Questions and Answers of Introduces Quantitative Finance

Consider a stochastic process such that the underlying security $S$ follows the model:\[d S_{t}=\mu S_{t} d t+\sigma_{t} S_{t} d Z_{t}\]where $Z$ is a standard Brownian motion. Suppose the
Calculate the solution to the following SDE:\[d X_{t}=\alpha\left(m-X_{t}\right) d t+\sigma d B_{t}\]with $X_{0}=x$. The process satisfying this equation is called the meanreverting
Let $B_{t}$ be a standard Brownian motion started at 0 . Use that for any function $f$ we have:\[\mathbf{E}\left[f\left(B_{t}\right)\right]=\frac{1}{\sqrt{2 \pi t}} \int_{-\infty}^{\infty} f(x)
Let $X_{t}, t \geq 0$, be defined as\[X_{t}=\left\{B_{t} \mid B_{t} \geq 0\right\}, \quad \forall t>0\]that is, the process has the paths of the Brownian motion conditioned by the current value
If $X_{t} \sim N(0, t)$, calculate the distribution of $\left|X_{t}\right|$. Calculate $\mathbf{E}\left|X_{t}\right|$ and $V\left(\left|X_{t}\right|\right)$.
If $X_{t} \sim N\left(0, \sigma^{2} t\right)$ and $Y_{t}=e^{X_{t}}$, calculate the pdf of $Y_{t}$. Calculate $\mathbf{E}\left[Y_{t}\right]$ and $V\left(Y_{t}\right)$. Calculate the
Prove by induction that\[\int_{0}^{T} B_{t}^{k} d B_{t}=\frac{B_{T}^{k+1}}{k+1}-\frac{k}{2} \int_{0}^{T} B_{t}^{k-1} d t\]
Solve the following SDEs using the general integrating factor method with $X_{0}=0$ :(a) $d X_{t}=\frac{X_{t}}{t} d t+\sigma t X_{t} d B_{t}$,(b) $d X_{t}=X_{t}^{\alpha}+\sigma X_{t} d B_{t}$.
Suppose $V=V(S)$. Find the most general solution of the Black-Scholes equation.
Suppose $V=\Lambda_{1}(t) \Lambda_{2}(S)$. Find the most general solution of the Black-Scholes equation.
Prove that for a European Call option on an asset that pays no dividends the following relations hold:\[C \leq S, \quad C \geq S-E \exp (-r(T-t))\]
Given the formulation of the free boundary problem for the valuation of an American Put option,\[\begin{aligned}& \frac{\partial P}{\partial t}+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2}
In a market with $N$ securities and $M$ futures, where $\left(S_{1}, \cdots, S_{N}\right)$ is the present values vector and $\left(S_{1}^{j}(T), \cdots, S_{N}^{j}(T)\right)$ the future values
Explain the reason why it is convenient to represent the bicubic spline in the form\[\operatorname{spline}(K, T)=\sum_{i=1}^{p} \sum_{j=1}^{q} c_{i j} M_{i}(K) N_{j}(T)\]where \(M_{i}(K), i=1,
(a) Find the solution to the following PDE:\[\begin{aligned}& \frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}} \\& u(0, t)=u(L, t)=0 \\& u(x, 0)=0 \\& \frac{\partial
Find the solution to the following PDE:\[\begin{aligned}& \frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}} \\& u(0, t)=u(\pi, t)=0 \\& u(x, 0)=0 \\& \frac{\partial
Find the solution to the Laplace equation in $R^{3}$ :\[\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0\]assuming that\[u(x, y,
Solve the following initial value problem using Laplace transform:(a)\[\frac{d y}{d t}=-y+e^{-3 t}, \quad y(0)=2\](b)\[\frac{d y}{d t}+11 y=3, \quad y(0)=-2\](c)\[\frac{d y}{d t}+2 y=2 e^{7 t}, \quad
Solve the following boundary value problem:\[y^{\prime \prime}(x)=f(x), \quad y(0)=0, \quad y(1)=0\]Hence solve $y^{\prime \prime}(x)=x^{2}$ subject to the same boundary conditions.
