# 金融数学 |MATH10003 Financial Mathematics代写

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The next block focuses on continuous time finance and contains an introduction to the basic ideas of Stochastic calculus. The last chapter is an overview of Actuarial Finance. This course is a great introduction to finance theory and its purpose is to give students a broad perspective on the topic.”

Once again, differentiating in $(b, c)$, we find that the random variable $\left(X_{T}, M_{T}^{X}\right)$ has the bivariate density
$$f^{X}(T, b, c)=\frac{2(2 c-b)}{T \sqrt{T}} \phi\left(\frac{2 c-b}{\sqrt{T}}\right) \cdot e^{\mu b-\frac{1}{2} \mu^{2} T} .$$
Note that the processes $\left(\mu t+\sigma B_{t}\right)$ and $\left(\mu t-\sigma B_{t}\right)$ have the same law. Hence we consider the process
$$Y_{t}=\mu t+\sigma B_{t} \text { for } \sigma>0 .$$
Write $F^{Y}(T, b, c)=P\left(Y_{T}<b, M_{T}^{Y}<c\right)$. Consider
$$\widehat{X}{t}=\sigma^{-1} Y{t}=\frac{\mu}{\sigma} t+B_{t}$$

## MATH10003  COURSE NOTES ：

Consider again the situation with two assets, the riskless bond
$$S_{t}^{0}=e^{r t}$$
and a risky asset $S^{1}$ with dynamics
$$d S_{t}^{1}=S_{t}^{1}\left(\mu d t+\sigma d B_{t}\right) .$$
$\left(B_{t}\right)$ is a standard Brownian motion on a probability space $(\Omega, \mathcal{F}, P)$. Consider the risk-neutral probability $P^{\theta}$ and the $P^{\theta}$-Brownian motion $W^{\theta}$ given by
$$d W_{t}^{\theta}=\theta d t+\sigma d B_{t}$$
Here $\theta=\frac{r-\mu}{\sigma}$. Under $P^{\theta}$,
$$d S_{t}^{1}=S_{t}^{1}\left(r d t+\sigma d W_{t}^{\theta}\right),$$

# 优化算法|STAT0025 Optimisation Algorithms代写

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This module aims to provide an introduction to the ideas underlying the optimal choice of component variables, possibly subject to constraints, that maximise (or minimise) an objective function. The algorithms described are both mathematically interesting and applicable to a wide variety of complex real-life situations.

there is a flow of diffeomorphisms $x \rightarrow \xi_{s, t}(x)$ associated with this system, together with their non-singular Jacobians $D_{s, t}$.

In the terminology of Harrison and Pliska , the return process $Y_{t}=\left(Y_{t}^{1}, \ldots, Y_{t}^{d}\right)$ is here given by
$$d Y_{t}=(\mu-\rho) d t+\Lambda d W_{t}$$

can be removed by applying the Girsanov change of measure. Write
$$\eta(t, S)=\Lambda(t, S)^{-1}(\mu(t, S)-\rho),$$
and define the martingale $M$ by
$$M_{t}=1-\int_{0}^{t} M_{s} \eta\left(s, S_{s}\right)^{\prime} d W_{s}$$

## STAT0025 COURSE NOTES ：

Consider a standard Brownian motion $\left(B_{t}\right){t \geq 0}$ defined on $(\Omega, \mathcal{F}, P)$. The filtration $\left(\mathcal{F}{t}\right)$ is that generated by $B$. Recall that $B_{t}$ is normally distributed, and
$$P\left(B_{t}<x\right)=\Phi\left(\frac{x}{\sqrt{t}}\right)$$
Therefore
$$P\left(B_{t} \geq x\right)=1-\Phi\left(\frac{x}{\sqrt{t}}\right)=\Phi\left(-\frac{x}{\sqrt{t}}\right)$$
For a real-valued process $X$, we shall write
$$M_{t}^{X}=\max {0 \leq s \leq t} X{s}, \quad m_{t}^{X}=\min {0 \leq s \leq t} X{s}$$

# 数值方法|MATH0033 Numerical Methods代写

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Many phenomena in engineering and the physical and biological sciences can be described using mathematical models. Frequently the resulting models cannot be solved analytically, in which case a common approach is to use a numerical method to find an approximate solution. The aim of this course is to introduce the basic ideas underpinning computational mathematics, study a series of numerical methods to solve different problems, and carry out a rigorous mathematical analysis of their accuracy and stability.

