LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Delta hedging sizes a share position so that a small move in the stock is offset inside an option book. The lesson builds that hedge for a short call when the research view is about volatility, and the desk has no shared view on whether the shares will rise or fall. In the worked case, a model price of about $36.81 and a delta near 0.5469 lead to a long hedge of 54,695 shares against 100,000 short calls.
Why it matters
The Black-Scholes equation, the 1973 model of Fisher Black, Myron Scholes, and Robert Merton, prices a call or a put from five inputs: strike, spot, volatility, time to expiration, and the riskless rate. In a classroom the formula is often run forward to a fair price. In a listed options market the price is already visible, and the missing input is volatility. Running the formula backward from the traded price produces the volatility the market is embedding in that price. The lesson calls this expected or implied volatility. A view that later realized volatility will differ from that embedded number is a view about the option being rich or cheap relative to the path the stock actually takes. Delta is the tool the lesson uses to stop the stock's direction from dominating that view.
The idea in plain words
Delta is the partial derivative of the option price with respect to the spot, holding the other inputs fixed. For a call, that derivative reduces to the cumulative normal function evaluated at the usual d1 term, so a call delta sits between zero and one. For a put, the same differentiation and the symmetry of the normal curve produce a delta that is negative or zero. The plain reading is local. If a call has delta 0.5 and the share price is 100, a one-dollar rise in the shares is associated with about a 50 cent rise in the call. A one-dollar drop is associated with about a 50 cent drop. A put with delta minus 0.5 moves the other way: a one-dollar rise in a share priced at 100 lines up with about a 50 cent fall in the put.
Because the option and the shares respond in a known ratio, a book can hold the opposite share exposure. The lesson's client wants to buy 100,000 calls. The writer is therefore short those calls. If the shares rise, the short calls lose. Buying delta-many shares is the offset the lesson computes.
How to use it
The case uses Apple calls with strike 350, spot 320, a riskless rate of 8 percent, one year to expiration, and an implied volatility of 30 percent. The lesson's pricing object, given those inputs, reports a call value of $36.81 and a delta of about 0.5469, which it also rounds in speech to about 0.55. For 100,000 short calls the share hedge is 54,695 shares bought. The arithmetic the lesson wants you to see is this. A one-dollar rise in the shares lifts each call by about 54 cents, so the short call book is down on the order of $54,695. The 54,695 shares are up by about the same dollar amount. The directional piece is offset on that local reading. What remains is the volatility view: the lesson's research says realized volatility will come in below the 30 percent embedded in the price, so the calls look rich against that research. The stated book is short 100,000 calls and long 54,695 shares.
What usually goes wrong
Delta is a slope on a curved price. After the spot leaves 320, the slope changes. Restoring the offset means trading more shares. Those trades have costs and a gain or loss that can land on either side. The lesson gives no clock for the reset and no cost function it will treat as the answer.
The 54,695 share count belongs to this case: strike 350, spot 320, rate 8 percent, one year, implied volatility 30 percent. A different input changes delta. The lesson frames any trading result as coming from the volatility view, after hedge costs, and it spends its close on the fact that the hedge is a linear patch on a curved price.
Questions people ask
What is implied volatility in this lesson?
It is the volatility number that, fed into the pricing model with the other four inputs, lines up with the option price already trading in the market.
Why buy shares after selling calls?
The short calls lose when the shares rise, by about delta per share per option. Buying delta-many shares creates an offsetting gain on that same small rise. The lesson's count is 54,695 shares against 100,000 calls.
Does the hedge stay correct forever?
The lesson says no schedule is given. Delta changes as the spot moves, so the share count has to be revised, and each revision has trading costs and its own gain or loss.