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