Step 1 . Evaluate the function
$$\boldsymbol{F}\left(\boldsymbol{p}{k}\right)=\left[\begin{array}{l} f{1}\left(p_{k}, q_{k}\right) \ f_{2}\left(p_{k}, q_{k}\right) \end{array}\right]$$
Step 2. Evaluate the Jacobian
$$\boldsymbol{J}\left(\boldsymbol{P}{k}\right)=\left[\begin{array}{ll} \frac{\partial}{\partial x} f{1}\left(p_{k}, q_{k}\right) & \frac{\partial}{\partial y} f_{\mathrm{I}}\left(p_{k}, q_{k}\right) \ \frac{\partial}{\partial x} f_{2}\left(p_{k}, q_{k}\right) & \frac{\partial}{\partial y} f_{2}\left(p_{k}, q_{k}\right) \end{array}\right]$$

Step 3. Solve the linear system
$$J\left(P_{k}\right) \Delta P=-F\left(P_{k}\right) \text { for } \Delta P$$
Step 4. Compute the next point:
$$P_{k+1}=P_{k}+\Delta P .$$

## MATH0033 COURSE NOTES ：

$$f(x)=P_{N}(x)+E_{N}(x),$$
where $P_{N}(x)$ is a polynomial that can be used to approximate $f(x)$ :
$$f(x) \approx P_{N}(x)=\sum_{k=0}^{N} \frac{f^{(k)}\left(x_{0}\right)}{k !}\left(x-x_{0}\right)^{k} .$$
The error term $E_{N}(x)$ has the form
$$E_{N}(x)=\frac{f^{(N+1)}(c)}{(N+1) !}\left(x-x_{0}\right)^{N+1}$$
for some value $c=c(x)$ that lies between $x$ and $x_{0}$.

# 图论|MATH0029 Graph Theory代写

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The course aims to introduce students to discrete mathematics, a fundamental part of mathematics with many applications in computer science and related areas. The course provides an introduction to graph theory and combinatorics, the two cornerstones of discrete mathematics. The course will be offered to third or fourth year students taking Mathematics degrees, and might also be suitable for students from other departments. There will be an emphasis on extremal results and a variety of methods.

$$(1-\epsilon) p q n=(1-\epsilon) c n \leq \sum_{i=1}^{r}\left|X_{i}\right|=r \ell \leq q \ell$$
whence $(1-\epsilon) n \leq \epsilon \ell$ and hence $j<n \leq 2 \epsilon \ell$. Because $\left|Y_{s}\right| \geq 4^{-d_{j, k} \ell} \geq 4^{-\Delta} \ell$ and $\left(\Delta^{2}+2\right) \epsilon<4^{-\Delta}$,

$$\left|Y_{s}\right|-\Delta^{2} \epsilon \ell-j>4^{-\Delta} \ell-\Delta^{2} \epsilon \ell-2 \epsilon \ell=\left(4^{-\Delta}-\left(\Delta^{2}+2\right) \epsilon\right) \ell>0$$
Also, because $\epsilon<\frac{1}{4}$ and $\left|Y_{t}\right| \geq 4^{-d_{j, k} \ell}$,
$$\left(\frac{1}{2}-\epsilon\right)\left|Y_{t}\right| \geq \frac{1}{4}\left(4^{-d_{j, k}} \ell\right)=4^{-d_{j, k}-1} \ell=4^{-d_{j+1, k} \ell}$$

## MATH0029 COURSE NOTES ：

For a pair ${X, Y}$ of disjoint sets of vertices of a graph $G$, we define its index of regularity by:
$$\rho(X, Y):=|X | Y|(d(X, Y))^{2}$$
This index is nonnegative. We extend it to a family $\mathcal{P}$ of disjoint subsets of $V$ by setting:
$$\rho(\mathcal{P}):=\sum_{X, Y \in \mathcal{P}} \rho(X, Y)$$
In the case where $\mathcal{P}$ is a partition of $V$, we have:
$$\rho(\mathcal{P})=\sum_{\substack{X \in Y \in \mathcal{P} \ X \neq Y}}|X | Y|(d(X, Y))^{2} \leq \sum_{\substack{X, Y \in \mathcal{P} \ X \neq Y}}|X||Y|<\frac{n^{2}}{2}$$

# 金融数学|MATH0031 Financial Mathematics代写

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This is a first course at the advanced undergraduate level in mathematical finance; centring on the mathematics of financial derivatives which relies on both probability theory and PDE based approaches. It assumes no prior knowledge of finance. The module begins with an introduction to the type of language and terminology used in the investment banking arena, followed by the essential elements of probability theory and stochastic calculus required for the pricing of options later in the course.

