The TI Nspire calculator is a great platform for visualizing data via interactive graphs. The built-in facility like input slider for variable value adjustment allowed dynamic visualization to complex equations, like the volatility sensitivity in delta-hedged gains used financial investment. Since this strategy involved a single call option, the volatility exposure equals the vega value of the option.
The following setup on the Nspire provided the functions to calculate the vega values.
This spreadsheet input screen stores the spot prices and the calculated Black Scholes vega values.
Finally, with the data plotting screen the graph of Delta hedged gains of volatility sensitivity is completed. An additional slider control can easily be added on it to adjust an offset variable so as to visualize scenarios under different spot price.
The Variance Inflation Factors function is available in R for determining existence of multicollinearity. The VIF function is given by:
And to use this built-in function is R:
The autocorrelation function is the ratio of the kth sample autocovariance to the sample covariance, i.e.,
A plot of rk against lag k is evaluated for discernible patterns, relationships, and absolute values (e.g., close to zero).
The Blockchain is expected to be the revolutionary technology to take the centre stage in our society where the traditional ledger system once dominates, from bitcoin that emerged in the finance sector to fields where transactions are dependent on authenticity, be it a paper document from bank, an import / export data exchange, or even documents in judicial systems, it is important to understand the principles of its fundamental roots in cryptography.
For example, to ensure the rightful spending of currency in bitcoins, there are a lot of technology being in place on the virtual money market. One being the Elliptic Curve Cryptography that is based on mathematics to ensure the identity of parties involved in any bitcoin transactions.
The gamma function is not a built-in function in the TI Nspire. Nevertheless this function can easily be defined and used for visualising the gamma probability distribution function.
It is the World Cup season. There are many prediction models, and one of the most widely used statistical technique is the Poisson distribution:
The historic match results are available in the public domain, for example, at http://eloratings.net/. The data are analysed to obtain an index referred to attack strength or goal expectancy. This can further elaborates into more complex data like home team and away team expectancy.
By using a matrix of score scales, usually from 0 to 9, all possible outcomes under 10 goals per team are defined with the respective probability. An example matrix with scores from 0 to 2 will look like one below:
This is the basics of predicting soccer in a quantitative way using the Poisson distribution. There are of course many other prediction methods and models, including organic method like asking the famous Paul the octopus 🙂
One of the predicted, and proved feasible, attack to the blockchain technology is the 51% attack. There are several forms of this attack including weakness in the blockchain algorithm itself that allowed easier than usual forking of chain to dishonestly win the popular vote mechanism. In other cases, the perpetrator has to amass more hashing power than all other nodes in the blockchain combined to corrupt the distributed ledger, and hence the name of “51%”.
An interesting mathematical study on the effectiveness of this attack is based on the regularized incomplete beta function to calculate the probability of success of a perpetrator.