### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Objective Functions: Expected Average Shortfall

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• 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|Expected Maximum Shortfall

Starting with an initial wealth $W_{0}$ and an annual nominal guarantee of $G$, the liability at the planning horizon at time $T$ is given by
$$W_{0}(1+G)^{T}$$
To price the liability at time $t<T$ consider a zero-coupon Treasury bond, which pays 1 at time $T$, 1.e. $L_{T}(\omega)=1$, tor all scenarıos $\omega \in \lesssim 2$. 1 he zero-coupon Treasury bond price at time $t$ in scenario $\omega$ assuming continuous compounding is given by
$$Z_{t}(\omega)=e^{-y_{t, T}(\omega)(T-t)}$$

where $y_{t, T}(\omega)$ is the zero-coupon Treasury yield with maturity $T$ at time $t$ in scenario $\omega$.

This gives a formula for the value of the nominal or fixed guarantee barrier at time $t$ in scenario $\omega$ as
$$L_{t}^{N}(\omega):=W_{0}(1+G)^{T} Z_{t}(\omega)=W_{0}(1+G)^{T} e^{-y_{t} T(\omega)(T-t)}$$
In a minimum guaranteed return fund the objective of the fund manager is twofold; firstly to manage the investment strategies of the fund and secondly to take into account the guarantees given to all investors. Investment strategies must ensure that the guarantee for all participants of the fund is met with a high probability.

In practice the guarantor (the parent bank of the fund manager) will ensure the investor guarantee is met by forcing the purchase of the zero coupon bond of (22) when the fund is sufficiently near the barrier defined by $(23)$. Since all upside potential to investors is thus foregone, the aim of the fund manager is to fall below the barrier with acceptably small if not zero probability.

Ideally we would add a constraint limiting the probability of falling below the barrier in a VaR-type minimum guarantee constraint, i.e.
$$P\left(\max {t \in T \text { loal }} h{t}(\omega)>0\right) \leq \alpha$$
for $\alpha$ small. However, such scenario-based probabilistic constraints are extremely difficult to implement, as they may without further assumptions convert the convex large-scale optimization problem into a non-convex one. We therefore use the following two convex approximations in which we trade off the risk of falling below the barrier against the return in the form of the expected sum of wealth.

Firstly, we look at the expected average shortfall (EAS) model in which the objective function is given by:
$\left{\max {\left{\begin{array}{l}x{t, a}(\omega), x_{t, \alpha}^{+}(\omega), x_{t, a}^{-}(\omega): \ a \in A, \omega \in \Omega, t \in T^{d} \cup[T]\end{array}\right.}\left{\sum_{\omega \in \Omega} \sum_{\left.t \in T^{d} \cup \mid T\right]} p(\omega)\left((1-\beta) W_{t}(\omega)-\beta \frac{h_{S}(\omega)}{\mid T^{d} \cup[T \mid}\right)\right}\right.$
$\left.-\beta\left(\sum_{\omega \in \Omega} p(\omega) \sum_{t \in T^{d} \cup{T]} \frac{h_{d}(\omega)}{\left|T^{w} \cup[T]\right|}\right)\right}$

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

In this section we present a three-factor term structure model which we will use to price both our bond portfolio and the fund’s liability. Many interest-rate models, like the classic one-factor Vasicek (1977) and Cox, Ingersoll, and Ross (1985) class of models and even more recent multi-factor models like Anderson and Lund (1997), concentrate on modeling just the short-term rate.

However for the minimum guaranteed retum funds we have to deal with a longterm liability and bonds of varying maturities. We therefore must capture the dynamics of the whole term structure. This has been achieved by using the economic factor model described below in Section 3.1. In Section $3.2$ we describe the pricing of coupon-bearing bonds and Section $3.3$ investigates the consequences of rolling the bonds on an annual basis.

