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## 金融代写|金融衍生品代写Financial Derivatives代考|FOREIGN EXCHANGE FORWARD AND FUTURES CONTRACTS

The foreign exchange forward market is an over-the-counter (OTC) market that trades on a worldwide basis and is dominated by large financial institutions. Typical contract amounts are quite large, and the market sees very few individual investors. By contrast, there are foreign exchange futures markets in many countries. In comparison with the forward market, these futures markets are quite small. The discussion of these markets will focus on the forward market, which will be contrasted with a single foreign exchange futures market, the Chicago Mercantile Exchange in the United States. ${ }^1$

## 金融代写|金融衍生品代写Financial Derivatives代考|FOREIGN EXCHANGE OPTIONS

\begin{aligned} C_t & =e^{-r_F(T-t)} F C N\left(d_1\right)-X e^{-r_d(T-t)} N\left(d_2\right) \ P_t & =X e^{-r_F(T-t)} N\left(-d_2\right)-F C e^{-r_F(T-t)} N\left(-d_1\right) \ d_1 & =\frac{\ln (F C / X)\left(r_D-r_F+0.5 \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}} \ d_2 & =d_1-\sigma \sqrt{T-t} \end{aligned}

\begin{aligned} C_t & =\text { price of a call option (priced in the domestic currency) on } \ & \text { foreign currency } F C \ P_t & =\text { price of a put option (priced in the domestic currency) on } \ & \text { foreign currency } F C \ F C & =\text { a quantity of the foreign currency } \ r_D, r_F & =\text { domestic and foreign interest rates, respectively } \ X & =\text { exercise price } \ T-t & =\text { time until expiration } \ \sigma^2 & =\text { variance of the foreign currency value } \ \mathrm{N}(\bullet) & =\text { cumulative normal function } \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 金融代写|金融衍生品代写Financial Derivatives代考|Single-Stock Futures

Single-stock futures have been developed as an alternative for managing the risk of investing in stocks. Single-stock futures offer a market to buy or sell the underlying stock at some future date, typically at some short horizon (within a year). They represent a somewhat lower-cost alternative to stock options, since entering into a futures contract requires only margins. Worldwide, and most predominantly in South Africa, single-stock futures trade on a wide variety of stocks. As with stock options, however, trading is likely to be concentrated in firms where uncertainty is greatest.

Although in the United States, single-stock futures typically are settled on a $T+3$ basis with the delivery of the underlying stock, the use of single-stock futures facilitates short positions in stock. By using single-stock futures, an investor avoids the costly logistics required for borrowing shares in a short sale arrangement. Further, the relatively low ( 20 percent) margins on U.S. single-stock futures make for a low-cost competitive alternative to options for hedging purposes.

Single-stock futures are priced with a simple present value/future value relation. More specifically, the price of a single-stock future should represent the present value of today’s stock price along with the present value of any future dividends that might be paid out before the futures contract expires. Mathematically, we can state that the value of a single-stock future $F$ is
$$F=\left(S_o-D\right) e^{r T}$$
where
$S_o=$ current stock price
$D=$ present value of dividends paid during the life of the futures contract
$T=$ time to maturity (in years)
$r=$ continuously compounded risk-free rate of interests over the life of the futures contract

## 金融代写|金融衍生品代写Financial Derivatives代考|Futures on Stock Indexes

Some of the most actively traded futures contracts involve stock index futures. Stock index futures range from broad-based to narrow-based indexes. The most broadly traded index futures are based on country-specific indexes such as the S\&P 500, the Financial Times Stock Index FTSE 100 (London-listed companies), the Deutschen Actien Index DAX 30 (Frankfurt-listed companies), and the Cotation Assistee en Continu CAC 40 (Euronext Paris-listed companies). Stock indexes based on other countries typically are narrower in scope. For instance, the Portugese Stock Index PSI-20 index includes 20 Euronext Lisbon-listed companies, with the top 5 firms representing approximately 75 percent of the market capitalization of the entire index.

In the United States, OneChicago lists a number of narrow-based index futures. The indexes traded here are typically comprised of portfolios of four to seven stocks, many slanted toward Canadian stocks.

