Quantitative Finance 1st Edition Maria Cristina Mariani, Ionut Florescu - Solutions

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A large retail store buys an item from a supplier in batches of 100. Because of the delicate nature of the item some are inevitably damaged in transit between the supplier and the store and the price the store pays to the supplier reflects this.Over the last few years an average of four items are
The finance department of a large organisation has responsibility for monitoring costs in other departments of the organisation. A photocopier facility is available for one department’s use and data has been collected which reveals that the number of photocopies made on the machine on a daily
Return to the example of Machine 2 in the shampoo bottle problem. Assume that management likewise wishes to ensure that the advertised contents satisfy some specific criteria. Determine what the advertised contents should be if management:(a) require no more than 1 per cent of bottles to be under
Determine for Machine 1 the proportion of bottles:(a) that will contain more than 515 ml(b) that will contain less than 475 ml(c) that will contain between 520 and 525 ml(d) that will contain between 490 and 480 ml.
If we define an event as the number six showing, what is the probability of this event happening? How do you know? Would you class this as theoretical, empirical or subjective?
A retail organisation has recently been investigating customer spending patterns at two of its stores, Store A and Store B, and has obtained the following data:Using these statistics, and any others you can derive, draft a short management report summarising the implications of this data. The Store
The data shown in Table 4.15 shows average weekly earnings for all adults, weekly rate of unemployment benefit for adults and the retail price index for the UK for the period 1994–2003.Using this data comment on any trends in earnings and benefits. Do you think the RPI is the most appropriate
The leisure services committee of a local authority has seen its budget decline over the last few years and is currently having to consider closing a number of its leisure facilities – sports halls, small parks, museums and so on. One closure being considered is that of a small branch library in
Your organisation has recently started advertising its services on the internet. The marketing manager has indicated that she wants to know how long it takes internet users to access your company’s internet screen, since there is concern it is taking too long and deterring interest. You have
Table 4.14 shows the size of the fishing fleet in England and Wales by size of vessel in 1992 and 2002 (data at 31 December each year). By calculating and using appropriate statistics comment on the changes that have taken place.
Table 4.13 shows the age distribution of males and females in the UK at the time of their marriage, for 1991, 1996 and 2001.134 4 MANAGEMENT STATISTICS Age distribution at marriage: UK Males Females 1991 1996 2001 1991 1996 2001 Under 21 years 13 271 5 497 4 625 38 305 18 485 13 874 21–24 79
How ‘reliable’ do you think the price index is as a measure of how inflation is affecting the consumer over time?
Consider the consumer price index which is published by the government (or its statistical agency). Find out:(a) what weights are used in the calculation of the price index;(b) how these weights are obtained;(c) whether the calculations follow the Laspeyres or Paasche method.
In the Laspeyres index in this chapter we used 2000 as the base year. Choose 2005 as the base year instead and recalculate the Laspeyres series for 2000, 2005 and 2006. Comment on these values compared with the index base 2000.
For the data in Table 4.10, plot the real and money values over time on the same graph. Comment on the potential for misleading the user of money values.
For the two data sets shown below calculate the mean and standard deviation and comment on the relationship.Set 1 Set 2 10 100 20 200 30 300 40 400 50 500
Is there any evidence that the finance director’s suspicions are correct and that costs have escalated?
Why do you think a reduction in LOS would be seen, in management terms, as an improvement in performance, all other things being equal? Based on these statistics would you conclude that, in this area of health care, the hospital has ‘improved’ its performance?
Do you think we have any evidence regarding the relationship between sales and profit that we assumed earlier? Is this relationship exactly the same for all stores? Are there any stores that appear to differ from the norm?
In what way do you think these two variables might be connected? What would you expect to happen to one store’s profit as its sales increaseddecreased?
Draw a comparable histogram for profits for Region B stores and compare it with Figure 3.8. What comments can you make about this distribution compared with that of Region A?
A firm has analysed its sales and profitability and found that its profit can be represented as:where F is profit (£000s) and Q is units sold (in 000s).It has also found that its costs are:where C is costs (£000s).(a) Obtain a third equation showing the firm’s revenue, R.
