### 统计代写|离散时间鞅理论代写martingale代考|Pricing and hedging without tears

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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

## 统计代写|离散时间鞅理论代写martingale代考|An insurance viewpoint

Let us assume that we have sold an option (at $t=0$ ) at the price $C$. An option, with payoff $F_{T}$, gives the right to the holder to exercise the payoff at a maturity $T$. Its gain at $T$ is $F_{T}$. For example, $F_{T}=\left(S_{T}-K\right)^{+}$for a call and $F_{T}=\left(K-S_{T}\right)^{+}$for a put where $S_{T}$ is the asset price at $T$ and $K$ is called the strike. Although in this part we will focus mainly on call options as a simple example, our discussion remains valid for a general (non American) option with payoff $F_{T} \equiv F\left(S_{T}\right)$. From $t=0$ up to the maturity $T$, we do not apply any hedging strategy (see Section 1.2). The cash $C$ is invested (at $t=0$ ) into a bank account with a fixed interest rate $r$. The value of our portfolio at $T$, composed of only our bank account, is $C e^{r T}$. At $T$, the client exercises his option and our portfolio value becomes finally
$$\pi_{T} \equiv e^{r T} C-F_{T}$$

which is equivalent to the discounted portfolio value:
$$e^{-r T} \pi_{T}=C-e^{-r T} F_{T}$$
How should we fix $C$ ?

## 统计代写|离散时间鞅理论代写martingale代考|Utility preference

A first approach is to choose $C$ such that the expectation of $\pi_{T}$ vanishes, $\mathbb{E}^{P^{\text {hint }}}\left[\pi_{T}\right]=0$, i.e.,
$$C_{\mathrm{ins}}=e^{-r T} \mathbb{E}^{\mathrm{P}^{\text {hiwt }}}\left[F_{T}\right]$$
The expectation is taken under a probability phist that describes the (historical) fluctuations of $S_{T}$. More precisely, uncertainty is described by a probability space $\left(\Omega, \mathcal{F}, \mathbb{P}^{\text {hist }}\right)$ where $\Omega$ is the set of possible outcomes, $\mathcal{F}$ is a $\sigma$-algebra that describes the set of all possible events (an event is a subset of $\Omega$ ) and $\mathbb{P}^{\text {hist }}$ is a probability describing the likelihoods of the various events.

Note that $S_{T}$ is a positive random variable, in principle atomic as $S_{T}$ take discrete values. Leaving apart the atomic nature of the distribution of $S_{T}$, we choose here for the sake of simplicity a log-normal density
$$p\left(S_{T}=K\right)=\frac{1}{\sqrt{2 \pi \sigma^{2} T} K} \exp -\frac{\left(\ln \frac{K}{S_{0}}+\frac{1}{2} \sigma^{2} T-\mu T\right)^{2}}{2 \sigma^{2} T}$$
Here $S_{0}$ denotes the asset value at $t=0$ (which is known). This density depends on two parameters: a drift $\mu$ and a (log-normal) volatility $\sigma$. Note that $\mu T$ and $\sigma \sqrt{T}$ are dimensionless and satisfies
$\mathbb{E}^{\mathbb{P}^{\text {hixt }}}\left[\frac{S_{T}}{S_{0}}\right]=e^{\mu T}$
$-\frac{2}{T} \mathbb{E}^{\mathbb{P h}^{\text {hït }}}\left[\ln \frac{S_{T}}{S_{0}}\right]=\sigma^{2}-2 \mu$
By direct integration of $(1.2)$ with $F_{T}=\left(S_{T}-K\right)^{+}$, we obtain for a call option with strike $K$ and maturity $T$ :
$$C_{\text {ins }}(T, K)=e^{(\mu-r) T} S_{0} N\left(d_{1}\right)-K e^{-r T} N\left(d_{2}\right)$$
with
$$d_{1}=\frac{1}{\sigma \sqrt{T}}\left(\ln \frac{S_{0}}{K}+\left(\mu+\frac{1}{2} \sigma^{2}\right) T\right), \quad d_{2}=d_{1}-\sigma \sqrt{T}$$
$N(x) \equiv \int_{-\infty}^{x} e^{-\frac{y^{2}}{2}} d y$ is the Gaussian cumulative distribution. Note that this formula depends here on the drift $\mu$ (see the scaling $e^{(\mu-r) T}$ and the expression of $\left.d_{1}\right)$, a priori different from $r$. It is at least debatable whether the drift is really observable or even statistically measurable. At this point, the price $C$ is strongly model-dependent as it depends on our choice of the density (through the two parameters $\mu$ and $\sigma$ ).