Portfolio managers at mutual funds, hedge funds, insurance companies, and other institutions face systematic portfolio risks when they hold equity portfolios. Likewise, market makers and dealers who sell index products to clients also can be exposed to systematic risk. Stock index futures provide an efficient mechanism for managing this portfolio risk at a relatively low cost. Since entering into a futures contract requires only margin payments, these institutions can avoid the premium payment that accompanies index options.

## 金融代写|金融衍生品代写Financial Derivatives代考|Single-Stock Futures

$$F=\left(S_o-D\right) e^{r T}$$

$S_o=$当前股价
$D=$在期货合约有效期内支付的股息的现值
$T=$成熟期(年)
$r=$在期货合约有效期内连续复合无风险利率

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 金融代写|金融衍生品代写Financial Derivatives代考|Theory of Normal Backwardation

According to the theory of normal backwardation, speculators in commodity futures will receive a positive return as compensation for the price risk transferred to them by the hedger. Keynes’s (1930) original theory assumed that the hedger, as a producer, would sell futures. The risk premium would be paid to the long speculator by setting the futures price below the expected future spot price: The futures prices would, on average, rise through time, resulting in a positive rate of return for the long speculator.

Note that this use of the term backwardation differs from that used in describing the basis or futures curve. Backwardation occurs when the current spot price is greater than a futures price. Normal backwardation, however, describes the relationship between the forward price and the expected spot price: The forward price is set below the expected spot price, and this persistent downward bias in the forward price rewards the long speculator with positive returns.

A modification of the normal backwardation model argues that not all markets will return a risk premium to the speculator (Telser 1958). Competition between speculators normally will drive the risk premium to zero, so only markets in which there are too few speculators willing to take on hedger risk will pay a risk premium to a long position. Such a thin market would be characterized by low liquidity, with few transactions, a low volume of trade in the commodity relative to its production, and consequently large price fluctuations.

Keynes’s original model has been modified to recognize that not all hedging interest will be short. Hedgers can be producers or consumers, and speculators will garner positive returns if they take a position opposite to the majority of hedgers in the market. If hedgers are net short, then a long speculative position should result in a positive return, as in the original backwardation model. If hedgers are net long in a particular market, then speculators are paid to go short, and a short position will reap positive returns. Thus, to reap consistently positive returns, speculators must be able to tell where hedging net interest will be in a particular market.
It is important to recognize that the risk premium on commodity futures contracts can be linked to convenience yield. We have seen that if futures prices are at less than full carry because of inadequate inventories, then convenience yield will be relatively large; this implies increased uncertainty and a demand for risk transference. If speculators are to be rewarded for providing a risk transfer function, then periods of high convenience yield (when futures are driven below the current spot price- the traditional definition of backwardation) may also be periods in which long futures positions exhibit positive returns. The size of the risk premium will depend on inventory levels, which drive the convenience yield, and the relative risk sensitivities of inventory holders (hedgers) and investors (speculators).

## 金融代写|金融衍生品代写Financial Derivatives代考|COMMODITY INVESTMENT STRATEGIES

Commodity indexes are used to track commodity prices and to represent a portfolio of commodity positions. Since commodities are extremely heterogeneous, the behavior of a particular index, and comparisons of index performance, can be very sensitive to how a given index is constructed. Indexes vary by the selection and weighting of constituent components, and how components are rebalanced.

The Standard \& Poor’s Goldman Sachs Commodity Index (SP-GSCI) and the Dow Jones-AIG Commodity Index (DJ-AIGCI) are the most widely used in structuring tradable commodity index products, and futures are traded on these indexes at the CMEG. Other indexes include the Reuters/Jeffries Commodity Research Bureau (CRB) and the Deutsche Bank Liquid Commodity Index (DBLCI).