Consider the vector $u$ that consists of 32 equally spaced samples of the function $f(t) \approx \cos (4 \pi t)$ on the interval [0,1]. That is, $u_{1}=f(0), u_{2}=$
Define the Haar's wavelets function $\psi(t)$ and verify that for every $t$,\[si(t)= \begin{cases}1, & 0 \leq t
For the scaling numbers\[\begin{aligned}& \alpha_{1}=\frac{1+\sqrt{3}}{4 \sqrt{2}} \\& \alpha_{2}=\frac{3+\sqrt{3}}{4 \sqrt{2}} \\& \alpha_{3}=\frac{3-\sqrt{3}}{4 \sqrt{2}} \\&
Show that the following eight vectors are pairwise orthogonal:\[\begin{aligned}& s 1=(1,1,0,0,0,0,0,0)^{T} \\& s 2=(0,0,1,1,0,0,0,0)^{T} \\& s 3=(0,0,0,0,1,1,0,0)^{T} \\& s 4=(0,0,,0,0,0,1,1)^{T} \\&
Haar transform(a) For an $N \times N$ Haar transformation matrix, the Haar basis functions are\[\psi_{k}(t)=\psi_{p q}(t)=\frac{1}{\sqrt{N}} \begin{cases}2^{p / 2}, & (q-1) / 2^{p} \leq t
Consider the periodic signal\[f(t)=\left\{\begin{array}{lr}\cos (t)+D, & -1 \leq t \leq 0 \\\sin \left(t^{6}\right) / t^{3}, & 0 \leq t\leq 2\end{array}\right.\]where \(D=\sin \left(2^{6}\right) /
Consider the signal $f(t)=\sin (7 t)$. Use Matlab to obtain the discrete wavelets transform of this signal.
Let $f(\theta)$ be the $2 \pi$-periodic function determined by the formula\[f(\theta)=\theta^{2}, \quad \text { for }-\pi \leq \theta \leq \pi\]Find the Fourier series for $f(\theta)$.
Let $f(\theta)$ be the $2 \pi$-periodic function determined by the formula\[f(\theta)=|\sin \theta|, \quad \text { for }-\pi \leq \theta \leq \pi\]Show that the Fourier series for $f$ is given
For a Markov process define its infinitesimal operator:\[T(t) f\left(X_{s}\right)=\int f(y) p\left(t, X_{s}, y\right) d y=E\left[f\left(X_{t+s}\right) \mid X_{s}\right]\]Show that this operator is a
(The Martingale problem on $L)$ Let $f, g)$ be a pair of functions in $L$. Find a process $\left\{X_{t}\right\}_{t}$ defined on $E$ such
Construct a binomial tree $\left(u=\frac{1}{d}\right)$ with three quarters for an asset with present value $\$ 100$. If $r=0.1$ and $\sigma=0.4$, using the tree compute the prices of:(a) A
In the binomial model obtain the values of $u, d$, and $p$ given the volatility $\sigma$ and the risk-free interest rate $r$ for the following cases:(a) $p=\frac{1}{2}$.(b)
Compute the value of an option with strike $\$ 100$ expiring in four months on underlying asset with present value by $\$ 97$, using the binomial model. The risk-free interest rate is $7 \%$
In a binomial tree with $n$ steps, let $f_{j}=f_{u \ldots u d \ldots d}(j$ times $u$ and $n-j$ times $d$ ). For a European Call option expiring at $T=n \Delta t$ with strike $K$, show
We know that the present value of a share is $\$ 40$ and that after one month it will be $\$ 42$ or $\$ 38$. The risk-free interest rate is $8 \%$ per year continuously compounded.(a) What is
The price of a share is $\$ 100$. During the following six months the price can go up or down in a $10 \%$ per month. If the risk-free interest rate is $8 \%$ per year, continuously
The price of a share is $\$ 40$, and it is incremented in $6 \%$ or it goes down in $5 \%$ every three months. If the risk-free interest rate is $8 \%$ per year, continuously compounded,
The price of a share is $\$ 25$, and after two months it will be $\$ 23$ or $\$ 27$. The risk-free interest rate is $10 \%$ per year, continuously compounded. If $S_{T}$ is the price of the
The price of a share is $\$ 40$. If $\mu=0.1$ and $\sigma^{2}=0.16$ per year, find a $95 \%$ confidence interval for the price of the share after six months (i.e. an interval
Suppose that the price of a share verifies that $\mu=16 \%$ and the volatility is $30 \%$. If the closing price of the share at a given day is $\$ 50$, compute:(a) The closing expected value of
American Down and Out Call. Given $K=100, T=\frac{1}{30}, S=100, \sigma=0.6, r=$ 0.01 , and $B=95$. Construct a tree for all nodes below the barrier value $=0$.
