# 金融代写|金融模型代写Modelling in finance代考|FINN3103

## 金融代写|金融模型代写Modelling in finance代考|Federal Funds futures

The 30-Day Federal Funds futures (simply called Fed Funds futures) are based on the monthly average of overnight Fed Funds rate for the contract month.

Let $0<t_0<t_1<\cdots<t_n$ be the relevant dates for the Fed Funds futures, with $t_0$ the first business day of the reference month, $t_{i+1}$ the business day following $t_i$ and $t_n$ the first business day of the following month. Let $\delta_i$ be the accrual factor between $t_i$ and $t_{i+1}(0 \leq i \leq n-1)$ and $\delta$ the accrual factor for the total period $\left[t_0, t_n\right]$.
The overnight rates fixing between $t_i$ and $t_{i+1}$ are denoted $I_X^O\left(t_i\right)$ with $F_X^O\left(t_i, t_i, t_{i+1}\right)=I_X^O\left(t_i\right)$. The future price on the final settlement date $t_n$ is
$$\Phi_{t_n}=1-\frac{1}{\delta}\left(\sum_{i=0}^{n-1} \delta_i I_X^O\left(t_i\right)\right) .$$
The margining is done on the price multiplied by the notional and divided by the one month accrual fraction (1/12).

The model used is the Hull-White model on the discounting curve as described in Appendix A. The result uses the deterministic spread hypothesis $\mathrm{S}^{\mathrm{CPN}}$ between the curves.

Theorem 6.6 In the HJM model on the discount curve in the multi-curve framework and with the deterministic hypothesis between spread and discount factor ratio $\mathrm{SO}^{C P N}$, the price of the average overnight future for the period $\left[t_0, t_n\right]$ is given for $t_j \leq t<t_{j+1}$ $(j \in{0,1, \ldots, n})$ by
$$\Phi_t=1-\frac{1}{\delta}\left(\sum_{i=1}^j \delta_i I_X^O\left(t_i\right)+\sum_{i=j+1}^{n-1} 1-\left(1+\delta_i F_X^O\left(t, t_i, t_{i+1}\right)\right) \gamma\left(t, t_i, t_i, t_{i+1}\right)\right)$$
where $\gamma$ is given in Appendix A.
The formula is divided into two parts to cope with the case where the averaging period has started already. In those cases the convexity adjustments $\gamma$ are very small.
Proof: The generic futures pricing formula applied to the instrument is
\begin{aligned} \Phi_t &=\mathrm{E}^N\left[1-\frac{1}{\delta}\left(\sum_{i=0}^{n-1} \delta_i F_X^O\left(t_i, t_i, t_{i+1}\right)\right) \mid \mathcal{F}t\right] \ &=1-\frac{1}{\delta}\left(\sum{i=1}^j \delta_i I_X^O\left(t_i\right)+\sum_{i=j+1}^{n-1} 1-\mathrm{E}^N\left[1+\delta_i F_X^O\left(t_i, t_i, t_{i+1}\right) \mid \mathcal{F}_t\right]\right) . \end{aligned}
We are left with the same technical computation as in Section 2.6, once for each period. The result follows immediately.

## 金融代写|金融模型代写Modelling in finance代考|Deliverable swaps futures

The arbitrage-free price of the future $\Phi_\theta$ in $\theta$ satisfies, for PV $_\theta$ the present value of the underlying swap without up-front payment in $\theta$ and with notional 1 to the long pärty,
$$\left(1-\Phi_\theta\right) N P^D\left(\theta, t_0\right)+\mathrm{PV}\theta N=0$$ or $$\Phi\theta=1+\frac{\mathrm{PV}_\theta}{P^D\left(\theta, t_0\right)}$$

After delivery, the swap is cleared on CME clearing. The swaps are collateralised according to CME variation margin methodology. Here we suppose that the methodology is equivalent to a discounting with a given discounting curve as described in the previous chapters of the book.

The results presented here are similar to the one presented in Kennedy (2010) for SwapNote futures and in Henrard (2006b) for bond futures. They are more general as they are for multi-factor models and in the multi-curve framework.

