I believe that volatility can´t be infinity given the Central Limit Theorem. For example, the implied volatility of a strike call goes to infinity as maturity goes to zero, and this implies that volatility is always finite prior to the expiration date.

**Question of the week: **When solving the last webwork, I noticed that Ce(0) = Pe(0). Is is due to the symmetry of the normal distribution?

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I think that financial derivatives trading makes the market more efficient because risks can be isolated and spread out independently in a more consistent and diversified way, and moreover, they help reducing uncertainty (if used properly). This relies on the fact that there is a whole market of derivatives, and thus, it is possible to attain to the same goal by using and mixing different products and choosing the best options available to maximize profits (or minimize risk). For example, given that a company has plenty of derivatives with different levels of risk, it will be able to transfer/sell those that do not fit the benchmark portfolio after some time T. This argument is further motivated by the fact that trading itself makes any market more efficient, as it if driven by the forces of supply and demand. These forces can definitely be applied to the situation since in the end derivatives are products, and as we all know, any product can be projected into a supply and demand analysis. Thus, in conclusion, derivatives makes the market better off because they provide an opportunity to reduce uncertainty and manage risks more efficiently.

**Question of the week: **I understand that financial derivatives are all based in a group of similar basic characteristics. The differences between derivatives rely on deeper details. Then, what do you think are the basic characteristics that give the name of a derivative to a financial product? Do you know which were the first types of derivatives that were used in the financial market and what was there main objective by then?

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E(K1K2)= 0.5(1)(2) + 0.5(2)(1)= 2

Cov(K1K2)= 2 – U1U2= 2-1.5^2= -0.25

Correlation= -0.25/(sigma1*sigma2)= -0.25/(sqrt(0.25)*sqrt(0.25))= -1

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**Question of the week: **Considering a 2×2 matrix to solve one constraint, what is the purpose of using the determinant to calculate the inverse matrix of A if in the end the determinant will cancel? In other words, can we just skip the whole process of having to write 1/detA if we know that it will get cancelled?

Also, a more conceptual question: are futures the same as forwards?

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I think that a terrorist attack, similar to 9/11, will cause another financial shock wave. The 9/11 attack had big economic effects that instigated a global drop in the stock markets. Furthermore, there were billions of dollars caused in insurance losses. For these reasons, a similar occurrence will certainly have equivalent consequences at least. Nevertheless, I argue that such an event will cause further losses and more extravagant effects. This is due to the fact that after 9/11, governments became very precautious about possible terrorism, and many resources were put towards security. As a result, people think that they are safer and that a new series of attacks would be very unlikely (an adverse selection event). In addition to this, the dead of the former leader of Al-Qaeda has surely lead people to feel safer, so another attack will definitely be very surprising. In the end, there would be more panic, which will further cause fear within the markets and thus, spreading it throughout the world and many other economic sectors. This could cause a new financial shock wave.

**Question of the week: **Is it possible to find the optimization values using eigenvalues from matrices? If so, how would it be possible?

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In first place, I would like to emphasize on the importance of calculating the correlation between a sample of stocks or other risky assets. Knowing the correlation is important in creating a portfolio because it helps to diversify risk. In other words, you do not want to own two positively correlated items because if the price of either falls, then the price of the other correlated asset will fall too.

Taking the previous into consideration, an example of two correlated stocks is Google and Microsoft. This could be explained by the fact that both corporations are big competitors of the “technology era.”

Actually, after doing some research, I found out that the correlation between these two corporations is 0.79.

Source: http://www.macroaxis.com/invest/menu/pitchletHome/marketCorrelation

**Question of the week: **What is considered to be a good range of correlation between risky assets in a portfolio? For example, would you say that a correlation of 0.5 (-0.5) is high? Is it possible to build a portfolio with correlation equal to 0?

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Nonetheless, I would like to argue that now day’s young investors are probably very cautious about their investments. With the recent disruption in the financial markets, many investors are shrinking investment accounts. In other words, younger investors are more risk averse than older investors.

Furthermore, I also believe that very wealthy investors will bear more risk than modest income investors because their budget constraint and their indifference curves are higher than those of modest income investors. Hence, wealthy investors could afford to invest risky, if they are willing to, because they have the money to afford any losses. In comparison, modest income investors will be more concerned about retaining and increasing profits by investing safe. Hence, they will use better statistical tools in order to spread out the risks.

**Question of the week: **I know there are different methods to calculate risk, one of them being volatility. What do you think is the best method to calculate the risks of a “worst-case scenario”?

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Nevertheless, investors in government securities still face a small amount of credit risk. To be more specific, there have been cases where governments have defaulted on their domestic debt, for example Russia in 1998 and Argentina between the years 1991-2001. Moreover, a perfect representation of this concern is being held right now in Greece. Even after receiving a rescue package due to the financial crisis of 2007, Greece is again facing a risk of default. As a result, every market is afraid of a contagion due to a possible default.

Likewise, governments are seeing credit rating downgrades that are also affecting the credibility of the markets. Even the “big” countries like the USA are facing this kind of situations, where debt constitutes much of the GDP. This is a result of macro effects that economists and financial analysts can´t predict; major economic forces that affect most of the financial institutions, if not all, and that may spread around the world, just like back in 1929 and just recently in 2007.

Following, we can definitely conclude that there aren´t any 100% credit worthy financial institutions, or at least not at this point of history.

**Question of the week:** When should you use the Poisson Distribution vs. the Binomial Distribution?

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Moreover, calculus is much diversified in the sense that one can obtain the same answer using different methods. Applied to life, this becomes very helpful when facing a difficult problem, since most of the times there are various ways to get with it. In other words, analysis, in addition to the fact that calculus also helps developing logic and reasoning skills, is probably the most valuable asset obtained by learning it. Together, analysis, logic, and reasoning, become very beneficial and crucial in the real world, when time management and quick decisions are fundamental.

In the end, I think it is not about just learning the math and playing around with equations and numbers, but realizing that learning it will improve the brain´s capacity to process data, ideas, and information.

**Question of the week:** How can Kelly´s strategy be helpful in trading (considering that it is a theory solely based on betting)?

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Investors have various risk management tools, like diversification, which permit them to minimize losses. Gamblers, however, lack such tools to avoid a total loss capital. For example, a gambler who bets $100 in a horse race might win or lose. If his horse losses, then he losses %100 of his capital. In comparison, a broker is able to spread out the risks in his portfolio, being able to make up for the losses with gains on other stocks. Additionally, there are certain types of investments, like annuities, that pay interest every specified period, so in short terminology, it is very difficult to lose everything.

Furthermore, investors make their decisions based on real time data, driven by market and information forces. Even though prices constantly fluctuate, investors also rely on a wide set of external factors, like interest rates, maturity dates, etc, and research which allows them to invest thinking in the long-term returns. Gamblers, nevertheless, rely only on short-term bases and immediate circumstances. Also, they can´t base their bets on previous research or information of their surroundings because they are either using probability theory (which only some professional gamblers do) or are driven by luck and/or emotions.

**Question of the week:** Why are the random return and log return different?

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