# 金融代写|利率建模代写Interest Rate Modeling代考|MTH5520

## 金融代写|利率建模代写Interest Rate Modeling代考|IMPLIED CAP AND CAPLET VOLATILITIES

As a preparation for subsequent sections, in this section, we consider calibrating the LIBOR market model based on cap prices only. Cap prices imply prices of caplets of various maturities, and the Black’s volatilities corresponding to the caplet prices will be taken to be the norms of the local volatility vectors. The volatility vectors of forward rates can be determined based on the norms and, in addition, information on forward-rate correlations.

Let $\operatorname{Cap}(n)$ denote the market price of a cap that has $n$ caplets, and let $K_n$ denote the strike rate. In the marketplace, a cap is quoted using the implied cap volatility, $\bar{\sigma}n$, defined as the single non-negative number that, when plugged in into Black’s formula for all caplets, reproduces the market price of the cap. That is, $\bar{\sigma}_n$ satisfies $$\operatorname{Cap}(n)=\sum{j=0}^{n-1} P_0^{T_{j+1}} \Delta T_j \cdot B C\left(f_j(0), K_n, T_j, \bar{\sigma}_n\right),$$
where $B C(f, K, T, \sigma)$ stands for Black’s call formula,
$$B C(f, K, T, \sigma)=f N\left(d_1\right)-K N\left(d_2\right)$$ with
$$d_1=\frac{\ln (f / K)+(1 / 2) \sigma^2 T}{\sigma \sqrt{T}}, \quad d_2=d_1-\sigma \sqrt{T} .$$
Given a sequence of implied cap volatilities, $\bar{\sigma}n, n=1,2, \ldots, N$, for increasing cap maturities, we can bootstrap the values of all caplets of maturities up to the longest maturity of the caps. The caplet prices can be quoted using their Black’s volatilities, $\left{\sigma_j\right}$. In terms of the Black’s volatilities of caplets, we can recast the cap price as \begin{aligned} \operatorname{Cap}(n) &=\sum{j=0}^{n-1} P_0^{T_{j+1}} \Delta T_j \cdot B C\left(f_j(0), K_n, T_j, \sigma_j\right) \ &=\sum_{j=0}^{n-1} \operatorname{Caplet}(j+1, n) \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|CALIBRATING THE LIBOR MARKET MODEL TO CAPS

Calibration here means to specify $\left{\gamma_j(t)\right}$, the local volatility vector of forward rates, so that the market prices of all caplets are reproduced. According to Black’s formula, the implied Black’s volatility, $\sigma_j$, relates to the local volatility vector of the $j$ th forward rate via
$$\sigma_j^2=\frac{1}{T_j} \int_0^{T_j}\left|\gamma_j(t)\right|^2 \mathrm{~d} t$$
The right-hand side is the mean variance of the $j$ th forward rate. Equation $7.12$ allows us to determine $\left|\gamma_j(t)\right|$. For a parametric calibration, we may take the functional specification of Equation $6.79$ and solve for the parameters through an optimization procedure that minimizes the aggregated squared errors between the two sides of Equation 7.12. The procedure is a rather typical one.

The focus of this section, however, is on nonparametric calibration. Suppose that $\gamma_j(t)$ is constant in $t$, we can simply take
$$\left|\gamma_j(t)\right|=\sigma_j, \quad 0 \leq t \leq T_j, \quad j=1, \ldots, N .$$
The plot of $\left{\left|\gamma_j(t)\right|\right}$ is called the local volatility surface of the LIBOR market model. Figure $7.2$ shows the local volatility surface for May 22, 1997, generated using the implied Black’s volatility of the caplets in Table 7.3. Note that the part of the surface over the area of $t \leq T$ is the only relevant part.

## 金融代写|利率建模代写利率建模代考|隐含的CAP和CAPLET波动率

，其中$B C(f, K, T, \sigma)$代表布莱克的赎回公式，
$$B C(f, K, T, \sigma)=f N\left(d_1\right)-K N\left(d_2\right)$$ with
$$d_1=\frac{\ln (f / K)+(1 / 2) \sigma^2 T}{\sigma \sqrt{T}}, \quad d_2=d_1-\sigma \sqrt{T} .$$

## 金融代写|利率建模代写利率建模代考|校准LIBOR市场模型到上限

$$\sigma_j^2=\frac{1}{T_j} \int_0^{T_j}\left|\gamma_j(t)\right|^2 \mathrm{~d} t$$

$$\left|\gamma_j(t)\right|=\sigma_j, \quad 0 \leq t \leq T_j, \quad j=1, \ldots, N .$$
$\left{\left|\gamma_j(t)\right|\right}$的图称为LIBOR市场模型的局部波动面。图$7.2$显示了1997年5月22日的局部波动率面，它是使用表7.3中小图的隐含布莱克波动率生成的。请注意，在$t \leq T$区域上的那部分表面是唯一相关的部分

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