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There is a huge strand of literature on graph theory in finance which analyzes networks summarized here: Allen, F., and A. Babus (2009): “Networks in Finance,” in Network-based Strategies and Competencies, ed. by P. Kleindorfer, and J. Wind, pp. 367–382. First applications of graph theory in networks focused on credit-risk in interconnected banks and how ...


4

You may show it as follows: \begin{align*} f_{t,T}&= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{T-t}}-1\\ &=e^{\frac{1}{T-t} \left[\ln (1+r_T)^T - \ln (1+r_t)^t \right]} -1\\ &\approx e^{\frac{1}{T-t} \left[(1+r_T)^T-1 - \big((1+r_t)^t -1\big)\right]} -1\\ &=e^{\frac{1}{T-t} \left[(1+r_T)^T - (1+r_t)^t\right]} -1\\ &\approx 1+ \...


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 ...


3

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 ...


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 ...


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 $...


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A famous example of using graph theory in finance is the detection of triangle arbitrage by finding a negative cycle in a graph. More on this problem and the solution on Math.SE.


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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 ...


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The allocation of regulatory capital to a set of trades that makes up the portfolios and sub portfolios of a bank. One theory is Shapley value which is combinatorial index, but complexity runs far deeper. Encryption one might consider inherent to a lot of finance. I suspect one might consider graph theory in terms of money flows / capital flows from ...


1

The formula is multiplicative because interests are compounded (re-invested). Let's say you have 10% interest on a 1 USD deposit, compounded (calculated and paid) twice a year. That would be 5 cents of interest for the first semester. Then you have a deposit of 1.05 USD during the next semester. After one year, your interests are 1 * 0.05 + 1.05 * 0.05 = 0....


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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