All the indexes just mentioned include a range of commodity sectors, but specific commodities and their weights differ. Component selection involves a trade-off between the index’s ability to represent commodities as an asset class and the ease with which the index can be replicated. For instance, the CRB gives a broad picture of overall commodity price movements while the DJ-AIG selects components based on the liquidity of the futures contract. TheSP-GSCI contains the largest number of commodities (25), while the DJ-AIG constrains the weighting that any individual sector can have in the index. Many indexes use only nearby prices, but others (CRB) include other delivery months. A different index approach is taken by the Nasdaq/OMX Global Agriculture Index, which tracks the performance of shares in companies with activity in agriculture or farming. Since a commodity index cannot use market capitalization for a weighting scheme (all futures involve an equal long and short position), weights are determined by other means. The CRB uses equal weighting, but other indexes use world production or consumption data to reflect relative economic importance.

Changes in relative prices can alter the original weighting scheme over time, so an index may be rebalanced periodically. However, how an index is rebalanced can affect its performance since rebalancing can imply a particular investment strategy for the index portfolio. Rebalancing an equally weighted portfolio, for instance, necessitates selling appreciating commodities and buying depreciating commodities. Similarly, infrequent rebalancing can mimic a momentum investment strategy.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 金融代写|金融衍生品代写Financial Derivatives代考|SYNTHETIC ASSET ALLOCATION

To understand whether someone is truly hedging, speculating, or engaging in some activity in between, it is useful to understand how one uses a derivatives contract to offset a cash market risk. As the CFTC definition implies, a hedge position is one that involves two positions having values that move in opposite directions. One of the positions is in the cash market and normally is held as a result of the ongoing operation of an enterprise. The other position is in a derivative, such as a futures or option contract, and is entered into for the purpose of hedging.

A classic example of a hedging transaction is the farmer trying to protect himself from a fall in the value of crops he is producing. Consider the risk a farmer faces when he plants corn in the spring. The farmer has committed a certain amount of resources in terms of seed, fertilizer, and time to producing the corn. If prices are low in the fall when the farmer goes to sell his corn, he faces the risk that he will not be able to cover the costs of his inputs. In the language of the CFTC definition, he faces financial losses from adverse price changes. This risk can be offset, however, through the use of a futures contract.

Suppose the farmer believes that he will be able to get $\$ 2.90$per bushel at harvest for his corn. Referring to Exhibit 3.1, on May 1 the farmer consults the Chicago Board of Trade price for December delivery of corn and sees that the contract is currently selling for$\$3.00$ per bushel. He decides to sell 10 futures contracts to cover his expected production of 50,000 bushels. On December 1 , the price of corn in his local area is $\$ 2.65$per bushel, insufficient to cover his production costs. However, at the same time, the price of a futures contract has fallen to$\$2.75$ per bushel. Since the farmer initially sold the futures contracts at $\$ 3.00$per bushel, he can now purchase them back at$\$2.75$ per bushel, thereby generating a profit of $\$ 0.25$per bushel, offsetting the lower price on his physical corn. The supplemental value from the futures transactions ultimately allows the farmer to net a sale price of$\$2.90$ per bushel, thereby covering his production costs.

Whether an individual or company is hedging agricultural commodities, energy commodities or financial assets, hedges using derivatives contracts, whether they are futures traded on an organized exchange or swaps traded off of an exchange, all function in generally the same fashion: They allow hedgers to hold a financial contract that fluctuates in value opposite that of the commodity or asset they are trying to hedge. In some cases, like the example of the farmer, the asset or commodity being hedged is already held by the hedger, who faces the financial risk that the commodity or asset being held will fall in value before it is sold. Under such circumstances, the hedger would sell a futures contract to hedge this exposure. Such hedges are referred to as short hedges because the hedger has sold, or “gone short,” the futures contract.

## 金融代写|金融衍生品代写Financial Derivatives代考|SPECULATION

If the classic depiction of hedging is the farmer locking in a sale price for a crop, the classic speculator is the commodities trader who buys a pork bellies contract for no other reason than he believes prices will rise. Years ago, such speculators were limited to agricultural products or precious metals. Today, individuals wanting to chase profits have a panoply of choices available to them, ranging from physical commodities-agricultural products, metals and energy-to financial products-interest rate instruments, stocks, and foreign currencies. In today’s markets, one can even take a view on temperature, rainfall, or even election results.
For the moment, as with the case of hedgers in the previous section, we will consider speculation in its purest form. That is, speculation is the taking on of a price risk for the simple purpose of trying to profit based on expectations of which way prices will move. Speculators who expect prices to rise will enter into long positions, while those believing they will fall short the market.

