统计代写|离散时间鞅理论代写martingale代考|Super-replication: Linear programming

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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
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• 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|Linear programming

We define seller’s super-replication price $C$ as
DEFINITION 1.1 Seller’s super-replication price
$$C_{\text {sel }} \equiv \inf \left{C: \exists H \text { s.t. } \pi_{T} \geq 0, \quad \mathbb{P}^{\text {hist }}-\text { a.s. }\right}$$
where the abbreviation a.s. means almost surely. $C$ and $H$ are chosen such that the portfolio value at $T$ is nonnegative for all realization of $S_{T}$ distributed according to the law $\mathbb{P}^{\text {hist }}$. This definition is clearly that of non-adverse risk traders.

Here, note that our definition depends weakly on our modeling assumption only through the negligible sets of $\mathbb{P}^{\text {hist }}$. $\mathbb{P h}^{\text {hist }}$ can be replaced by any probability $\mathbb{Q}$ equivalent to $\mathbb{P}^{\text {hist }}$ – see Definition 1.2.
For completeness, we recall the definition of equivalent probabilities:
DEFINITION 1.2 Equivalent probabilities $\mathbb{P} \sim \mathbb{Q}$ – we say $\mathbb{P}$ and $\mathbb{Q}$ are equivalent on a sigma field $\mathcal{F}$ – if $\mathbb{P}$ and $\mathbb{Q}$ have the same negligible sets: $\mathbb{P}(A)=0$ if and only if $\mathbb{Q}(A)=0$ for all $A \in \mathcal{F}$.

For example if $\mathbb{P}^{\text {hist }}$ is an atomic probability supported on the points $\left(s_{i}\right){i=1, \ldots, N}$ with probabilities $\left(p{i} \equiv \mathbb{P}^{\text {hist }}\left(S_{T}=s_{i}\right) \neq 0\right){i=1, \ldots, N}, \mathbb{Q} \sim \mathbb{P}^{\text {hist }}$ means that $\mathbb{Q}$ is also an atomic probability supported on the same points $\left(s{i}\right){i=1, \ldots, N}$ with probability $\left(q{i} \neq 0\right){i=1, \ldots, N}$. $C{\text {sel }}$ defines a so-called infinite-dimensional linear programming problem: we need to compute the infimum of a linear cost, i.e., $C$, with respect to the real variables $C$ and $H$, subject to an infinite number of constraint inequalities, parameterized by $S_{T}$. If we assume that $\mathbb{P}^{\text {phist }}$ is an atomic probability supported on $N$ points $\left(s_{i}\right){i=1, \ldots, N}$ – this is the case in practice as the unit price is one cent – the super-replication price can be stated as a more conventional finite-dimensional linear programming (written here for $F{T}=\left(S_{T}-K\right)^{+}$)
$$\begin{array}{r} C_{\text {sel }}^{N} \equiv \inf \left{C: \exists H \text { s.t. } H\left(s_{i} e^{-r T}-S_{0}\right)+C-e^{-r T}\left(s_{i}-K\right)^{+} \geq 0,\right. \ i=1, \ldots, N} \end{array}$$
$C_{\text {sel }}^{N}$ can then be solved numerically using a simplex algorithm. We will now present a dual formulation (Monge-Kantorovich dual) of $C_{\text {sel }}$ that will be fundamental for understanding the notion of arbitrage-free prices and riskneutral probabilities.

统计代写|离散时间鞅理论代写martingale代考|Arbitrage-free prices and bounds

DEFINITION 1.4 Arbitrage opportunity $\mathbb{P}^{\text {hist }}$ is arbitrage-free if there does not exist $H$ and $\pi_{0}<0$ for which
$$\pi_{0}+H\left(S_{T} e^{-r T}-S_{0}\right) \geq 0, \quad \mathbb{P}^{\text {hist }}-\text { a.s. }$$
This means that starting with a strictly negative portfolio value at $t=0$, a positive profit can be locked in without any downside risk. We say that we have an arbitrage opportunity. It is clear that from a modeling point of view, such arbitrage opportunity should be disregarded. Indeed, if such an opportunity would show up, it would generate a large demand on the underlying $S$, and at the equilibrium the arbitrage would disappear. Similarly, we define arbitrage-free prices as

DEFINITION $1.5$ Arbitrage-free prices We say that $C$ is an arbitragefree price if there does not exist $H$ and $\pi_{0}<0$ for which
$$\pi_{0}+H\left(S_{T} e^{-r T}-S_{0}\right)+C-e^{-r T} F_{T} \geq 0, \quad \mathbb{P}^{\text {hist }}-\text { a.s. }$$

In fact these two definitions are consistent if we consider an extended market with two assets $S_{T}$ and $F_{T}$ and prices $S_{0}$ and $C$ at $t=0$. Throughout this book, we will assume
Assumption $2 \mathbb{P}^{\text {hist }}$ is arbitrage-free.

