scholarly journals Valuation of cliquet-style guarantees with death benefits

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yaodi Yong ◽  
Hailiang Yang
Keyword(s):  
Brownian Motion ◽  
Closed Form ◽  
Discrete Time ◽  
Asset Price ◽  
Random Variable ◽  
Residual Lifetime ◽  
Exponential Distributions ◽  
Numerical Examples ◽  
Price Process ◽  
Insurance Product

<p style='text-indent:20px;'>In this paper, we consider the problem of valuing an equity-linked insurance product with a cliquet-style payoff. The premium is invested in a reference asset whose dynamic is modeled by a geometric Brownian motion. The policy delivers a payment to the beneficiary at either a fixed maturity or the time upon the insured's death, whichever comes first. The residual lifetime of a policyholder is described by a random variable, assumed to be independent of the asset price process, and its distribution is approximated by a linear sum of exponential distributions. Under such characterization, closed-form valuation formulae are derived for the contract considered. Moreover, a discrete-time setting is briefly discussed. Finally, numerical examples are provided to illustrate our proposed approach.</p>

scholarly journals Valuation of Cliquet-Style Guarantees with Death Benefits in Jump Diffusion Models

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 2011
Author(s):  
Yaodi Yong ◽  
Hailiang Yang
Keyword(s):  
Independent Random Variable ◽  
Linear Combination ◽  
Random Variable ◽  
Jump Diffusion ◽  
Diffusion Models ◽  
Financial Asset ◽  
Exponential Distributions ◽  
Inverse Transform ◽  
Insurance Product

This paper aims to value the cliquet-style equity-linked insurance product with death benefits. Whether the insured dies before the contract maturity or not, a benefit payment to the beneficiary is due. The premium is invested in a financial asset, whose dynamics are assumed to follow an exponential jump diffusion. In addition, the remaining lifetime of an insured is modelled by an independent random variable whose distribution can be approximated by a linear combination of exponential distributions. We found that the valuation problem reduced to calculating certain discounted expectations. The Laplace inverse transform and techniques from existing literature were implemented to obtain analytical valuation formulae.

scholarly journals General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory

Risks ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 11
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Sergiy Shklyar
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Fractional Brownian Motion ◽  
Covariance Matrix ◽  
Discrete Time ◽  
Asset Price ◽  
Cholesky Decomposition ◽  
Additive Scheme ◽  
With Memory ◽  
General Conditions

We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space D [ 0 , T ] . Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process.

ON THE VARIANCES OF SYSTEM SIZE AND SOJOURN TIME IN A DISCRETE-TIME DAR(1)/D/1 QUEUE

2011 ◽  
Vol 25 (4) ◽  
pp. 519-535 ◽  
Author(s):  
Daniel Wei-Chung Miao ◽  
Hung Chen
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Performance Measures ◽  
Discrete Time ◽  
Sojourn Time ◽  
Blow Up ◽  
System Size ◽  
Batch Size ◽  
Numerical Examples ◽  
Cross Terms

We consider a discrete-time DAR(1)/D/1 queue and provide an analysis on the variances of both its system size and sojourn time. Our approach is simple, but the results are nice, as these variances are found in closed form. We first establish the relation between these variances, based on which we then use the conditioning technique to analyze the expected cross terms that come from its system recurrence relation. The closed-form results allow us to explicitly examine the effect from the batch size distribution and the autocorrelation parameter p. It is observed that as p grows toward 1, the standard deviations of the two performance measures will blow up in same asymptotic order of O(1/(1−p)) as their means. These are demonstrated through numerical examples.

scholarly journals What is the time value of a stream of investments?

2005 ◽  
Vol 42 (3) ◽  
pp. 861-866 ◽  
Author(s):  
Ragnar Norberg ◽  
Mogens Steffensen
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Integral ◽  
Asset Price ◽  
Expected Value ◽  
Integral Expression ◽  
Time Value ◽  
Price Process

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

scholarly journals Phase-Type Models in Life Insurance:Fitting and Valuation of Equity-Linked Benefits

Risks ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 17 ◽  
Author(s):  
Søren Asmussen ◽  
Patrick Laub ◽  
Hailiang Yang
Keyword(s):  
Life Insurance ◽  
Asset Price ◽  
High Water ◽  
Insurance Risk ◽  
Exponential Distributions ◽  
Price Process ◽  
Phase Type ◽  
Dense Class ◽  
Coxian Distributions ◽  
Matrix Exponentials

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance. The advantage is that PH distributions form a dense class and that problems having explicit solutions for exponential distributions typically become computationally tractable under PH assumptions. In the first part of this paper, fitting of PH distributions to human lifetimes is considered. The class of generalized Coxian distributions is given special attention. In part, some new software is developed. In the second part, pricing of life insurance products such as guaranteed minimum death benefit and high-water benefit is treated for the case where the lifetime distribution is approximated by a PH distribution and the underlying asset price process is described by a jump diffusion with PH jumps. The expressions are typically explicit in terms of matrix-exponentials involving two matrices closely related to the Wiener-Hopf factorization, for which recently, a Lévy process version has been developed for a PH horizon. The computational power of the method of the approach is illustrated via a number of numerical examples.

scholarly journals What is the time value of a stream of investments?

2005 ◽  
Vol 42 (03) ◽  
pp. 861-866 ◽  
Author(s):  
Ragnar Norberg ◽  
Mogens Steffensen
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Integral ◽  
Asset Price ◽  
Expected Value ◽  
Integral Expression ◽  
Time Value ◽  
Price Process

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

scholarly journals Stochastic sensitivity of bull and bear states

2021 ◽  
Author(s):  
Jochen Jungeilges ◽  
Elena Maklakova ◽  
Tatyana Perevalova
Keyword(s):  
Additive Noise ◽  
Asset Price ◽  
Market Model ◽  
Asset Market ◽  
Stochastic Version ◽  
Market Participants ◽  
Price Process ◽  
Stochastic Sensitivity ◽  
Stable Equilibria ◽  
Parametric Noise

AbstractWe study the price dynamics generated by a stochastic version of a Day–Huang type asset market model with heterogenous, interacting market participants. To facilitate the analysis, we introduce a methodology that allows us to assess the consequences of changes in uncertainty on the dynamics of an asset price process close to stable equilibria. In particular, we focus on noise-induced transitions between bull and bear states of the market under additive as well as parametric noise. Our results are obtained by combining the stochastic sensitivity function (SSF) approach, a mixture of analytical and numerical techniques, due to Mil’shtein and Ryashko (1995) with concepts and techniques from the study of non-smooth 1D maps. We find that the stochastic sensitivity of the respective bull and bear equilibria in the presence of additive noise is higher than under parametric noise. Thus, recurrent transitions are likely to be observed already for relatively low intensities of additive noise.

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

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