Speculators can be categorized in several different ways. One common way is to classify them by how they form their price expectations. Those relying on basic economic conditions to form expectations are referred to as fundamental traders. Traders in an alternate group, which form expectations based on analyses of price patterns and other market statistics, are called technical traders.

Fundamental traders operate on the premise that futures prices reflect the underlying conditions related to the supply and demand of commodities and the valuation of financial assets. The goal of the fundamental trader is first to identify the key economic conditions and variables that affect prices and second to observe changes in those conditions, it is hoped, before they are incorporated into the market price. For example, a fundamental trader interested in trading Eurodollar futures would be concerned with Fed policy, inflation rates, and other economic indicators that would signal upcoming changes to Eurodollar rates. A fundamental trader of physical commodities similarly would be concerned with factors that would increase or decrease the supply or demand for a commodity. If the trader can gain an edge in terms of gathering information on these factors and take a position in the market before the market has incorporated the information, she stands to gain from her efforts.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 衍生品, 金融代写, 金融衍生品

## avatest™帮您通过考试

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## 金融代写|金融衍生品代写Financial Derivatives代考|HEDGING TRANSACTIONS

To understand whether someone is truly hedging, speculating, or engaging in some activity in between, it is useful to understand how one uses a derivatives contract to offset a cash market risk. As the CFTC definition implies, a hedge position is one that involves two positions having values that move in opposite directions. One of the positions is in the cash market and normally is held as a result of the ongoing operation of an enterprise. The other position is in a derivative, such as a futures or option contract, and is entered into for the purpose of hedging.

A classic example of a hedging transaction is the farmer trying to protect himself from a fall in the value of crops he is producing. Consider the risk a farmer faces when he plants corn in the spring. The farmer has committed a certain amount of resources in terms of seed, fertilizer, and time to producing the corn. If prices are low in the fall when the farmer goes to sell his corn, he faces the risk that he will not be able to cover the costs of his inputs. In the language of the CFTC definition, he faces financial losses from adverse price changes. This risk can be offset, however, through the use of a futures contract.

Suppose the farmer believes that he will be able to get $\$ 2.90$per bushel at harvest for his corn. Referring to Exhibit 3.1, on May 1 the farmer consults the Chicago Board of Trade price for December delivery of corn and sees that the contract is currently selling for$\$3.00$ per bushel. He decides to sell 10 futures contracts to cover his expected production of 50,000 bushels. On December 1 , the price of corn in his local area is $\$ 2.65$per bushel, insufficient to cover his production costs. However, at the same time, the price of a futures contract has fallen to$\$2.75$ per bushel. Since the farmer initially sold the futures contracts at $\$ 3.00$per bushel, he can now purchase them back at$\$2.75$ per bushel, thereby generating a profit of $\$ 0.25$per bushel, offsetting the lower price on his physical corn. The supplemental value from the futures transactions ultimately allows the farmer to net a sale price of$\$2.90$ per bushel, thereby covering his production costs.

Whether an individual or company is hedging agricultural commodities, energy commodities or financial assets, hedges using derivatives contracts, whether they are futures traded on an organized exchange or swaps traded off of an exchange, all function in generally the same fashion: They allow hedgers to hold a financial contract that fluctuates in value opposite that of the commodity or asset they are trying to hedge. In some cases, like the example of the farmer, the asset or commodity being hedged is already held by the hedger, who faces the financial risk that the commodity or asset being held will fall in value before it is sold. Under such circumstances, the hedger would sell a futures contract to hedge this exposure. Such hedges are referred to as short hedges because the hedger has sold, or “gone short,” the futures contract.

