随机微积分作业代写stochastic calculus代考| PRICING OF RANDOM PAYOFFS AT FIXED FUTURE DATES

随机微积分作业代写stochastic calculus代考|  PRICING OF RANDOM PAYOFFS AT FIXED FUTURE DATES

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
  • 随机控制
  • Ito积分
  • black-Scholes-Merton option pricing formula
  • Fokker–Planck equation
  • 布朗运动 Brownian motion
随机微积分作业代写stochastic calculus代考

微积分作业代写calclulus代考|Higher dimensional PDEs

The above approach can also be extended to options $h=f\left(S_{1}(T), \ldots, S_{n}(T)\right)$ depending on a vector $S(t)=\left(S_{1}(t), \ldots, S_{n}(t)\right)^{\prime}$ of assets. Starting from
$$
d S_{j}^{A}(t)=S_{j}^{A}(t) \sigma_{j}(t) \cdot d W_{t}^{T}, \quad 1 \leq j \leq n
$$
where $W_{t}^{T}$ is an $n$-dimensional $P_{T}$-Brownian motion and the $\sigma_{j}:[0, T] \rightarrow R^{n}$ are continuous functions, note $d\left\langle S_{i}^{A}, S_{j}^{A}\right\rangle_{t}=S_{i}^{A}(t) S_{j}^{A}(t) \sigma_{i}(t) \cdot \sigma_{j}(t) d t$ and try a representation
$$
\pi_{t}^{A}(h)=F\left(t, S^{A}(t)\right)=F\left(t, S_{1}^{A}(t), \ldots, S_{n}^{A}(t)\right)
$$
for some function $F=F(t, s)=F\left(t, s_{1}, \ldots, s_{n}\right) \in C^{1,2}\left([0, T] \times R_{+}^{n}\right)$. This yields the PDE
$$
\frac{\partial F}{\partial t}+\frac{1}{2} \sum_{i j} s_{i} s_{j} \sigma_{i}(t) \cdot \sigma_{j}(t) \frac{\partial^{2} F}{\partial s_{i} \partial s_{j}}=0
$$
with boundary condition
$$
F(T, s)=f(s),
$$
where $s=\left(s_{1}, \ldots, s_{n}\right)$. An argument similar to the above shows that conversely (12), (13) combined with the boundedness of all the partial derivatives $\partial F / \partial s_{j}$ suffices to establish the representation (11). Note that we have already solved the

PDE $(12),(13)$ in 4.d.eq. $(7)$ above. We have $d\left\langle\log \left(S_{i}^{A}\right), \log \left(S_{j}^{A}\right)\right\rangle_{t}=\sigma_{i}(t) \cdot \sigma_{j}(t) d t$ which follows from $(10)$. Thus the process $\sigma_{i}(t) \cdot \sigma_{j}(t)$ is the instantaneous covariation of $\log \left(S_{i}^{A}\right)$ and $\log \left(S_{j}^{A}\right)$. A replicating strategy $\theta$ for $h$ invests in the vector $\left(A(t), S_{1}(t), \ldots, S_{n}(t)\right)$ with weights $\theta(t)=\left(K(t), H_{1}(t), \ldots, H_{n}(t)\right)$ given by
$$
H_{j}(t)=\frac{\partial F}{\partial s_{j}}\left(t, S^{A}(t)\right) \quad \text { and } \quad K(t)=F\left(t, S^{A}(t)\right)-\sum_{j} H_{j}(t) S_{j}^{A}(t)
$$

微积分作业代写calclulus代考|Continuous dividends

Assume now that the asset $S_{j}$ pays a continuous dividend $D_{j}(t)$ satisfying $d D_{j}(t)=q_{j}(t) S_{j}(t) d t$ and let $h=f\left(S_{1}(T), \ldots, S_{n}(T)\right)$ be as above. From the discussion in 2.c we know that the option price remains unaffected if the assets $S_{j}(t)$ are replaced with their dividend-free reductions $\bar{S}{j}(t)=S{j}(t) C_{j}(t)$ with $C_{j}(t)=\exp \left(-\int_{t}^{T} q_{j}(s) d s\right)$. In other words
$$
\pi_{t}(h)=A(t) F\left(t, S_{1}^{A}(t), \ldots, \bar{S}{n}^{A}(t)\right)=A(t) F\left(t, S{1}^{A}(t) C_{1}(t), \ldots, S_{n}^{A}(t) C_{n}(t)\right)
$$
where the function $F=F\left(t, \tilde{s}{1}, \ldots, \tilde{s}{n}\right) \in C^{1,2}\left([0, T] \times R_{+}^{n}\right)$ satisfies
$$
\frac{\partial F}{\partial t}+\frac{1}{2} \sum_{i j} \tilde{s}{i} \tilde{s}{j} \sigma_{i}(t) \cdot \sigma_{j}(t) \frac{\partial^{2} F}{\partial \tilde{s}{i} \partial \tilde{s}{j}}=0
$$
with boundary condition $F(T, \tilde{s})=f(\tilde{s})$. This appears to be the most efficient approach to dealing with continuous dividends as it shows that the same valuation PDE can be used as in the dividend-free case if the formula for the option price is adjusted accordingly. However if it is desired to write the option price in terms of the dividend paying assets $S_{j}(t)$ as $\pi_{t}(h)=A(t) G\left(t, S_{1}^{A}(t), \ldots, S_{n}^{A}(t)\right)$, the corresponding PDE for the function $G=G\left(t, s_{1}, \ldots, s_{n}\right)$ can also be derived.

