### 数学代考|计算复杂性理论代写computational complexity theory代考|The Kim and Markowitz Portfolio Insurers Model

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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
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|计算复杂性理论代写computational complexity theory代考|The Kim and Markowitz Portfolio Insurers Model

Harry Markowitz is very well known for being one of the founders of modern portfolio theory, a contribution for which he has received the Nobel Prize in economics. It is less well known, however, that Markowitz is also one of the pioneers in employing agent based simulations in economics.

During the October 1987 crash markets all over the globe plummeted by more than $20 \%$ within a few days. The surprising fact about this crash is that it appeared to be spontaneous – it was not triggered by any obvious event. Following the 1987 crash researchers started to look for endogenous market features, rather than external forces, as sources of price variation. The Kim-Markowitz [15] model explains the 1987 crash as resulting from investors? “Constant Proportion Portfolio Insurance” (CPPI) policy. Kim and Markowitz proposed that market instabilities arise as a consequence of the individual insurers’ efforts to cut their losses by selling once the stock prices are going down.

The Kim Markowitz agent based model involves two groups of individual investors: rebalancers and insurers (CPPI investors). The rebalancers are aiming to keep a constant composition of their portfolio, while the insurers make the appropriate operations to insure that their eventual losses will not exceed a certain fraction of the investment per time period.

The rebalancers act to keep a portfolio structure with (for instance) half of their wealth in cash and half in stocks. If the stock price rises, then the stocks weight in the portfolio will increase and the rebalancers will sell shares until the shares again constitute $50 \%$ of the portfolio. If the stock price decreases, then the value of the shares in the portfolio decreases, and the rebalancers will buy shares until the stock again constitutes $50 \%$ of the portfolio. Thus, the rebalancers have a stabilizing influence on the market by selling when the market rises and buying when the market falls.

A typical CPPI investor has as his/her main objective not to lose more than (for instance) $25 \%$ of his initial wealth during a quarter, which consists of 65 trading days. Thus, he aims to insure that at each cycle $75 \%$ of the initial wealth is out of reasonable risk. To this effect, he assumes that the current value of the stock will not fall in one day by more than a factor of 2 . The result is that he always keeps in stock twice the difference between the present wealth and $75 \%$ of the initial wealth (which he had at the beginning of the 65 days investing period). This determines the amount the CPPI agent is bidding or offering at each stage. Obviously, after a price fall, the amount he wants to keep

in stocks will fall and the CPPI investor will sell and further destabilize the market. After an increase in the prices (and personal wealth) the amount the CPPI agent wants to keep in shares will increase: he will buy, and may support a price bubble.

The simulations reveal that even a relatively small fraction of CPPI investors (i. e. less than $50 \%$ ) is enough to destabilize the market, and crashes and booms are observed. Hence, the claim of Kim and Markowitz that the CPPI policy may be responsible for the 1987 crash is supported by the agent based simulations. Various variants of this model were studied intensively by Egenter, Lux and Stauffer [5] who find that the price time evolution becomes unrealistically periodic for a large number of investors (the periodicity seems related with the fixed 65 days quarter and is significantly diminished if the 65 day period begins on a different date for each investor).

## 数学代考|计算复杂性理论代写computational complexity theory代考|The Arthur, Holland, Lebaron, Palmer and Tayler Stock Market Model

Palmer, Arthur, Holland, Lebaron and Tayler [30] and Arthur, Holland, Lebaron, Palmer and Tayler 3 construct an agent based simulation model that is focused on the concept of co-evolution. Each investor adapts his/her investment strategy such as to maximally exploit the market dynamics generated by the investment strategies of all others investors. This leads to an ever-evolving market, driven endogenously by the everchanging strategies of the investors.

The main objective of AHLPT is to prove that market fluctuations may be induced by this endogenous coevolution, rather than by exogenous events. Moreover, AHLPT study the various regimes of the system: the regime in which rational fundamentalist strategies are dominating vs. the regime in which investors start developing strategies based on technical trading. In the technical trading regime, if some of the investors follow fundamentalist strategies, they may be punished rather than rewarded by the market. AHLPT also study the relation between the various strategies (fundamentals vs. technical) and the volatility properties of the market (clustering, excess volatility, volume-volatility correlations, etc.).

