3. Theta: the toll the passing of time charges

Derivatives 4: The Greeks

The previous module left a question open: if having gamma in your favor means every market move benefits you, why does not everybody have it? Because there is a bill, it is paid every day and no deferral is allowed.

What theta measures

Theta is the change in the premium from the simple passing of one day, with the underlying, implied volatility and rates exactly the same. It is the only Greek whose movement is known in advance: time only goes one way.

For whoever buys options, theta is negative: they lose value with every day that passes. For whoever sells them, positive: every day that passes takes a slice of what they collected. And it affects only time value, not intrinsic. A call with a strike of 70 with the share at 100 has 30 euros of intrinsic value the clock cannot touch; what evaporates is the part that pays for the possibility of things getting better still.

The decay is not linear: it accelerates

This is the point that costs the most money to learn. Our one-month 100-strike call is worth 4.00 euros and will be worth zero if the share ends at 100 or below. It is tempting to divide four euros by thirty days and conclude that around 13 cents a day are lost. The reality, on the same assumptions as the rest of the level, is this:

• With 30 days ahead, the premium is 4.00 euros and the next day costs about 7 cents.
• With 15 days, the premium is around 2.84 euros and the next day costs about 10 cents.
• With 5 days, the premium is around 1.64 euros and the next day costs about 17 cents.
• With 2 days, the premium is around 1.04 euros and the next day costs about 30 cents.
• On the last day everything left evaporates, about 73 cents.

The pattern matters more than the particular figures. The time value of an at-the-money option does not fall in a straight line: it falls with the square root of the time remaining. Put in a way you can check against the numbers above, halving the time left does not take half the value away, it takes around 29%. From 30 to 15 days, the premium goes from 4.00 to 2.84. From 4 days to 2, from 1.47 to 1.04. Always the same percentage, and that is why the daily decay, in euros, shoots up at the end.

A qualification worth adding that almost never gets said: that acceleration is a phenomenon of options at the money. A one-month call with a strike of 130 has little time value to lose and loses it early, sitting at almost nothing long before expiry; its daily decay in the last week is minuscule, simply because there is nothing left to decay. What accelerates at the end is the theta of whatever is still alive on the border.

One operational detail: the models discount calendar time, not sessions. On a Friday afternoon the option has a whole weekend ahead during which the price cannot move but the clock most certainly does. Monday dawns with three days less of value. Many traders adjust their volatility calendars to spread the decay less naively, but that is a management correction, not a law of the market.

Gamma and theta are the same coin

Here the idea that organizes these two modules comes together. Gamma and theta always run in opposite directions. Whoever buys options has gamma in their favor and theta against them: they pay every day for the privilege of having every move benefit them. Whoever sells them collects every day and in exchange puts up with every move hurting them.

It is neither a coincidence nor a stylistic rule: it is what makes the option's price make sense. If you could have gamma in your favor without paying theta, you would have a money-making machine for any market that moved.

The useful question is how much the underlying has to move for the exchange to come out level. On our call, gamma is 0.04 and theta about 7 cents a day. The gain gamma contributes grows with the square of the move, so a small move does not cover it and a large one covers it easily. Doing the sum with those two numbers, break-even sits at a daily move of around 1.8 euros, that is, 1.8%.

And that number did not come out of nowhere: it is the daily move implied by an annual volatility of 35% when it is spread across calendar days, which is how the model we have valued the option with discounts time. Counting sessions only, the divide-by-16 rule from level 3 would give 2.2%. It could not have been any other number. The option's price already carries inside it the amount of movement needed to justify it.

Worked case: two traders, a quiet month

Ana buys one contract of our call for 400 euros and does nothing else. Bruno sells that same contract and hedges the delta daily by buying and selling shares. The month passes without incident: no session with a move above 1%, and the share ends at 100.00.

• Ana loses the whole 400 euros. Her right expires worthless because the share did not clear the strike.
• Bruno has collected 400 euros and has been readjusting his hedge. Since the daily moves stayed well below the 1.8% break-even, that rebalancing cost him considerably less than what he was paid for time.

Bruno wins this month. What this example does not say is what happens to Bruno in the month the underlying opens with a 12% gap: there is a whole level further on for that. Theta is collected daily and in comfortable amounts; gamma against you is paid all at once. That the exchange is fair does not mean the two faces look alike.