Using the generic pricing future price process theorem (Hunt and Kennedy, 2004, Theorem 12.6),
$$\Phi_0=\mathrm{E}^{\mathbb{N}}\left[\Phi_\theta\right] .$$
Using the swap pricing formula,
$$\Phi_\theta=1+\sum_{i=0}^n d_\theta^i \frac{P_X^D\left(\theta, t_i\right)}{P_X^D\left(\theta, t_0\right)} .$$
The pricing formula uses the description of the swap in terms of equivalent cashflows as provided in Section 2.4.
Using Lemma A.1, if $d_\theta^i$ is independent of the ratio of discount factors, the futures can be written as a function of a random variable $X_i$ :
$$1+\sum_{i=0}^n d_t^i \frac{P_X^D\left(0, t_i\right)}{P_X^D\left(0, t_0\right)} \exp \left(-\alpha\left(0, \theta, t_0, t_i\right) X_i-\frac{1}{2} \alpha^2\left(0, \theta, t_0, t_i\right)\right) \gamma\left(0, \theta, t_0, t_i\right) .$$
The exponential terms have a expectation of 1 . We thus obtain the following result.

# 金融模型代考

## 金融代写|金融模型代写在金融建模代考|凸性调整

$$\sum_{i=1}^n \delta_i I_X^O\left(t_{i-1}\right)$$

$N_0 \mathrm{E}^N\left[\left(N_{t_n}\right)^{-1} \sum_{i=1}^n \delta_i I_X^O\left(t_{i-1}\right)\right]$
$=\sum_{i=1}^n \mathrm{E}^N\left[\left(N_{t_n}\right)^{-1} \delta_i F_i^O\left(t_{i-1}\right)\right]$
$=\sum_{i=1}^n \mathrm{E}^N\left[\left(N_{t_n}\right)^{-1}\left(\left(1+\delta_i F_X^D\left(t_{i-1}, t_{i-1}, t_i\right)\right) \beta^O-1\right)\right]$ .

$$N_{t_n}^{-1}=P_X^D\left(0, t_n\right) \exp \left(-\int_0^{t_n} v\left(s, t_n\right) \cdot d W_s-\frac{1}{2} \int_0^{t_n}\left|v\left(s, t_n\right)\right|^2 d s\right)$$和
\begin{aligned} 1+\delta_i F^D\left(t_{i-1}, t_{i-1}, t_i\right)=&\left(1+\delta_i F_i^D\left(0, t_{i-1}, t_i\right)\right) \ & \times \exp \left(-\int_0^{t_{i-1}}\left(v\left(s, t_i\right)-v\left(s, t_{i-1}\right)\right) \cdot d W_s\right.\ &\left.-\frac{1}{2} \int_0^{t_{i-1}}\left(\left|v\left(s, t_i\right)\right|^2-\left|v\left(s, t_{i-1}\right)\right|^2\right) d s\right) . \end{aligned}

## 金融代写|金融模型代写在金融建模代考|凸性调整和近似

$$A_a \simeq \ln \left(\prod_{i=1}^n\left(1+\delta_i I^O\left(t_i\right)\right)=\ln \left(1+A_c\right) .\right.$$当贴现曲线为隔夜远期利率曲线时，离散复利利率本身可近似为连续组合
$$A_c \simeq \exp \left(\int_{t_0}^{t_n} r_\tau d \tau\right)-1 .$$因此，这种息票的现值近似为
$$M_0 \mathrm{E}^M\left[\left(M_{t_n}\right)^{-1} \int_{t_0}^{t_n} r_\tau d \tau\right]$$ 对于任何数字 $M$。这里我们选择了 $t_n$-forward numeraire，也就是说 $P_X^D\left(., t_n\right)$。数字的变化 $\mathrm{HJM}$ 模型由

\begin{aligned} \int_{t_0}^{t_n} r_\tau d \tau=& \ln \left(\frac{P^D\left(0, t_0\right)}{P^D\left(0, t_n\right)}\right)\left(\int_0^{t_0} v\left(s, t_0\right) \cdot d W_s^{t_n}+\int_0^{t_n} v\left(s, t_n\right) \cdot d W_s^{t_n}\right.\ &-\frac{1}{2} \int_0^{t_n} \mid\left(v\left(s, t_n\right)-\left.v\left(s, t_0\right)\right|^2 d s\right) \end{aligned}两个随机积分的期望为0，最终结果如下。

$$P^D\left(0, t_n\right)\left(\ln \left(\frac{P^D\left(0, t_0\right)}{P^D\left(0, t_n\right)}\right)+\zeta\right)$$

$$\zeta=\frac{1}{2} \int_0^{t_n} \mid\left(v\left(s, t_n\right)-\left.v\left(s, t_0\right)\right|^2 d s .\right.$$

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