随机微积分（stochastic calculus），数学概念，是高等数学中研究函数的微分(Differentiation)、积分(Integration)以及有关概念和应用的数学分支。它是数学的一个基础学科，内容主要包括极限、微分学、积分学及其应用。微分学包括求导数的运算，是一套关于变化率的理论。它使得函数、速度、加速度和曲线的斜率等均可用一套通用的符号进行讨论。积分学，包括求积分的运算，为定义和计算面积、体积等提供一套通用的方法
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- 随机偏微分方程
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- Ito积分
- 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|Derivatives of securities paying dividends
2.c Derivatives of securities paying dividends. So far we have assumed that the stock $S$ does not pay a dividend. This was important in the definition of a selffinancing trading strategy, which in turn is the basis for arbitrage arguments. When considering opportunities for riskless profit the dividend stream accruing to the holder of an asset paying dividends must be taken into account.
Let us assume now that the asset $S$ pays a dividend continuously at the yield $q(t)$ assumed to be a deterministic function of time. In other words we assume that the dividend process $D_{t}$ satisfies
$$
d D_{t}=q(t) S_{t} d t .
$$
Here $D_{t}$ is to be interpreted as the total amount of dividend received up to time $t$. An example of such an asset is a foreign currency. In this case $q(t)$ is the foreign risk-free rate.
A trading strategy $\phi_{t}=\left(K_{t}, H_{t}\right)$ holding $K_{t}$ units of the riskless bond and $H_{t}$ units of the dividend paying stock $S$ at time $t$ will now be called sel $f$-financing, if the corresponding portfolio price process $V_{t}(\phi)=K_{t} B_{t}+H_{t} S_{t}$ satisfies
$$
d V_{t}(\phi)=K_{t} d B_{t}+H_{t} d S_{t}+q(t) H_{t} S_{t} d t
$$
Set $Q(t, T)=\int_{t}^{T} q(s) d s, A_{t}=e^{-Q(t, T)}$ and $S_{t}^{1}=A_{t} S_{t}=S_{t} e^{-Q(t, T)}$. Then $A_{t}$ is a nonstochastic, bounded variation process satisfying $d A_{t}=q(t) A_{t} d t$ and $A_{T}=1$. Using the stochastic product rule and 1.a.eq.(1) we have
$$
\begin{aligned}
d S_{t}^{1} &=A_{t} d S_{t}+S_{t} d A_{t}=A_{t} d S_{t}+q(t) A_{t} S_{t} d t \
&=A_{t} S_{t}\left[\mu(t) d t+\sigma(t) d W_{t}\right]+q(t) A_{t} S_{t} d t \
&=S_{t}^{1}\left[(\mu(t)+q(t)) d t+\sigma(t) d W_{t}\right] .
\end{aligned}
$$
Thus $S_{t}^{1}$ follows a geometric Brownian motion with time dependent nonstochastic parameters. We can thus view $S_{t}^{1}$ as the price process of a new stock. Let $\psi_{t}=$ $\left(K_{t}, H_{t} / A_{t}\right)$ be a trading strategy taking positions in the bond and the new stock $S_{t}^{1}$. The corresponding price process $V_{t}(\psi)$ satisfies
$$
V_{t}(\psi)=K_{t} B_{t}+\left(H_{t} / A_{t}\right) S_{t}^{1}=K_{t} B_{t}+H_{t} S_{t}=V_{t}(\phi)
$$
that is, $\phi$ and $\psi$ have the same price process. From $(0)$ it follows that $d S_{t}^{1}=$ $A_{t}\left(d S_{t}+q(t) S_{t} d t\right)$. If $\phi$ is self-financing, then
$$
\begin{aligned}
d V_{t}(\psi) &=d V_{t}(\phi)=K_{t} d B_{t}+H_{t}\left(d S_{t}+q(t) S_{t} d t\right) \
&=K_{t} d B_{t}+\left(H_{t} / A_{t}\right) d S_{t}^{1} .
\end{aligned}
$$
This is the self-financing condition for the trading strategy $\psi$ if the new stock $S_{t}^{1}$ is assumed to pay no dividend. Let us call the asset $S^{1}$ the dividend-free reduction of $S$. Then we have seen:
微积分作业代写calclulus代考|Equivalent martingale measure
In the case of a dividend-free asset the equivalent martingale measure $P_{B}$ made the discounted price process of the asset into a martingale with the crucial consequence being that the discounted price process of a self-financing trading strategy $\phi$ is a $P_{B^{-}}$local martingale (and a martingale under suitable, not very restrictive conditions on the coefficients $K_{t}, H_{t}$ of $\phi$ ). Thus an appropriate notion for an equivalent martingale measure in the case of a dividend paying asset is as an equivalent probability with respect to which the discounted price processes of self-financing trading strategies taking positions in this asset are local martingales.
Because of the equality $V_{t}^{B}(\phi)=V_{t}^{B}(\psi)$, where $\phi$ and $\psi$ are as above, this will be true of an equivalent martingale measure for the dividend-free reduction $S^{1}$ of $S$. Since the price process $S_{t}^{1}$ satisfies
$$
d S_{t}^{1}=S_{t}^{1}\left((\mu(t)+q(t)) d t+\sigma(t) d W_{t}\right)
$$
this equivalent martingale measure $P_{B}$ replaces the yield $\mu(t)+q(t)$ of the dividendfree reduction $S_{t}^{1}$ of $S_{t}$ with the risk-free yield $r(t)$. Equivalently, $P_{B}$ replaces the yield $\mu(t)$ of $S_{t}$ with the difference $r(t)-q(t)$.
