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By choosing a log-normal distribution (with drift $\mu$ and volatility $\sigma$ ) for $\mathbb{P}^{\text {hist }}$, we can show that the seller’s super-replication price of a call option is $S_{0}$. The reader should remark that from Jensen’s inequality we have also for all $\mathbb{Q} \in \mathcal{M}{1}$, $$C{\mathrm{buy}}=\left(S_{0}-K e^{-r T}\right)^{+} \leq \mathbb{E}^{Q}\left[e^{-r T}\left(S_{T}-K\right)^{+}\right] \leq C_{\mathrm{sel}}=S_{0}$$
This seller super-replication’s price is very expensive as it is identical to the price of a forward contract that it is the option that delivers $S_{T}$ at the maturity. It is therefore fairly unexpected that a (reasonable) client is willing to accept to pay a call option at the same price as a forward contract for which the payoff super-replicates at maturity those of a call option $\left(S_{T}-K\right)^{+} \leq S_{T}$. In this section, we disregard the super-replication approach in this respect and fix $C$ and $H$ such that the variance of $\pi_{T}$ for a payoff $F_{T}$ is minimised under the constraint $\mathbb{E}^{\mathbb{P}^{\text {hist }}}\left[\pi_{T}\right]=0$ (i.e., fixed return):

DEFINITION 1.8 Mean-variance hedging The mean-variance hedging is defined as the following quadratic programming problem:
$$P_{\text {quad }} \equiv \inf {C, H \text { s.t. } \mathbb{E}^{\text {phiat }}}[\pi T]=0 \mathbb{E}^{\mathbb{P}^{\text {hiat }}}\left[\pi{T}^{2}\right]$$
As the cost $\mathbb{E}^{\mathrm{Phist}^{h}}\left[\pi_{T}^{2}\right]$ (resp. the constraint $\mathbb{E}^{\mathrm{Ph}^{\text {int }}}\left[\pi_{T}\right]=0$ ) is a quadratic (resp. linear) form with respect to $C$ and $H,(1.16)$ defines a quadratic programming problem. The constraint $\mathbb{E}^{\mathbb{P}^{\text {hint }}}\left[\pi_{T}\right]=0$ gives
$$C_{\text {quad }}=e^{-r T} \mathbb{E}^{\mathrm{p}^{\text {hint }}}\left[F_{T}\right]-H \mathbb{E}^{\mathbb{P}^{\text {hiat }}}\left[\left(S_{T} e^{-r T}-S_{0}\right)\right]$$
where $C_{\text {quad }}$ is the minimizer in (1.16). The first term corresponds to our insurance price $C_{\text {ins. }}$. Computing the infimum over $H$ of $E\left[\pi_{T}^{2}\right]$ with $C=C_{\text {quad }}$, we obtain
$$\mathbb{E}^{\mathbb{P}^{\text {hint }}}\left[\pi_{T}\left(S_{T}-\mathbb{E}^{\mathbb{P}^{\text {hist }}}\left[S_{T}\right]\right)\right]=0$$
where we have used $e^{-r T} \partial_{H} \pi_{T}=e^{-r T}\left(S_{T}-\mathbb{E}^{\mathbb{P}^{\text {hist }}}\left[S_{T}\right]\right)$. This is equivalent to
Finally,

## 统计代写|离散时间鞅理论代写martingale代考|Quantile hedging

Quantile hedging consists in replacing Definition $1.1$ of the seller’s price by the following:
$$C_{p}=\inf \left{C: \exists H \text { s.t. } \mathbb{P h}^{\text {ist }}\left[\pi_{T} \geq 0\right] \geq p\right.$$
and $C+H\left(S_{T} e^{-r T}-S_{0}\right) \geq 0, \quad \mathbb{P}^{\text {hist }}-$ a.s. $}$
$p \in[0,1]$ is interpreted as the probability of super-replicating the claim $F_{T}$ under the historical measure. Here we have added the constraint that the trader’s portfolio should be greater than a threshold $-L$ :
$$C+H\left(S_{T} e^{-r T}-S_{0}\right) \geq-L$$
For convenience, we have chosen $L=0$, this can be easily modified.
By definition, $C_{1}=C_{\text {sel }}$, the super-replication price for $F_{T} \geq 0$. $C_{\text {sel }}$ can be very high – recall for instance that the super-replication price of a call option equals $S_{0}$. In the quantile hedging approach, we only impose that the payoff can be super-replicated with a probability $p$. In their seminal paper [75], Föllmer and Leukert show that the corresponding optimal strategy consists in superhedging a modified payoff. More precisely, we have.