## 金融代写|金融衍生品代写Financial Derivatives代考|SPECULATION

If the classic depiction of hedging is the farmer locking in a sale price for a crop, the classic speculator is the commodities trader who buys a pork bellies contract for no other reason than he believes prices will rise. Years ago, such speculators were limited to agricultural products or precious metals. Today, individuals wanting to chase profits have a panoply of choices available to them, ranging from physical commodities-agricultural products, metals and energy-to financial products-interest rate instruments, stocks, and foreign currencies. In today’s markets, one can even take a view on temperature, rainfall, or even election results.
For the moment, as with the case of hedgers in the previous section, we will consider speculation in its purest form. That is, speculation is the taking on of a price risk for the simple purpose of trying to profit based on expectations of which way prices will move. Speculators who expect prices to rise will enter into long positions, while those believing they will fall short the market.

Speculators can be categorized in several different ways. One common way is to classify them by how they form their price expectations. Those relying on basic economic conditions to form expectations are referred to as fundamental traders. Traders in an alternate group, which form expectations based on analyses of price patterns and other market statistics, are called technical traders.

Fundamental traders operate on the premise that futures prices reflect the underlying conditions related to the supply and demand of commodities and the valuation of financial assets. The goal of the fundamental trader is first to identify the key economic conditions and variables that affect prices and second to observe changes in those conditions, it is hoped, before they are incorporated into the market price. For example, a fundamental trader interested in trading Eurodollar futures would be concerned with Fed policy, inflation rates, and other economic indicators that would signal upcoming changes to Eurodollar rates. A fundamental trader of physical commodities similarly would be concerned with factors that would increase or decrease the supply or demand for a commodity. If the trader can gain an edge in terms of gathering information on these factors and take a position in the market before the market has incorporated the information, she stands to gain from her efforts.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 金融代写|金融衍生品代写Financial Derivatives代考|Swap Contracts

A swap contract is a contract in which two counterparties agree to make periodic payments that differ in a fundamental way from each other until some future date. The terms of a swap contract, besides the maturity and notional value of the contract, can include the currencies to be exchanged (foreign currency swap), the rate of interest applicable to each counterparty (interest rate swap), and the timetable by which payments are made. Swap contracts are an over-the-counter, negotiated derivative contract like a forward contract rather than an exchangetraded instrument like a futures contract. The swap contract counterparties must be classified as Eligible Contract Participants, as defined by the Commodity Exchange Act. $^3$ Although foreign currency swaps predate interest rate swaps, interest rate swaps are economically most important today.

Consider the uses for an interest rate swap. Suppose the swap contract specifies the exchange of floating rate cash flows for fixed rate cash flows. That is, a counterparty agrees to pay a fixed cash flow (based on a fixed rate) to another counterparty and in return receive a variable cash flow (based on a floating rate). The exchange of cash flows occurs periodically, say every six months; the cash flows are netted against each other so that whichever counterparty’s cash flow is larger, that counterparty pays the difference to the other. A swap of fixed rate for floating rate cash flows reduces the fixed rate payer’s exposure to unexpected rate increases, which is an important commercial risk for the holder of existing fixed-income securities or firms that anticipate the issuance of debt in the future. If rates rise during the contract’s life, the fixed rate payer will receive cash flows that offset the loss of value in existing securities or the increase in debt issuance cost. Similarly, the swap of floating rate for fixed rate cash flows reduces the floating-rate payer’s exposure to unexpected rate decreases.

Suppose in 1999, Maytag issues a three-year, \$100 million par floating rate note with semiannual interest payments at 80 basis points over the London Interbank Offered Rate (LIBOR), which is at 3.2 percent per annum. Maytag’s interest expense floats with LIBOR but at the current rate, Maytag will pay \$2.0 million as interest to the debt holders every six months, in March and September. Maytag is exposed to the commercial risk of unexpected increases in LIBOR. To transfer the risk, Maytag enters into a swap agreement with a U.S. commercial bank to pay a fixed 5 percent per annum on the $\$ 100$million until maturity. In return, the bank agrees to pay Maytag a variable amount based on LIBOR plus 80 basis points. Only the cash flow differential is exchanged in the agreement. Exhibit 1.10 is a table of the swap cash flows as LIBOR rises. Maytag makes increasing cash payments to debt holders as the rate floats higher; completely offsetting the interest expense are equivalent cash inflows from the U.S. commercial bank. Maytag still pays a fixed rate cash flow to the U.S. commercial bank of$\$2.5$ million every six months. Note that Maytag has credit risk exposure from the bank only when the value of the interest rate swap is positive (last column of Exhibit 1.10).