The equality $F\left(t, \bar{S}{1}^{A}(t), \ldots, \bar{S}{n}^{A}(t)\right)=\pi_{t}^{A}(h)=G\left(t, S_{1}^{A}(t), \ldots, S_{n}^{A}(t)\right)$ suggests that we should have $F\left(t, \tilde{s}{1}, \ldots, \tilde{s}{n}\right)=G\left(t, s_{1}, \ldots, s_{n}\right)$, where the variables $\tilde{s}$ and $s$ are related by
$$
s_{j}=C_{j}(t)^{-1} \tilde{s}{j}=\exp \left(\int{t}^{T} q_{j}(s) d s\right) \tilde{s}{j} $$ From this it follows that $\partial s{j} / \partial t=-s_{j} q_{j}(t)$ and $\partial s_{j} / \partial \tilde{s}{j}=C{j}(t)^{-1}$ and so
$$
\begin{aligned}
&\frac{\partial F}{\partial t}=\frac{\partial G}{\partial t}+\sum_{j} \frac{\partial G}{\partial s_{j}} \frac{\partial s_{j}}{\partial t}=\frac{\partial G}{\partial t}-\sum_{j} s_{j} q_{j}(t) \frac{\partial G}{\partial s_{j}} \
&\frac{\partial F}{\partial s_{j}}=\frac{\partial G}{\partial s_{j}} \frac{\partial s_{j}}{\partial \tilde{s}{j}}=\frac{\partial G}{\partial s{j}} C_{j}(t)^{-1}
\end{aligned}
$$
It follows that $\partial^{2} F / \partial \tilde{s}{i} \partial \tilde{s}{j}=\left(\partial^{2} G / \partial s_{i} \partial s_{j}\right) C_{i}(t)^{-1} C_{j}(t)^{-1}$ and consequently $\tilde{s}{i} \bar{s}{j} \partial^{2} F / \partial \tilde{s}{i} \partial \tilde{s}{j}=s_{i} s_{j} \partial^{2} G / \partial s_{i} \partial s_{j}$. Entering this into the PDE for $F$ yields
$$
\frac{\partial G}{\partial t}-\sum_{j} s_{j} q_{j}(t) \frac{\partial G}{\partial s_{j}}+\frac{1}{2} \sum_{i j} s_{i} s_{j} \sigma_{i}(t) \cdot \sigma_{j}(t) \frac{\partial^{2} G}{\partial s_{i} \partial s_{j}}=0
$$
with boundary condition $G(T, s)=f(s)$.

随机微积分作业代写stochastic calculus代考| PRICING OF RANDOM PAYOFFS AT FIXED FUTURE DATES

微积分作业代写calclulus代考|Higher dimensional PDEs

上述方法也可以扩展到optionsH=F(小号1(吨),…,小号n(吨))取决于向量小号(吨)=(小号1(吨),…,小号n(吨))′的资产。从…开始
d小号j一种(吨)=小号j一种(吨)σj(吨)⋅d在吨吨,1≤j≤n
在哪里在吨吨是一个n维磷吨——布朗运动和σj:[0,吨]→Rn是连续函数,注意d⟨小号一世一种,小号j一种⟩吨=小号一世一种(吨)小号j一种(吨)σ一世(吨)⋅σj(吨)d吨并尝试代表
圆周率吨一种(H)=F(吨,小号一种(吨))=F(吨,小号1一种(吨),…,小号n一种(吨))
对于某些功能F=F(吨,s)=F(吨,s1,…,sn)∈C1,2([0,吨]×R+n). 这产生了 PDE
∂F∂吨+12∑一世js一世sjσ一世(吨)⋅σj(吨)∂2F∂s一世∂sj=0
有边界条件
F(吨,s)=F(s),
在哪里s=(s1,…,sn). 与上述类似的论证表明,反过来,(12),(13)结合所有偏导数的有界性∂F/∂sj足以建立表示 (11)。请注意,我们已经解决了

偏微分方程(12),(13)在 4.d.eq.(7)更多。我们有d⟨日志⁡(小号一世一种),日志⁡(小号j一种)⟩吨=σ一世(吨)⋅σj(吨)d吨紧随其后的是(10). 因此过程σ一世(吨)⋅σj(吨)是的瞬时协变日志⁡(小号一世一种)和日志⁡(小号j一种). 复制策略θ为了H投资于向量(一种(吨),小号1(吨),…,小号n(吨))带重物θ(吨)=(到(吨),H1(吨),…,Hn(吨))由
Hj(吨)=∂F∂sj(吨,小号一种(吨)) 和 到(吨)=F(吨,小号一种(吨))−∑jHj(吨)小号j一种(吨)

