Dynamic hedging, and where the premium goes

Every other page here describes a position and what it is worth. This one is about what happens next, because an option is not a bet you place and leave. Sell one and you have taken on an exposure that changes every time the underlying moves, and the only way to see what that costs is to run it forward.

What the premium actually buys

Sell a thirty-day at-the-money call at 25% volatility and roughly 300 lands in your account. It is tempting to read that as income. It is not. It is the market's estimate of what it will cost you to stay flat until expiry.

Being short the call leaves you short delta as the underlying rises, so you buy to stay hedged. When it falls you sell. Buy high, sell low, thirty times, and the losses accumulate — that is what being short gamma means, and it is the only reason anyone pays you the premium in the first place.

Run the figure above a few times. The premium is always about 300. The cost of hedging is always about 300. What is left over changes sign from path to path and is small either way. That is not the hedge failing. That is the option being correctly priced.

When you keep it

The premium is the price of a forecast: 25% annualised. The market either delivers that or it does not.

The underlying barely leaves a six-point range, the hedge rarely has to chase it, and the hedging bill comes to about 180 against the 300 you took in. You keep the difference. This is the entire business model of a short-volatility book, and it is also why those books look brilliant for long stretches and then do not.

Same position, same discipline, same 300 collected. The underlying travels to 125 instead of 106, the hedge has to chase every swing, and the hedging bill comes to nearly 380 — the premium gone, and about 75 of your own money with it.

The asymmetry worth noticing is not in these three numbers, which are close to symmetric around the break-even. It is in the distribution behind them. Realised volatility cannot fall below zero, so the best case is bounded: you keep the premium and no more. There is no matching ceiling on the other side. The quiet path pays you a known maximum; the violent path has no maximum at all, and a thirty-day chart of a market that merely doubled its volatility is nowhere near the bad end of that distribution.

The part the model does not tell you

Black-Scholes assumes you rebalance continuously. Nobody does. Hedge thirty times instead of infinitely often and you are left with an error that has nothing to do with whether your volatility forecast was right — it is the residual of a discrete approximation to a continuous instruction.

That error shrinks as you hedge more often, roughly with the square root of the number of rebalances. Going from six hedges to sixty cuts the spread of outcomes by about two thirds. It also multiplies your transaction costs, which the model prices at zero and your broker does not.

So the real question on a hedging desk is never "how do I eliminate this error". It is "where does the marginal hedge stop paying for itself" — and that is a question about spread, commission and market impact, not about the Greeks.

Why this page exists

The risk systems I have spent most of my career on exist to answer this question across a book rather than a single position: what am I short, what will it cost me to stay flat, and what happens to that number if the world moves. SPAN answers a version of it for margin. VAR answers a version of it for capital. This is the version a trader lives with every day.

The figures above use the same Black-Scholes implementation as the options P&L calculator — no separate demo model, and no market data. The paths are generated and seeded, so the same button always produces the same story.