The data in Table 2.1 show the amount of consumer credit in the UK for each year from 1992 to 2002, together with the Retail Price Index.(a) From the Office for National Statistics website, update the series for the latest years.(b) Using the RPI, deflate each of the three series.(c) Graph the
An airline company is considering providing a new daily service between Edinburgh and Copenhagen. The aircraft has a maximum capacity of 200 passengers and each flight incurs a fixed cost of £25 000 (regardless of the number of passengers). In addition, a cost is also incurred of £75 per
Next week you have to travel around parts of the country on business and have decided to hire a car from a car-hire company. You have contacted two companies which offer different services. The first company will rent you a car for £45 per day.The second company will charge you £30 per day but
A firm finds that the demand for its product can be represented mathematically as:where Qd is the number of units of the product demanded by customers and P is the selling price.(a) Construct a graph for a price between 0 and 200.(b) Comment on the shape of the graph. Why does the line slope
A company is thinking of investing £15 000 in a savings fund. The fund runs for a ten-year period and is expected to generate a return each year of 8 per cent of the amount invested. All annual returns remain in the fund until the end of the period.(a) Construct a table showing the value of the
1 In the chapter we formulated a break-even problem with:Find the new break-even level if:(a) overheads increase by 15 per cent(b) costs increase by 1.50 per item(c) the selling price increases to 11.99.
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 variance $v(t)=\sigma_{t}^{2}$ also follows a stochastic process given by\[d v_{t}=\kappa(\theta-v(t)) d
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 Ørnstein-Uhlenbeck process.
Let $B_{t}$ be a standard Brownian motion started at 0 . Use that for any function $f$ we have:\[\mathbf{E}\left[f\left(B_{t}\right)\right]=\frac{1}{\sqrt{2 \pi t}} \int_{-\infty}^{\infty} f(x) e^{-\frac{x^{2}}{2 t}} d x\]to calculate:\[\mathbf{E}\left[B_{t}^{2 k}\right]\]for some $k$, an
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 being positive.(a) Show that the pdf of $X_{t}$ is\[f_{X_{t}}(x)=2 f_{B_{t}}(x), \forall x \geq
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 transition probability\[\mathbf{P}\left(Y_{t}=y \mid Y_{t_{0}}=y_{0}\right)\]and give the density of this
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) $d X_{t}=X_{t}^{\alpha}+\sigma X_{t} d B_{t}$. For what values of $\alpha$ the solution of the equation explodes (is equal to $\infty$ or
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} P}{\partial S^{2}}+r S \frac{\partial P}{\partial S}-r P \leq 0 \\& \frac{\partial P}{\partial
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 vector $(j=$ $1, \cdots, M)$, we say that a portfolio \(\pi=\left(\pi_{1}, \cdots,
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 $M_{i}(K), i=1, \ldots, p, N_{j}(T)$, and $j=1, \ldots, q$ are normalized $B$-splines instead
(a) Find the solution to the following PDE:\[\begin{aligned}& \frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}} \\& u(0, t)=u(L, t)=0 \\& u(x, 0)=0 \\& \frac{\partial u}{\partial t}(x, 0)=f(x)\end{aligned}\]where $L$ is a constant and $f$ is a given function.(b) Find
Find the solution to the following PDE:\[\begin{aligned}& \frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}} \\& u(0, t)=u(\pi, t)=0 \\& u(x, 0)=0 \\& \frac{\partial u}{\partial t}(x, 0)=g(x)\end{aligned}\]
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\[u(x, y, z)=f(x) g(y) h(z)\]
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)\[\frac{d y}{d t}+2 y=2 e^{7 t}, \quad y(0)=0\]
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, $u_{1}=f(0), u_{2}=$ $f\left(\frac{1}{32}\right), \ldots, u_{3}=f\left(\frac{31}{32}\right)$. Compute the Haar wavelet transform $y=W_{32} u$
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\[\begin{aligned}& \alpha_{1}=\frac{1+\sqrt{3}}{4 \sqrt{2}} \\& \alpha_{2}=\frac{3+\sqrt{3}}{4 \sqrt{2}} \\& \alpha_{3}=\frac{3-\sqrt{3}}{4 \sqrt{2}} \\& \alpha_{4}=\frac{1-\sqrt{3}}{4\sqrt{2}}\end{aligned}\]verify the following identity:(a)
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} \\& s 5=(1,1,1,1,0,0,0,0)^{T} \\& s 6=(0,0,0,0,1,1,1,1)^{T} \\& s 7=(1,1,1,1,1,1,1,1)^{T} \\& s
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 $D=\sin \left(2^{6}\right) / 2^{3}-\cos (1)$, sampled at 1024 equidistant points in $[-1,2]$. Use the wavelet transform to
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 by\[\frac{2}{\pi}-\frac{4}{\pi} \sum_{n=1}^{\infty} \frac{\cos 2 n \theta}{4 n^{2}-1}\]From this,
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 contraction with respect to:(a) Norm 0: $\|f(x)\|_{0}=\max f(x)$(b) Norm 1:
(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 that\[M_{t}=f\left(X_{t}\right)-f\left(X_{0}\right)-\int_{0}^{t} g\left(X_{0}\right) d s\]is a martingale.
