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

my-assignmentexpert™ 随机微积分stochastic calculus作业代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。my-assignmentexpert™， 最高质量的随机微积分stochastic calculus作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于随机微积分stochastic calculus作业种类很多，难度波动比较大，同时其中的大部分作业在字数上都没有具体要求，因此随机微积分stochastic calculus作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

my-assignmentexpert™ 为您的留学生涯保驾护航 在经济学作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的微积分calculus代写服务。我们的专家在随机微积分stochastic calculus 代写方面经验极为丰富，各种随机微积分stochastic calculus相关的作业也就用不着 说。

• 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

## 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 $G(T, s)=f(s)$.

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

d小号j一种(吨)=小号j一种(吨)σj(吨)⋅d在吨吨,1≤j≤n

∂F∂吨+12∑一世js一世sjσ一世(吨)⋅σj(吨)∂2F∂s一世∂sj=0

F(吨,s)=F(s),

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

\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)
$$其中函数 F=F\left(t, \tilde{s} {1}, \ldots, \tilde{s} {n}\right) \in C^{1 ,2}\left([0, T] \times R_{+}^{n}\right)s一种吨一世sF一世和s \frac{\partial F}{\partial t}+\frac{1}{2} \sum_{ij} \tilde{s} {i} \tilde{s} {j} \sigma_{i}(t ) \cdot \sigma_{j}(t) \frac{\partial^{2} F}{\partial \tilde{s} {i} \partial \tilde{s} {j}}=0$$

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