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

To capture the dynamics of the whole term structure, we will use a Gaussian economic factor model (EFM) (see Campbell $(2000)$ and also Nelson and Siegel (1987)) whose evolution under the risk-neutral measure $Q$ is determined by the stochastic differential equations
\begin{aligned} &\mathbf{d X} \mathbf{X}{t}=\left(\mu{X}-\lambda_{X} X_{t}\right) d t+\sigma_{X} \mathbf{d} \mathbf{W}{t}^{X} \ &\mathbf{d Y { t }}=\left(\mu_{Y}-\lambda_{Y} Y_{t}\right) d t+\sigma_{Y} \mathbf{d} \mathbf{W}{t}^{Y} \ &\mathbf{d R}{t}=k\left(X_{t}+Y_{t}-R_{t}\right) d t+\sigma_{R} \mathbf{d W}{t}^{R} \end{aligned} where the $d W$ terms are correlated. The three unobservable Gaussian factors $\mathbf{R}, \mathbf{X}$ and $\mathbf{Y}$ represent respectively a short rate, a long rate and (minus) the slope between an instantaneous short rate and the long rate. Solving these equations the following formula for the yield at time $t$ with time to maturity equal to $T-t$ is obtained (for a derivation, see Medova et $a l ., 2005$ ) $$y{t, T}=\frac{A(t, T) R_{t}+B(t, T) X_{t}+C(t, T) Y_{t}+D(t, T)}{T}$$
where
\begin{aligned} &B(t, T):=\frac{k}{k-\lambda_{X}}\left{\frac{1}{\lambda_{X}}\left(1-e^{-\lambda x(T-t)}\right)-\frac{1}{k}\left(1-e^{-k(T-t)}\right)\right} \ &C(t, T):=\frac{k}{k-\lambda_{Y}}\left{\frac{1}{\lambda_{Y}}\left(1-e^{-\lambda_{Y}(T-t)}\right)-\frac{1}{k}\left(1-e^{-k(T-t)}\right)\right} \end{aligned}
$D(t, T):=\left(T-t-\frac{1}{k}\left(1-e^{-k(T-t)}\right)\right)\left(\frac{\mu_{X}}{\lambda_{X}}+\frac{\mu_{Y}}{\lambda_{Y}}\right)-\frac{\mu_{X}}{\lambda_{X}} B(t, T)-\frac{\mu_{Y}}{\lambda_{Y}} C(t, T)$
$-\frac{1}{2} \sum_{i=1}^{3}\left{\frac{m_{X_{i}}}{2 \lambda_{X}}\left(1-e^{-2 \lambda x(T-t)}\right)+\frac{m_{Y_{i}}}{2 \lambda_{Y}}\left(1-e^{-2 \lambda \lambda_{Y}(T-t)}\right)\right.$
$+\frac{n_{i}^{2}}{2 k}\left(1-e^{-2 k(T-t)}\right)+p_{i}^{2}(T-t)+\frac{2 m X_{i} m_{Y_{i}}}{\lambda_{X}+\lambda_{Y}}\left(1-e^{-(\lambda x+\lambda y)(T-t)}\right)$
$+\frac{2 m_{X_{i}} n_{i}}{\lambda_{X}+k}\left(1-e^{-\left(\lambda \lambda_{X}+k\right)(T-t)}\right)+\frac{2 m_{X_{i}} p_{i}}{\lambda_{X}}\left(1-e^{-\lambda_{X}(T-t)}\right)$
$+\frac{2 m_{Y_{i}} n_{i}}{\lambda Y+k}\left(1-e^{-(\lambda \gamma+k)(T-t)}\right)+\frac{2 m_{Y_{j}} p_{i}}{\lambda Y}\left(1-e^{-\lambda \gamma(T-t)}\right)$
$\left.+\frac{2 n_{i} p_{i}}{k}\left(1-e^{-k(T-t)}\right)\right}$

and
$m_{X_{i}}:=-\frac{k \sigma_{X_{i}}}{\left.\lambda x(k-\lambda)^{\prime}\right)}$
$m_{Y_{i}}:=-\frac{k \sigma_{Y_{i}}}{\lambda \gamma(k-\lambda y)}$
$n_{i}:=\frac{\sigma_{X_{i}}}{k-\lambda x}+\frac{\sigma \gamma_{i}}{k-\lambda_{Y}}-\frac{\sigma_{R_{i}}}{k}$
$p_{i}:=-\left(m X_{i}+m Y_{i}+n_{i}\right)$.
Bond pricing must be effected under the risk-neutral measure $Q$. However, for the model to be used for forward simulation the set of stochastic differential equations must be adjusted to capture the model dynamics under the real-world or market measure $P$. We therefore have to model the market prices of risk which take us from the risk-neutral measure $Q$ to the real-world measure $P$.

Under the market measure $P$ we adjust the drift term by adding the risk premium given by the market price of risk $\gamma$ in terms of the quantity of risk. The effect of this is a change in the long-term mean, e.g. for the factor $\mathbf{X}$ the long-term mean now equals $\frac{\mu x+\gamma x \sigma x}{\lambda x}$. It is generally assumed in a Gaussian world that the quantity of risk is given by the volatility of each factor.

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

$\left{\max {\left{X吨,一种(ω),X吨,一种+(ω),X吨,一种−(ω): 一种∈一种,ω∈Ω,吨∈吨d∪[吨]\right.}\left{\sum_{\omega \in \Omega} \sum_{\left.t \in T^{d} \cup \mid T\right]} p(\omega)\left((1 -\beta) W_{t}(\omega)-\beta \frac{h_{S}(\omega)}{\mid T^{d} \cup[T \mid}\right)\right}\right .\left.-\beta\left(\sum_{\omega \in \Omega} p(\omega) \sum_{t \in T^{d} \cup{T]} \frac{h_{d}(\欧米茄)}{\left|T^{w} \cup[T]\right|}\right)\right}$

