# 随机微积分作业代写stochastic calculus代考| THE SIMPLE BLACK SCHOLES MARKET

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• 随机偏微分方程
• 随机控制
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• black-Scholes-Merton option pricing formula
• Fokker–Planck equation
• 布朗运动 Brownian motion

## 微积分作业代写calclulus代考|Trading strategies and absence of arbitrage

1.A trading strategy (dynamic portfolio) is a predictable $R^{2}$-valued process $\phi_{t}=\left(K_{t}, H_{t}\right), t \in[0, T]$, satisfying $K \in L(B)$ and $H \in L(S)$. This requirement ensures that the stochastic differentials $K d B$ and $H d S$ are defined. Because of the special nature of our processes $B$ and $S$ it is easily seen to be equivalent with the condition
$$\int_{0}^{T}\left|K_{t}\right| d t+\int_{0}^{T} H_{t}^{2} d t<\infty, P \text {-as. }$$ The process $\phi$ is to be interpreted as a continuously changing portfolio holding $K_{t}$ units of the bond and $H_{t}$ units of the stock at time $t$. The coefficients $K$ and $H$ are called the portfolio weights. Thus $$V_{t}(\phi)=K_{t} B_{t}+H_{t} S_{t}$$ is the price of this portfolio at time $t$. Such a portfolio is called tame, if the price process $V_{t}(\phi)$ is bounded below, that is, $V_{t}(\phi) \geq C>-\infty$, for all $0 \leq t \leq T$ and some constant $C$. Consider now a portfolio $\phi=(K, H)$ which holds a constant number of shares of the bond and the stock except for a single rebalancing of the portfolio at time $t_{0}$; in other words assume that
$$K_{t}=\left{\begin{array}{ll} a_{-}, & t \leq t_{0}, \ a_{+}, & t>t_{0}, \end{array} \quad \text { and } \quad H_{t}= \begin{cases}b_{-}, & t \leq t_{0} \ b_{+}, & t>t_{0}\end{cases}\right.$$
where $a_{-}, a_{+}, b_{-}, b_{+}$are constants. The rebalancing at time $t_{0}$ will be called selffinancing, if money is neither injected nor withdrawn from the portfolio. In other words the position at time $t_{0}$ is sold and the proceeds immediately invested into the new position. This is of course equivalent with the requirement $V_{t_{0}}(\phi)=a_{+} B_{t_{0}}+$ $b_{+} S_{t_{0}}$ which in turn is equivalent with the equality
$$\Delta V_{t}(\phi)=V_{t}(\phi)-V_{t_{0}}(\phi)=a_{+}\left(B_{t}-B_{t_{0}}\right)+b_{+}\left(S_{t}-S_{t_{0}}\right)=K_{t} \Delta B_{t}+H_{t} \Delta S_{t}, \quad t>t_{0}$$
This leads to the following fundamental definition:

## 微积分作业代写calclulus代考|Definition

1.c.1. let $\phi_{t}=\left(K_{t}, H_{t}\right)$ be a tame trading strategy. Then $\phi$ is self-financing if and only if the discounted portfolio price process satisfies $d V_{t}^{B}(\phi)=H_{t} d S_{t}^{B}$.

Proof. Write $V_{t}=V_{t}(\phi)$. From $V_{t}^{B}=K_{t}+H_{t} S_{t}^{B}$ and the stochastic product rule it follows that
$$d V_{t}^{B}=d K_{t}+H_{t} d S_{t}^{B}+S_{t}^{B} d H_{t}+d\left\langle H, S^{B}\right\rangle_{t} .$$
Observing that $d S_{t}^{B}=-r S_{t} B_{t}^{-1} d t+B_{t}^{-1} d S_{t}$ we can write $S_{t}^{B}=S_{0}+\int_{0}^{t} d S_{r}^{B}=$ $A_{t}+\left(B^{-1} \cdot S\right){t}$, where $A$ is a continuous bounded variation process. Consequently $\left\langle H, S^{B}\right\rangle=\left\langle H, B^{-1} \bullet S\right\rangle=B^{-1} \bullet\langle H, S\rangle$, in other words $d\left\langle H, S^{B}\right\rangle{t}=B_{t}^{-1} d\langle H, S\rangle_{t}$. Thus (1) can be rewritten as
$$d V_{t}^{B}=H_{t} d S_{t}^{B}+\left[d K_{t}+S_{t}^{B} d H_{t}+B_{t}^{-1} d\langle H, S\rangle_{t}\right] .$$
Similarly from $V_{t}=K_{t} B_{t}+H_{t} S_{t}$, the stochastic product rule and the bounded variation property of $B$,
\begin{aligned} d V_{t} &=K_{t} d B_{t}+B_{t} d K_{t}+H_{t} d S_{t}+S_{t} d H_{t}+d\langle H, S\rangle_{t} \ &=K_{t} d B_{t}+H_{t} d S_{t}+\left[B_{t} d K_{t}+S_{t} d H_{t}+d\langle H, S\rangle_{t}\right] \ &=K_{t} d B_{t}+H_{t} d S_{t}+B_{t}\left[d K_{t}+S_{t}^{B} d H_{t}+B_{t}^{-1} d\langle H, S\rangle_{t}\right] \end{aligned}
Using (2) the last bracket can be rewritten as $d V_{t}^{B}-H_{t} d S_{t}^{B}$. Consequently
$$d V_{t}=K_{t} d B_{t}+H_{t} d S_{t}+B_{t}\left(d V_{t}^{B}-H_{t} d S_{t}^{B}\right)$$
and 1.c.1 follows.
Remark. Let us call a trading strategy $\phi$ static if it is (pathwise) constant, that is, the coefficients $K_{t}=K_{0}$ and $H_{t}=H_{0}$ do not depend on $t$. Then, obviously,
$$d V_{t}(\phi)=d\left(K_{0} B_{t}+H_{0} S_{t}\right)=K_{0} d B_{t}+H_{0} d S_{t},$$
that is, $\phi$ is self-financing. Let us now gather the crucial facts about self-financing trading strategies:

