Mar 18

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

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.

Call option intrinsic value

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.

Call option time value

Figure 2. Call option time value

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

Intrinsic value plus time value

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.


Feb 18

Call options and probabilistic intuition - dependence on strike

Call options and probabilistic intuition - dependence on strike

Awhile ago I gave a course in options following the excellent book by Hull. Frankly, at that time I didn't understand a fraction of what a trader needs to understand about options. I plan to make several posts to explain the intuition behind the Black-Scholes formula for options valuation. The geniality of this formula consists in the fact that it correctly reflects the likelihood of various events. More precisely, the more likely a certain event, the higher the call option price. The level of explanation will be midway between the theoretical view of Hull and practical view of option traders. We start with a series of definitions.

Why do we want options?

A stock price moves up and down, and this movement is pretty erratic. You don’t want to buy the stock, because it’s an expensive investment and the price may fall right after you buy the stock. However, in case the price goes up, you want to gain from that movement, without the financial commitment required by the stock purchase. You are willing to pay a relatively small amount for the guarantee that if the stock goes up, you will be able to buy it at a fixed (presumably, lower) price. At the same time, you don’t want to buy the stock if it goes down. This is the idea behind the call option.

Basic definitions

A call option gives its buyer the right, but not the obligation, to buy the stock at the price fixed in the option contract. This price is called a strike price and denoted K. The contract is valid until a certain date called an expiration date and denoted T. More generally, T may denote any time before expiration and S(T) will then denote the stock price at that time (it is random and depends on time, while the strike price is fixed for the life of the contract and is deterministic).

There are two parties in the transaction: the call buyer owns a long call and the call seller owns a short call. We discuss only long positions; for short positions most of the time the opposite is true.

At any point in time during the life of the option, the strike price K may be higher than the stock price. In this case people say that it is out of the money (OTM). If K=S, the option is at the money (ATM). If K<S, the option is in the money (ITM). The call buyer wants the call to be ITM at expiration.

If at expiration the call option is ITM, the long call holder buys the stock at price K and sells at S(T), profiting S(T)-K-c, where c is the call price paid at the outset. If at expiration the call option is ATM or OTM, the holder doesn't gain from buying the stock and gives up the right to buy it, because buying is not an obligation. The option expires worthless and the call holder’s loss is c.

The market price c of a call option depends on 5 variables: the strike price K, stock price S, time to expiration T (measured in years), volatility of the stock price \sigma and risk-free interest rate r (annualized). Our purpose is to understand the option price dynamics when one of the arguments of the call option changes, while others are kept fixed.

The call price changes during the life of the option, depending on market conditions. The call holder can sell it at any time prior to expiration. The call price at that time depends on the event \{S(T)>K\} that the option is in the money. In all our explanations we will use this event. The higher its likelihood, the higher the call price.

So, how does the call price depend on the strike price?

In the money ranges for two strikes

Figure 1. In the money ranges for two strikes

Consider two strike prices K_1<K_2. Since \{K_2<S(T)\}\subset\{K_1<S(T)\} (see Figure 1), the K_1 call should be more expensive than the K_2 call because the market values higher events with higher probabilities. As the strike increases, the call price falls. This is illustrated in Figure 2.

Call as a function of strike price

Figure 2. Call as a function of strike price

Figure 2 was produced in Mathematica using the Black-Scholes formula, which is valid for European options that can be exercised only at expiration. Our simple probabilistic argument is true also for American options that can be exercised any time prior to expiration. In Figure 2 we assume that the stock price is $50 (the other assumptions are Expiration=0.5, Interest Rate=0.1, Volatility=0.5, Dividend=0.05). The red part of the curve corresponds to in the money strikes, and the out of the money strikes are shown by the blue part of the curve.

Note also that in Mathematica it's better to use the command FinancialDerivative than to program the Black-Scholes formula directly. The latter for some reason sometimes doesn't work.