### 金融代写|波动率模型代写Market Volatility Modelling代考|FE720

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|波动率模型代写Market Volatility Modelling代考|Replicating Strategies

The Black-Scholes analysis of a European-style derivative yields an explicit trading strategy in the underlying risky asset and riskless bond whose terminal payoff is equal to the payoff $h\left(X_T\right)$ of the derivative at maturity, no matter what path the stock price takes. Thus, selling the derivative and holding a dynamically adjusted portfolio according to this strategy “covers” an investor against all risk of eventual loss, because a loss incurred at the final time from one part of this portfolio will be exactly compensated by a gain in the other part. This replicating strat$e g y$, as it is known, therefore provides an insurance policy against the risk of being short the derivative. It is called a dynamic hedging strategy since it involves continuous trading, where to hedge means to eliminate risk. The essential step in the Black-Scholes methodology is the construction of this replicating strategy and arguing, based on no arbitrage, that the value of the replicating portfolio at time $t$ is the fair price of the derivative. We develop this argument in the following sections.

We consider a European-style derivative with payoff $h\left(X_T\right)$, a function of the underlying asset price at maturity time $T$. Assume that the stock price $\left(X_t\right)$ follows the geometric Brownian motion model (1.20), a solution of the stochastic differential equation (1.2). A trading strategy is a pair $\left(a_t, b_t\right)$ of adapted processes specifying the number of units held at time $t$ of the underlying asset and the riskless bond, respectively. We suppose that $\mathbb{E}\left{\int_0^T\left(a_t\right)^2 d t\right}$ and $\int_0^T\left|b_t\right| d t$ are finite so that the stochastic integral involving $\left(a_t\right)$ and the usual integral involving $\left(b_t\right)$ are well-defined.

Assuming, as in (1.1), that the price of the bond at time $t$ is $\beta_t=e^{r t}$, the value at time $t$ of this portfolio is $a_t X_t+b_t e^{\prime t}$. It will replicate the derivative at maturity if its value at time $T$ is almost surely equal to the payoff:
$$a_T X_T+b_T e^{r T}=h\left(X_T\right) .$$
In addition, this portfolio is to be self-financing, meaning that the variations of its value are due only to the variations of the market – that is, the variations of the stock and bond prices. No further funds are required after the initial investment,yields an instant profit with no exposure to future loss, since the terminal payoff of the trading strategy is equal to the payoff of the derivative.

## 金融代写|波动率模型代写Market Volatility Modelling代考|Self-Financing Portfolios

As in Section 1.3.1, a portfolio comprises $a_i$ units of stock and $b_t$ in bonds; we denote by $V_t$ its value at time $t$ :
$$V_t=a_t X_t+b_t e^{r t} .$$
The self-financing property (1.28), namely $d V_t=a_t d X_t+r b_t e^{r t} d t$, implies that the discounted value of the portfolio, $\widetilde{V}_t=e^{-r t} V_t$, is a martingale under the risk-neutral probability $\mathbb{P}^{\star}$. This important property of self-financing portfolios is cbtained as follows:
\begin{aligned} d \tilde{V}_t & =-r e^{-r t} V_t d t+e^{-r t} d V_t \ & =-r e^{-r t}\left(a_t X_t+b_t e^{r t}\right) d t-e^{-r t}\left(a_t d X_t+r b_t e^{r t} d t\right) \ & =-r e^{-r t} a_t X_t d t+e^{-r t} a_t d X_t \ & =a_t d\left(e^{-r t} X_t\right) \ & =a_t d \tilde{X}_t \ & =\sigma a_t \widetilde{X}_t d W_t^{\star} \quad(\text { by }(1.46)), \end{aligned}
which shows that $\left(\tilde{V}_t\right)$ is a martingale under $\mathbb{P}^{\star}$ as a stochastic integral with respect to the Brownian motion $\left(W_t^*\right)$. Indeed, the same computation shows that if a portfolio satisfies $d \tilde{V}_t=a_t d \widetilde{X}_t$ then it is self-financing.

A simple calculation demonstrates the connection between martingales and no arbitrage. Suppose that $\left(a_t, b_t\right)_{0 \leq t \leq T}$ is a self-financing arbitrage strategy; that is,
$$V_T \geq e^{r T} V_0 \quad(I P \text {-a.s. }),$$
with
$$\mathbb{P}\left{V_T>e^{r T} V_0\right}>0,$$
so that the strategy never makes less than money in the bank and there is some chance of making more. But
$$\mathbb{E}^\left{V_T\right}=e^{r T} V_0$$ by the martingale property, so (1.51) and (1.52) cannot hold. This is because $\mathbb{P}$ and $\mathbb{P}^$ are equivalent and so (1.51) and (1.52) also hold with $\mathbb{P}$ replaced by $\mathbb{P}^{\star}$.

# 波动率模型代考

## 金融代写|波动率模型代写Market Volatility Modelling代考|Replicating Strategies

$$a_T X_T+b_T e^{r T}=h\left(X_T\right) .$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Self-Financing Portfolios

$$V_t=a_t X_t+b_t e^{r t} .$$

$$V_T \geq e^{r T} V_0 \quad(I P \text {-a.s. }),$$

Imathbb ${P} \backslash l$ eft $\left{V_{-} T>e^{\wedge}{r \mathrm{~T}} V_{-} \backslash \backslash\right.$ ight $}>0$,

Imathbb{E $}^{\wedge} \backslash l$ eft $\left{V_{-} T \backslash r i g h t\right}=e \wedge{r ~ T} \vee_{-} 0$

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