## 02 May 2008

### Time & Time Decay

The third input into Option Pricing Models is time until expiry. The key to understanding this principle is understanding that the option buyer is paying the option writer to take on risk. The buyer has the right to transact shares, but the writer is obligated to transact shares if the buyer exercises that right.

The call writer stands to lose a substantial amount of money if the stock rises past the strike price. Likewise, the put writer can lose a substantial amount of money if the stock drops below the strike price.

The general principle is very similar to insurance premiums. The more insurance you buy in terms of time, the more expensive it is. So it is with options, the more time you buy, the more premium you pay.

On the option writer’s side of the equation, the more time the option writer sells, the higher the chance of the option moving ITM, therefore the more you get paid for taking on this higher risk.

Therefore the general rule of thumb is, all things being equal, more time equals higher option premium, less time means lower option premiums.

This has implications for option holders. Options are a depreciating asset, often described as like holding melting ice in your hand. As the life of an option is finite and fixed, the option loses time value the closer to expiry it gets. This is known as “time decay”.

The graph below is of a call option with 30 days until expiry plotted as the uppermost line in red, with the other colored lines at six-day intervals with the black line at expiry. This clearly shows the time decay in long options.

On the other hand, this is to the benefit of option writers, who in return for assuming the bulk of the risk, get to profit from time decay.

For options that are ATM, time decay accelerates as expiry approaches. Have a look at the graph below. It is a graph of the extrinsic value only of the call option in the above graph.

It is very clear from this graph the accelerating nature of time decay. At 30 days from expiry, the ATM option has \$295 (in this example) of extrinsic value, yet according to our model, at 6 days from expiry, the option still retains \$130, with approximately \$90 at 3 days.

It is often graphically represented as follows with time elapsed on the x axis:

But what not many option resourses will tell you though, is that this is a representation of time decay ATM or quite close to it. As the option gets further ITM or OTM, this characteristic changes. Time decay, depending how far away from the money it is, will actually decelerate into expiry, as shown in the graph below:

This is useful knowledge when considering strategies where deep ITM or deep OTM options are used.

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