### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Pricing Coupon-Bearing Bonds

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Pricing Coupon-Bearing Bonds

As sufficient historical data on Euro coupon-bearing Treasury bonds is difficult to obtain we use the zero-coupon yield curve to construct the relevant bonds. Coupons on newly-issued bonds are generally closely related to the corresponding spot rate at the time, so the current zero-coupon yield with maturity $T$ is used as a proxy for the coupon rate of a coupon-bearing Treasury bond with maturity $T$. For example, on scenario $\omega$ the coupon rate $\delta_{2}^{B^{10}}(\omega)$ on a newly issued 10-year Treasury bond at time $t=2$ will be set equal to the projected 10 -year spot rate $y_{2,10}(\omega)$ at time $t=2$.
Generally
$$\begin{array}{llr} \delta_{t}^{B^{(T)}}(\omega)=y_{t, T}(\omega) & \forall t \in T^{d} & \forall \omega \in \Omega \ \delta_{t}^{B^{(T)}}(\omega)=\delta_{\lfloor t\rfloor}^{(T)}(\omega) & \forall t \in T^{i} & \forall \omega \in \Omega, \end{array}$$
where L.」 denotes integral part. This ensures that as the yield curve falls, coupons on newly-issued bonds will go down correspondingly and each coupon cash flow will be discounted at the appropriate zero-coupon yield.

The bonds are assumed to pay coupons semi-annually. Since we roll the bonds on an annual basis, a coupon will be received after six months and again after a year just before the bond is sold. This forces us to distinguish between the price at which the bond is sold at rebalancing times and the price at which the new bond is purchased.

Let $P_{t, B^{(T)}}^{(\text {sell }}$ denote the selling price of the bond $B^{(T)}$ at time $t$, assuming two coupons have now been paid out and the time to maturity is equal to $T-1$, and let $P_{t, B^{(t)}}^{(\text {tuy })}$ denote the price of a newly issued coupon-bearing Treasury bond with a maturity equal to $T$.
The “buy’ bond price at time $t$ is given by
\begin{aligned} B_{t}^{T}(\omega)=& F^{B^{T}} e^{-(T+\lfloor t]-t) y_{t, T+\lfloor t \mid-t}(\omega)} \ &+\sum_{s=\frac{|2|}{2}+\frac{1}{2}, \frac{\lfloor 2\rfloor \mid}{2}+1, \ldots,[t]+T} \frac{\delta^{A^{T}}(\omega)}{2} F^{B^{T}} e^{-(s-t) y_{t, t x-t)}(\omega)} \end{aligned}
where the principal of the bond is discounted in the first term and the stream of coupon payments in the second.
At rebalancing times $t$ the sell price of the bond is given by
$$B_{t}^{T}(\omega)=F^{B^{T}} e^{-(T-1) y_{i, T-1}(\omega)}+\sum_{s=\frac{1}{2}, 1, \ldots, T-1} \frac{\delta_{i-1}^{B^{T}}(\omega)}{2} F^{B^{T}} e^{-(s-t) y,(\omega-t)(\omega)}$$
$\omega \in \Omega \quad t \in\left{T^{d} \backslash{0}\right} \cup{T}$with coupon rate $\delta_{t-1}^{B^{T}}(\omega)$. The coupon rate is then reset for the newly-issued Treasury bond of the same maturity. We assume that the coupons paid at six months are re-invested in the off-the-run bonds. This gives the following adjustment to the amount held in bond $B^{T}$ at time $t$

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Historical Backtests

We will look at an historical backtest in which statistical models are fitted to data up to a trading time $t$ and scenario trees are generated to some chosen horizon $t+T$. The optimal root node decisions are then implemented at time $t$ and compared to the historical returns at time $t+1$. Afterwards the whole procedure is rolled forward for T trading times.
Our backtest will involve a telescoping horizon as depicted in Figure $4 .$
At each decision time $t$ the parameters of the stochastic processes driving the stock return and the three factors of the term structure model are re-calibrated using historical data up to and including time $t$ and the initial values of the simulated scenarios are given by the actual historical values of the variables at these times. Re-calibrating the simulator parameters at each successive initial decision time $t$ captures information in the history of the variables up to that point.

Although the optimal second and later-stage decisions of a given problem may be of “what-if” interest, manager and decision maker focus is on the implementable first-stage decisions which are hedged against the simnlated future uncertainties The reasons for implementing stochastic optimization programmes in this way are twofold. Firstly, after one year has passed the actual values of the variables realized may not coincide with any of the values of the variables in the simulated scenarios. In this case the optimal investment policy would be undefined, as the model only has

optimal decisions defined for the nodes on the simulated scenarios. Secondly, as one more year has passed new information has become available to re-calibrate the simulator’s parameters. Relying on the original optimal investment strategies will ignore this information. For more on backtesting procedures for stochastic optimization models see Dempster et al. (2003).

For our backtests we will use three different tree structures with approximately the same number of scenarios, but with an increasing initial branching factor. We first solve the five-year problem using a $6.6 .6 .6 .6$ tree, which gives 7776 scenarios. Then we use $32.4 .4 .4 .4=8192$ scenarios and finally the extreme case of $512.2 .2 .2 .2=8192$ scenarios.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Robustness of Backtest Results

Empirical equity returns are now well known not to be normally distributed but rather to exhibit complex behaviour including fat tails. To investigate how the EMS MC model performs with more realistic asset return distributions we report in this section experiments using a geometric Brownian motion with Poisson jumps to model equity returns. The stock price process $\mathbf{S}$ is now assumed to follow
$$\frac{d \mathbf{S}{t}}{\mathbf{S}{t}}=\bar{\mu}{S} d t+\tilde{\sigma}{S} d \overline{\mathbf{W}}{t}^{S}+\mathbf{J}{t} d \mathbf{N}_{t}$$
where $\mathbf{N}$ is an independent Poisson process with intensity $\lambda$ and the jump saltus $\mathbf{J}$ at Poisson epochs is a normal random variable.

As the EMS MC model and the $512.2 .2 .2 .2$ tree provided the best results with Gaussian returns the backtest is repeated for this model and treesize. Figure 12 gives the historical backtest results and Tables 5 and 6 represent the allocations for the $512.2 .2 .2 .2$ tests with the EMS MC model for the original GBM process and the GBM with Poisson jumps process respectively. The main difference in the two tables is that the investment in equity is substantially lower initially when the equity index volatility is high (going down to $0.1 \%$ when the volatility is $28 \%$ in 2001 ), but then increases as the volatility comes down to $23 \%$ in 2003 . This is born out by Figure 12 which shows much more realistic in-sample one-year-ahead portfolio wealth predictions (cf. Figure 10$)$ and a 140 basis point increase in terminal historical fund return over the Gaussian model. These phenomena are the result of the calibration of the normal jump saltus distributions to have negative means and hence more downward than upwards jumps resulting in downwardly skewed equity index return distributions, but with the same compensated drift as in the GBM case.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Pricing Coupon-Bearing Bonds

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## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Robustness of Backtest Results

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## 广义线性模型代考

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## MATLAB代写

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