r/options thread heading - the Black Scholz PDE and its significance in options trading

If you're on r/options you've probably seen this in the sidebar:

∂V/∂t + ½σ²S²(∂²V/∂S²) + rS(∂V/∂S) − rV = 0

V = option value, S = stock price, t = time, σ = volatility, r = risk-free rate.

The Black-Scholz PDE. Looks scary. It's not — every term is a Greek you already trade.

∂V/∂t is theta. The thing that bleeds out of your long calls overnight.

∂V/∂S is delta. How much your option moves per $1 in the stock. Multiply by S and r and that's the carry on your hedge.

∂²V/∂S² is gamma. How fast delta itself moves. The σ²S² sitting in front of it is where volatility actually shows up in your P&L — variance times gamma is the dollar amount you make from the stock moving around if you're hedged.

rV at the end is just the financing cost of holding the option.

Spent way too long ignoring this thing before I bothered to actually read it. Big mistake.

The whole equation says one thing: if you're delta-hedged, what you make from gamma when the stock moves has to equal what you lose to theta. Othewise free money exists and the market eats it.

This is why gamma and theta are inseperable.

Long options = you collect gamma, you pay theta.

Short options = the reverse.

Market makers are trying to find gamma that's cheaper than the theta they're paying, or theta that's richer than the gamma they're giving up.

The equationdoes have has flaws. Constant vol (lol, skew exists), continuous trading (gaps happen), log-normal returns (tails fatter than this assumes), no dividends in this form, constant rates. So you get local vol models, Heston, jump-diffusion, all stacked on top.

BUT THIS IS ALWAYS THE STARTING POINT.