## 金融代写|金融衍生品代写Financial Derivatives代考|Option Contracts

Option contracts fall into one of two basic categories: calls or puts. In a call (put) option contract the contract buyer has the right but not the obligation to purchase (sell) a fixed quantity from (to) the seller at a fixed price before a certain date. Every option contract has both a buyer and a seller. The contract buyer has a right but not an obligation to initiate an exchange; the seller is obligated to perform, however, should the buyer exercise the contract rights. The fixed price in an option contract is the exercise or strike price-the price at which the contract buyer either purchases from the contract seller (call option) or sells to the contract seller (put option). The contract maturity date is also called the contract expiration date. Finally, the option buyer makes a nonrefundable payment to the option seller, called the option premium, to obtain the rights of the option contract. The purpose of an option pricing model, such as the Black-Scholes model or the binomial model, is to estimate a “fair” option contract premium.

In general, a call option buyer (seller) expects the price of the underlying security to increase (decrease or stay steady) above the option exercise price. If not, the call option seller keeps the nonrefundable payment, the call option premium. A put option buyer (seller) expects the price of the underlying security to decrease (increase or stay steady) below the option exercise price. If so, the put option buyer can exercise the right to sell the underlying instrument to the put option seller at the relatively high exercise price. If an option contract is held to expiration, the option may expire worthless, be exercised by the contract buyer, or be sold for the difference between the contract exercise price and the market price of the underlying.

Consider the call option risk profile in Exhibit 1.12. The buyer of an option contract, call or put option, is called the option long; the option seller is called the option short. In Exhibit 1.12, if the unexpected change in the underlying instrument’s price, $\Delta P$, at option expiration is negative (or prices fall), the long call position is worthless and the call option buyer forfeits the call premium. At the same time, the short call option position is profitable by the amount of the premium. The horizontal, dashed lines to the left of the vertical axis illustrate the returns. If the unexpected change in the underlying instrument’s price, $\Delta P$, at option expiration is positive (or prices rise), the long call position increases the value of the option buyer, $\Delta V$. Before the option buyer can break even, however, the price must rise sufficiently to cover the nonrefundable option premium paid to the option short. At the same time, the short call position keeps part of the premium paid by the call long until prices rise sufficiently. The sloping, dashed lines to the right of the vertical axis illustrate the returns. Exhibit 1.12 shows that the risk profile of a long call position is similar to a long forward or long futures contract position. The risk profile of a short call option position is similar to a short forward or futures contract position but only if underlying prices rise.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 衍生品, 金融代写, 金融衍生品

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## 金融代写|金融衍生品代写Financial Derivatives代考|Hull and White

Hull and White have extended the Vasicek model to incorporate time dependent parameters.
$$d r=(\eta(t)-\gamma r) d t+c d W$$
Under this risk-neutral process the value of zero-coupon bonds
$$Z(r, t ; T)=e^{A(t ; T)-r B(t ; T)}$$
where $A(t ; T)=\int_t^T \eta(s) B(s ; T) d s+\frac{c^2}{2 \gamma^2}\left(T-t+\frac{2}{\gamma} e^{-\gamma(T-t)}-\frac{1}{2 \gamma} e^{-2 \gamma(T-t)}-\frac{3}{2 \gamma}\right)$

and
$$B(t ; T)=\frac{1}{\gamma}\left(1-e^{-\gamma(T-t)}\right)$$
Here the time-dependent parameter $\eta(t)$ can also be identified from the technique of yield curve fitting given as follows.

