Delta hedging
Why the option price is what it is: replicate it, and see what you earn when the world disagrees with your volatility.
The argument behind the price
Black-Scholes is not a guess about what options are worth; it is the cost of manufacturing one. Sell a call, hold \(\Delta\) shares against it, and adjust \(\Delta\) as the spot moves. If you could adjust continuously and the spot really had volatility \(\sigma\), the position would be riskless and would earn the risk-free rate, so the premium must equal the cost of the hedge. The simulator does exactly this, except that it hedges at discrete times and lets the true volatility differ from the one you sold at.
The profit of a hedged option
Hedge at the implied volatility \(\sigma_{imp}\) while the spot moves with realised volatility \(\sigma_{real}\). Over a short interval the hedged book earns
$$ dP \approx \tfrac{1}{2}\left(\sigma_{imp}^2 - \sigma_{real}^2\right) S^2 \Gamma\, dt $$Sold too dear, the book collects the difference every day, weighted by Gamma: most when the spot is near the strike and expiry is close. Sold too cheap, it bleeds the same way. This is the sense in which an option is a bet on volatility, and it is why traders speak of "buying" and "selling" volatility.
Hedging error
Hedging only every \(\Delta t\) leaves the book exposed to the spot's move in between, which Gamma turns into profit or loss. The result is noise with zero mean and a standard deviation that grows like \(\sqrt{\Delta t}\): hedge four times as often and the noise halves. Transaction costs pull the other way, since every rebalance costs money, so there is an optimal frequency, and the answer depends on Gamma and on the cost per trade.
To see it happen, watch the hedge on the hedging page: one path plays in time, the hedge trades back to the Black-Scholes delta on each rebalancing date, and the gap between the book and theory, the hedging error, builds up beside the same path hedged almost continuously.
What does not matter
The drift of the spot barely affects the hedged result. Whether the stock trends up or down, the hedge neutralises the direction and leaves only the volatility exposure. That is the practical face of risk-neutral pricing: the expected return of the stock does not appear in the option price because the hedge removes it.
Check your understanding
4 questions. Pick an answer for each, then check. Nothing leaves your browser session.
1. A short option hedged at its implied volatility earns, per unit of time, roughly
2. Rebalancing the hedge four times as often
3. The drift of the stock affects the hedged book
4. Transaction costs make the optimal hedging frequency