Under the equivalent martingale measure $\mathcal{Q}$, the one-factor Vasicek (1FV) model is given by
$$d r_{t}=\chi_{r}\left(\bar{r}-r_{t}\right) d t+\sigma_{r} d W_{r, t}^{\mathcal{Q}}$$

Under this specification bond prices are defined by $D(t, T)=\exp (A(t, T)-$ $\left.B(t, T) r_{t}\right)$, where $B(t, T) \equiv \frac{1-e^{-x_{f} t}}{x_{t}}$,
$$A(t, T) \equiv \frac{(B(t, T)-\tau)\left(x_{r}^{2} \bar{r}-\frac{\sigma_{r}^{2}}{2}\right)}{x_{r}^{2}}-\frac{\sigma_{r}^{2} B^{2}(t, T)}{4 \chi_{r}}$$
and, for notational convenience, $\tau \equiv T-t$.
Let $\tilde{X}{t} \equiv\left[y{l, L}(t) r_{t}\right]^{\prime}$ and $\tau_{l} \equiv T_{l}$, the term-to-maturity of the swaptions contract to be priced. The associated transform of the state vector $\tilde{X}{t}$ is given by $$\psi^{\mathcal{Q}{l+\perp L}}\left(\tilde{u} \equiv(\tilde{u} 0)^{\prime}, \tilde{X}{t}, 0, T{l}\right)=\exp \left[\alpha\left(\tau_{l}\right)+\tilde{u} y_{l, L}(0)\right]$$

## MATH0031 COURSE NOTES ：

The one-factor generalized Vasicek (1FGV) model defines the short rate $r_{t}=$ $\delta+x_{1, t}$, where $\delta \in \mathbb{R}$ is constant, and
$$d x_{1, t}=-x_{1} x_{1, t} d t+\sigma_{1} d W_{1, t^{*}}^{\mathcal{Q}}$$
Bond prices are given by $D(t, T)=\exp \left(A(t, T)+B_{x_{1}}(t, T) x_{1, t}\right)$, where, in general, $B_{x}(t, T) \equiv \frac{1-e^{-x t}}{x}$ and
$$A(t, T) \equiv-\delta \tau+\frac{1}{2} \frac{\sigma_{1}^{2}}{x_{1}^{2}}\left[\tau-2 B_{x_{1}}(t, T)+B_{2 x_{1}}(t, T)\right]$$

# 金融数学 Maths for Finance FINANCE 2110

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Not surprisingly, we will examine the characteristic function of $S$ as a means to determining its probability distribution. For matrix-valued random variables we consider
$$f(\Theta)=E e^{i \operatorname{Tr}\left(X X^{T} \Theta\right)}=E e^{i x_{k}^{j} x_{l}^{j} \Theta_{\mathbf{k}}}=E e^{i x_{k}^{j} \Theta_{\mathbf{k}} x_{l}^{j}}=\left(E e^{i x_{k}^{1} \Theta_{\mathbf{k}} x_{l}^{1}}\right)^{n}$$

$$g=E_{t} e^{i \Theta_{i j} z_{i}(T) z_{j}(T)}$$
with $z$ a correlated, driftless Brownian motion (with covariance structure represented by $\Sigma$ ). We have that
$$g_{t}+\frac{1}{2} \Sigma_{i j} g_{z_{i} z_{j}}=0$$

Consider a sample of size $T$, denoted by
$$X=\left(\begin{array}{ccc} z_{1} & \cdots & z_{T} \ v_{1} & \cdots & v_{T} \end{array}\right)$$
Now, divide this sample into $T-p$ overlapping blocks of size $p+1$ and define the vector $y$ via $y_{j}=\left(z_{j}^{T}, \ldots, z_{j+\rho}^{T}\right)^{T}$. The empirical characteristic function $(\mathrm{ECF})$ is then given by
$$g(\phi ; y)=\frac{1}{T-p} \sum_{j=1}^{T-p} e^{i \phi^{T} y}$$
Denoting the (unconditional) characteristic function in by $f(\phi)$ and the underlying vector of model parameters by $\theta$. An ECF estimator can then be crafted à la GMM as
$$\hat{\theta}=\arg \min {\theta} \int{-\infty}^{\infty}|f(\phi)-g(\phi ; y)|^{2} w(\phi) d \phi$$