## 金融代写|金融衍生品代写Financial Derivatives代考|Yield Curve Fitting

Let us consider the Ho and Lee short term model. It is the simplest that can be used to fit the yield curve. It will be useful to examine this model in detail to see one way in which fitting is done in practice. In this model, the solution of bond pricing equation for a zerocoupon bond is simply
\begin{aligned} Z(r, t ; T) & =e^{A(t ; T)-r B(t ; T)} \ A & =\int_t^T \eta(s)(T-s) d s+\frac{1}{6} c^2(T-t)^3 \end{aligned}

## 金融代写|金融衍生品代写Financial Derivatives代考|Hull and White

Hull 和 White 扩展了 Vasicek 模型以包含时间相关参数。
$$d r=(\eta(t)-\gamma r) d t+c d W$$

$$Z(r, t ; T)=e^{A(t ; T)-r B(t ; T)}$$

$$B(t ; T)=\frac{1}{\gamma}\left(1-e^{-\gamma(T-t)}\right)$$

## 金融代写|金融衍生品代写Financial Derivatives代考|Yield Curve Fitting

$$Z(r, t ; T)=e^{A(t ; T)-r B(t ; T)} A \quad=\int_t^T \eta(s)(T-s) d s+\frac{1}{6} c^2(T-t)^3$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 金融代写|金融衍生品代写Financial Derivatives代考|STRESS TESTING AND BACK TESTING

In addition to calculating a VaR, many companies carry out what is known as a stress test of their portfolio. Stress testing involves estimating how the portfolio would have performed under some of the most extreme market moves seen in the last 10 to 20 years.
For example, to test the impact of an extreme movement in U.S. equity prices, a company might set the percentage changes in all market variables equal to those on October 19, 1987 (when the S\&P 500 moved by 22.3 standard deviations). If this is considered to be too extreme, the company might choose January 8,1988 (when the S\&P 500 moved by 6.8 standard deviations). To test the effect of extreme movements in U.K. interest rates, the company might set the percentage changes in all market variables equal to those on April 10, 1992 (when 10-year bond yields moved by 7.7 standard deviations).

Stress testing can be considered as a way of taking into account extreme events that do occur from time to time but that are virtually impossible according to the probability distributions assumed for market variables. A five-standard-deviation daily move in a market variable is one such extreme event. Under the assumption of a normal distribution, it happens about once every 7,000 years, but, in practice, it is not uncommon to see a five-standard-deviation daily move once or twice every 10 years. Whatever the method used for calculating VaR, an important reality check is back testing. It involves testing how well the VaR estimates would have performed in the past. Suppose that we are calculating a 1-day $99 \%$ VaR. Back testing would involve looking at how often the loss in a day exceeded the 1-day $99 \%$ VaR calculated for that day. If this happened on about $1 \%$ of the days, we can feel reasonably comfortable with the methodology for calculating VaR. If it happened on, say, $7 \%$ of days, the methodology is suspect.

## 金融代写|金融衍生品代写Financial Derivatives代考|PRINCIPAL COMPONENTS ANALYSIS

One approach to handling the risk arising from groups of highly correlated market variables isprincipal components analysis. This takes historical data on movements in the market variablesand attempts to define a set of components or factors that explain the movements. The approach is best illustrated with an example. The market variables we will consider are tenU.S. Treasury rates with maturities between three months and 30 years.results produced by Frye for these market variables using 1,543 daily observations between 1989 and 1995.u The first column in Table 16.3 shows the maturities of the rates that were considered. The remaining ten columns in the table show the ten factors (or principal components) describing the rate moves. The first factor, shown in the column labeled PCI, corresponds to a roughly parallel shift in the yield curve. When we have one unit of that factor, the three-month rate increases by 0.21 basis points, the six-month rate increases by 0.26 basis points, and so on. The second factor is shown in the column labeled PC2. It corresponds to a “twist” or “steepening” of the yield curve. Rates between 3 months and 2 years move in one direction; rates between 3 years and 30 years move in the other direction. The third factor corresponds to a “bowing” of the yield curve. Rates at the short end and long end of the yield curve move in one direction; rates in the middle move in the other direction. The interest rate move for a particular factor is known as factorloading. In our example, the first factor’s loading for the three-month rate is 0.21 .15 Because there are ten rates and ten factors, the interest rate changes observed on any given day can always be expressed as a linear sum of the factors by solving a set of ten simultaneous equations. The quantity of a particular factor in the interest rate changes on a particular day is known as the factor score for that day.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 衍生品, 金融代写, 金融衍生品

## avatest™帮您通过考试

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 金融代写|金融衍生品代写Financial Derivatives代考|Proof of the sufficient condition

Prove it by change of probability measure

If $\mathcal{E}(-\theta \cdot B)$ is a martingale, then we can define a new probability $\mathbb{Q}$ by $\frac{d \mathbb{Q}}{d \mathbb{P}}=\mathcal{E}(-\theta \cdot B)$.

