# 金融代写|金融衍生品代写Financial derivatives代考|FIN4250

## 金融代写|金融衍生品代写Financial derivatives代考|Equity Index

A stock market index is a measurement targeting to describe the performance of the stock market. It is computed from the price of selected stocks. An index is a mathematical construct which does not allow direct investment. People may invest through index futures, mutual funds or exchange-traded funds (ETFs) tracking the index.

An equity index may be classified according to its calculation method. Let $k$ be the number of stocks included in the index, $n_t^i$ the total number of outstanding (freefloat) shares for stock $i$ on day $t$, and $S_t^i$ the stock price. The two main classes of index are as follows:

• Capitalization Weighted Index:
The majority of the equity indices belongs to this type, e.g. S\&P 500, Euro Stoxx 50. The calculation for each index is a variation of the below general method:
$$\text { Index }t=\text { Index }{t-1} \times \frac{\sum_{i=1}^k n_t^i S_t^i}{\sum_{i=1}^k n_t^i S_{t-1}^i}$$
• Price Weighted Index:
There are only a few equity indices of this type. Dow Jones Industrial Average and Nikkei Index are the most well-known ones. The calculation is as follows:
$$\text { Index }t=\frac{\sum{i=1}^k S_t^i}{\text { Divisor }_t},$$
where Divisor $_t$ serves for scaling the price continuity by adjusting its value in case of some corporate actions and other external factors. As the stock price alone does not contain any particular information, the methodology of relating the weight to its price level is obviously not a rational approach.

## 金融代写|金融衍生品代写Financial derivatives代考|Equity Forward, Cash & Carry

An equity forward on a single stock is an OTC contract between two parties where the long party is engaged to buy from the short party a certain quantity $N$ (called notional) of the stock at a pre-agreed delivery price $K$ at the expiry of the contract. The payoff of the long party is
$$\text { Payoff }=N \times\left(S_T-K\right),$$
where $S_T$ is the stock price at expiry $T$.
The market forward price (denoted as $F$, the value of $K$ such that the forward contract value is zero) for equities can be derived with no-arbitrage condition. In its general form, a stock is assumed to pay both cash and proportional dividend. The stock borrow/lending fee rate is called repo rate $^2$ in modelling for equities.
Proposition 3.1 Let $S_0$ be stock price at time 0.
(1) For a non-dividend paying stock with zero repo rate and constant interest rate $r$, the forward price $F$ is
$$F=S_0 e^{r T}$$
(2) For the general case, let $r_t$ be the instantaneous interest rate, $\lambda_t$ the instantaneous repo rate and $d_{t_k}$ the rate for the proportional dividend at time $t_k{ }^3$ The forward price $F$ is
$$F=S_0 e^{\int_0^T\left(r_t-\lambda_t\right) d t-\sum_{t_k \leq T} d_{t_k}} .$$
Proof
(1) The short party can hedge the forward contract by the Cash \& Carry Strategy which consists of buying one stock at time 0 financed by a loan and holding the stock until the expiry. The table shows that, by setting $K=S_0 e^{r T}$, the total $\mathrm{P} / \mathrm{L}$ of the short forward position and the long stock position with a loan is zero which means perfect hedging.

# 金融衍生品代考

## 金融代写|金融衍生品代写Financial derivatives代考|Equity Index

• Capitalization Weighted Index:
大多数股票指数属于此类，例如 S\&P 500、Euro Stoxx 50。每个指数的计算都是以下一般方法 的变体:
$$\text { Index } t=\operatorname{Index} t-1 \times \frac{\sum_{i=1}^k n_t^i S_t^i}{\sum_{i=1}^k n_t^i S_{t-1}^i}$$
• 价格加权指数:
这种类型的股票指数很少。道琼斯工业平均指数和日经指数是最著名的。计算如下:
$\$ \$$\backslash text { Index } t=\backslash frac \left{\right. sum \left.{i=1}^{\wedge} \mathrm{k} S_{-} t^{\wedge} i\right} \backslash text \left.{\text { Divisor }}_{-} t\right}_{,} \ \$$
where Divisor $t_t$ 在某些公司行为和其他外部因嫊的情况下，通过调整其价值来扩大价格连续性。
由于股价本身不包含任何特定信息，因此将权重与其价格水平相关联的方法显然不是一种合理
的方法。

## 金融代写|金融衍生品代写Financial derivatives代考|Equity Forward, Cash & Carry

$$\text { Payoff }=N \times\left(S_T-K\right),$$

(1) 对于回购利率为零且利率不变的非股息支付股票 $r$ ，远期价格 $F$ 是
$$F=S_0 e^{r T}$$
(2) 对于一般情况，让 $r_t$ 是瞬时利率， $\lambda_t$ 瞬时回购利率和 $d_{t_k}$ 当时的比例股息率 $t_k{ }^3$ 远期价格 $F$ 是
$$F=S_0 e^{\int_0^T\left(r_t-\lambda_k\right) d t-\sum_{t_k \leqslant T} d_{t_k}} .$$

（1）空方可以通过现金、携带策略对远期合约进行套期保值，该策略包括在时间 0 购买一只股票并持 有该股票直至到期。该表表明，通过设置 $K=S_0 e^{r T}$ ，总数 $\mathrm{P} / \mathrm{L}$ 空头远期头寸和带贷款的多头股 票头寸的比例为零，这意味着完美的对冲。

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