## 金融代写|利率建模代写Interest Rate Modeling代考|CALIBRATION TO CAPS, SWAPTIONS

In the fixed-income market, caps, floors, and ATM swaptions dominate liquidity, and hence they are considered to be benchmark derivatives. For hedging purposes, swaptions are used more often than are caps and floors. Hence, it is highly desirable to have a LIBOR model calibrated to swaptions, in addition to caps and floors. If we take a parametric approach for the calibration, we will have a trivial numerical problem for which we do not have much to say. All we need to do is to adopt the parameterization for $\left|\gamma_j(t)\right|$ like Equation 6.79, and solve for the parameters through a minimization procedure. In this section, we instead will take a nonparametric approach and avoid putting a structure on the solution, in hopes of gaining insights into the “objective” local volatility function.

Unlike conventional approaches of option calibrations that match prices, we instead take an alternative approach to match Black’s volatilities. The calibration to ATM swaptions is described as follows. Let $\left{\sigma_{m, n}\right}$ be a set of implied Black’s volatilities of benchmark swaptions. We look for $\left{\gamma_j(t)\right}$, which makes the Black’s volatilities of the model equal to the implied Black’s volatilities:
$$\sigma_{m, n}^2=\frac{1}{T_m-t} \int_t^{T_m} \sum_{j, k=m+1}^n \omega_j \omega_k \gamma_j^{\mathrm{T}}(s) \boldsymbol{\gamma}_k(s) \mathrm{d} s, \quad \text { for }(m, n) \in \Gamma .$$
Here, we use $\Gamma$ to denote the index set of the input swaptions. Note that when $n=m+1$, the swaption reduces to a caplet and Equation $7.23$ reduces to Equation 7.12, so Equation $7.23$ applies to both caplets and swaptions. When there are only caps (or equivalently, caplets) in the input set, the local volatility can be time-independent, corresponding to a volatility function that is flat in the direction of time, as was shown in Figure 7.2. If there are, in addition, swaptions in the input set, we will need to bend the volatility function or, in other words, make the local volatility function time-dependent as well.

## 金融代写|利率建模代写Interest Rate Modeling代考|Rank-Reduction Algorithm

To fit a LIBOR market model optimally to input correlations, we must first find the low-rank approximations to the input correlation matrices, as was explained in the last section. This is formulated as a constrained minimization problem 7.28, which is a special case of the so-called rank reduction problem for matrices. A somewhat more general rank-reduction problem can be formulated as follows. For a symmetric matrix, $C \in R^{N \times N}$, where $R^{N \times N}$ represents the collection of $N$-by- $N$ real matrices, the rank- $n$ approximation is defined as the solution to the following problem:
\begin{aligned} &\min _X|C-X|_F \ &\text { s.t. } \operatorname{rank}(X) \leq n<N, \quad \operatorname{diag}(X)=\operatorname{diag}(C) . \end{aligned}
We denote any solution to problem $7.36$ as $C^$. Note that problem $7.36$ does not impose the explicit condition of the non-negativity of a solution. This, however, is not a concern, because we can prove that $C^$ will be automatically a non-negative matrix provided that $C$ is one. The proof is beyond the scope of this book and we refer to Zhang and Wu (2003) for details. For later use, we denote the feasible set of problem $7.36$ as
$$\mathcal{H}=\left{X \in R^{N \times N} \mid \operatorname{rank}(X) \leq n, \quad \operatorname{diag}(X)=\operatorname{diag}(C)\right} .$$
Following the general approach of Lagrange methods, we transform the above constrained minimization problem into an equivalently min-max problem. Let $\mathcal{R}_n$ be the subset of $R^{N \times N}$ for matrices with ranks less or equal to $n$.

## 金融代写|利率建模代写Interest Rate Modeling代考|校准瓶盖、交换器

$$\sigma_{m, n}^2=\frac{1}{T_m-t} \int_t^{T_m} \sum_{j, k=m+1}^n \omega_j \omega_k \gamma_j^{\mathrm{T}}(s) \boldsymbol{\gamma}_k(s) \mathrm{d} s, \quad \text { for }(m, n) \in \Gamma .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|降秩算法

\begin{aligned} &\min _X|C-X|_F \ &\text { s.t. } \operatorname{rank}(X) \leq n<N, \quad \operatorname{diag}(X)=\operatorname{diag}(C) . \end{aligned}

$$\mathcal{H}=\left{X \in R^{N \times N} \mid \operatorname{rank}(X) \leq n, \quad \operatorname{diag}(X)=\operatorname{diag}(C)\right} .$$
，按照拉格朗日方法的一般方法，我们将上述约束最小化问题转化为等价的最小-极大问题。对于秩小于或等于$n$的矩阵，设$\mathcal{R}_n$为$R^{N \times N}$的子集。

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