By Girsanov’s theorem , $\hat{B}(t)=B(t)+\int_0^t \theta(s) d s$ is a Brownian motion under $\mathbb{Q}$.

For any portfolio $\pi(\cdot)$, the discounted wealth process $\bar{X}(t)$ satisfies
$d \bar{X}(t)=\bar{\pi}(t)^{\prime} u(t) d t+\bar{\pi}(t)^{\prime} \sigma(t) d B(t)=\bar{\pi}(t)^{\prime} \sigma(t) d \hat{B}(t)$.

$\bar{X}$ is a $\mathbb{Q}$-local martingale, hence supermartingale.

So $\mathbb{E}_{\mathbb{Q}}[\bar{X}(T)] \leq x(0)=0$. which means $\pi(\cdot)$ can not be an arbitrage opportunity.

Equivalent martingale measure (EMM)

The market is called standard if the condition in Th 1 holds.

In this case, any $\theta(\cdot)$ satisfying $\sigma(\cdot) \theta(\cdot)=u(\cdot)$ is called a risk premium (process)

Any probability measure $\mathbb{Q}$ defined by $\frac{d \mathbb{Q}}{d \mathbb{P}}=\mathcal{E}(-\theta \cdot B)$ is called an EMM. Under $\mathbb{Q}$,

discounted wealth process for a tame portfolio is a local martingale, therefore a supermartingale;
discounted prices of assets are local martingales.

## 金融代写|金融衍生品代写Financial Derivatives代考|Replication and pricing

Definition 2: A contingent claim $\xi$ at time $T$ is replicable if it can be replicated by some tame portfolio and the initial wealth $S_0(0) \mathbb{E}_{\mathbb{Q}}\left[\xi / S_0(T)\right]$ for some $E M M \mathbb{Q}$.

For $\forall \xi$ at time $T$ replicated by some tame portfolio, then the replication cost at time 0 satisfies $V_{\xi}(0) \geq S_0(0) \mathbb{E}_{\mathrm{Q}}\left[\xi / S_0(T)\right]$.

If $V_{\xi}(0)=S_0(0) \mathbb{E}_{\mathbb{Q}}\left[\xi / S_0(T)\right]$, then the discounted wealth process is a $\mathbb{Q}$-martingale.

This definition excludes those contingent claim achieved by portfolios with discounted wealth process being strict local martingale.

Replication and pricing
Theorem 2: For any replicable $\xi, \mathbb{E}_{\mathbb{Q}}\left[\xi / S_0(T)\right]$ is invariant for any EMM $\mathbb{Q}$.

Given a replicable $\xi$,

For any EMM $\mathbb{Q}$, calculate $\bar{X}t:=\mathbb{E}{\mathbb{Q}}\left[\xi / S_0(T) \mid \mathcal{F}_t\right]$;

Calculate $d \bar{X}_t=[\cdots] d t+H_t^{\top} d \hat{B}(t)$.

According to Theorem 2, $\bar{X}_t$ is invariant over all EMM.
Solve $\sigma(t)^{\top} \pi_t e^{-\int_0^t r_s d s}=H(t)$.

$\pi_t$ is the replicating portfolio.

If $\xi$ is replicable, its replication cost is $S_0(0) \mathbb{E}_{\mathbb{Q}}\left[\frac{\xi}{S_0(T)}\right]$.

## 金融代写|金融衍生品代写Financial Derivatives代考|Proof of the sufficient condition

$d \bar{X}(t)=\bar{\pi}(t)^{\prime} u(t) d t+\bar{\pi}(t)^{\prime} \sigma(t) d B(t)=\bar{\pi}(t)^{\prime} \sigma(t) d \hat{B}(t)$.
$\bar{X}$ 是一个Q-本地鞅，因此超级鞅。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。