Category: math

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 $$t_i$$ then $$ \tau – time of default

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ito lemma in finance

to derive equations for asset price evolution one uses Ito lemma: $$ d f(X_t,Y_t) = f_{x}(X)dX+f_{y}(Y)dY+f_{xy}dXdY + \frac{1}{2} f_{xx}(X) dXdX+ \frac{1}{2} f_{yy}(Y) dY dY $$ basically its the same as Taylor formula for 2 variables developed until 2nd order. when

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Girsanov theorem finance

to change the probability measure one uses Girsanov theorem (formula): $$ \frac{N_a(0)}{N_a(T)} d \mathbb{P}_a=\frac{N_b(0)}{N_b(T)} d \mathbb{P}_b$$ $$N_a$$ is numeraire (price of any non dividend paying asset, usually bond or bank account) and it’s correspoding probability measure is $$P_a$$ $$N_b$$ is

Posted in math