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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 ...
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
$$
\...
4
OK, here is a simplified demonstration:
Before we consider swaps, let us consider very simple bonds. Suppose that you have a choice of two zero-coupon bonds. A riskless one costs 95 and is certain to pay 100 in 1 year. A risky one costs 90, is expected to also pay 100 in 1 year, but with some probability $p$ will default and only pay some $R<100$ on the ...
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