With this notion of equivalent martingale measure $P_{B}$ the forward price $F$ of $S$ for delivery at time $T$ can again be written as
$$
F_{t}=F_{t}^{1}=E_{P_{B}}\left[S_{T}^{1} \mid \mathcal{F}{t}\right]=E{P_{B}}\left[S_{T} \mid \mathcal{F}{t}\right], $$ exactly as in the dividend-free case (recall $S{T}^{1}=S_{T}$ ).
Put call parity. Since European puts and calls on $S$ have the same price processes as the corresponding puts and calls on $S^{1}$, the put call parity assumes the form $S_{t}^{1}+P_{t}-C_{t}=K e^{-R(t, T)}$, that is,
$$
C_{t}-P_{t}=S_{t} e^{-Q(t, T)}-K e^{-R(t, T)}
$$
微积分作业代写calclulus代考|Derivatives of securities paying dividends
2.c 支付股息的证券衍生品。到目前为止，我们假设股票小号不支付股息。这在定义自筹资金交易策略时很重要，而这反过来又是套利论点的基础。在考虑获得无风险利润的机会时，必须考虑到资产持有人支付股息的股息流。
现在让我们假设资产小号以收益率连续支付股息q(吨)假定为时间的确定性函数。换句话说，我们假设股息过程D吨满足
dD吨=q(吨)小号吨d吨.
这里D吨被解释为截至时间收到的股息总额吨. 这种资产的一个例子是外币。在这种情况下q(吨)是国外无风险利率。
一种交易策略φ吨=(到吨,H吨)保持到吨无风险债券的单位和H吨股息支付股票的单位小号有时吨现在将被称为 selF-融资，如果相应的投资组合价格过程五吨(φ)=到吨乙吨+H吨小号吨满足
d五吨(φ)=到吨d乙吨+H吨d小号吨+q(吨)H吨小号吨d吨
放问(吨,吨)=∫吨吨q(s)ds,一种吨=和−问(吨,吨)和小号吨1=一种吨小号吨=小号吨和−问(吨,吨). 然后一种吨是一个非随机的有界变化过程，满足d一种吨=q(吨)一种吨d吨和一种吨=1. 使用随机乘积规则和 1.a.eq.(1) 我们有
d小号吨1=一种吨d小号吨+小号吨d一种吨=一种吨d小号吨+q(吨)一种吨小号吨d吨 =一种吨小号吨[μ(吨)d吨+σ(吨)d在吨]+q(吨)一种吨小号吨d吨 =小号吨1[(μ(吨)+q(吨))d吨+σ(吨)d在吨].
因此小号吨1遵循具有时间相关非随机参数的几何布朗运动。因此我们可以查看小号吨1作为一只新股票的价格过程。让ψ吨= (到吨,H吨/一种吨)成为在债券和新股票中持仓的交易策略小号吨1. 对应的价格流程五吨(ψ)满足
五吨(ψ)=到吨乙吨+(H吨/一种吨)小号吨1=到吨乙吨+H吨小号吨=五吨(φ)
那是，φ和ψ有相同的价格过程。从(0)它遵循d小号吨1= 一种吨(d小号吨+q(吨)小号吨d吨). 如果φ是自筹资金，那么
d五吨(ψ)=d五吨(φ)=到吨d乙吨+H吨(d小号吨+q(吨)小号吨d吨) =到吨d乙吨+(H吨/一种吨)d小号吨1.
这是交易策略的自筹资金条件ψ如果新股小号吨1假定不支付股息。让我们称资产小号1无股息减少小号. 然后我们看到了：
微积分作业代写calclulus代考|Equivalent martingale measure
在无股息资产的情况下，等效鞅测度磷乙将资产的贴现价格过程变为鞅，其关键结果是自筹资金交易策略的贴现价格过程φ是一个磷乙−局部鞅（以及在适当的、对系数不是非常严格的条件下的鞅到吨,H吨的φ）。因此，在支付股息资产的情况下，等效鞅测度的一个适当概念是，在该资产中持仓的自筹资金交易策略的贴现价格过程是局部鞅的等效概率。
因为平等五吨乙(φ)=五吨乙(ψ)， 在哪里φ和ψ如上所述，对于无股息减少的等效鞅测度，这将是正确的小号1的小号. 由于价格过程小号吨1满足
d小号吨1=小号吨1((μ(吨)+q(吨))d吨+σ(吨)d在吨)
这个等价的鞅测度磷乙代替产量μ(吨)+q(吨)无股息减少小号吨1的小号吨无风险收益r(吨). 等效地，磷乙代替产量μ(吨)的小号吨有区别r(吨)−q(吨).
有了这个等价鞅测度的概念磷乙远期价格F的小号准时交货吨又可以写成
$$
F_{t}=F_{t}^{1}=E_{P_{B}}\left[S_{T}^{1} \mid \mathcal{F} {t}\ right]=E {P_{B}}\left[S_{T} \mid \mathcal{F} {t}\right], $$ 与无股息情况完全相同（回忆 $S {T}^{ 1}=S_{T}).磷你吨C一种一世一世p一种r一世吨和.小号一世nC和和你r○p和一种np你吨s一种ndC一种一世一世s○n小号H一种v和吨H和s一种米和pr一世C和pr○C和ss和s一种s吨H和C○rr和sp○nd一世nGp你吨s一种ndC一种一世一世s○nS^{1},吨H和p你吨C一种一世一世p一种r一世吨和一种ss你米和s吨H和F○r米S_{t}^{1}+P_{t}-C_{t}=K e^{-R(t, T)},吨H一种吨一世s,C吨−磷吨=小号吨和−问(吨,吨)−到和−R(吨,吨)$
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