### 金融代写|金融数学代写Financial Mathematics代考|STAT2032

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## 金融代写|金融数学代写Financial 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.

## 金融代写|金融数学代写Financial 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 $\mathcal{F}_{n-1}$ 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 $$is the scalar product of x and y, and S_{n}=\left(S_{n}^{0}, S_{n}^{1}, \ldots, S_{n}^{d}\right). The above notation may be ambiguous. Indeed, using our notations, V_{n}(\Phi) represents the wealth at time n, just after the asset prices have been updated and before the portfolio is redistributed for the next period. We may also wish to consider the wealth at the instant n after the redistribution of the portfolio. In fact, in most cases that we will consider, these two quantities are equal. DEFINITION 5.4.-An investment strategy \Phi is said to be self-financed if for any 1 \leq n \leq N-1, we have$$
\sum_{i=1}^{d} \phi_{n}^{i} S_{n}^{i}=\sum_{i=1}^{d} \phi_{n+1}^{i} S_{n}^{i},
$$or \left\langle\Phi_{n}, S_{n}\right\rangle=\left\langle\Phi_{n+1}, S_{n}\right\rangle, using condensed notation. A self-financed strategy is, therefore, a strategy where at each step, the entire wealth is reinvested without withdrawal or an exogenous infusion of money. We will now describe self-financing strategies through a specific decomposition of the realized wealth. PROPOSITION 5.1.-An investment strategy \Phi is self-financed if and only if, for any 1 \leq n \leq N-1, we have$$
\widetilde{V}{n}(\Phi)=\widetilde{V}{0}(\Phi)+\sum_{k=1}^{n}<\Phi_{k},\left(\widetilde{S}{k}-\widetilde{S}{k-1}\right)>.
$$PROOF.- We proceed using double implication. Let us first assume that \Phi is self-financed. We then have, by definition, \left\langle\Phi_{n}, S_{n}>\right. =\left\langle\Phi_{n+1}, S_{n}\right\rangle. Upon dividing by S_{n}^{0}, we obtain \left\langle\Phi_{n}, S_{n}\right\rangle=\left\langle\Phi_{n+1}, \widetilde{S}{n}\right\rangle. Therefore, using the self-financing relation, we obtain$$ \begin{aligned} \tilde{V}{n}(\Phi) &=\left\langle\Phi_{n}, \tilde{S}{n}>\right.\ &=\left\langle\Phi{0}, \widetilde{S}{0}>+\sum{k=1}^{n}<\Phi_{k}, \widetilde{S}{k}>-<\Phi{k-1}, \tilde{S}_{k-1}>\right.
\end{aligned}
$$## 金融代写|金融数学代写Financial Mathematics代考|Arbitrage We will now establish a link between viable financial markets and martingales. The term viable is used here to signify the impossibility of a definite increase in wealth with an initial investment of zero. Let us formalize this. DEFINITION 5.6.-An investment strategy is said to be an arbitrage strategy or arbitrage opportunity if it is an admissible strategy, with an initial value of zero and a final, non-zero value: -V_{0}(\Phi)=0 -V_{n}(\Phi) \geq 0 for any 0 \leq n \leq N - there exist n and \omega \in \Omega such that V_{n}(\omega)>0 Let us note that the last condition would simply translate to \mathbb{P}\left(V_{n}>0\right)>0 if the universe \Omega was not finite. DEFINITION 5.7.-A financial market is said to be viable if there are no arbitrage opportunities. We will now see that the absence of an arbitrage opportunity translates into the fact that discounted risky assets are martingales, except for a suitably changed probability. DEFINITION 5.8. – A probability \mathbb{P}^{*} on (\Omega, \mathcal{F}) is a risk-neutral probability if, under this probability, all discounted risky assets are martingales. THEOREM 5.1.- A financial market is viable if and only if there exists a risk-neutral probability \mathbb{P}^{*} equivalent { }^{l} to \mathbb{P}. PROOF. – It is assumed that \mathbb{P}^{} exists and that there exists an admissible strategy \Phi such that V_{0}(\Phi)=0. Under \mathbb{P}^{},\left(\widetilde{S}{\mathrm{r}}\right) is a martingale and since \Phi is self-financed, we have$$ \tilde{V}{n}(\Phi)=\tilde{V}{0}(\Phi)+\sum{k=1}^{n}<\Phi_{k},\left(\widetilde{S}{k}-\widetilde{S}{k-1}\right)>
$$It can thus be seen that \left(V_{n}(\Phi)\right) is a martingale transform, since \Phi is predictable; therefore, it is a martingale under \mathbb{P}^{*}. In particular, we have$$
\mathbb{E}\left[\tilde{V}{n}(\Phi)\right]=\mathbb{E}\left[\tilde{V}{0}(\Phi)\right]=0
$$for any n. In addition, since \widetilde{V}{n}(\Phi) \geq 0 for any n, because \Phi is admissible, we necessarily have that \widetilde{V}{n}(\Phi(\omega)=0 for all \omega \in \Omega. Thus, no arbitrage opportunity exists. ## 金融数学代考 ## 金融代写|金融数学代写Financial Mathematics代考|Financial assets 让我们考虑一个由无风险资产组成的金融市场(小号n0)0≤n≤ñ和d≥1风险资产(小号n0)0≤n≤ñ为了1≤一世≤d. 风险资产的价格由随机过程表示。无风险资产动态是 小号00=1 小号n0=(1+r)n,n≥1 在哪里r是市场上的利率。风险资产具有随机动态，目前未指定。由于我们需要比较不同日期的资产价格，我们有时可能希望忽略货币贬值的影响，这被认为与无风险利率有关。 定义 5.1.-资产的折现价格是价格除以无风险资产的现值 小号~n一世=小号n一世小号n0=小号n一世(1+r)n,n≥0 在数学上，贴现价格等同于考虑无风险利率为零。 ## 金融代写|金融数学代写Financial Mathematics代考|Investment strategies 现在让我们考虑一位希望在市场上投资的投资者。为了定义他们的投资策略，他们必须在每一刻都知道投资于每种资产的股票数量。 定义 5.2.-投资策略是一个过程披=(披n)1≤n≤ñ对应于数量披n=(φn0,φn1,…,φnd)∈Rd+1投资者持有的每项资产在瞬间之间n−1和瞬间n. 这是一个可预测的过程：对于任何1≤n≤ñ， 和0≤一世≤d,φn一世是个Fn−1可衡量的。 投资策略必须是可预测的，因为决定如何在瞬间分配投资组合n−1和n只能基于当前可用的信息n−1. 换句话说，没有内幕交易，投资者也没有关于市场未来的信息。 给定当前的资产价格和投资策略，可以计算出投资者每时每刻的投资组合价值。 定义 5.3.- 瞬间的价值n, 应用投资策略的投资组合披是 在n(披)=∑一世=1dφn一世小号n一世=<披n,小号n> 其中$$ 是的标量积X和是， 和小号n=(小号n0,小号n1,…,小号nd).

∑一世=1dφn一世小号n一世=∑一世=1dφn+1一世小号n一世,

## 金融代写|金融数学代写Financial Mathematics代考|Arbitrage

−在0(披)=0 −在n(披)≥0对于任何0≤n≤ñ −存在n和ω∈Ω这样在n(ω)>0

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