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

dXX吨=(μX−λXX吨)d吨+σXd在吨X d是吨=(μ是−λ是是吨)d吨+σ是d在吨是 dR吨=到(X吨+是吨−R吨)d吨+σRd在吨R在哪里d在项是相关的。三个不可观测的高斯因子R,X和是分别表示短期利率、长期利率和（减去）瞬时短期利率和长期利率之间的斜率。求解这些方程，得到以下时间产量的公式吨到期时间等于吨−吨获得（关于推导，参见 Medova et一种一世.,2005)是吨,吨=一种(吨,吨)R吨+乙(吨,吨)X吨+C(吨,吨)是吨+D(吨,吨)吨

\begin{aligned} &B(t, T):=\frac{k}{k-\lambda_{X}}\left{\frac{1}{\lambda_{X}}\left(1-e^{ -\lambda x(Tt)}\right)-\frac{1}{k}\left(1-e^{-k(Tt)}\right)\right} \ &C(t, T):=\ frac{k}{k-\lambda_{Y}}\left{\frac{1}{\lambda_{Y}}\left(1-e^{-\lambda_{Y}(Tt)}\right)- \frac{1}{k}\left(1-e^{-k(Tt)}\right)\right} \end{对齐}\begin{aligned} &B(t, T):=\frac{k}{k-\lambda_{X}}\left{\frac{1}{\lambda_{X}}\left(1-e^{ -\lambda x(Tt)}\right)-\frac{1}{k}\left(1-e^{-k(Tt)}\right)\right} \ &C(t, T):=\ frac{k}{k-\lambda_{Y}}\left{\frac{1}{\lambda_{Y}}\left(1-e^{-\lambda_{Y}(Tt)}\right)- \frac{1}{k}\left(1-e^{-k(Tt)}\right)\right} \end{对齐}
D(吨,吨):=(吨−吨−1到(1−和−到(吨−吨)))(μXλX+μ是λ是)−μXλX乙(吨,吨)−μ是λ是C(吨,吨)
-\frac{1}{2} \sum_{i=1}^{3}\left{\frac{m_{X_{i}}}{2 \lambda_{X}}\left(1-e^{ -2 \lambda x(Tt)}\right)+\frac{m_{Y_{i}}}{2 \lambda_{Y}}\left(1-e^{-2 \lambda \lambda_{Y}( Tt)}\right)\right.+\frac{n_{i}^{2}}{2 k}\left(1-e^{-2 k(Tt)}\right)+p_{i }^{2}(Tt)+\frac{2 m X_{i} m_{Y_{i}}}{\lambda_{X}+\lambda_{Y}}\left(1-e^{-(\ lambda x+\lambda y)(Tt)}\right)+\frac{2 m_{X_{i}} n_{i}}{\lambda_{X}+k}\left(1-e^{- \left(\lambda \lambda_{X}+k\right)(Tt)}\right)+\frac{2 m_{X_{i}} p_{i}}{\lambda_{X}}\left(1 -e^{-\lambda_{X}(Tt)}\right)+\frac{2 m_{Y_{i}} n_{i}}{\lambda Y+k}\left(1-e^ {-(\lambda \gamma+k)(Tt)}\right)+\frac{2 m_{Y_{j}} p_{i}}{\lambda Y}\left(1-e^{-\lambda \gamma(Tt)}\right)\left.+\frac{2 n_{i} p_{i}}{k}\left(1-e^{-k(Tt)}\right)\right }-\frac{1}{2} \sum_{i=1}^{3}\left{\frac{m_{X_{i}}}{2 \lambda_{X}}\left(1-e^{ -2 \lambda x(Tt)}\right)+\frac{m_{Y_{i}}}{2 \lambda_{Y}}\left(1-e^{-2 \lambda \lambda_{Y}( Tt)}\right)\right.+\frac{n_{i}^{2}}{2 k}\left(1-e^{-2 k(Tt)}\right)+p_{i }^{2}(Tt)+\frac{2 m X_{i} m_{Y_{i}}}{\lambda_{X}+\lambda_{Y}}\left(1-e^{-(\ lambda x+\lambda y)(Tt)}\right)+\frac{2 m_{X_{i}} n_{i}}{\lambda_{X}+k}\left(1-e^{- \left(\lambda \lambda_{X}+k\right)(Tt)}\right)+\frac{2 m_{X_{i}} p_{i}}{\lambda_{X}}\left(1 -e^{-\lambda_{X}(Tt)}\right)+\frac{2 m_{Y_{i}} n_{i}}{\lambda Y+k}\left(1-e^ {-(\lambda \gamma+k)(Tt)}\right)+\frac{2 m_{Y_{j}} p_{i}}{\lambda Y}\left(1-e^{-\lambda \gamma(Tt)}\right)\left.+\frac{2 n_{i} p_{i}}{k}\left(1-e^{-k(Tt)}\right)\right }

n一世:=σX一世到−λX+σC一世到−λ是−σR一世到
p一世:=−(米X一世+米是一世+n一世).

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