数学代考|金融数学代考Financial Mathematics代写|Evaluation of Variable Annuity Guarantees with the Effect

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代考|金融数学代考Financial Mathematics代写|Underlying asset price dynamics

We propose a new model, jump-diffusion model, which takes into account the large, infrequent and abnormal variations of asset price caused by market news arrival in addition to normal changes of asset price due to disequilibrium in market demand and supply. We extend the work of Milevsky and Posner (2001) by allowing jumps in the underlying asset price process.

Compared to Black-Scholes model which considers only normal asset price changes, jump-diffusion model considers both normal and abnormal asset price changes and hence makes it a better model. Different from stochastic volatility models, jump-diffusion models have better analytical tractability specifically for path-dependent options and capture short term characteristics or behaviour of the financial market better (Yan \& Hanson, 2006). In other words, they handle short-term smiles better.

We evaluate the guarantee fees using this proposed model and compare to the fees evaluated using the Black-Scholes model which only considers risk to the life insurer brought by normal changes of asset price caused by the imbalance of market demand and supply.

In the proposed new model, jump-diffusion model, the value of an asset at time $t,\left(S_{t}\right)$ follows a $G B M$ and a jump process outlined by Poisson process, $\mathrm{N}{\mathrm{t}}$. The asset price follows GBM between jumps. $$\mathrm{d} \mathrm{S}{\mathrm{t}}=\mu \mathrm{S}{\mathrm{t}} \mathrm{d} t+\sigma \mathrm{S}{\mathrm{t}} \mathrm{dB} \mathrm{B}{t}+J \mathrm{~S}{\mathrm{t}} \mathrm{d} N_{\mathrm{t}}, \quad \mathrm{S}(0)=\mathrm{S}{0}$$ where $\mathrm{N}{\mathrm{t}}$ is a Poisson process with an intensity of $\lambda$ given by:
$P{N(t)=n}=\frac{(\lambda t)^{n}}{n !} \mathrm{e}^{-\lambda t}$
$S_{t}$ is the asset price at time $t, B_{t}$ is a standard Brownian motion, $J$ (is a function which causes a jump of stock price) is the jump size or magnitude which are i.i.d r.vs. The study by Merton (1976) considers the case where the jump sizes are normally distributed $(\mathrm{J} \sim \mathrm{N}(\mu, \sigma))$.

数学代考|金融数学代考Financial Mathematics代写|Jump-diffusion model vs. Black–Scholes model

Figures 9 and 10 show M\&E fee for Malaysian male and female annuitants, respectively, where the FPVA premiums are invested in Malaysia stock market. The fees are calculated using Black-Scholes model and jump-diffusion model.
Black-Scholes model ignores the jumps in the asset prices caused by over or under reaction due to good or bad news coming from the market or an individual company. Hence it doesn’t consider the impact of information arrival on the asset price changes. It only considers normal asset price changes but not abnormal. By assuming jumps in asset prices, the risks modelled in the jump-diffusion models are higher compared to the risks modelled in the Black-Scholes model. This makes the prices obtained using jump-diffusion model to be higher than the prices obtained using BlackScholes model, refer Figures 9 and 10 .

The nature of the changes of the asset prices determines the various features of the financial asset returns (returns distribution, asymmetry, volatility smile). There are many sources which determine the nature of the change of the asset prices. Some of these include: normal asset price changes due to disequilibrium in supply and demand on the market; abnormal asset price changes due to infrequent events caused by large-scale imbalance in the national or international market economy. VA pricing models strive to capture the various important features of the asset returns. Indirectly pricing models quantify the risks presented by the various sources.

Some models consider only one category of risk, while other models consider a combination of many sources of risk which an investor is exposed to in the stock market. In our study, we consider two models: the BlackScholes model considers the risk of loss to the insurers due to normal changes of asset price; and the jump-diffusion model considers the risk of losses from normal and abnormal changes in the asset price.

Many sources of risks expose the investor to a significant risk of loss compared to few sources of risks. This in turn will compel the life insurance company issuing the VA with guarantees to charge high guarantee fees when using a pricing framework which accounts for many sources of risk of loss compared to a pricing model which takes into account few sources of risks of loss. Black-Scholes model produces lower guarantee fees than the jumpdiffusion model because it accounts for only sources of risk of loss due to normal changes of the asset price, while jump-diffusion model accounts for risks of loss due to normal and abnormal changes in the asset price, refer Figures 9 and 10 .

数学代考|金融数学代考Financial Mathematics代写|Issuing VA to annuitants of different countries

Figures 11 and 12 show M\&E fee for male and female annuitants, respectively, of Malaysia, Tanzania and Canada when the investment of the contributions is done in Malaysia stock exchange market. Mortality rates explain the level of mortality risk which individual annuitants bring into the group. To be fair, the life insurance company should charge the annuitants according to the risk they bring into the group. When the life insurance company charges same price to all annuitants of the same age regardless the region or country they come from, good risks (annuitants with lower mortality rate at same age) will feel they are overcharged and hence terminate the contract and surrender it while bad risks (annuitants with high mortality rates at same age) will feel that they are undercharged. In addition the company will be attracting bad risks and chasing away good risks. This will lead to an adverse selection problem which puts the company in a high risk of losses.

To avoid this, the M \&E fees should depend on the mortality rates of an individual person, refer Figures 11 and 12 and explained in detail in Juma, Lee, Goh, Chin, and Liew (2016) and Juma and Lee (2017). The higher the mortality rate of an individual, the higher the fees irrespective of the region although the mortality table of the region is used in the calculation of the fees.

数学代考|金融数学代考Financial Mathematics代写|Underlying asset price dynamics

d小号吨=μ小号吨d吨+σ小号吨d乙乙吨+Ĵ 小号吨dñ吨,小号(0)=小号0在哪里ñ吨是强度为的泊松过程λ给出：

数学代考|金融数学代考Financial Mathematics代写|Jump-diffusion model vs. Black–Scholes model

Black-Scholes 模型忽略了由于来自市场或个别公司的好消息或坏消息导致的过度或反应不足导致的资产价格跳跃。因此没有考虑信息到达对资产价格变化的影响。它只考虑正常的资产价格变化，不考虑异常。通过假设资产价格跳跃，跳跃扩散模型中建模的风险与布莱克-斯科尔斯模型中建模的风险相比更高。这使得使用跳跃扩散模型获得的价格高于使用 BlackScholes 模型获得的价格，参见图 9 和图 10。

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