Use the implicit finite difference method to solve the heat conduction problem on the unit square:\[\begin{aligned}& \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t} \\& u(x, 0)=x
Find the finite difference solution of the problem\[\begin{aligned}& \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t} \quad 0by the explicit method. Plot the solution using
Prove that the order of convergence of the Crank-Nicolson finite difference method is\[O\left(\Delta x^{2}+\left(\frac{\Delta t}{2}\right)^{2}\right)\]
Given the Irwin-Hall distribution with pdf,\[f_{Y}(y)=\frac{1}{2(n-1) !} \sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}n \\k\end{array}\right)(y-k)^{n-1} \operatorname{sign}(y-k)\]where the sign
A call option pays an amount $V(S)=\frac{1}{1+\exp (S(T) K)}$ at time $T$ for some predetermined price $K$. Discuss what you would use for a control variate and generate a simulation to
Consider the following normal mixture density:\[f(x)=0.7 \frac{1}{\sqrt{2 \pi 9}} e^{-\frac{(x-2)^{2}}{18}}+0.3 \frac{1}{\sqrt{2 \pi 4}} e^{-\frac{(x+1)^{2}}{8}}\]
Calculate the solution to the following mean-reverting ØrnsteinUhlenbeck SDE:\[d X_{t}=\mu X_{t} d t+\sigma d B_{t}\]with $X_{0}=x$.
Show that the stochastic process\[e^{\int_{0}^{t} c(s) d B_{s}-\frac{1}{2} \int_{0}^{t} c^{2}(s) d s}\]is a martingale for any deterministic function $c(t)$. Does the result change if \(c(t,
Give an explicit solution for the mean-reverting Ørnstein-Uhlenbeck SDE\[d X_{t}=\alpha\left(\mu-X_{t}\right) d t+\sigma d B_{t}\]with $X_{0}=x$.
Give the Euler-Milstein approximation scheme for the following SDE:\[d S_{t}=\mu_{S} S_{t} d t+\sigma S_{t}^{\beta} d B_{t}\]where $\beta \in(0,1]$. Generate five paths and plot them for the
Consider the [102] model\[\begin{aligned}d S_{t} & =\mu_{S} S_{t} d t+\sqrt{Y_{t}} S_{t} d B_{t} \\d Y_{t} & =\mu_{Y} Y_{t} d t+\xi Y_{t} d W_{t}\end{aligned}\]The volatility process is
Prove directly that, for each $t \geq 0$,\[\mathbb{E}\left(e^{-z T(t)}\right)=\int_{0}^{\infty} e^{-z s} f_{T(t)}(s) d s=e^{-t z^{1 / 2}}\]where $(T(t), t \geq 0)$ is the Lévy subordinator.
What is a unit root test and what are its consequences?
Suppose the total volatility of returns on a stock is 25%. A linear model with two risk factors indicates that the stock has betas of 0.8 and 1.2 on the two risk factors. The factors have volatility
Consider two identical call options, with strikes $K_{1}$ and $K_{2}$; show that:(a) If $K_{2}>K_{1}$ then $C\left(K_{2}\right)>C\left(K_{1}\right)$(b) If $K_{2}>K_{1}$ then
Consider a European Call option with strike $K$ and time to expiration $T$. Denote the price of the call for $C(S, T)$ and let $B(T)$ the price of one unit of a zero coupon bond maturing at
A happy Call option is a Call option with payoff $\max (\alpha S, S-K)$. So we always get something with a happy Call! If $C_{1}$ and $C_{2}$ are the prices of two call options with strikes \(n
If we have a model for the short rate $r(t)$, show that(a) the zero coupon bond price can be calculated as:\[P(t, s)=\mathbf{E}_{t}\left[e^{\int_{t}^{s} r(u) d
Briefly explain the main difference between exchange-traded funds (ETFs) and mutual funds?
Explain the difference between Inverse ETFs and Leverage ETFs.
Briefly describe what a credit default swap is and how it is generally used.
A credit default swap requires a premium of 70 basis points per year paid semiannually. The principal is $$350$ million and the credit default swap is settled in cash. A default occurs after five
Consider an infinite Bernoulli process with $p=0.5$, that is, an infinite sequence of random variables $\left\{Y_{i}, i \in \mathbb{Z}\right\}$ with
Let $Z$ be a Brownian motion defined in [0,T]. Given a partition $\mathscr{P}$ such that \(0=t_{0}
Suppose that the price of an asset follows a Brownian motion :\[d S=\mu S d t+\sigma S d z\](a) What is the stochastic process for $S^{n}$ ?(b) What is the expected value for $S^{n}$ ?
