these models can be represented within our framework. Moreover, the setup is not restricted to the valuation of one “entire” insurance contracts, but, on one hand, it can also be used to determi... |

54 | An analysis of a least squares regression method for American option pricing.
- Clément, Lamberton, et al.
- 2002
(Show Context)
Citation Context ...a simple surrender option. Subsequently, problems for the application of this method to more general embedded options as well as potential solutions are identified. As pointed out by Clément et al. (=-=[17]-=-), the algorithm consists of two different types of approximations. Within the first approximation step, the continuation value function is replaced by a finite linear combination of certain “basis” f... |

46 | The Subjective and Objective Evaluation of Incentive Stock Options,
- Ingersoll
- 2004
(Show Context)
Citation Context ...spective, one could assume that policyholders will maximize their personal utility, which would lead to a non-trivial control problem similar as for the valuation of employee stock options (see [16], =-=[28]-=-, or references therein). However, the assumption of homogenous policyholders does not seem proximate. In particular, the implied assertion that options within contracts with the same characteristics ... |

42 |
Affine processes for dynamic mortality and actuarial valuations.
- Biffis
- 2005
(Show Context)
Citation Context ...n by the short rate process. 3In order to include the mortality component, we fix another probability space ( !M,GM,PM ) and a homogenous population of x-year old individuals at inception. Similar to =-=[12,20]-=-, we assume that a q2-dimensional vector of locally bounded, adapted Lévy processes (YMt )t∈[0,T ] = (YM,(q1+1)t , . . . ,Y M,(q) t )t∈[0,T ], q = q1 + q2, on ( !M,GM,PM ) is given. Now let !(·, ·) :... |

36 | A fast and accurate fft-based method for pricing early-exercise options under Lévy processes
- Lord, Fang, et al.
- 2008
(Show Context)
Citation Context ... solved. Several numerical methods have been proposed for the solution, e.g. based on finite difference schemes (see e.g. [4,19]), based on wavelet methods ([37]), or Fourier transform based methods (=-=[30,36]-=-). While, in comparison to Monte Carlo simulations, the complexity does not increase exponentially in time, the high number of P(I)DEs needing to be solved may slow down the algorithm considerably. 93... |

36 | Fast deterministic pricing of options on Lévy driven assets. RiskLab research report
- Matache, Petersdorff, et al.
- 2002
(Show Context)
Citation Context ...PIDEs with non-local integral terms must be solved. Several numerical methods have been proposed for the solution, e.g. based on finite difference schemes (see e.g. [4,19]), based on wavelet methods (=-=[37]-=-), or Fourier transform based methods ([30,36]). While, in comparison to Monte Carlo simulations, the complexity does not increase exponentially in time, the high number of P(I)DEs needing to be solve... |

32 | Guarantees Investment Contracts: Distributed and Undistributed Excess Return, Scandinavian Actuarial Journal,
- Miltersen, Persson
- 2003
(Show Context)
Citation Context ...any. Furthermore, these contracts usually contain a surrender option, i.e. the policyholder is allowed to lapse the contract at time " ∈ {1, . . . ,T}. Such contracts are, for instance, considered in =-=[15,25,38]-=-. All these models can be represented within our framework. Moreover, the setup is not restricted to the valuation of one “entire” insurance contracts, but, on one hand, it can also be used to determi... |

24 |
A Levy Process-Based Framework for the Fair Valuation of Participating Life Insurance Contracts.
- BALLOTTA
- 2005
(Show Context)
Citation Context ... model better represents the statistical properties of empirical log returns. Similar exponential Lévy models have been applied to the valuation of insurance contracts by different authors (see e.g. =-=[6]-=- or [31]). 10 By the regulation on investments ([1]), German insurers are obligated to keep the proportion of stocks within their asset portfolio below 35%. For example, the German“Allianz Lebensversi... |

22 | K.: Minimum rate of return guarantees: The Danish case
- Hansen, Miltersen
- 2002
(Show Context)
Citation Context ...ramework. Moreover, the setup is not restricted to the valuation of one “entire” insurance contracts, but, on one hand, it can also be used to determine the value of multiple contracts at a time (see =-=[26]-=-) or, on the other hand, parts of insurance contracts, such as embedded options. Clearly, we can determine the value of an arbitrary option by computing the value of the same 5contract in- and excludi... |

22 |
Risk-Minimizing Hedging Strategies for Insurance Payment Processes,
- Møller
- 2001
(Show Context)
Citation Context ...omplete financial market, i.e. if QF is unique, with a deterministic evolution of mortality and under the assumption of risk-neutrality of an insurer with respect to mortality risk (cf. [2]), Møller (=-=[39]-=-) points out that if PM denotes the physical measure, Q as defined above is the so-called Minimal Martingale Measure (see [44]). This result can be extended to incomplete market settings when choosing... |

