3) using simple but mind-numbing algebra.

You know the story about Gauss? If not, he was about 11 or 12, and the teacher asked the boys to evaluate the sum 1+2+3+...100.

The boys all did it the long way and came to various different answers. Gauss said 5050, which was right. The teacher asked how he did it so quickly. Gauss replied that you could split the sequence into pairs: (1+100) + (2+99) + ... (50+51). 50 such pairs, each the same value 101, 50 x 101 = 5050.

The second method is an example of mathematical insight, but hard to encode 'insight'.

Something else (thinking out loud) is that the SDEs

dS/S  = rdt + sig dW
dS/S = rdt - sig dW

give the same option value 

Interesting. But what does that tell us?

Poulson writes

[$]BS^{call} (S(t), \tau, K, r, -\sigma) = S N(-d1) - e^{-r\tau} K N (-d2)[$]

This is because [$]-\sigma[$] is in the denominator.

Something else (thinking out loud) is that the SDEs

dS/S  = rdt + sig dW
dS/S = rdt - sig dW

give the same option value 

Interesting. But what does that tell us?

Getting a negative price is hard to get here(?)
Maybe BS was not built for [$] \sigma < 0 [$] and be done with it.. a pathological case?

Something else (thinking out loud) is that the SDEs

dS/S  = rdt + sig dW
dS/S = rdt - sig dW

give the same option value 

Interesting. But what does that tell us?

Getting a negative price is hard to get here(?)
Maybe BS was not built for [$] \sigma < 0 [$] and be done with it.. a pathological case?

Better/standard interpretation of second SDE: 
[$]W^*(t) \equiv -W(t)[$] is also a standard Brownian motion (keeping [$]\sigma > 0[$]),
 [$]\Rightarrow[$] same option values.   (think Monte Carlo if not obvious).

Interesting. But what does that tell us?

Getting a negative price is hard to get here(?)
Maybe BS was not built for [$] \sigma < 0 [$] and be done with it.. a pathological case?

Better/standard interpretation of second SDE: 
[$]W^*(t) \equiv -W(t)[$] is also a standard Brownian motion (keeping [$]\sigma > 0[$]),
 [$]\Rightarrow[$] same option values.   

I see, so the increments are [$]\sigma \sqrt{T} (- \varepsilon)[$] where [$]\varepsilon[$] ~ [$]N(0,1)[$]

edit:
I ran MC pricer and both cases [$]\sigma = {0.1, -0.1}[$] give the same price.

"dS/S  = rdt + sig dW
dS/S = rdt - sig dW
give the same option value"

Of course, but this isn't a SDE/PDE thing, just look at the physics.
(1) starts at S0 and jiggles around like up-down-up-down, and (2) starts at S0 and jiggles along like down-up-down-up. The dynamics they describe are identical, they can't give different option prices.

"dS/S  = rdt + sig dW
dS/S = rdt - sig dW
give the same option value"

Of course, but this isn't a SDE/PDE thing, just look at the physics.
(1) starts at S0 and jiggles around like up-down-up-down, and (2) starts at S0 and jiggles along like down-up-down-up. The dynamics they describe are identical, they can't give different option prices.

I don't like physics in this context. aka hand-waving. It's just a simulation using RNG.
I need a mathematical proof. 

I disagree fairly profoundly. my progression was theoretical physics -> quant -> trader, and the longer I spent in the industry the more convinced I became about the importance of a feel for the physics of option trading.

I actually became slightly notorious when interviewing people for trading desks - I would ask someone if they really understood (for example) what a quanto drift adjustment was all about, and if an answer came back that yacked on about measure-change the interview was over. The idea that you can blame Girsanov for losing money on your position is ridiculous.

Certain corners of the trading world are dominated by option traders who have come up through academia, a stint in a quant group, and then into the real world. They are mostly terrible, although je ne serais pas assez indiscret pour pointer du doigt une nationalité en particulier.

So, we can blame Bourbaki?
And what's a "jiggle"?

1. yes, probably
2. technical term for a harmonic oscillator

1. yes, probably
2. technical term for a harmonic oscillator

1. My hero was J.L. Lions
2.simple, damped, 2d 3d??