### 统计代写|金融统计代写financial statistics代考| Introduction to Option Management

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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

## 统计代写|金融统计代写financial statistics代考|Arbitrage Relations

In this section we consider the fundamental notion of no-arbitrage. An arbitrage opportunity arises if it is possible to make a riskless profit. In an ideal financial market, in which all investors dispose of the same pieces of information and in which all investors can react instantaneously, there should not be any arbitrage opportunity. Since otherwise each investor would try to realize the riskless profit instantaneously. The resulting transactions would change the prices of the involved financial instruments such that the arbitrage opportunity disappears.

Additionally to no-arbitrage we presume in the remaining chapter that the financial market fulfills further simplifying assumptions which are in this context of minor importance and solely serve to ease the argumentation. If these assumptions hold we speak of a perfect financial market.

Assumption (Perfect Financial Market) There are no arbitrage opportunities, no transaction costs, no taxes, and no restrictions on short selling. Lending rates equal borrowing rates and all securities are perfectly divisible.

The assumption of a perfect financial market is sufficient to determine the value of future and forward contracts as well as some important relations between the prices of some types of options. Above all no mathematical model for the price of the financial instrument is needed. However, in order to determine the value of options more than only economic assumptions are necessary. A detailed mathematical modelling becomes inevitable. Each mathematical approach though has to be in line with certain fundamental arbitrage relations being developed in this chapter. If the model implies values of future and forward contracts or option prices which do not fulfil these relations the model’s assumptions must be wrong.

An important conclusion drawn from the assumption of a perfect financial market and thus from no-arbitrage will be used frequently in the proofs to come. It is the fact that two portfolios, which have at a certain time $T$ the same value, must

have the same value at a prior time $t<T$ as well. Due to its importance we will further illustrate this reasoning. We proceed from two portfolios $A$ and $B$ consisting of arbitrary financial instruments. Their value in time $t$ will be denoted by $W_{A}(t)$ and $W_{B}(t)$, respectively. For any fixed point of time $T$, we assume that $W_{A}(T)=W_{B}(T)$ independently of the prior time $T$ values of each financial instrument contained in $A$ and $B$. For any prior point of time $t<T$ we assume without loss of generality that $W_{A}(t) \leq W_{B}(t)$. In time $t$ an investor can construct without own financial resources a portfolio which is a combination of $A$ and $B$ by buying one unit of every instrument of $A$, selling one unit of every instrument of $B$ (short selling) and by investing the difference $\Delta(t)=W_{B}(t)-W_{A}(t) \geq 0$ at a fixed rate $r$. The combined portfolio has at time $t$ a value of
$$W_{A}(t)-W_{B}(t)+\Delta(t)=0,$$

## 统计代写|金融统计代写financial statistics代考|Portfolio Insurance

A major purpose of options is hedging, i.e. the protection of investments against market risk caused by random price movements. An example for active hedging with options is the portfolio insurance. That is to strike deals in order to change at

a certain point of time the risk structure of a portfolio such that at a future point of time

• the positive profits are reduced by a small amount (which can be interpreted as an insurance premium) and in that way
• the portfolio value does not drop below a certain floor.
The portfolio insurance creates a risk structure of the portfolio which prevents extreme losses. For illustration purposes we consider at first a simple example.
Example 2.4 An investor has a capital of 10,500 EUR at his disposal to buy stocks whose current price is 100 EUR. Furthermore, put options on the same stock with a delivery price of $K=100$ and a time to maturity of 1 year are quoted at a market price of 5 EUR per contract. We consider two investment alternatives.
Portfolio B: Buying 100 stocks for 10,000 EUR and buying 100 put options for 500 EUR.

## 统计代写|金融统计代写financial statistics代考|Binary One-Period Model

The simplest of the option pricing formulae is the binomial option pricing formula. Here we take a look at a very simple model: the binary one-period model. The material in this section is only intended to be introductory. More details on the use of numerical procedures involving binomial trees are given in Chap. $7 .$

Consider a stock with a price of $S_{0}$ and a European call option on the stock with a strike price $K$ where the current price is $C_{0}$. Assume that the call is being valued one period before expiration $(T=1)$ and that the interest rate $r$ is equal to 0 in the one-period model. We let the future stock price be one of only two values: the stock price can either increase from $S_{0}$ to $S^{u}$ with probability $p$ or decrease from $S_{0}$ to $S^{d}$ with probability $(1-p)$. If the stock price moves up to $S^{u}$, the payoff will be $S_{T}-K$; if the stock price moves down to $S^{d}$, the payoff will be 0 , see Fig. 2.2.
Our goal is to determine the value $C_{0}$ of the call. The following different investment possibilities exist:

1. zerobond (with interest rate $r=0$ ),
2. $S_{0}$ the current value of the stock,
3. $C_{0}\left(C_{T}\right)$ the price of European call at time $0(T)$ with strike price $K$.
In order to value the call correctly, we examine two strategies. The first one is simply to buy the call. The second strategy is to choose a certain number of stocks $x$ and a decisive amount of a zerobond $y$ in a way that ensures the same payoff as the call at time $T$. Table $2.10$ shows the cash flows for both strategies.

In order to duplicate the payoff of the “buy-a-call” strategy, both cash flows must match whether the stock price goes up or down.

## 统计代写|金融统计代写financial statistics代考|Portfolio Insurance

• 正利润减少了少量（可以解释为保险费）并且以这种方式
• 投资组合价值不会低于某个下限。
投资组合保险创建了一个防止极端损失的投资组合风险结构。为了说明的目的，我们首先考虑一个简单的例子。
例 2.4 一位投资者有 10,500 欧元的资本可供他购买当前价格为 100 欧元的股票。此外，以交割价格为同一股票看跌期权ķ=100到期时间为 1 年，按每份合约 5 欧元的市场价格报价。我们考虑两种投资选择。
投资组合 A：买入 105 只股票。
投资组合 B：以 10,000 欧元买入 100 股股票，以 500 欧元买入 100 股看跌期权。

## 统计代写|金融统计代写financial statistics代考|Binary One-Period Model

1. zerobond（含利率r=0 ),
2. 小号0股票的当前价值，
3. C0(C吨)当时欧洲看涨期权的价格0(吨)以行使价ķ.
为了正确评估调用，我们研究了两种策略。第一个是简单地买入看涨期权。第二种策略是选择一定数量的股票X和决定性数量的零债券是以确保与通话时相同的回报的方式吨. 桌子2.10显示两种策略的现金流。

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## MATLAB代写

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