## PCA simple example (principle component analisys)

simple examples of what PCA does with 2D set of points: ellipsoid like set of points: upper graph is original set of points red vectors are PCA components scaled by explained variance buttom graph is transformed set of points so

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## how to value Credit Default Swap default leg and default probabilities

how to value CDS (credit default swap) default leg with following time structure: 0—-t1—–t2—-t3—–t4—–….—T Suppose that default (at time ) can only occur at discrete times t1,t2,t3,.. and Qi=survival probability until time then default leg of Credit Default Swap pays:

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## ito lemma in finance

to derive equations for asset price evolution one uses Ito lemma: basically its the same as Taylor formula for 2 variables developed until 2nd order. when applying Ito’s lema the following formulas are useful: Applications it’s used for example to

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## Girsanov theorem finance

to change the probability measure one uses Girsanov theorem (formula): is numeraire (price of any non dividend paying asset, usually bond or bank account) and it’s correspoding probability measure is is another numeraire with it’s probability measure Applications expectation of

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