统计代写|离散时间鞅理论代写martingale代考|A worked-out example: The binomial model

Let us assume that the historical probability $\mathbb{P}^{\text {hist }}$ is supported on two points $S_{u}=u S_{0}$ and $S_{d}=d S_{0}$ with probabilities $p_{u}$ and $p_{d}$ and $u>d$ without loss of generality. Note that $e^{-r T} S_{T}$ is not required to be a Phist -martingale. The price obtained using an insurance point of view is (see Formula (1.2))
$$C_{\mathrm{ins}}=e^{-r T}\left(p_{u}\left(u S_{0}-K\right)^{+}+p_{d}\left(d S_{0}-K\right)^{+}\right)$$
The super-replication price for a call can be obtained easily using our dual formulation. Let us characterize the convex set $\mathcal{M}{1}: \mathbb{Q} \in \mathcal{M}{1}$ if and only if $\mathbb{Q}$ is supported on the points $S_{u}$ and $S_{d}\left(\right.$ as $\left.\mathbb{Q} \sim \mathbb{P}^{\text {hist }}\right)$ and satisfies $\mathbb{E}^{\mathbb{Q}}\left[e^{-r T} S_{T}\right]=$ $S_{0}:$
\begin{aligned} q_{u}+q_{d} &=1 \ q_{u} S_{u}+q_{d} S_{d} &=e^{r T} S_{0} \end{aligned}

where $q_{u} \equiv \mathbb{Q}\left(S_{T}=S_{u}\right) \neq 0$ and $q_{d} \equiv \mathbb{Q}\left(S_{T}=S_{d}\right) \neq 0$. There is a (unique) solution if and only if $d<e^{r T}<u$ for which
$$q_{u}=\frac{e^{r T}-d}{u-d}, \quad q_{d}=\frac{u-e^{r T}}{u-d}$$
From Corollary 1.1, the binomial model is arbitrage-free if and only if $d<$ $e^{r T}<u$. The super-replication price is then
$$C=e^{-r T}\left(q_{u}\left(u S_{0}-K\right)^{+}+q_{d}\left(d S_{0}-K\right)^{+}\right)$$
Note that as explained previously, $C$ depends only on the negligible set of $\mathbb{P}^{\text {hist }}$ (through the points $S_{d}$ and $S_{u}$ ) and does not depend on $p_{u}$ and $p_{d}$. As the set $\mathcal{M}_{1}$ is a singleton, the price of this option is unique from Corollary $1.2$ (in particular the super and sub-replication prices coincide).

统计代写|离散时间鞅理论代写martingale代考|Linear programming

C_{\text {sel }} \equiv \inf \left{C: \exists H \text { st } \pi_{T} \geq 0, \quad \mathbb{P}^{\text {hist }}- \text { 作为 }\right}C_{\text {sel }} \equiv \inf \left{C: \exists H \text { st } \pi_{T} \geq 0, \quad \mathbb{P}^{\text {hist }}- \text { 作为 }\right}

\begin{array}{r} C_{\text {sel }}^{N} \equiv \inf \left{C: \exists H \text { st } H\left(s_{i} e^{-r T}-S_{0}\right)+Ce^{-r T}\left(s_{i}-K\right)^{+} \geq 0,\right. \ i=1, \ldots, N} \end{数组}\begin{array}{r} C_{\text {sel }}^{N} \equiv \inf \left{C: \exists H \text { st } H\left(s_{i} e^{-r T}-S_{0}\right)+Ce^{-r T}\left(s_{i}-K\right)^{+} \geq 0,\right. \ i=1, \ldots, N} \end{数组}
C这个 ñ然后可以使用单纯形算法进行数值求解。我们现在将提出一个对偶公式（Monge-Kantorovich dual）C这个 这对于理解无套利价格和风险中性概率的概念至关重要。

统计代写|离散时间鞅理论代写martingale代考|A worked-out example: The binomial model

C一世ns=和−r吨(p在(在小号0−ķ)++pd(d小号0−ķ)+)

q在+qd=1 q在小号在+qd小号d=和r吨小号0

q在=和r吨−d在−d,qd=在−和r吨在−d

C=和−r吨(q在(在小号0−ķ)++qd(d小号0−ķ)+)

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