# 金融代写|利率建模代写Interest Rate Modeling代考|MATH4513

## 金融代写|利率建模代写Interest Rate Modeling代考|SENSITIVITY WITH RESPECT TO THE INPUT PRICES

In this section, we discuss the pricing and hedging of a LIBOR derivative with a model calibrated to the prices of benchmark instruments and correlation matrices. The theory we develop is in the spirit of Avellaneda et al. (1998) for calibrating an equity derivatives model via relative-entropy minimization. Without loss of generality, we suppose that a derivative is sold at time $t=0$, which promises to receive a sequence of cash flows, $\left{F_i\right}$, at time $T_i, i=1, \ldots, N$, and the cash flows are contingent on the forward rates, $\left{f_j\right}_{j=1}^N$, in the future. By using cash flow measures, we can express the value of the derivative as
$$\varphi(0, \mathbf{f}(0))=\sum_{i=1}^N P\left(0, T_i\right) E^{Q_j}\left[F_i\right] .$$
The expectation in Equation $7.71$ can be calculated by Monte Carlo simulations.

Consider hedging the short position using benchmark instruments, for which we need to calculate the hedge ratios. Let the prices of the benchmarks be $\left{C_j\right}$. By the chain rule, we have
$$\frac{\partial \varphi}{\partial C_j}=\sum_{i=1}^I \frac{\partial \varphi}{\partial d_i} \frac{\partial d_i}{\partial h_j} \frac{\partial h_j}{\partial C_j} .$$
Hence, we must calculate the three derivative chains as follows:
$$\left{\frac{\partial h_j}{\partial C_j}\right},\left{\frac{\partial d_i}{\partial h_j}\right}, \text { and }\left{\frac{\partial \varphi}{\partial d_i}\right} .$$
The first two derivative chains are not difficult to obtain. The partial derivative of volatilities with respect to prices can be calculated by making use Black’s formula (which includes a caplet as a special case).

## 金融代写|利率建模代写Interest Rate Modeling代考|Futures Price versus Forward Price

The value of a forward contract with maturity, $T$, and strike, $K$, on a tradable asset is
$$V_0=E^{\mathbb{Q}}\left[B_T^{-1}\left(S_T-K\right) \mid \mathcal{F}_0\right]$$ where $\mathbb{Q}$ stands for the risk-neutral measure, $S_T$ the asset price at maturity, and $B_T$ the balance of the money market account at $T$ :
$$B_T=\exp \left(\int_0^T r_s \mathrm{~d} s\right) .$$
The forward price is defined as the strike price that nullifies the value of the forward contract. In view of Equation 8.1, we know that the forward price for the contract satisfies
$$F_0^T=\frac{E_0^{\mathbb{Q}}\left[B_T^{-1} S_T\right]}{E_0^{\mathbb{Q}}\left[B_T^{-1}\right]}=E_0^{\mathbb{Q} T}\left[S_T\right],$$
that is, it is the expectation of the terminal asset price under the forward measure, $\mathbb{Q}_T$. We have already learned in Chapter 4 that the above expectation equals
$$F_0^T=\frac{\hat{S}_0}{P(0, T)},$$
where $\hat{S}_t$ is the stripped-dividend price of the asset at time $t$.
With everything else the same, a futures contract differs from a forward contract by “marking to market,” meaning that the P\&L from holding the contract is credited to or debited from the holder’s margin account on a daily basis. The margin account, meanwhile, is accrued using risk-free interest rates. A futures price parallels the forward price and nullifies the value of a futures contract. Let $\tilde{F}_t^T$ be the futures price observed at time $t \leq T$. At the maturity of the futures contract, the futures price is fixed or set to the price of the underlying security, that is, $\tilde{F}_T^T=S_T$.

## 金融代写|利率建模代写利率建模代考|敏感性与输入价格

$$\varphi(0, \mathbf{f}(0))=\sum{i=1}^N P\left(0, T_i\right) E^{Q_j}\left[F_i\right] .$$

$$\frac{\partial \varphi}{\partial C_j}=\sum_{i=1}^I \frac{\partial \varphi}{\partial d_i} \frac{\partial d_i}{\partial h_j} \frac{\partial h_j}{\partial C_j} .$$

$$\left{\frac{\partial h_j}{\partial C_j}\right},\left{\frac{\partial d_i}{\partial h_j}\right}, \text { and }\left{\frac{\partial \varphi}{\partial d_i}\right} .$$

## 金融代写|利率建模代写利率建模代考|期货价格与远期价格

$$V_0=E^{\mathbb{Q}}\left[B_T^{-1}\left(S_T-K\right) \mid \mathcal{F}_0\right]$$，其中$\mathbb{Q}$表示风险中性度量，$S_T$表示到期时的资产价格，$B_T$表示货币市场账户在$T$处的余额:
$$B_T=\exp \left(\int_0^T r_s \mathrm{~d} s\right) .$$

$$F_0^T=\frac{E_0^{\mathbb{Q}}\left[B_T^{-1} S_T\right]}{E_0^{\mathbb{Q}}\left[B_T^{-1}\right]}=E_0^{\mathbb{Q} T}\left[S_T\right],$$
，即为远期度量下终端资产价格$\mathbb{Q}_T$的期望。我们已经在第4章中了解到，上述预期等于
$$F_0^T=\frac{\hat{S}_0}{P(0, T)},$$
，其中$\hat{S}_t$是资产在$t$时刻的去股息价格。

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