MF is linked with physics mostly because it solves the same PDEs (Black-Scholes equation is a certain type of Schrödinger equation for instance). As for the specific links you mentioned : Lie Algebra ...

Since $W_{2t}-W_{t}$ is independent of $W_t$ and has the same law as $W_{2t-t}=W_t$ we only have to compute $$P(X(X+Y)<0)$$ where $(X,Y)$ follows a bivariate normal distribution (with zero ...

The local volatility is just a $\mathbb{R}_+\times[0,T]\mapsto \mathbb{R}_+$ function where $T$ is some time horizon. It is the solution of a simple equation so it expression is written as $\sigma(K,t)... View answer 3 votes I assume no interest rates to clarify the approach. The Heston model is written under the risk-neutral probability as $$\frac{dS_t}{S_t} = \sqrt{v_t}dW_t$$ $$dv_t = -\kappa(v_t-\eta)dt + \theta \... View answer Accepted answer 1 votes$$ Z_t = f(S_t) := \left( \frac{S_t}{H} \right)^p  dZ_t = \partial_x f(S_t) dS_t + \frac{1}{2} \partial^2_{xx} f(S_t) d\langle S \rangle_t = p\frac{S_t^{p-1}}{H^p} dS_t + \frac{1}{2} p(p-1) \frac{... View answer 0 votes No. In practice the local volatility model has a finite number of slices, so a single slice works as well. Now the problem is : how to compute the time derivative ? Well without adding any ... View answer 0 votes To me this aims at computing a daily implied volatility surface. For some stocks/indices you may have either vanillas options or american options quoted in the market. If your implied volatility is ... View answer 0 votes Say you have a portfolio with$\alpha$dollars in cash and$\beta$stocks at time$t=0$. The value of your portfolio at time$t\$ is $$P_t = \alpha e^{rt} + \beta S_t \tag{1}$$ Black-Scholes ...