20 |
Pricing of unit-linked life insurance policies.
- Aase, Persson
- 1994
(Show Context)
Citation Context ... ofPM . In a complete financial market, i.e. if QF is unique, with a deterministic evolution of mortality and under the assumption of risk-neutrality of an insurer with respect to mortality risk (cf. =-=[2]-=-), Møller ([39]) points out that if PM denotes the physical measure, Q as defined above is the so-called Minimal Martingale Measure (see [44]). This result can be extended to incomplete market setting... |

20 |
Fourier space time stepping for option pricing with Lévy models
- Jackson, Jaimungal, et al.
(Show Context)
Citation Context ... solved. Several numerical methods have been proposed for the solution, e.g. based on finite difference schemes (see e.g. [4,19]), based on wavelet methods ([37]), or Fourier transform based methods (=-=[30,36]-=-). While, in comparison to Monte Carlo simulations, the complexity does not increase exponentially in time, the high number of P(I)DEs needing to be solved may slow down the algorithm considerably. 93... |

19 |
Indifference Pricing of Insurance Contracts in a Product Space Model: Applications, Insurance:
- Møller
- 2003
(Show Context)
Citation Context .../or the insurer’s 3 In what follows, we write !(x + t,Yt ) := !(x + t,YMt ) and r (t,Yt ) := r ( t,YFt ) , where (Yt)t∈[0,T ] :=( YFt ,Y M t ) t∈[0,T ] is the state process. 4preferences (see [10] or =-=[40]-=-). In what follows, we assume that the insurer has chosen a measure PM for valuation purposes, so that a particular choice for the valuation measureQ is given. To obtain a model for our generic life i... |

18 |
Risk-neutral valuation of participating life insurance contracts
- Bauer, Kiesel, et al.
- 2006
(Show Context)
Citation Context ...such as in [23] or exchange options such as in [41]. Alternatively, the value of a certain embedded option may be determined by isolating the cash-flows corresponding to the considered guarantee (see =-=[8]-=-). In [9], Bauer et al. consider Variable Annuities including so-called Guaranteed Minimum Death Benefits (GMDBs) and/or Guaranteed Minimum Living Benefits (GMLBs). Again, their model structure fits i... |

16 |
On systematic mortality risk and riskminimization with survivor swaps
- Dahl, Melchior, et al.
(Show Context)
Citation Context ...ssumption of risk-neutrality with respect to mortality risk may not be adequate. Then, the measure choice depends on the availability of adequate mortality-linked securities traded in the market (see =-=[21]-=- for a particular example and [13] for a survey on mortality-linked securities) and/or the insurer’s 3 In what follows, we write !(x + t,Yt ) := !(x + t,YMt ) and r (t,Yt ) := r ( t,YFt ) , where (Yt)... |

16 |
Fair valuation of path-dependent participating life insurance contracts
- Taskanen, Lukkarinen
- 2003
(Show Context)
Citation Context ...ce contracts. The idea for this algorithm is based on solving the corresponding control problem on a discretized state space and, for special insurance contracts, was originally presented in [25] and =-=[45]-=-. The value Vt of our generic insurance contract depends on t, the state process Yt , and the state variables Dt . By arbitrage arguments, it can be shown that the value function is almost surely left... |

8 | The interaction of guarantees, surplus distribution, and asset allocation in with-profit life insurance policies
- Kling, Richter, et al.
- 2007
(Show Context)
Citation Context ...o German regulatory and legal requirements, whereas the IS-case models the typical behavior of German insurance companies in the past; this distribution rule was first introduced by Kling et al. (see =-=[32]-=-). 4.1.1 The MUST-case In Germany, insurance companies are obligated to guarantee a minimum rate of interest g on the policyholder’s account, which is currently fixed at 2.25%. Furthermore, according ... |

7 |
T.: Fair valuation of insurance contracts under Lévy process specifications
- Kassberger, Kiesel, et al.
- 2008
(Show Context)
Citation Context ...better represents the statistical properties of empirical log returns. Similar exponential Lévy models have been applied to the valuation of insurance contracts by different authors (see e.g. [6] or =-=[31]-=-). 10 By the regulation on investments ([1]), German insurers are obligated to keep the proportion of stocks within their asset portfolio below 35%. For example, the German“Allianz Lebensversicherungs... |

7 |
Risk-neutral valuation of participating life insurance contracts in a stochastic ineterest rate environment’, Insurance
- Zaglauer, Bauer
- 2008
(Show Context)
Citation Context ...dimensional heat equation, from which an integral representation can be derived when the terminal condition is given. If a modified Black-Scholes model with stochastic interest rates is assumed as in =-=[47]-=-, the situation gets more complex: The PDE is no longer analytically solvable and one has to resort to numerical methods. For a general exponential Lévy process driving the financial market, PIDEs wi... |