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 European Call option with strike 110(b) A European Put option with strike 110(c) A American Put option
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) $u=\frac{1}{d}$. Hint: $E\left(S_{t+\Delta t}^{2}\right)=S_{t}^{2} e^{\left(2 r+\sigma^{2}\right) \Delta t}$
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 $7 \%$ per year and the volatility is $20 \%$. Assume that $u=\frac{1}{d}$.
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 that\[f_{j}=\max \left\{S u^{j} d^{n-j}, 0\right\}\]In particular, if $u=\frac{1}{d}$, we have
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 value of a European Call option that expires in one month, with strike price $\$ 39$ ?(b)
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 compounded,(a) what is the value of a European Call option expiring in one year with strike price $\$ 100$
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, compute:(a) The price a European Put option expiring in six months with a strike price of $\$ 42$(b)
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 share after two months, what is the value of an option with the same expiration date (i.e. two
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 $I_{0.95}=(\underline{S}, \bar{S})$ so that $\left.p\left(S \in I_{0.95}\right)=0.95\right)$. Hint: use that if
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 the share for the next day(b) The standard deviation of the closing price of the share for the next
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:\[\begin{aligned}& \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t} \\& u(x, 0)=x e^{x} \\& u(0, t)=u(1, t)=0\end{aligned}\]
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 $h=0.1$ and $k=0.005$ for the first three time levels $(j=0,1,2)$.
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 function $\operatorname{sign}(x)$ is defined on $(-\infty, 0) \cup(0, \infty)$
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 determine how it performs, assuming geometric Brownian motion for the stock price, interest rate of \(6
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 $c(t, \omega)$ is a stochastic process such that the stochastic integral is well defined?
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 following parameter values:\[\mu_{S}=0.1, \quad \sigma=0.7, \quad \beta=\frac{1}{3}, \quad S_{0}=100\]
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 $\sqrt{Y_{t}}$. Show that the volatility process has moments\[\begin{aligned}&
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 15 and 20%, respectively, with correlation of 0.5. How much of the stocks volatility can be
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 $K_{2}-K_{1}>C\left(K_{2}\right)-C\left(K_{1}\right)$
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 time $T$.\[C(S, T) \geq \max \{0, S-K B(T)\}\]
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 $n K$ and $m K$, show that the price of a happy Call is given by the formula \[C_{H}=a_{1}
If we have a model for the short rate $r(t)$, show that(a) the zero coupon bond price can be calculated as:\[P(t, s)=\mathbf{E}_{t}\left[e^{\int_{t}^{s} r(u) d u}\right]=\mathbf{E}\left[e^{\int_{t}^{s} r(u) d u} \mid \mathcal{F}_{t}\right]\](b) the price of an option that pays $C(T)$ at time
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 years and two months, and the calculation agent estimates that the price of the reference bond is 50%
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 $\mathbf{P}\left(Y_{i}=0\right)=\mathbf{P}\left(Y_{i}=1\right)=0.5$, for all $i \in \mathbb{Z}$. We would like to study the length of the
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 $\theta_{1}=2, \theta_{2}=0,7, \theta_{3}=0.8$, and $\Delta t=0.001$.(a) Generate a single path of the
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 $U_{t}=\sqrt{V_{t}}$. Under which conditions on the parameters $\kappa, \bar{V}$, and $\sigma$ the
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 future values $S_{1}^{1}>S_{1}^{2}>S_{1}^{3}$.i) Give an "if and only if" condition for no
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 circumstances will the option be exercised?(c) Draw a diagram showing the variation of the trader's
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 circumstances will the option be exercised?(c) Draw a diagram showing how the profit from a long
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\[d S_{t}=\alpha\left(\beta-S_{t}\right) d t+\sigma \sqrt{S_{t}} d Z_{t}\]where $\alpha, \beta$ and $\sigma$ are

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Quantitative Finance 1st Edition Maria Cristina Mariani, Ionut Florescu - Solutions

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