Posted on Categories:固定收益与信贷, 数学代写, 金融代写

金融代写|固定收益与信贷代写Fixed Income and Credit代考|FM225 The Model

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金融代写|固定收益与信贷代写Fixed Income and Credit代考|The Model

We consider a financial market with $N$ different financial assets. These assets could in principle be almost anything, like bonds, stocks, options, or whatever financial instrument that is traded on a liquid market. The market only exists at the two points in time $t=0$ and $t=1$, and the price per unit of asset No. $i$ at time $t$ will be denoted by $S_t^i$. We thus have a price vector process $S_t, t=0,1$ and we will view the price vector as a column vector, i.e.
$$S_t=\left[\begin{array}{c} S_t^1 \ \vdots \ S_t^N \end{array}\right]$$
The randomness in the system is modeled by assuming that we have a finite sample space $\Omega=\left{\omega_1, \ldots, \omega_M\right}$ and that the probabilities $p_j=P\left(\omega_j\right), j=1, \ldots, M$ are all strictly positive. The price vector $S_0$ is assumed to be deterministic and known to us, but the price vector at time $t=1$ depends upon the outcome $\omega \in \Omega$, and $S_1^i\left(\omega_j\right)$ denotes the price per unit of asset No. $i$ at time $t=1$ if $\omega_j$ has occured.

金融代写|固定收益与信贷代写Fixed Income and Credit代考|Absence of Arbitrage

We now define a portfolio as an $N$ dimensional row vector $h=\left[h^1, \ldots, h^N\right]$ with the interpretation that $h^i$ is the number of units of asset No. $i$ that we buy at time $t=0$ and keep until time $t=1$.

Since we are buying the assets with deterministic prices at time $t=0$ and selling them at time $t=1$ at stochastic prices, the value process of our portfolio will be a stochastic process $V_t^h$ defined by
$$V_t^h=\sum_{i=1}^N h^i S_t^i=h S_t, \quad t=0,1$$
and in more detail we can write this as
$$V_t^h\left(\omega_i\right)=h S_t\left(\omega_i\right)=h d_i=(h D)_i .$$
There are various similar, but not equivalent, variations of the concept of an arbitrage portfolio. The standard one is the following.

Definition 3.1 The portfolio $h$ is an arbitrage portfolio if it satisfies the conditions
\begin{aligned} V_0^h & =0 \ P\left(V_1^h \geq 0\right) & =1 \ P\left(V_1^h>0\right) & >0 . \end{aligned}
In more detail we can write this as
$$\begin{gathered} V_0^h<0, \ V_1^h\left(\omega_i\right) \geq 0, \quad \text { for all } i=1, \ldots, M \ V_1^h\left(\omega_i\right)>0, \quad \text { for some } i=1, \ldots, M \end{gathered}$$

金融代写|固定收益与信贷代写Fixed Income and Credit代考|The Model

$$S_t=\left[S_t^1: S_t^N\right]$$

金融代与写固定收益与信贷代写Fixed Income and Credit代考|Absence of Arbitrage

$$V_t^h=\sum_{i=1}^N h^i S_t^i=h S_t, \quad t=0,1$$

$$V_t^h\left(\omega_i\right)=h S_t\left(\omega_i\right)=h d_i=(h D)_i .$$

$$V_0^h=0 P\left(V_1^h \geq 0\right) \quad=1 P\left(V_1^h>0\right)>0 .$$

$$V_0^h<0, V_1^h\left(\omega_i\right) \geq 0, \quad \text { for all } i=1, \ldots, M V_1^h\left(\omega_i\right)>0, \quad \text { for some } i=1, \ldots, M$$

MATLAB代写

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