8

As Sanjay said, you can apply Itô's Lemma to $f(t,x)=x^2$ and obtain \begin{align*} \mathrm{d} S^2_t=\left(2\mu S_t^2+\sigma^2S_t^2\right)\mathrm{d}t+\left(2\sigma S_t^2\right)\mathrm{d}W_t. \end{align*} Thus, $(S_t^2)$ is again a geometric Brownian motion and hence, for each time point $t$ log-normally distributed with drift $2\mu+\sigma^2$ and volatility $...


5

As you mentioned, we have $$\sum_{l=0}^{k}{p}=\frac{k(k+1)}{2}$$ You want to know $$\sum_{k=0}^{n}{\sum_{l=0}^{k}{p}}=\sum_{k=0}^{n}{\frac{k(k+1)}{2}}=\frac{1}{2}\sum_{k=0}^{n}{k^2}+\frac{1}{2}\sum_{k=0}^{n}{k}$$ you know that $$\sum_{k=0}^{n}{k^2}=\frac{n(n+1)(2n+1)}{6}$$ Therefore $$\sum_{k=0}^{n}{\sum_{l=0}^{k}{p}}=\frac{n(n+1)(2n+1)}{12}+\frac{n(n+...


4

We assume that the stock price process $\{S_t,\,t>0\}$ satisfies, under the real-world probability measure $P$, an SDE of the form \begin{align*} dS_t=S_t\big((\mu-q)dt+\sigma dW_t\big), \end{align*} where $\{W_t, \, t >0\}$ is a standard Brownian motion. Here, we need to consider the total return asset $e^{qt}S_t$, that is, the asset with the dividend ...


3

Classic asset price model in the continuous-time limit using a Wiener process notation can be written as $$ dS_t=\mu S_tdt+\sigma S_t dX $$ where $S_t$ is the stock price (not the stock return) and $dX$ is an independent random variable with normal distribution. If we eliminate the drift ($\mu = 0$) and only focus on randomness as asked in your question we ...


3

(these are just my random opinions. Sorry, I expect many people to disagree with some of them.) Bloomberg makes no effort to make its terminals available to students - quite the opposite. Hence lots of recent graduates don't know the terminal, but do know how to get things done without the terminal, and don't think they need the terminal. It's a problem for ...


2

The first is something of a theoretical question. It's widely held/assumed that stocks follow a BM process, it appears as though the author is setting the table for the subsequent statement. The second is an artifact of applying Ito's lemma...the $dW_tdt$ and $dtdt$ terms both equal 0, hence fall out, leaving only $dW_t^2$ = dt. Thus, the variance ...


2

This is pretty standard fare for a Stats 101 course, so as to rationale, etc. you might benefit from picking up a textbook or otherwise do some reading on this. In brief though, hypothesis testing allows us to assess the likelihood sample estimates are different than theorized values in the absence of actual population values. In the cases above, with a ...


2

At $t_1$, this payoff can be priced using the Margrabe formula as used for pricing an exchange option. See Margrabe Formula here Using the notations in the question and those used the hyperlinked document above - $Price_{t_1} = P_{t_1}e^{(\mu_P-r)\tau}\Phi(d_+) -HR \times G_{t_1}e^{(\mu_G-r)\tau}\Phi(d_-) \tag{1}$ $Price_0$ is the discounted value of $...


2

Kelly DOES reflect the odds! The simple binary bet form of Kelly is: Kelly Fraction = (p(win) * (odds + 1) - 1) / odds So for a 60% chance of a 50% risk, ie 1:1 equals odds 1, that’s 20% of your capital at risk. More formally, Kelly seeks to maximise log-wealth (LW) LW = sum ( Pi * ln(1 + Stake * Payoffi) Maximise LW, then dLW/dStake = 0 For each ...


2

The only difference in the derivation when you have a dividend-yield paying stock lies in the value of the Riskless Portfolio $\Pi_t$. The financial meaning here is the key: to delta-hedge your option you buy a quantity $\Delta$ of the stock $S$, and only the stock is paying you the dividend, so you have to add this contribution in time to your hedge. The ...


1

For the first question, it is the standard assumption to make for stock returns if no other information is given. That's not to say it's a great assumption, but there it is clearly the only one that can be justified in this context. For the second part, independence of returns tells you that investment for T years has cumulative variance $T \sigma^2$ (when ...


1

Based on the way the question is phrased, the simplest approach is: 1) Estimate the gravitational potential energy of the Earth-Moon system. 2) Use result from 1) to estimate an average power as Energy/time. 3) Estimate the solar power on the illuminated half of the Earth. 4) The comparison of the result in 2) and 3) allows you to judge relative magnitude. ...


1

Does optimizing a solution for a given set of parameters sound like something quants would need to know how to do? Yep! Indeed many things quants do revolve around optimization, and linear programming (and integer programming, multi-integer programming, quadratic programming, etc.) is all about finding the optimal solution to something given some set of ...


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