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2

Your integration of 1/S dS is incorrect for a stochastic process. You must use stochastic calculus. That would give you the adjustment term, somewhat like a convexity adjustment.

0

I verified numerically, $\lambda$ is the eigenvalue of $A$. Also, if you looked at this paper: http://liu.diva-portal.org/smash/get/diva2:1140151/FULLTEXT02.pdf, equation 11 is a special case of the above general formula in your question. It is clearly showing that $\lambda$ is the eigenvalue.

6

write down Ito's lemma for the function X: $$dX=\frac{\partial X}{\partial Y}dY+\frac{1}{2}\frac{\partial^2 X}{\partial Y^2}(dY)^2+\frac{\partial X}{\partial c}dc+\frac{1}{2}\frac{\partial^2 X}{\partial c^2}(dc)^2+\frac{\partial^2 X}{\partial Y \partial c}dYdc+\frac{\partial^2 X}{\partial c \partial Y}dcdY$$ Using the following: $\frac{\partial X}{\... 2 If you apply Ito to$Y_t=aZ_t^2$it simple to arrive at$dY_t$: $$f(t,z):=az^2 \\ dY_t = f_t(t,Z_t)dt+f_z(t,Z_t)dZ_t+\frac{1}{2}f_{zz}(t,Z_t)*[dZ_t]^2 \\ dY_t = 0*dt+ 2aZ_t dZ_t+\frac{1}{2}2a[dZ_t]^2 \\ dY_t = 2aZ_t dZ_t+adt \\$$ Which is your desired result. Recall that$dZ_t^2=dt\$.

1

volatility of the volatility controls convexity of the skew/smile => more vol of vol generates more convex function ( = more smile) mean revertion and correlation between brownian motions both control ATM skew. long term variance controls overall level of skew (moves whole skew graph higher) In practice these parameters are calibrated to market quotes of ...

2

Actually, all investments, retirement accounts, mutual fund accounts, utility bills, supermarket price listings are reported or stated in the Constant Numeraire, which may also be called Dollar-kept-under-the-mattress Numeraire It is the most widely (indeed the only) Numeraire used in real life. How nice it would be if my retirement account or mutual fund ...

2

It is a big topic but here is a simplistic recipe! The starting point would be to check the distribution of the historical returns. Histogram would give an idea of how the shifts are distributed. Have a look at the tails, if the tails are fat or don’t ‘tail-off’ then that would be indicative of jumps or non constant volatility. If you decide that a simple ...

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