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…

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…

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…

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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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…

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