# Introduction

No free lunch

Story would be too good to be true if holding a call/put and dynamically rebalancing gave a profit due to gamma.

We will now explain where the paradox is coming from.

We left out one very important feature:

- trader who owned the option and was making free gamma money out of it had to pay the premium in the first place.

- the profit he makes afterwards has to make up for this premium payment.

The longer the lifetime of the option --> the more time to move further ITM --> the more expensive the option should be.

So we know that the value of the option becomes worth less and less with every day that passes by.

This process is inevitable. So little by little the option loses its value.

The change in value from one day to the next = theta.

Clearly this number is expected to be negative for both ATM call and ATM put option.

Let us turn to an example that will unfold the finesse of the Greek that makes you lose.

S0 = 20 with = 35%. r = 2.5%. No dividend. --> 1Y ATM put = 2.50€.

Suppose there are 250 trading days in a year --> we assume the option only loses value on working days.

We could write a fixed loss of 0.01€/day in our book. It would make the loss of time value completely predictable.

However, there would be an inconsistency between the market value and the booked value.

By accepting a MtM valuation, we have to accept that:

- we only know the theta today

- we also know all time value will have disappeared when we reach maturity

- but we cannot predict how the time value will behave over time

In fact, the theta exhibits 2 features:

1. The option loses little value at the start and this process speeds up as maturity approaches.

2. Dependency on the moneyness.

The real potential of an option is found in the ATM range.

Suppose the following scenario happens: put option starts ATM --> stock moves down --> put price rises.

Although value increased, the extrinsic value (added value over intrinsic value = IV) is less than for ATM option.

Clearly, if the time value is less altogether, there is less value to lose.

In fact one can think of scenarios where the time value actually first decreases, and afterwards increases again.

Suppose the trader owns an ATM put option.

First week --> stock does not move --> both time value and option value decrease.

Second week --> stock drops by 10% --> option price increases (due to IV) and time value decreases.

We know that if the time value is smaller, the daily decrease in time value is lower than before the actual move.

If the stock moves back up to the ATM level, the option value goes down but the time value increases.

As of then, the daily decrease in time value is more substantial than before this move back up.

How all the changes in time value add up?

This is the topic of the next section, where we see the theta in relation to the gamma.

Over the lifetime the sum of all changes in time value on a daily basis just add up to the option premium of the first day.

The reason behind this is nothing else but the fact that an option converges to its intrinsic value.

The creation of extra time value is a temporary effect.

Note that at first sight the theta is negative, there are 2 situations where it turns positive.

1. ITM put options are known to have a positive time decay (see section: Time Value and Extrinsic Value).

2. ITM options, the theta can turn positive as well, in the case where the dividend yield > IR --> forward < current level. Impact of Volatiliy on Theta: Theta Strengthened by Volatility

Higher volatility --> higher option price --> higher time value --> more time value to lose --> bigger | Theta |

As the vega is higher in ATM region --> effect is larger in this region --> theta change most noticeable when ATM.