# 数学代写|离散数学作业代写discrete mathematics代考|MPCS50103

## 数学代写|离散数学作业代写discrete mathematics代考|Portfolio optimization with withdrawal

In this section, the investor is allowed to withdraw a proportion $c_n$ of their wealth at the instant $n$, after updating the asset prices, but before the redistribution of their portfolio for the next investment period. Therefore, they only reinvest the non-withdrawn part. The corresponding investment-withdrawal strategy $\left(\pi_n, c_n\right)$ is no longer self-financed, but it must remain predictable and the wealth after the withdrawal must be positive or zero at each instant. Therefore, it can thus be shown that the new wealth at the time $n$ after the evolution of the prices and when the value of the withdrawal is
$$V_n(\pi, c)=\prod_{i=1}^n\left(1-c_i\right)\left(\pi_i T_i+\left(1-\pi_i\right)(1+r)\right),$$
such that the value of the wealth withdrawn at the instant $n$ is $R_n(\pi, c)$ with
$$R_n(\pi, c)= \begin{cases}\frac{c_n V_n(\pi, c)}{1-c_n} & \text { if } c_n \neq 1, \ V_{n-1}(\pi, c)\left(\pi_n T_n+\left(1-\pi_n\right)(1+r)\right) & \text { if not. }\end{cases}$$
1) Graphically represent the evolution of the wealth for the investment strategy with the following withdrawal policy:
a) The proportion of the wealth invested in the risky asset is fixed over time, at $1 / 4$,
b) The proportion of the wealth withdrawn at each instant is equal to $1 \%$ over the interval $[1,80], 5 \%$ over the interval $] 80,90], 10 \%$ over the interval $] 90,95], 25 \%$ over the interval $] 95,100$ [ and upon maturity, all the remaining wealth is withdrawn.
We will take the following parameters: the initial wealth $V_0=1$, the risky asset evolves as per the Cox, Ross and Rubinstein model with parameters $d=-2 \%$, $u=10 \%$ and $q=0.52$, interest rate $r=4 \%$, duration of investment: $N=100$ periods.

## 数学代写|离散数学作业代写discrete mathematics代考|Complete markets

We now wish to see whether it is possible to construct admissible strategies that allow us to obtain a European option $X_N$ at the date of maturity $N$.

DEFINITION 6.2.- A European option $X_N$ is said to be simulable if there exists an admissible strategy whose value at the date of maturity $N$ is exactly $X_N$.

DEFINITION 6.3.-A market is said to be complete if any European option is simulable.

We will now see that it is possible to characterize complete markets using martingales.

THEOREM 6.1.-A market is viable and complete if and only if there exists a unique risk-neutral probability $\mathbb{P}^*$ equivalent to $\mathbb{P}$.

PROOF. – The characterization of a viable market through the existence of viable, risk-neutral probabilities has already been seen in Theorem $5.2$.

Let us assume that the market is viable and complete and that there exist two risk-neutral probabilities $\mathbb{P}1$ and $\mathbb{P}_2$. Let $X_N$ be a European option. Since the market is complete, there exists an admissible strategy $\Phi$ such that $V_N(\Phi)=X_N$. Furthermore, under a risk-neutral probability, the discounted value of the portfolio $\left(\widetilde{V}_n(\Phi)\right)$ is a martingale. In particular, we have \begin{aligned} &\mathbb{E}_1\left[V_N(\Phi)\right]=\frac{\mathbb{E}_1\left[X_N\right]}{S_N^0}=\mathbb{E}_1\left[\widetilde{V}_0(\Phi)\right]=V_0(\Phi) \ &\mathbb{E}_2\left[V_N(\Phi)\right]=\frac{\mathbb{E}_2\left[X_N\right]}{S_N^0}=\mathbb{E}_2\left[\widetilde{V}_0(\Phi)\right]=V_0(\Phi) \end{aligned} Thus, for any random $X_N \mathcal{F}_N=\mathcal{F}$-measurable variable, we have $\mathbb{E}_1\left[X_N\right]=$ $\mathbb{E}_2\left[X_N\right]$. Taking $X_N=1\omega$, for example, we obtain $\mathbb{P}_1(\omega)=\mathbb{P}_2(\omega)$ for any $\omega \in \Omega$; therefore, $\mathbb{P}_1=\mathbb{P}_2$. Hence, we have unicity.

The converse is based on the orthogonality properties of the sub-spaces of random variables and sheds no particular light on the concepts of martingales or financial markets. We have therefore omitted it here and the reader who wishes to learn more about them may wish to consult, among others, [LAM 97, Theorem 3.4] for more details.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|Portfolio optimization with withdrawal

$$V_n(\pi, c)=\prod_{i=1}^n\left(1-c_i\right)\left(\pi_i T_i+\left(1-\pi_i\right)(1+r)\right)$$

$$R_n(\pi, c)=\left{\frac{c_n V_n(\pi, c)}{1-c_n} \quad \text { if } c_n \neq 1, V_{n-1}(\pi, c)\left(\pi_n T_n+\left(1-\pi_n\right)(1+r)\right) \quad\right. \text { if not. }$$
1) 以图形方式表示具有以下退出策略的投赆策略的财富演变:
a) 投资于风险资产的财富比例随着时间的推移是固定的，在 $1 / 4$ ，
b) 每一时刻提取的财富比例等于 $1 \%$ 在区间内 $[1,80], 5 \%$ 在区间内 $] 80,90], 10 \%$ 在区间内 $] 90,95], 25 \%$ 在区间内] 95,100 [ 到期后，所有剩余的财富将被提取。

## 数学代写|离散数学作业代写discrete mathematics代考|Complete markets

$$\mathbb{E}_1\left[V_N(\Phi)\right]=\frac{\mathbb{E}_1\left[X_N\right]}{S_N^0}=\mathbb{E}_1\left[\tilde{V}_0(\Phi)\right]=V_0(\Phi) \quad \mathbb{E}_2\left[V_N(\Phi)\right]=\frac{\mathbb{E}_2\left[X_N\right]}{S_N^0}=\mathbb{E}_2\left[\tilde{V}_0(\Phi)\right]=V_0(\Phi)$$

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