1. Delta: how the option moves with the underlying

Derivatives 4: The Greeks

The price of an option does not depend on one thing alone. It depends on the underlying's price, on the time left to expiry, on the volatility the market is pricing in, on the interest rate and on the strike price, which is the only item on the list that never moves. Four live variables, a single output number: the premium.

The Greeks answer a very narrow question: if I move one of those variables and leave all the others exactly where they were, how much does the premium change. That final clause, leaving the others still, is the small print of the whole level. In the real market it never holds: the underlying moves and, at the same time, the implied reacts and the clock ticks on. Almost every surprise with options is born of forgetting that each Greek is a snapshot with the other variables frozen.

The example we shall carry through the whole level

We shall always work on the same case, so that the sums can be followed from module to module. It is a constructed example, not a real quote: the numbers come from a standard valuation model, rounded:

• A share trades at 100 euros. It pays no dividend during the life of the option.
• A 100-strike call with one month to run, that is, right at the money.
• The market's implied volatility, as assumed, is 35% a year.
• On those assumptions, that call's premium is around 4.00 euros a share.
• One contract is equivalent to 100 shares, so that contract costs around 400 euros.

Four euros on an underlying of a hundred: the option costs 4% of the share and gives the right, for thirty days, to everything above 100.

What delta measures exactly

Delta is the change in the premium for every euro the underlying rises, with everything else held still. In our example that call's delta is approximately 0.52.

Read it literally: if the share goes from 100 to 101 in the same instant, with no change in the implied and no time passing, the premium goes from 4.00 to around 4.52. If it falls to 99, the premium drops to around 3.48. Delta is the slope of the curve relating the two prices, measured right at the point where you are standing.

There is a second reading, more useful for managing a position: delta translates the option into equivalent shares. One contract of that call, with a delta of 0.52 over 100 shares, behaves in the face of small moves just like 52 shares. It is not the same thing in any other sense, but it reacts the same way to the next euro, and that lets you add very different positions up into a single figure.

The range, and what each extreme means

The delta of a bought call lives between 0 and 1. That of a bought put, between minus 1 and 0. The extremes have meaning:

• Delta close to 0. The option is so far out of the money that a euro barely affects it. A one-month call with a strike of 130 would have a delta of a few hundredths: the market has all but written that right off.
• Delta close to 1. The option is so deep in the money that it behaves like the share itself: a call with a strike of 70 moves almost euro for euro, because its exercise is practically certain.
• Delta around 0.50. This is the at-the-money zone, where the market is genuinely undecided about whether that right will be worth anything.

Selling reverses the sign. Whoever sells that call has a delta of minus 0.52: he gains when the underlying falls. And a share has a delta of 1 by definition, because it rises a euro when the underlying rises a euro.

Delta looks like a probability, but it is not one

A very widespread shortcut does the rounds: delta is the probability that the option ends up in the money. It is a convenient approximation and it is worth knowing why it is only that.

On our example's assumptions, the same model that gives a delta of 0.52 assigns that call a probability of ending in the money of around 48%. Four points of difference. It is not a rounding error: they are two different quantities. The probability answers whether it ends in the money; delta answers what the right is worth in the face of the next move, and that also weighs how far in the money it might end up, not just whether it does. The gap between the two grows with volatility and with the horizon, and it is small in options close to expiry.

There is a second caveat, and a more important one: those probabilities are the model's, calculated in a world where nobody demands a premium for taking on risk. They are not the real probability of anything. Using delta as a quick estimate is fine; presenting it as a measured probability is going too far.

A portfolio's delta and what being neutral means

Deltas add up. That is where their real usefulness lies: a portfolio of shares, calls and puts on the same underlying is summed up in a single number that says how much it gains or loses for every euro that underlying moves.

Suppose you hold 500 shares at 100 euros. Your delta is 500: every euro of rise gives you 500 euros. You want to neutralize it with one-month 100-strike puts, whose delta on our assumptions is around minus 0.48 a share, that is, minus 48 a contract.

• You buy 10 contracts: they contribute minus 480 of delta.
• Your total delta comes to plus 20, not zero.
• To nail zero you would need 10.4 contracts, and contracts come whole.

Being delta neutral means that, in the face of small moves, the portfolio's value does not change. It does not mean being without risk, and the next module explains why: that neutrality has a very short sell-by date. As soon as the share moves a few euros, the puts' delta will have changed on its own and that plus 20 will be something else without your having touched a thing.