vanna
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Why Lie groups, differential geometry and string theory relate to MF?
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6 votes

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 ...

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probability question about brownian motion
6 votes

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 ...

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Commonly used vol surface calibration model in the industry
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6 votes

The most used equity volatility models in the industry are the Black-Scholes model (including its time dependent version) and the local volatility model. It always come along with stochastic rates, ...

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In Dupire's paper, why is $(S_t, t)$ in the $(K, T)$ space?
3 votes

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)...

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SKEW and VIX relations?
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 \...

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Show that Z(t)/Z(0) is a positive mean-1 martingale
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{...

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Time-independent local volatility
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 ...

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How should I calculate the implied volatility of an American option in a real-time production environment?
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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 ...

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Basic question about Black Scholes derivation
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 ...

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