5 | Valuing the surrender options embedded in a portfolio of italian life guaranteed participating policies: a least squares monte carlo approach
- Andreatta, Corradin
- 2003
(Show Context)
Citation Context ...sMonte Carlo (LSM) approach by Longstaff and Schwartz ([35]) was originally presented for pricing American options but has recently also been applied to the valuation of insurance contracts (see e.g. =-=[5]-=- and [41]). We present the algorithm for life insurance contracts with a simple surrender option. Subsequently, problems for the application of this method to more general embedded options as well as ... |

5 | H.: Assessing the risk potential of premium payment options in participating life insurance contracts
- Gatzert, Schmeiser
- 2006
(Show Context)
Citation Context ...rical valuation approaches. Moreover, some studies do not apply methods from financial mathematics appropriately to the valuation of life insurance products (e.g. questionable worst-case scenarios in =-=[23]-=- and [33]; see Sec. 3.1 below for details). The objective of this article is to formalize the valuation problem for insurance contracts in a general way and to provide a survey on concrete valuation m... |

4 |
Monte Carlo Methods in Financial Engineering”, ser.Stochastic Modelling and Applied Probability
- Glasserman
(Show Context)
Citation Context ...s K→' almost surely. Hence, fixing b, we can construct an asymptotically valid (1− ( ) confidence interval for EQ [V̂0]. But this estimator for the risk-neutral value V0 =V (0,X0) is biased high (see =-=[24]-=-, p. 433), i.e. EQ [ V̂0 ]≥V (0,X0), where, in general, we have a sharp inequality. However, under some integrability conditions the estimator is asymptotically unbiased and hence, we can reduce the b... |

4 |
Valuation of life insurance surrender and exchange options
- Nordahl
(Show Context)
Citation Context ...e option. For example, the generic model can be used in this way to analyze paid-up and resumption options within participating life insurance contracts such as in [23] or exchange options such as in =-=[41]-=-. Alternatively, the value of a certain embedded option may be determined by isolating the cash-flows corresponding to the considered guarantee (see [8]). In [9], Bauer et al. consider Variable Annuit... |

3 |
Ein allgemeines Modell zur Analyse und Bewertung von Guaranteed Minimum Benefits in Fondspolicen
- Bauer, Kling, et al.
- 2007
(Show Context)
Citation Context ...ract and, hence, should not be disregarded by insurance companies. Moreover, for different kinds of non-European options and/or contracts, the influence may be significantly more pronounced (see e.g. =-=[9]-=- for Guaranteed Minimum Benefits within Variable Annuities). 4.6 Asset models (II) Although the Black-Scholes model is still very popular in practice, numerous empirical studies suggest that it is not... |

3 |
H.: Analysis of embedded options in individual pension schemes in Germany. The Geneva Risk and Insurance Review 31
- Kling, Ruß, et al.
- 2006
(Show Context)
Citation Context ...uation approaches. Moreover, some studies do not apply methods from financial mathematics appropriately to the valuation of life insurance products (e.g. questionable worst-case scenarios in [23] and =-=[33]-=-; see Sec. 3.1 below for details). The objective of this article is to formalize the valuation problem for insurance contracts in a general way and to provide a survey on concrete valuation methodolog... |

3 |
Hedging life insurance contracts in a Lévy process financial market
- Riesner
- 2006
(Show Context)
Citation Context ...the so-called Minimal Martingale Measure (see [44]). This result can be extended to incomplete market settings when choosing QF to be the Minimal Martingale Measure for the financial market (see e.g. =-=[42]-=-). However, in [22], Delbaen and Schachermayer quote “the use of mortality tables in insurance” as “an example that this technique [change of measure] in fact has a long history” in actuarial sciences... |

2 |
M.: Relative hedging of systematic mortality risk (2007). Working paper
- Bayraktar, Ludkovski
(Show Context)
Citation Context ...ies) and/or the insurer’s 3 In what follows, we write !(x + t,Yt ) := !(x + t,YMt ) and r (t,Yt ) := r ( t,YFt ) , where (Yt)t∈[0,T ] :=( YFt ,Y M t ) t∈[0,T ] is the state process. 4preferences (see =-=[10]-=- or [40]). In what follows, we assume that the insurer has chosen a measure PM for valuation purposes, so that a particular choice for the valuation measureQ is given. To obtain a model for our generi... |

2 |
de Varenne, F.: On the risk of insurance liabilities: Debunking some common pitfalls
- Briys
- 1997
(Show Context)
Citation Context ...any. Furthermore, these contracts usually contain a surrender option, i.e. the policyholder is allowed to lapse the contract at time " ∈ {1, . . . ,T}. Such contracts are, for instance, considered in =-=[15,25,38]-=-. All these models can be represented within our framework. Moreover, the setup is not restricted to the valuation of one “entire” insurance contracts, but, on one hand, it can also be used to determi... |