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

We introduce an utility function $U$, which is strictly increasing and concave. We consider the value $u(x, 0)$ of the supremum over all hedging portfolios starting from the initial capital $x$ of the expectation of the utility of the discounted final wealth under the historical measure $\mathbb{P}^{h i s t}$ :
$$u(x, 0) \equiv \sup {H} \mathbb{E}^{\mathrm{P}^{\text {hint }}}\left[U\left(x-H\left(e^{-r T} S{T}-S_{0}\right)\right)\right]$$
Similarly, the value $u\left(x-C, F_{T}\right)$ is defined for a claim $F_{T}$ as
$$u\left(x-C, F_{T}\right) \equiv \sup {H} \mathbb{E}^{\mathbb{P h}^{\mathrm{iixt}}}\leftU\left(x-C+e^{-r T} F{T}-H\left(e^{-r T} S_{T}-S_{0}\right)\right)\right$$
The utility indifference buyer’s price, as introduced by Hodges-Neuberger $[107]$, is the quantity $C_{\mathrm{HN}}$ such that
DEFINITION 1.10 Utility indifference buyer’s price
$$u(x, 0)=u\left(x-C_{\mathrm{HN}}, F_{T}\right)$$
This means that a buyer should accept quoting a price for the claim $F_{T}$ when buying and delta-hedging this derivative becomes as profitable as setting up a pure delta strategy. The expression $u\left(x-C, F_{T}\right)$ can be dualized into
THEOREM $1.5$
$$u\left(x-C, F_{T}\right)=\inf {\mathbb{Q} \in \mathcal{M}{1}} \mathbb{E}^{\mathrm{Q}}\left[\left(e^{-r T^{T}} F_{T}+x-C\right)+\frac{d \mathbb{P}^{\text {Pist }}}{d \mathbb{Q}} U^{}\left(\frac{d \mathbb{Q}}{d P^{2} i s t}\right)\right]$$ with $U^{}(p) \equiv \sup {x \in \mathbb{R}}{U(x)-p x}$ the Legendre-Fenchel transform of $U$. The functions $U$ and $U^{}$ also satisfy the conjugate relation: $U(x)=\inf {p \in \mathbb{R}_{+}}{p x+$ $\left.U^{}(p)\right}$

## 离散时间鞅理论代考

Cb在是=(小号0−ķ和−r吨)+≤和问[和−r吨(小号吨−ķ)+]≤Cs和l=小号0

C四边形 =和−r吨和p暗示 [F吨]−H和磷嗨 [(小号吨和−r吨−小号0)]

## 统计代写|离散时间鞅理论代写martingale代考|Quantile hedging

C_{p}=\inf \left{C: \exists H \text { st } \mathbb{P h}^{\text {ist }}\left[\pi_{T} \geq 0\right] \geq p\对。$$和 C+H\left(S_{T} e^{-r T}-S_{0}\right) \geq 0, \quad \mathbb{P}^{\text {hist }}-因为 } p \in[0,1] 被解释为在历史度量下超级复制声明 F_{T} 的概率。在这里，我们添加了交易者的投资组合应大于阈值 -L 的约束：C_{p}=\inf \left{C: \exists H \text { st } \mathbb{P h}^{\text {ist }}\left[\pi_{T} \geq 0\right] \geq p\对。$$ 和 $C+H\left(S_{T} e^{-r T}-S_{0}\right) \geq 0, \quad \mathbb{P}^{\text {hist }}-$因为 $}$ $p \in[0,1]$ 被解释为在历史度量下超级复制声明 $F_{T}$ 的概率。在这里，我们添加了交易者的投资组合应大于阈值 $-L$ 的约束：
C+H\left(S_{T} e^{-r T}-S_{0}\right) \geq-L
$$为方便起见，我们选择大号=0, 这可以很容易地修改。 根据定义，C1=C这个 ，超复制价格为F吨≥0. C这个 可能非常高——例如，看涨期权的超复制价格等于小号0. 在分位数对冲方法中，我们只强加收益可以以一定的概率进行超级复制p. 在他们的开创性论文 [75] 中，Föllmer 和 Leukert 表明，相应的最优策略包括对修改后的收益进行超对冲。更准确地说，我们有。 ## 统计代写|离散时间鞅理论代写martingale代考|Utility indifference price 我们引入一个效用函数在，它是严格递增和凹的。我们考虑价值在(X,0)从初始资本开始的所有套期保值投资组合的上限X在历史测度下，折现后的最终财富的效用预期磷H一世s吨 : 在(X,0)≡支持H和磷暗示 [在(X−H(和−r吨小号吨−小号0))] 同样，值在(X−C,F吨)为索赔定义F吨as$$
u\left(xC, F_{T}\right) \equiv \sup {H} \mathbb{E}^{\mathbb{P h}^{\mathrm{iixt}}}\left U\left (x-C+e^{-r T} F{T}-H\left(e^{-r T} S_{T}-S_{0}\right)\right)\right

u(x, 0)=u\left(x-C_{\mathrm{HN}}, F_{T}\right)

u\left(xC, F_{T}\right)=\inf {\mathbb{Q} \in \mathcal{M}{1}} \mathbb{E}^{\mathrm{Q}}\left[\左(e^{-r T^{T}} F_{T}+xC\right)+\frac{d \mathbb{P}^{\text {Pist }}}{d \mathbb{Q}} U ^{}\left(\frac{d \mathbb{Q}}{d P^{2} ist}\right)\right]  与在(p)≡支持X∈R在(X)−pXLegendre-Fenchel 变换在. 功能在和在也满足共轭关系：U(x)=\inf {p \in \mathbb{R}_{+}}{p x+\left.U^{}(p)\right}

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