# 金融代写|金融模型代写Modelling in finance代考|FIN280

In this section we compute the Fed Fund swaps value including the convexity adjustment. The amount
$$\sum_{i=1}^n \delta_i I_X^O\left(t_{i-1}\right)$$
is paid in $t_n$.
We work in the multi-curve framework under the assumption $\mathrm{S}^{\mathrm{CPN}}$ of constant spread $\beta^O\left(t_{i-1}, t_i\right)$ between the discounting curve and the overnight forward curve. This assumption is trivially satisfied if the discounting curve is equal to the overnight forward curve. The model used is the multi-factor HJM model on the discounting curve as described in Appendix A. The forward overnight rate seen from $t$ is denoted $F_X^O\left(t, t_{i-1}, t_i\right)$. We will abbreviate it as $F_i^O(t)$.
The value of the coupon is, using the cash account numeraire,
$N_0 \mathrm{E}^N\left[\left(N_{t_n}\right)^{-1} \sum_{i=1}^n \delta_i I_X^O\left(t_{i-1}\right)\right]$
$=\sum_{i=1}^n \mathrm{E}^N\left[\left(N_{t_n}\right)^{-1} \delta_i F_i^O\left(t_{i-1}\right)\right]$
$=\sum_{i=1}^n \mathrm{E}^N\left[\left(N_{t_n}\right)^{-1}\left(\left(1+\delta_i F_X^D\left(t_{i-1}, t_{i-1}, t_i\right)\right) \beta^O-1\right)\right]$.
We compute the value of one of these expectations. For that we use standard HJM results given by Lemma A.1 and A.3:
$$N_{t_n}^{-1}=P_X^D\left(0, t_n\right) \exp \left(-\int_0^{t_n} v\left(s, t_n\right) \cdot d W_s-\frac{1}{2} \int_0^{t_n}\left|v\left(s, t_n\right)\right|^2 d s\right)$$ and
\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}
In what follows, we use the usual extension of $v(s, t)$ for values $s>t$ with 0 .

## 金融代写|金融模型代写Modelling in finance代考|Convexity adjustment and approximation

We can combine the approximations discussed above with a convexity adjustment. The amount paid can be approximated by
$$A_a \simeq \ln \left(\prod_{i=1}^n\left(1+\delta_i I^O\left(t_i\right)\right)=\ln \left(1+A_c\right) .\right.$$
When the discounting curve is the overnight forward rate curve, the discretely compounded rate itself can be approximated by a continuous composition
$$A_c \simeq \exp \left(\int_{t_0}^{t_n} r_\tau d \tau\right)-1 .$$
The present value of such a coupon is thus approximated by
$$M_0 \mathrm{E}^M\left[\left(M_{t_n}\right)^{-1} \int_{t_0}^{t_n} r_\tau d \tau\right]$$ for any numeraire $M$. Here we chose the $t_n$-forward numeraire, that is $P_X^D\left(., t_n\right)$. The change of numeraire in the $\mathrm{HJM}$ model is given by
$$d W_t^{t_n}=d W_t+v\left(t, t_n\right) d t .$$
We use the result on the dynamic of the cash account twice (once to $t_0$ and once to $t_n$ ) and the above change of numeraire to obtain
\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}
The expectation of the two stochastic integrals is 0 and the final result is as follows.
Theorem 6.5 In the multi-curve framework under hypothesis $S 0^{C P N}$ on the basis between discounting and overnight forwards, the approximated present value of a Fed Fund swap coupon in the multi-factor HJM model on the discounting curve is
$$P^D\left(0, t_n\right)\left(\ln \left(\frac{P^D\left(0, t_0\right)}{P^D\left(0, t_n\right)}\right)+\zeta\right)$$
with
$$\zeta=\frac{1}{2} \int_0^{t_n} \mid\left(v\left(s, t_n\right)-\left.v\left(s, t_0\right)\right|^2 d s .\right.$$

# 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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