数学代写|离散数学作业代写discrete mathematics代考|Math1030

数学代写|离散数学作业代写discrete mathematics代考|Financial assets

Let us consider a financial market comprising of a risk-free asset $\left(S_n^0\right){0 \leq n \leq N}$ and $d \geq 1$ risky assets $\left(S_n^0\right){0 \leq n \leq N}$ for $1 \leq i \leq d$. The price of the risky assets is represented by stochastic processes. The risk-free asset dynamic is
\begin{aligned} &S_0^0=1, \ &S_n^0=(1+r)^n, n \geq 1, \end{aligned}
where $r$ is the interest rate on the market. The risky assets have a random dynamic that is unspecified for the moment. As we need to compare the prices of the assets on different dates, we may sometimes wish to ignore the effect of the depreciation of currency, which is considered to be linked to the risk-free rate of interest.

DEFINITION 5.1.- The discounted prices of assets are the prices divided by the current value of the risk-free asset
$$\widetilde{S}_n^i=\frac{S_n^i}{S_n^0}=\frac{S_n^i}{(1+r)^n}, \quad n \geq 0 .$$
In mathematical terms, discounting prices is equivalent to considering that the risk-free interest rate is zero.

数学代写|离散数学作业代写discrete mathematics代考|Investment strategies

Let us now consider an investor who wishes to invest money in the market. In order to define their investment strategy, they must know, at each instant, the number of shares invested in each asset.

DEFINITION 5.2.-An investment strategy is a process $\Phi=\left(\Phi_n\right){1 \leq n \leq N}$ that corresponds to the quantities $\Phi_n=\left(\phi_n^0, \phi_n^1, \ldots, \phi_n^d\right) \in \mathbb{R}^{d+1}$ of each asset held by an investor between the instant $n-1$ and the instant $n$. It is a predictable process: for any $1 \leq n \leq N$, and $0 \leq i \leq d, \phi_n^i$ is the $\bar{F}{n-1}$ measurable

An investment strategy must be predictable, since decisions on how to distribute the portfolio between the instants $n-1$ and $n$ can be based only on the information available up to the instant $n-1$. In other words, there is no insider trading and the investor has no information on the future of the market.

Given the current prices of assets and an investment strategy, it is possible to calculate the value of the investor’s portfolio at every instant.

DEFINITION 5.3.- The value, at an instant $n$, of a portfolio that applies the investment strategy $\Phi$ is
$$V_n(\Phi)=\sum_{i=1}^d \phi_n^i S_n^i=<\Phi_n, S_n>,$$
where $\langle x, y\rangle$ is the scalar product of $x$ and $y$, and $S_n=\left(S_n^0, S_n^1, \ldots, S_n^d\right)$.
The above notation may be ambiguous. Indeed, using our notations, $V_n(\Phi)$ represents the wealth at time $n$, just after the asset prices have been updated and before the portfolio is redistributed for the next period. We may also wish to consider the wealth at the instant $n$ after the redistribution of the portfolio. In fact, in most cases that we will consider, these two quantities are equal.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Financial assets

$$S_0^0=1, \quad S_n^0=(1+r)^n, n \geq 1,$$

$$\widetilde{S}_n^i=\frac{S_n^i}{S_n^0}=\frac{S_n^i}{(1+r)^n}, \quad n \geq 0 .$$

数学代写|离散数学作业代写discrete mathematics代考|Investment strategies

$$V_n(\Phi)=\sum_{i=1}^d \phi_n^i S_n^i=<\Phi_n, S_n>$$

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