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3

Maybe this deck by Jim Gatheral would help get the intuition, see slides 10 and following. The dynamics you mentioned is obtained by: Looking at the Bergomi dynamics for the forward variance process; Assuming there is only one factor driving the dynamics; Noticing a similarity with a rough process when replacing the Bergomi exponential kernel by a Riemann-...

6

I'll give a heuristic "proof" for general European claims which will cause mathematicians to feel sick, but which physicists / practitioners would probably be quite happy work with: Write the Black-Scholes PDE as $$\frac{\partial F}{\partial\tau}(\tau) = \mathcal{A} F(\tau)$$ with $\tau = T- t$, and the operator $\mathcal A$ is defined as $$\... 9 Consider any option, vanilla or exotic. In between fixing dates it satisfies the Black & Scholes PDE (for simplicity zero interest rate and dividends)$$ \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 U}{\partial S^2}(S,t)+\frac{\partial U}{\partial t}(S,t)=0 $$Let {\cal V}(S,t) = \frac{\partial U}{\partial \sigma}(S,t) be the option vega. Differentiating ... 5 As shown in Credit Risk Modeling Notes (Bielecki, Jeanblanc, Rutkowski), Corollary 1.3.1, for t < s, we have:$$ P(\tau \leq s | {\cal F}_t) = N\left( -Y_t \sigma^{-1}(s-t)^{-1/2}- \nu(s-t)^{1/2}\right ) + {\rm e}^{-2\nu \sigma^{-2}Y_t} N\left( -Y_t \sigma^{-1}(s-t)^{-1/2}+ \nu(s-t)^{1/2}\right ),$$where$$ Y_t = y_0+ \nu t +\sigma W_t, \: \sigma >0,...

2

I have just finished working on this exercise, it is rather simple and this is also an old question, but why not share my reasoning here anyway. As far as the first point, performing antithetic sampling obviously speeds up convergence of the MonteCarlo method by picking two numbers for each draw, instead of one, $x$ and $-x$ with mean $= 0$. Hence the given ...

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