金融代写|利率建模代写Interest Rate Modeling代考|MATH3075

Vasicek利率模型一词是指一种对利率的运动和演变进行建模的数学方法。它是一种基于市场风险的单因素短利率模型。瓦西克利率模型常用于经济学中，以确定利率在未来的移动方向。

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

金融代写|利率建模代写Interest Rate Modeling代考|The Black-Scholes Formula

Consider the pricing of a European call option on an asset, $S_{t}$, which has the payoff
$$C_{T}=\left(S_{T}-K\right)^{+}$$
at time $T$. Assume that the short rate is a constant, $r_{t}=r$. According to the Black-Scholes-Merton Equation $2.57$ and the terminal condition $2.58$, the value of the call option satisfies
$$\left{\begin{array}{l} \frac{\partial C_{t}}{\partial t}+\frac{1}{2} \sigma_{t}^{2} S^{2} \frac{\partial^{2} C_{t}}{\partial S^{2}}+r S \frac{\partial C_{t}}{\partial S}-r C_{t}=0 \ C_{T}=(S-K)^{+} \end{array} .\right.$$
By solving this terminal-value problem of the partial differential equation (PDE), we can obtain the price of the option.

Alternatively, we can derive the formula for the call options by working out the expectation in Equation $2.71$ directly. We write
\begin{aligned} C_{t} &=\mathrm{e}^{-r(T-t)} E_{t}^{\mathrm{Q}}\left[\left(S_{T}-K\right)^{+}\right] \ &=\mathrm{e}^{-r(T-t)}\left(E_{t}^{\mathrm{Q}}\left[S_{T} 1_{S_{T}>K}\right]-K E_{t}^{\mathrm{Q}}\left[1_{S_{T}>K}\right]\right) \end{aligned}
Since
$$S_{T}=S_{t} \exp \left[\left(r-\frac{1}{2} \bar{\sigma}^{2}\right) \tau+\bar{\sigma} \sqrt{\tau} \cdot \varepsilon\right], \quad \varepsilon \sim N(0,1),$$
where $\tau=T-t$ and $\bar{\sigma}$ is the mean volatility,
$$\bar{\sigma}=\sqrt{\frac{1}{\tau} \int_{0}^{\tau} \sigma_{s}^{2} \mathrm{~d} s,}$$
we have
$$E_{t}^{\mathrm{Q}}\left[1_{S_{T}>K}\right]=\operatorname{Prob}\left(\varepsilon>-\frac{\ln \left(S_{t} / K\right)+\left(r-(1 / 2) \bar{\sigma}^{2}\right) \tau}{\bar{\sigma} \sqrt{\tau}}\right)=\Phi\left(d_{2}\right),$$
with
$$d_{2}=\frac{\ln \left(S_{t} / K\right)+\left(r-(1 / 2) \bar{\sigma}^{2}\right) \tau}{\bar{\sigma} \sqrt{\tau}}$$
Meanwhile,
\begin{aligned} E_{t}^{\mathrm{Q}}\left[S_{T} 1_{S_{T}>K}\right] &=\frac{1}{\sqrt{2 \pi}} \int_{-d_{2}}^{\infty} S_{t} \exp \left[\left(r-\frac{1}{2} \bar{\sigma}^{2}\right) \tau+\bar{\sigma} \sqrt{\tau} x-\frac{1}{2} x^{2}\right] \mathrm{d} x \ &=\frac{S_{t} \mathrm{e}^{r \tau}}{\sqrt{2 \pi}} \int_{-d_{2}-\bar{\sigma} \sqrt{\tau}}^{\infty} \exp \left(-\frac{1}{2} y^{2}\right) \mathrm{d} y \ &=S_{t} \mathrm{e}^{r \tau} \Phi\left(d_{1}\right), \end{aligned}
where
$$d_{1}=d_{2}+\bar{\sigma} \sqrt{\tau} .$$
By substituting Equations $2.77$ and $2.79$ into $2.74$, we arrive at the celebrated Black-Scholes formula:
$$C_{t}=S_{t} \Phi\left(d_{1}\right)-\mathrm{e}^{-r(T-t)} K \Phi\left(d_{2}\right) .$$
By direct verification, we can show that the hedge ratio, $\varphi_{t}$, is
$$\frac{\partial C_{t}}{\partial S_{t}}=\Phi\left(d_{1}\right) .$$
Next, we proceed to derive the formula for a put option, which has the payoff function
$$P_{T}=\left(K-S_{T}\right)^{+} .$$

