Delta Hedging
You've seen duration used extensively with bond portfolios. The idea is straightforward: since the portfolio's price changes somewhat predictably as measured by delta, just match delta where needed, since bonds can't always be used for transactions.
If a manager uses duration targeting of some sort and yields decline, the price generally increases approximately as predicted. What happens to duration?
No.
Absolutely!
Duration is an estimate; it changes with time, and it changes as rates change. In the options world, the rate of change is measured by __delta__, and similar rules apply.
The delta of an option is
$$\displaystyle \mbox{Delta} = \frac{\mbox{Change in option price}}{\mbox{Change in underlying price}}$$.
Now if the option is close to expiration and out of the money, how much do you think the option price is going to change when the price of the underlying moves up by 1?
Exactly.
If it's out of the money, it's about to be worth 0, and some small change in the underlying won't change that.
No.
Consider the value of the option at this point.
Not really.
Think about the option price in this case.
And suppose the call is deep in the money. How much would it move, then?
Probably not.
It should move more than that.
Yes!
That's really the range of deltas: 0 to 1. It should be somewhere in there. So if a dealer sells an illiquid call option and wants to approximate the position with a delta hedge, the answer is just buying shares of the underlying.
Unlikely.
Think about intrinsic value as the time value approaches 0.
Suppose you're a dealer who sold 30 call option contracts, each covering 100 shares of a stock. Your position is short 3,000 shares... sort of. But then you need to know how much those option prices will really change in price. Maybe the delta is estimated at 0.8. Will you need to buy more or fewer than 3,000 shares to be hedged?
No, fewer.
Good!
In fact, you may be guessing that it's 0.8 times 3,000, or 2,400 shares. That's right, and here's why. First, a portfolio of options and stocks has a value based on the number of shares times the share price plus the number of calls times the call price.
$$\displaystyle V = N_SS + N_CC$$
Any change in value is due to a change in share price and a change in the price of the calls.
$$\displaystyle \Delta V = N_S \Delta S + N_C \Delta C$$
What ratio from these change variables would give you your delta?
No.
Value isn't included in a delta calculation. It's a ratio of option price and underlying price; you can look back again to see it.
Sure.
So just divide everything by the change in _S_ in order to get that ratio.
$$\displaystyle \frac{\Delta V}{\Delta S} = N_S + N_C \frac{\Delta C}{\Delta S}$$
No.
The delta is a measure of the option price sensitivity.
Then take a step back, and remember why you are doing this. If you end up with the right number for a good delta hedge, what should the change in value be?
Exactly.
The change in value should be 0. The whole point here is determining that number so that you can perform a delta hedge. So just solve for that number, and you're done.
$$\displaystyle N_S = -N_C \frac{\Delta C}{\Delta S} = -N_C (\mbox{Delta})$$
There you go. So the short calls representing 3,000 shares is a "minus minus" here, and a delta of 0.8 leads you to needing 2,400 shares.
$$\displaystyle N_S = -(-3{,}000)(0.8) = 2{,}400$$
No.
Delta is a measure of price sensitivity of the call option. You can't change that just by hedging.
No.
That would suggest that you're not well hedged.
Now what will happen to the delta as the options get close to expiration?
No.
Delta doesn't have this much freedom. Think back to the range of deltas and how you get to the extremes.
No.
That's definitely not a requirement. Think back to the range of deltas and how you get to the extremes.
Yes.
It absolutely must. Recall the range of deltas and how you get to the extremes: in the money near expiration, it's 1. Out of the money near expiration, it's 0. It will end up as one of the two. This means that the delta hedge will fail, eventually. This is a short-term strategy right now.
Other issues with delta are that the 0.8 was an estimate to begin with, and smaller numbers require rounding for the number of shares that can make the hedge imperfect. But as long as this is a short-term thing, the delta hedge will work pretty well for taking the bulk of risk out of a naked short option position like this.
To summarize:
[[summary]]
It changes
It stays constant
About 1
Not at all
Maybe 0.5 or so
About 0.5
Close to 1.0
More than 1.0
More
Fewer
Change in _V_ over change in _S_
Change in _C_ over change in _S_
Change in _S_ over change in _C_
0
1
Delta
It must rise or fall
It must stay the same
It could rise, fall, or stay the same
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