微积分作业代写calclulus代考|Continuous dividends

现在假设资产小号j持续派发股息Dj(吨)令人满意的dDj(吨)=qj(吨)小号j(吨)d吨然后让H=F(小号1(吨),…,小号n(吨))如上。从 2.c 中的讨论我们知道,如果资产小号j(吨)被他们的无股息减少所取代 $\bar{S} {j}(t)=S {j}(t) C_{j}(t)在一世吨HC_{j}(t)=\exp \left(-\int_{t}^{T} q_{j}(s) ds\right).一世n○吨H和r在○rds$
\pi_{t}(h)=A(t) F\left(t, S_{1}^{A}(t), \ldots, \bar{S} {n}^{A}(t) \right)=A(t) F\left(t, S {1}^{A}(t) C_{1}(t), \ldots, S_{n}^{A}(t) C_{n }(t)\right)
$$
其中函数 $F=F\left(t, \tilde{s} {1}, \ldots, \tilde{s} {n}\right) \in C^{1 ,2}\left([0, T] \times R_{+}^{n}\right)s一种吨一世sF一世和s$
\frac{\partial F}{\partial t}+\frac{1}{2} \sum_{ij} \tilde{s} {i} \tilde{s} {j} \sigma_{i}(t ) \cdot \sigma_{j}(t) \frac{\partial^{2} F}{\partial \tilde{s} {i} \partial \tilde{s} {j}}=0
$$
有边界健康)状况F(吨,s~)=F(s~). 这似乎是处理连续股息的最有效方法,因为它表明,如果期权价格的公式相应调整,则可以使用与无股息情况相同的估值 PDE。但是,如果希望以派息资产的形式写出期权价格小号j(吨)作为圆周率吨(H)=一种(吨)G(吨,小号1一种(吨),…,小号n一种(吨)), 函数对应的 PDEG=G(吨,s1,…,sn)也可以导出。

等式 $F\left(t, \bar{S} {1}^{A}(t), \ldots, \bar{S} {n}^{A}(t)\right)=\pi_{ t}^{A}(h)=G\left(t, S_{1}^{A}(t), \ldots, S_{n}^{A}(t)\right)s你GG和s吨s吨H一种吨在和sH○你一世dH一种v和F\left(t, \tilde{s} {1}, \ldots, \tilde{s} {n}\right)=G\left(t, s_{1}, \ldots, s_{n}\right ),在H和r和吨H和v一种r一世一种b一世和s\波浪号{s}一种nds一种r和r和一世一种吨和db和$
s_{j}=C_{j}(t)^{-1} \tilde{s} {j}=\exp \left(\int {t}^{T} q_{j}(s) ds\对) \tilde{s} {j} $$ 由此得出 $\partial s {j} / \partial t=-s_{j} q_{j}(t)一种nd\partial s_{j} / \partial \tilde{s} {j}=C {j}(t)^{-1}一种nds○$
\begin{对齐}
&\frac{\partial F}{\partial t}=\frac{\partial G}{\partial t}+\sum_{j} \frac{\partial G}{\partial s_{ j}} \frac{\partial s_{j}}{\partial t}=\frac{\partial G}{\partial t}-\sum_{j} s_{j} q_{j}(t) \frac {\partial G}{\partial s_{j}} \
&\frac{\partial F}{\partial s_{j}}=\frac{\partial G}{\partial s_{j}} \frac{\部分 s_{j}}{\partial \tilde{s} {j}}=\frac{\partial G}{\partial s {j}} C_{j}(t)^{-1}
\end{对齐}
$$ 由此
得出 $\partial^{2} F / \partial \tilde{s} {i} \partial \tilde{s} {j}=\left(\partial^{2} G / \partial s_ {i} \partial s_{j}\right) C_{i}(t)^{-1} C_{j}(t)^{-1}一种ndC○ns和q你和n吨一世和\tilde{s} {i} \bar{s} {j} \partial^{2} F / \partial \tilde{s} {i} \partial \tilde{s} {j}=s_{i} s_ {j} \partial^{2} G / \partial s_{i} \partial s_{j}.和n吨和r一世nG吨H一世s一世n吨○吨H和磷D和F○rF和一世和一世ds∂G∂吨−∑jsjqj(吨)∂G∂sj+12∑一世js一世sjσ一世(吨)⋅σj(吨)∂2G∂s一世∂sj=0在一世吨Hb○你nd一种r和C○nd一世吨一世○nG(T, s)=f(s)$。

随机微积分作业代写stochastic calculus代考| PRICING OF RANDOM PAYOFFS AT FIXED FUTURE DATES
微积分作业代写calclulus代考

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