金融代写|利率建模代写Interest Rate Modeling代考|Short Rate and Money Market Accounts

The short rate is associated with a savings account in a bank. The short rate at time $t$ is conventionally denoted as $r_{t}$. Interest on a savings account is accrued daily, using the actual $/ 365$ convention. Let $B_{t}$ denote the account balance at time (or date) $t$, and let $\Delta t=1$ day $=1 / 365$ year. Then the new balance the next day at $t+\Delta t$ is
$$B_{t+\Delta t}=B_{t}\left(1+r_{t} \Delta t\right)$$

Because $\Delta t \leqslant 1$, daily compounding is very well approximated by continuous compounding: in the limit of $\Delta t \rightarrow 0$, Equation $3.1$ becomes
$$\mathrm{d} B_{t}=r_{t} B_{t} \mathrm{~d} t$$
Because $r_{t}$ is applied to $(t, t+d t)$, an infinitesimal interval of time, it is also called the instantaneous interest rate. As a mathematical approximation and idealization, continuous compounding is necessary to continuous-time finance. Suppose that a sum of money is deposited at $t=0$ into a savings account and that there has not been a deposit or withdrawal since. Then the balance at a later time, $t$, is
$$B_{t}=B_{0} \mathrm{e}^{\int_{0}^{t} r_{s} \mathrm{~d} s}$$
In the real world, the balance, $B_{t}$, is not known in advance due to the stochastic nature of the short rate. Nonetheless, the deposit in the savings account is considered a risk-free security, and its return is used as a benchmark to measure the profits and losses of other investments.

In reality, savings accounts for institutions and for individuals offer different interest rates, which reflect different overhead management costs for institutional and individual clients. To distinguish from an individual’s account, we call the savings account for an institution a money market account. Note that this is somewhat an abuse of terminology. In the United States, a money market account is also a type of savings account for retail customers, which offers higher interest rates under some restrictions, including minimum balances and limited numbers of monthly withdrawals. Its compounding rule is also different from continuous compounding. Hence, we need to emphasize here that, in fixed-income modeling, a money market account means a savings account for institutions that compounds continuously. Such a money market account plays an important role in continuous-time modeling of finance.

金融代写|利率建模代写Interest Rate Modeling代考|Term Rates and Certificates of Deposit

Term rates are associated to certificates of deposit (CD). A CD is a deposit that is committed for a fixed period of time, and the interest rate applied to the CD is called a term rate. For retail customers, the available terms are typically one month, three months, six months, and one year. Usually, the longer the term, the higher the term rate, as investors are awarded a higher premium for committing their money for a longer period of time. The interest payments of CDs use simple compounding. Let $r_{t, \Delta t}$ be the interest rate for the term $\Delta t$ and $I_{t}$ be the value of the deposit at time $t$. Then the balance at the maturity of the CD is
$$I_{t+\Delta t}=I_{t}\left(1+r_{t, \Delta t} \Delta t\right) .$$
Investors of CDs often roll over their CDs, meaning that after a CD matures, the entire amount (principal plus interest) is deposited into another CD with

the same terms but with the prevailing term rate at the time when the rolling over takes place. Suppose that a CD is rolled over $n$ times. Then the terminal balance at time $t+n \Delta t$ is
$$I_{t+n \Delta t}=I_{t} \cdot \prod_{i=1}^{n}\left(1+r_{t+(i-1)} \Delta t, \Delta t \Delta t\right) .$$
If the $\Delta t$ term rate remains unchanged over the investment horizon, that is, $r_{t+(i-1) \Delta t, \Delta t}=r_{t, \Delta t}, i=1, \ldots, n$, then there is
$$I_{t+n \Delta T}=I_{t}\left(1+r_{t, \Delta t} \Delta t\right)^{n},$$
and we say that the deposit is compounded $n$ times with interest rate $r_{t, \Delta t}$. We call $\omega=1 / \Delta t$ the compounding frequency, which is the number of compoundings per year. For example, when $\Delta t=3$ months or $0.25$ year, we have $\omega=1 / \Delta t=4$, corresponding to the so-called quarterly compounding. By the way, a savings account is compounded daily, corresponding to $\omega=365$.

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