Strike options, time value

A number of factors dictate whether an option is exercised or not. The first is the asymmetric profit-loss profile of option holders their potential gain is theoretically unlimited when the price of the underlyir asset rises, but they lose only their initial investment if the price falls. This asymmetry fevors runnir an option position. Another consideration favorir holdir is the fact that the options time value is lost if it is exercised early. In callable bonds, the call price often decreases as the bond approaches mamriry. This creates an incentive to delay exercise until a lower strike price is available. Coupon payments, on the other hand, may fevor earlier exercise, since, in the case of a normal, nonembedded option, this allows the holder to earn interest sooner. 

The time value of at-the-money options is the greatest. This is not so apparent from Exhibit 17.8 which shows the total premiums, comprising intrinsic value and time value. Exhibit 17.9 shows the time value component by itself, which makes this feature obvious. With OTM options, the entire premium is time value—this applies to the 117-strike calls and the 116-strike puts, for example. With ITM options, the time value is the total premium less the intrinsic value. For example, the intrinsic value for the April 2003 call option struck at 115 is 1.42 (116.42 less 115). Subtracting this from the total premium of 1.53 gives the time value of 0.11 shown in the table. 

Why is time value greatest for ATM options Think first abont OTM and ITM options. For deep ITM options, the owner is virtn-ally certain to exercise the option, so the right to tear up the option instead is minimal. In a similar way, the owner of a deep OTM option is virtually certain to let the option expire unexercised, so the right to exercise the option instead is also minimal. For ATM options there is no snch certainty on the contrary, uncertainty is at its greatest, so the right to decide later whether or not to exercise is also highest at this strike level. 

Time value is greatest for the ATM options struck at 97.750, equalling 0.100 for both puts and calls. The 97.875 strike options have a time value of 0.055, while the 97.625 strike options have a time value of only 0.050. As we will see when we examine hedging applications, this makes the economic cost of ATM options the highest. 

The option premium has two constituents intrinsic value and time value. Intrinsic value equals the difference berween the strike price and the underlying assets current price. It is what the options holders would realize if they were to exercise it immediately. Say a call option on a bond futures contract has a strike price of 100 and the contract is trading at 105. A holder who exercises the option, buying the futures at 100 and selling the contract immediately at 105, earns a profit of 5 that is the options intrinsic value. A put options intrinsic value is the amount by which the current underlying assets price is below the strike. Because an option holder will exercise it only if there is a benefit to so doing, the intrinsic value will never be less than zero. Thus, if the bond future in the example were trading at 95, the intrinsic value of the call option would be zero, not —5. 

As noted in chapter 8, the value of an option is a function of five factors The price of the underlying asset The option s strike price The options time to expiry... 

Some option strategies exploit theta. Writing a short-term option and simultaneously purchasing a longer-term one with the same strike price, for instance, represents a play on the options theta if the time value of the longer option decays at a slower rate than that of the short-dated option, the trade will be profitable. 

An option s value, or price, is composed of two elements its intrinsic value and its time value. The intrinsic value is what the holder would realize if the option were exercised immediately—that is, the difference between the strike price and the current price of the underlying asset. To illustrate, if a call option on a bond has a strike price of 100 and the underlying bond is currently trading at 103, the option has an intrinsic value of 3. The holder of an option will exercise it only if it has intrinsic value. The intrinsic value is never less than zero. An option with intrinsic value greater than zero is in the money. An option whose strike price is equal to the price of the underlying is at the money one whose strike price is above (in the case of a call) or below (in the case of a put) the underlying s price is out of the money. 

Several factors affect the decision if exercising the option or not. The first one is the asymmetric profit-loss profile. The potential gain of the option holder is unlimited when the price of the underlying asset rises, and losing only the initial investment if the price decreases. The second one is the time of value. In fact, in callable bonds, usually the price decreases as the bmid approaches maturity. This incentives the option holder to delay the exercise for a lower strike price. However, coupon payments with lower interest rates can favour the early exercise. 

Hence, the value at time 0 of a European call option with maturity Tq and strike price K on the coupon bearing bond, under the one-factor HJM model described above, is given by... 

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