In the first paper quoted above, the authors simulated a single stock and further limited the bid/offer decision to a ternary choice of: i) bid to buy one share, ii) offer to sell one share, or: iii) do nothing. Each agent had a collection of rules which described how he should behave (i, ii or iii) in various market conditions. If the current market conditions were not covered by any of the rules, the default was to do nothing. If more than one rule applied in a certain

market condition, the rule to act upon was chosen probabilistically according to the “strengths” of the applicable rules. The “strength” of each rule was determined according to the rule’s past performance: rules that “worked” became “stronger”. Thus, if a certain rule performed well, it became more likely to be used again.

The price is updated proportionally to the relative excess of offers over demands. In [3], the rules were used to predict future prices. The price prediction was then transformed into a buy/sell order through the use of a Constant Absolute Risk Aversion (CARA) utility function. The use of CARA utility leads to demands which do not depend on the investor’s wealth.

The heart of the AHLPT dynamics are the trading rules. In particular, the authors differentiate between “fundamental” rules and “technical” rules, and study their relative strength in various market regimes. For instance, $a^{\text {“fundamental }}$ ” rule may require a market conditions of the type:
$$\frac{\text { dividend }}{\text { current price }}>0.04$$
in order to be applied. A “technical” rule may be triggered if the market fulfills a condition of the type:
current price $>10$-period moving-average of past prices.
The rules undergo genetic dynamics: the weakest rules are substituted periodically by copies of the strongest rules and all the rules undergo random mutations (or even versions of “sexual” crossovers: new rules are formed by combining parts from 2 different rules). The genetic dynamics of the trading rules represent investors’learning: new rules represent new trading strategies. Investors examine new strategies, and adopt those which tend to work best. The main results of this model are:

For a Few Agents, a Small Number of Rules, and Small Dividend Changes

• The price converges towards an equilibrium price which is close to the fundamental value.
• There are no bubbles, crashes or anomalies.
• Agents follow homogeneous simple fundamentalist rules.

## 数学代考|计算复杂性理论代写computational complexity theory代考|The Lux and Lux and Marchesi Model

Lux [27] and Lux and Marchesi [28] propose a model to endogenously explain the heavy tail distribution of returns and the clustering of volatility. Both of these phenomena emerge in the Lux model as soon as one assumes that in addition to the fundamentalists there are also chartists in the model. Lux and Marchesi [28] further divide the chartists into optimists (buyers) and pessimists (sellers). The market fluctuations are driven and amplified by the fluctuations in the various populations: chartists converting into fundamentalists, pessimists into optimists, etc.
In the Lux and Marchesi model the stock’s fundamental value is exogenously determined. The fluctuations of the fundamental value are inputted exogenously as a white noise process in the logarithm of the value. The market price is determined by investors’ demands and by the market clearance condition.
Lux and Marchesi consider three types of traders:

• Fundamentalists observe the fundamental value of the stock. They anticipate that the price will eventually converge to the fundamental value, and their demand for shares is proportional to the difference between the market price and the fundamental value.
• Chartists look more at the present trends in the market price rather than at fundamental economic values; the chartists are divided into
• Optimists (they buy a fixed amount of shares per unit time)
• Pessimists (they sell shares).
Transitions between these three groups (optimists, pessimists, fundamentalists) happen with probabilities depending on the market dynamics and on the present numbers of traders in each of the three classes:The transition probabilities of chartists depend on the majority opinion (through an “opinion index” measuring the relative number of optimists minus the relative number of pessimists) and on the actual price trend (the current time derivative of the current market price), which determines the relative profit of the various strategies.
• The fundamentalists decide to turn into chartists if the profits of the later become significantly larger than their own, and vice versa (the detailed formulae used by Lux and Marchesi are inspired from the exponential transition probabilities governing statistical mechanics physical systems).
• The main results of the model are:
• No long-term deviations between the current market price and the fundamental price are observed.
• The deviations from the fundamental price, which do occur, are unsystematic.
• In spite of the fact that the variations of the fundamental price are normally distributed, the variations of the market price (the market returns) are not. In particular the returns exhibit a frequency of extreme events which is higher than expected for a normal distribution. The authors emphasize the amplification role of the market that transforms the input normal distribution of the fundamental value variations into a leptokurtotic (heavy tailed) distribution of price variation, which is encountered in the actual financial data.
• clustering of volatility.

