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Mar 18

## Intro to option greeks: theta, intrinsic value and time value

This world is full of surprises and uncertainties. One thing is for sure: time is unidirectional and nobody grows younger. This is true also for options: other things being equal, the value of the call option decreases with time. The dependence of option prices on time is the simplest but it allows us to discuss important related notions - the intrinsic value and time value.

## Exercising options

American options can be exercised before expiration. If you hold a long American call and it is in the money, you can request your broker that your option be exercised. That is, you buy the stock at the strike from the holder of the short call and gain the difference between the stock and strike prices. European options can be exercised only at expiration. Most of the time, exercising an option is not good because by acquiring stock you tie your money up in a larger investment and you lose what is called a time value.

For an ITM call, the difference "current stock price minus strike" is called its intrinsic value. This is how much you would profit if the call was exercised right away. For an OTM call, the intrinsic value is zero because you would not buy the stock at the price higher than the market price.

Figure 1. Call option intrinsic value

Thus, the intrinsic value has a kinked graph depicted in Figure 1 (the strike is assumed to be \$50 and instead of a fixed "current" price a range of stock prices is used on the horizontal axis). The intrinsic value equals $\max\{S-K,0\}$ where $S$ is the stock price and $K$ is the strike.

Figure 2. Call option time value

The difference "call price minus intrinsic value" is called time value, see Figure 2.

Figure 3. Intrinsic value plus time value

We obtain the representation illustrated in Figure 3:

call price = intrinsic value + time value.

## Dependence of the call price on time

The derivative of the call price with respect to time is called theta. Put it simply, in one day the call price declines by theta. For a given theta from the option chain, this kind of calculation cannot be extrapolated for more than a couple of days because theta itself changes with time. See the figures posted previously to satisfy yourself that:

• The absolute value of theta is the highest for at the money strikes. This is because if the call's strike is far away from the at the money one, the call is more likely to stay either OTM or ITM as time passes.
• As expiration approaches, the time decay increases around the at the money strike. The options that are far out of the money or deep in the money have theta equal to zero because in the time remaining until expiration their OTM or ITM status is not likely to change.

One has to remember that we concentrate on long call options. Since theta is always negative, time is working against long option holders. If you have a long call and the stock price is not moving much, the intrinsic value stays the same, while the time value goes to zero. It is better to close your position earlier. Conversely, time decay is favorable to an investor who writes (=sells) options. The call option seller hopes that by expiration the call will be OTM, the intrinsic and time values will both vanish and the option will expire worthless.

Remark. Stock options that are \$0.01 or more in the money are automatically exercised by the Options Clearing Corporation after the market close on the expiration date.