# 金融代写|金融衍生品代写Financial derivatives代考|FINA5220

## 金融代写|金融衍生品代写Financial derivatives代考|Overnight Indexed Swap

An overnight indexed swap (OIS) is an interest rate swap where the cash-flow of the floating leg is based on a floating rate calculated from the overnight rates such as Eonia or effective fed funds rate for the concerned period.

Consider the example of Eonia which is quoted on ACT/360. The floating rate for OIS of the $k$-th period $\left(t_{k-1}, t_k\right)$ will be calculated as the geometric average of Eonia fixings:

$$R_k=\left[\prod_{i=1}^{N_{f i x}^k}\left(1+\frac{r_i \times n_i}{360}\right)-1\right] \frac{360}{N_k},$$
where
$R_k$ : rate for OIS of the $k$-th period
$N_{f i x}^k$ : total number of fixings of the overnight rate for the $k$-th period
$r_i$ : Eonia fixing rate for day $i$
$n_i$ : number of calendar days covered by $r_i$
$N_k$ : total number of calendar days for the $k$-th period
The rate of OIS implies less risk than the corresponding interbank lending rate such as Libor and Euribor because OIS rate is based on actual trades, which makes it more difficult to be manipulated.

Because of counterparty risk, the derivatives trades are normally collateralized. The cash collaterals are in general paid with overnight interest rate. As a consequence, it is considered as a secured lending rate, i.e. a proxy of risk-free rate. It is natural to apply OIS implied rates for discounting in financial calculation. Even though Libor and Euribor are widely used as reference for swaps, OIS is gaining increasing popularity. The OIS yield curve construction is similar to that for Libor rate based instruments.

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

A stock dividend is a payment (e.g. yearly or quarterly) made by a corporation to its shareholders as a distribution of profits. The dividend amount is decided by the Board of the company at each time. For a detailed discussion on the dividend policy and practice, we refer to [72].

When a dividend is declared, an ex-dividend date is specified. It is the date on which shares bought no longer come attached with the right to be paid the declared dividend. Other factors remaining equal, the share price will normally drop on the ex-dividend date by the amount equivalent to the after-tax dividend. Before exdividend date, the stock is said to trade cum dividend.

A stock dividend can be either a cash dividend in monetary form, or a stock dividend which involves the company issuing more shares to its existing shareholders. For example, a 5\% stock dividend means that the shareholders receive one new share for every twenty already owned shares. Unlike the cash dividend, the stock dividend does not change the company’s market capitalization. In modelling, stock dividends are also called proportional dividends.

Suppose that a stock pays a dividend value of $D$ with ex-dividend date $t_d$. Let $S_{t_d-}$ be the stock price just before the ex-dividend date. The quantity $d=D / S_{t_d}$ is called the dividend yield.

Obviously, $S_{t_d}$ cannot be equal to $S_{t_d}$. If it was, the strategy of buying the stock immediately before $t_d$, collecting the dividend, and selling straight away, would yield a risk-free profit. In fact, in the absence of other factors such as taxes, the asset price must fall by exactly the amount of the dividend payment. That is, for a stock paying both proportional dividend at rate $d^{\text {prop }}$ and cash dividend amount $D^{\text {cash }}$, we have
$$S_{t_d}=S_{t_{\bar{d}}}\left(1-d^{\text {prop }}\right)-D^{\text {cash }} .$$
In continuous-time modelling, the proportional divided rate is the value $d$ such that $e^{-d}=1-d^{\text {prop }}$. Hence,
$$S_{t_d}=S_{t_d^{-}} e^{-d}-D^{c a s h}$$

# 金融衍生品代考

## 金融代写|金融衍生品代写Financial derivatives代考|Overnight Indexed Swap

$$R_k=\left[\prod_{i=1}^{N_{f i z}^k}\left(1+\frac{r_i \times n_i}{360}\right)-1\right] \frac{360}{N_k},$$

$R_k$ : OIS 的速率 $k$-th期
$N_{f i x}^k$ : 隔夜利率定价的总数 $k$-th期
$r_i$ : Eonia 日定盘价 $i$
$n_i$ : 涵盖的日历天数 $r_i$
$N_k$ : 日历的总天数 $k$-th 期
OIS 利率比相应的银行同业拆借利率如 Libor 和 Euribor 意味着风险更小，因为 OIS 利率基于实际 交易，这使得它更难被操纵。

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

$$S_{t_d}=S_{t_d}\left(1-d^{\text {prop }}\right)-D^{\text {cash }}$$

$$S_{t_d}=S_{t_d^{-}} e^{-d}-D^{c a s h}$$

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