The Hull White model is\[d X_{t}=a(t)\left(b(t)-X_{t}\right) d t+\sigma(t) d W_{t}\]In this problem take $a(t)=\theta_{1} t, b(t)=\theta_{2} \sqrt{t}$ and $\sigma(t)=\theta_{3} t$ where
The variance process in the Heston model satisfy a CIR process:\[d V_{t}=\kappa\left(\bar{V}-V_{t}\right)+\sigma \sqrt{V_{t}} d W_{t}\]Use Ito to calculate the dynamics of the volatility process
Consider a market with two securities, one of them risk free with rate of return $R$, and three future states. Suppose that the risky security has a present value $S_{1}$ and can take three
Suppose a trader buys a European Put on a share for $\$ 4$. The stock price is $\$ 53$ and the strike price is $\$ 49$.(a) Under what circumstances will the trader make a profit?(b) Under what
Suppose that an August Call option to buy a share for $\$ 600$ costs $\$60.5$ and is held until August.(a) Under what circumstances will the holder of the option make a profit?(b) Under what
Use Itô's lemma to express $d F$ given that $F(x)=x^{1 / 2}$, where the stochastic process $\left\{S_{t}, t \geq 0\right\}$ satisfies the stochastic differential equation\[d
Find the explicit solutions of the following stochastic differential equations:(a) Ornstein-Uhlenbeck process:\[d X_{t}=\mu X_{t} d t+\sigma d B_{t}\](b) Mean reverting Ornstein-Uhlenbeck process:\[d
For an estimator based on n data points, with a decay factor of π, prove that the half-life, h, is given by: h= In(0.5 +0.58") In(8)
Calculate the sample standard deviation for the data set in Problem 7 using no decay factor, a decay factor of 0.99, and a decay factor of 0.90.
Using the same data as in the previous question, calculate the one-day 95% VaR using the hybrid method with a window of 256 days and a decay factor of 0.99.
This question is the same as the previous question, but rather than owning a put and call option on XYZ, you have sold a put and call on XZY. As before, the strike of both options is $105 and time to
Prove that standard deviation displays positive homogeneity.
The PDF for daily profits at Euler Fund is given by the following equation:where π is the profits and c is a constant. Calculate the one-day 95% expected shortfall for Euler Fund. f(x)=ce -10% <
In the proceeding example, if instead of 17% and 14%, the two worst losses were 27% and 24%, what would the one-day 99% VaR and expected shortfall be? Losses 3 through 12 in the table remain the same.
In the previous question, what is the critical value for the F-test? Is the F-statistic significant at the 95% confidence level?
Using all the information from the previous two questions, determine the correlation between the index returns and the returns of XYZ.
In addition to the IBMshares fromQuestion 2, assume you are short 3million shares of MSFT.MSFT’s current price is $50 and the average daily volume of MSFT is 6 million shares. Construct a liquidity
In the previous question, what would you have been willing to pay for insurance if the factory was expected to earn $$1,000$ if there is no flood. In the event of a flood, you would still earn
Compute the sample mean and the standard deviation of the following returns: 7% 2% 6% -4% -4% 3% 0% 18% -1%
Calculate the population mean, standard deviation, and skewness of each of the following two series: Series #1 Series #2 -51 -61 -21 -7 21 33 51 35
Calculate the population mean, standard deviation, and kurtosis for each of the following two series: Series #1 Series #2 -23 -17 -7 -17 7 17 23 17
Given the probability density function for a random variable X,find the variance of X. f(x) = X 18 for 0 x 6
Using a decay factor of 0.95, calculate the mean, sample variance, and sample standard deviation of the following series. Assume t = 7 is the most recent data point and use all eight points: x 0 11 1
Given the following set of data, calculate the mean using no decay factor (rectangular window), a decay factor of 0.99, and a decay factor of 0.90. Assume time t = 10 is the most recent data point
You are estimating the expected value of the annual return of a stock-market index using an EWMA estimator with a decay factor of 0.98. The current estimate of the mean is 10%. Over the next three
What is the half-life for an estimator with a decay factor of 0.95 and 200 data points?What is the half-life for the same decay factor with 1,000 data points?
What is the half-life of an EWMA estimator with a decay factor of 0.96 and 32 data points? What is the length of a rectangular window with the most similar half-life?
Assume that the mean of a data-generating process is known and equal to zero. Your initial estimate of the standard deviation is 10%, after which you observe the following returns (t = 6 is the most
Assume we have an EWMA estimator with a decay factor of 0.96 and 50 data points.What percentage of the weight is captured with this estimator, compared to an estimator with the same decay factor and
Prove the formula for the first Cornish-Fisher moment. That is, givenprove thatRemember that R is normally distributed with a mean of zero. dV R + TR + Odt TR 2
You are asked to calculate the one-day 95% VaR for a portfolio using the historical method, with a window of 256 days. The table below contains the 20 worst backcast returns for the portfolio along
You are the risk manager for a currency trading desk.The desk had a VaR exceedance today. What is the most likely day for the next VaR exceedance?
You are the risk manager for a portfolio with a mean daily return of 0.40% and a daily standard deviation of 2.3%. Assume the returns are normally distributed (not a good assumption to make, in
The probability density function (PDF) for daily profits at Box Asset Management can be described by the following function (see Figure 3.3),Below −100 and above 100, the PDF is zero.What is the

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