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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 $$Let {\cal V}(S,t) = \frac{\partial U}{\partial \sigma}(S,t) be the option vega. Differentiating ... 2 Exact replicating portfolio for constant product AMM here: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3550601 It's irritating that people use a new term 'impermanent loss' for something that happens with any option and has been well understood for decades! 2 I haven't checked your numbers, but the delta hedging principle is that if the realized stock volatility is equal to the pricing/hedging volatility, then theta and gamma compensate each other. Since you have assumed r = q = 0 the pricing PDE is$$ \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} = 0  which you ...