At this point, the seller of our previous option (with payoff $F_{T}$ ) learns from his management that it can buy and sell the asset on the market. If we denote by $H$ the initial number of assets bought by the trader, the portfolio value at $t=0$ is
$$\pi_{0}=-H S_{0}+C$$
$H S_{0}$ corresponds to the price of a portfolio consisting of $H$ assets with unit price $S_{0}$ at $t=0$. Note that $H$ could be positive or negative. If $H>0$, we say that we have a long position, this means that the trader has bought $H$ shares. How to achieve $H<0$ (we say that we have a short position) as the trader does not hold the asset at $t=0$ ? The trader borrows at $t=0$ the share from a counterparty and sell it on the market at $t=0$, generating a profit $-H S_{0}>0$. He gives back the share to the counterparty at the maturity $T$ at the price $S_{T}$ with a small premium, called the repo. For the sake of simplicity, we consider here that the repo is null. The portfolio value at $T$ is then
$$\pi_{T} \equiv\left(-H S_{0}+C\right) e^{r T}+H S_{T}-F_{T}$$
which is equivalent to
$$e^{-r T} \pi_{T}=H\left(S_{T} e^{-r T}-S_{0}\right)+C-e^{-r T} F_{T}$$
$\left(-H S_{0}+C\right) e^{r T}$ is the value of our bank account, $H S_{T}$ the value of $H$ units of our asset (at the price $S_{T}$ ) and $F_{T}$ the payoff exercised at the maturity $T$. Note that in comparison with Equation (1.1), we have the additional term $H\left(S_{T} e^{-r T}-S_{0}\right)$, generated from our delta-hedging strategy.

In the next section, we focus first on question (A). As in the previous section, the price will be fixed by maximizing an utility function depending of our portfolio value $\pi_{T}$. These different optimizations will be framed as linear, quadratic and convex programming problems. As in the first section, we assume that.

## 统计代写|离散时间鞅理论代写martingale代考|Utility preference

C一世ns=和−r吨和磷希特 [F吨]

p(小号吨=ķ)=12圆周率σ2吨ķ经验−(ln⁡ķ小号0+12σ2吨−μ吨)22σ2吨

ï−2吨和磷H打 [ln⁡小号吨小号0]=σ2−2μ

C插件 (吨,ķ)=和(μ−r)吨小号0ñ(d1)−ķ和−r吨ñ(d2)

d1=1σ吨(ln⁡小号0ķ+(μ+12σ2)吨),d2=d1−σ吨
ñ(X)≡∫−∞X和−是22d是是高斯累积分布。请注意，此公式在此处取决于漂移μ（见缩放和(μ−r)吨和表达d1), 先验不同于r. 漂移是否真的可以观察到，甚至可以在统计上测量，至少还有待商榷。此时，价格C强烈依赖于模型，因为它取决于我们对密度的选择（通过两个参数μ和σ ).

H小号0对应于由以下组成的投资组合的价格H单价资产小号0在吨=0. 注意H可以是正面的或负面的。如果H>0，我们说我们有一个多头头寸，这意味着交易者已经买入H分享。如何实现H<0（我们说我们有空头头寸）因为交易者不持有资产吨=0? 交易者借入吨=0从对手方获得的份额，并在市场上以吨=0, 产生利润−H小号0>0. 他在到期时将股份返还给交易对手吨以价格小号吨以少量溢价，称为回购。为了简单起见，我们在这里认为 repo 是空的。投资组合价值吨那么是

(−H小号0+C)和r吨是我们银行账户的价值，H小号吨的价值H我们资产的单位（按价格小号吨） 和F吨到期行使的收益吨. 请注意，与等式（1.1）相比，我们有附加项H(小号吨和−r吨−小号0)，由我们的 delta 对冲策略生成。

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## MATLAB代写

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