# 数学代写|离散数学作业代写discrete mathematics代考|CS3653

## 数学代写|离散数学作业代写discrete mathematics代考|The Cox, Ross and Rubinstein model

We will now illustrate the different concepts introduced above using a specific case of financial market. This is a discretized version of the Black and Scholes model.
The market is considered to be made up of a risk-free asset $S_n^0=(1+r)^n$ and a single risky asset $S^1$ with the dynamic
$$S_0^1=1, \quad S_{n+1}^1=S_n^1 T_{n+1}, n \geq 0,$$
where $\left(T_n\right)_{1 \leq n \leq N}$ is a sequence of random variable taking only two values $1+d$ and $1+u$ with $-1<d<u$. In other words, $d$ and $u$ are interpreted as interest rates. We thus write
$$\Omega={1+d, 1+u}^N, \mathcal{F}_0={\emptyset, \Omega}, \mathcal{F}_n=\sigma\left(T_1, \ldots, T_n\right), 1 \leq n \leq N$$
In particular, $\mathcal{F}_N=\sigma\left(T_1, \ldots, T_N\right)=\mathcal{P}(\Omega)$ is the set of subsets of $\Omega$.
We will now characterize viable markets in this model. In order to do this, we start by studying risk-neutral probabilities.

PROPOSITION 5.2.- The discounted prices $\left(\widetilde{S}n^1\right){0 \leq n \leq N}$ are a martingale under a probability $\mathbb{P}^$ if and only if, for any $0 \leq n \leq N-1$, we have $$\mathbb{E}^\left[T_{n+1} \mid \mathcal{F}_n\right]=1+r$$
with $\mathbb{E}^$ denoting the expectation for the probability $\mathbb{P}^$.
PROOF. – Let us proceed through double implication.

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

We now study a portfolio optimization problem in the Cox, Ross and Rubinstein model.

Let $V_0$ be the wealth of an investor at the time 0 . The investor can invest their money either in a risky asset or in a risk-free asset, following an admissible strategy. We use $\phi_n^0$ and $\phi_n^1$ to denote the number of shares in the risk-free asset and the number of shares in the risky asset, respectively, held between the time $n-1$ and $n$. Let $\pi_n$ be the proportion of the wealth invested in the risky asset between the instants $n-1$ and $n$, that is,
$$\pi_n=\frac{\phi_n^1 S_{n-1}^1}{V_{n-1}} .$$
1) Express $\phi_n^0$ and $\phi_n^1$ as the functions of $\pi_n, S_{n-1}^0, S_{n-1}^1$ and $V_{n-1}$ for any $n$.
2) Derive from this that for any $n$, the wealth at the time $n$, after the evolution of the prices and before the redistribution of the portfolio has the value:
$$V_n=\left(\pi_n T_n+\left(1-\pi_n\right)(1+r)\right) V_{n-1} .$$
3) On the same graph and for the same random sampled trajectory, represent the evolution of the risk-free asset and the evolution of the wealth for the following two strategies:

a) The proportion of the wealth invested in the risky asset is fixed over time, at $1 / 4$.
b) The proportion of the wealth invested in the risky asset only takes the values 0 and 1 . It takes the value 1 when the price of a risky asset strictly exceeds that of the risk-free asset, and takes the value 0 when the risky asset is small than or equal to the risk-free asset, while remaining predictable.

We will take the following parameters: 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 $r=4 \%$ and duration of investment: $N=100$ periods.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|The Cox, Ross and Rubinstein model

$$S_0^1=1, \quad S_{n+1}^1=S_n^1 T_{n+1}, n \geq 0,$$

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