## 微积分作业代写calclulus代考|Trading strategies and absence of arbitrage

1.交易策略（动态投资组合）是可预测的R2价值过程φ吨=(到吨,H吨),吨∈[0,吨], 满足到∈一世(乙)和H∈一世(小号). 这一要求确保了随机微分到d乙和Hd小号被定义。由于我们流程的特殊性乙和小号很容易看出它与条件等价
∫0吨|到吨|d吨+∫0吨H吨2d吨<∞,磷-作为。 过程φ被解释为不断变化的投资组合持有到吨债券单位和H吨当时的股票单位吨. 系数到和H称为投资组合权重。因此五吨(φ)=到吨乙吨+H吨小号吨是该投资组合当时的价格吨. 这样的投资组合被称为驯服，如果价格过程五吨(φ)有界于下，即五吨(φ)≥C>−∞， 对所有人0≤吨≤吨和一些常数C. 现在考虑一个投资组合φ=(到,H)它持有固定数量的债券和股票，但投资组合的一次重新平衡除外吨0; 换句话说，假设
$$K_{t}=\left{一种−,吨≤吨0, 一种+,吨>吨0,\quad \text { 和 } \quad H_{t}={b−,吨≤吨0 b+,吨>吨0\正确的。 在H和r和一种−,一种+,b−,b+一种r和C○ns吨一种n吨s.吨H和r和b一种一世一种nC一世nG一种吨吨一世米和吨0在一世一世一世b和C一种一世一世和ds和一世FF一世n一种nC一世nG,一世F米○n和和一世sn和一世吨H和r一世nj和C吨和dn○r在一世吨Hdr一种在nFr○米吨H和p○r吨F○一世一世○.一世n○吨H和r在○rds吨H和p○s一世吨一世○n一种吨吨一世米和吨0一世ss○一世d一种nd吨H和pr○C和和ds一世米米和d一世一种吨和一世和一世nv和s吨和d一世n吨○吨H和n和在p○s一世吨一世○n.吨H一世s一世s○FC○你rs和和q你一世v一种一世和n吨在一世吨H吨H和r和q你一世r和米和n吨五吨0(φ)=一种+乙吨0+$$b+小号吨0$在H一世CH一世n吨你rn一世s和q你一世v一种一世和n吨在一世吨H吨H和和q你一种一世一世吨和 \Delta V_{t}(\phi)=V_{t}(\phi)-V_{t_{0}}(\phi)=a_{+}\left(B_{t}-B_{t_{0} }\right)+b_{+}\left(S_{t}-S_{t_{0}}\right)=K_{t} \Delta B_{t}+H_{t} \Delta S_{t}, \quad t>t_{0}$\$

## 微积分作业代写calclulus代考|Definition

1.c.1。让φ吨=(到吨,H吨)做一个温顺的交易策略。然后φ当且仅当折现的投资组合价格过程满足时，是自筹资金的d五吨乙(φ)=H吨d小号吨乙.

d五吨乙=d到吨+H吨d小号吨乙+小号吨乙dH吨+d⟨H,小号乙⟩吨.

d五吨乙=H吨d小号吨乙+[d到吨+小号吨乙dH吨+乙吨−1d⟨H,小号⟩吨].

d五吨=到吨d乙吨+乙吨d到吨+H吨d小号吨+小号吨dH吨+d⟨H,小号⟩吨 =到吨d乙吨+H吨d小号吨+[乙吨d到吨+小号吨dH吨+d⟨H,小号⟩吨] =到吨d乙吨+H吨d小号吨+乙吨[d到吨+小号吨乙dH吨+乙吨−1d⟨H,小号⟩吨]

d五吨=到吨d乙吨+H吨d小号吨+乙吨(d五吨乙−H吨d小号吨乙)

d五吨(φ)=d(到0乙吨+H0小号吨)=到0d乙吨+H0d小号吨,