Dynamic delta hedging

So, how do we make money from this?

We get market prices for different options. Such prices correspond to a specific volatility.
In a world as described by Black and Scholes, all strikes for a given maturity (should) have the same volatility.

Say we buy 100 ATM call. If I do nothing until expiration, what matters for my PL is only if the stock went up high enough to compensate the price I paid for these calls. Volatility of the stock doesn't directly matter, just the price at expiration.
If the volatility is high, I have more chances to have big moves in my stock but if it moves like crazy but ends up at the same price as when I bought the option, I don't gain anything. Note that high volatility doesn't mean more chances to make money but rather larger gains in the case I make money (as a first approximation, in details it also changes the expected value of the stock at expiration because the distribution is lognormal, but let's not go into that).
Drift matters though, if the drift is high, I have more chances to make money.
Remember that the volatility is taken into account in the pricing but the drift isn't. So if you could predict the drift to be higher than the risk free rate, you might have an hedge. On the other hand, if you can predict the drift, why not just trade the stock.

Now, what if we want to bet on the volatility directly instead. Using the implied volatility calculator that I described in the previous post, we could get implied volatility. With it, we can calculate the delta and perform dynamic delta hedging.

What is dynamic delta hedging? Let's see with an example.
An ATM call has roughly a 0.5 delta. This means that when the stock goes up by 1 euro, the call goes up by 0.5 euros. So if we are to delta hedge these 100 ATM call, we would sell 50 stocks. Each stock has a delta of 1, so the total position is delta flat (0 delta).
Then every time the underlying moves, the delta of the call changes. We know exactly how much because of gamma (the derivative of delta with respect to the stock), we can also reuse the calculator and calculate the new delta. Given that gamma is always positive for that option, when the stock goes up, call gains deltas. So my portfolio becomes delta long (delta>0). In order to become flat again, we sell more stocks.
If the stock starts going down, the delta of the call will reduce and to flatten the portfolio we need to buy stocks.
If instead, the stock goes further up, we sell more stocks.

Say we do that until expiration of the option. At expiration, the option settles in cash (it doesn't change the picture if it settles physically), we sell/buy the remaining stocks.
-If the stock went straight up, we kept on selling stocks while it was raising and then bought them back at higher price, that's a loss. But our call makes good money.
-If the stocks went straight down, the call is worth nothing but we sold stocks and bought them latter at lower price.
-If the stock oscillate, we buy low and sell stock higher. This generate money.

The very important result from Black and Scholes is that if you perform such dynamic hedging and the realized volatility is equal to the implied, the money you will make selling/buy stock (+ the value of option at expiration) will be exactly equal to the price you bought the option. No matter what price the stock trades at expiration.
You removed that risk and have a direct exposure to the volatility instead. You don't have exposure to the drift anymore though.

So now you know a way to trade the volatility of the stock:

If you think it will be higher than the implied: buy options and delta hedge them. The profit from (stock trading + value of the option at expiration) will be higher than the cost of buying the option.
If you think the opposite, you can sell option and delta hedge them and the loss from (stock trading + value of the option at expiration) will be smaller than the profit from selling the option (if you are right of course, otherwise you will loose money).

Note:
-This is true under the Black and Scholes assumptions, namely costless trading, possibility of trading fractions of stocks, continuity of the stock path (no jumps)...
-Should you use a call or a put? That doesn't make a difference in theory if the options are european as the only difference in terms of greeks is delta. In practice it could: 100 atm call delta hedged means selling 50 stocks, 100 atm put delta hedge means buying 50 stocks. The costs/availability of both could be different if you are not a big player.
-I mentioned that under BS, options with different strikes should have the same volatility. It is not the case because the market realizes that some BS assumption are not valid and there is more risk involved than it seems. However taking these risks into account, trading implied vol vs realized vol is a very good way to make money. (If you can predict future realized vol of course).