## 数学代考|计算复杂性理论代写computational complexity theory代考|The Kim and Markowitz Portfolio Insurers Model

Harry Markowitz 以现代投资组合理论的创始人之一而闻名，他因此获得了诺贝尔经济学奖。然而，鲜为人知的是，马科维茨也是在经济学中采用基于代理的模拟的先驱之一。

## 数学代考|计算复杂性理论代写computational complexity theory代考|The Arthur, Holland, Lebaron, Palmer and Tayler Stock Market Model

Palmer、Arthur、Holland、Lebaron 和 Tayler [30] 以及 Arthur、Holland、Lebaron、Palmer 和 Tayler 3构建了一个基于代理的仿真模型，该模型专注于协同进化的概念。每个投资者都会调整他/她的投资策略，以最大限度地利用所有其他投资者的投资策略产生的市场动态。这导致了一个不断发展的市场，由投资者不断变化的策略内生驱动。

AHLPT 的主要目标是证明市场波动可能是由这种内生的协同进化引起的，而不是由外生事件引起的。此外，AHLPT 研究了系统的各种制度：理性的原教旨主义策略占主导地位的制度与投资者开始基于技术交易制定策略的制度。在技​​术交易体系中，如果部分投资者遵循原教旨主义策略，他们可能会受到市场的惩罚而不是奖励。AHLPT 还研究各种策略（基本面与技术面）与市场波动性特性（聚类、过度波动、交易量-波动率相关性等）之间的关系。

AHLPT 动态的核心是交易规则。特别是，作者区分了“基本”规则和“技术”规则，并研究了它们在各种市场制度中的相对优势。例如，一种“基本的 ”规则可能需要以下类型的市场条件：
股利  目前的价格 >0.04

• 价格趋于接近基本价值的均衡价格。
• 交易量低。
• 没有气泡、崩溃或异常。
• 代理人遵循同质的简单原教旨主义规则。

## 数学代考|计算复杂性理论代写computational complexity theory代考|The Lux and Lux and Marchesi Model

Lux [27] 和 Lux 和 Marchesi [28] 提出了一个模型来内生地解释收益的重尾分布和波动性的聚类。一旦人们假设除了原教旨主义者之外，模型中还有图表主义者，这两种现象就会出现在勒克斯模型中。Lux 和 Marchesi [28] 进一步将图表分析师分为乐观主义者（买家）和悲观主义者（卖家）。市场波动是由不同人群的波动驱动和放大的：图表主义者转变为原教旨主义者，悲观主义者转变为乐观主义者，等等。

Lux 和 Marchesi 考虑了三种类型的交易者：

• 基本面主义者观察股票的基本价值。他们预计价格最终会收敛到基本价值，他们对股票的需求与市场价格和基本价值之间的差额成正比。
• 图表专家更多地关注市场价格的当前趋势，而不是基本经济价值；图表专家分为
• 乐观主义者（他们每单位时间购买固定数量的股票）
• 悲观主义者（他们出售股票）。
这三组（乐观主义者、悲观主义者、基本面主义者）之间的转变发生的概率取决于市场动态和三个类别中每个类别的当前交易者数量：图表师的​​转变概率取决于多数意见（通过“意见指数” ”衡量乐观者的相对数量减去悲观者的相对数量）和实际价格趋势（当前市场价格的当前时间导数），这决定了各种策略的相对利润。
• 如果后者的利润明显大于他们自己的利润，则原教旨主义者决定变成图表主义者，反之亦然（Lux 和 Marchesi 使用的详细公式的灵感来自于管理统计力学物理系统的指数转换概率）。
• 该模型的主要结果是：
• 没有观察到当前市场价格与基本价格之间的长期偏差。
• 确实发生的与基本价格的偏差是非系统性的。
• 尽管基本价格的变化是正态分布的，但市场价格（市场收益）的变化却不是。特别是回报表现出极端事件的频率高于正态分布的预期。作者强调了市场的放大作用，它将基本价值变化的输入正态分布转换为价格变化的细峰（重尾）分布，这在实际金融数据中会遇到。
• 波动性的聚类。

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## MATLAB代写

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