Options at-the-money

Overall, the lEE performs accurately for in-the-money and at-the-money options (see figure (4.3)). The relative and absolute deviation from the cdf is only about Arei K) th Aabs K) w 10 — 10 . We obtain less accurate figures only for far-out-of-the money options with an absolute approximation error of about Aabs K) w 10" - 10" , together with a relative error of A,/(/T) 10-4-10"2. 

First, the option price increases with the correlation for in-the-money and at-the-money options (moneyness < 1-075). By contrast, we observe a decrease in the price coming along with a higher correlation for out-of-the-money options (figure (6.3)). 

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. 

The market uses implied volatilities to gauge the volatility of individual assets relative to the market. The price volatility of an asset is not constant. It fluctuates with the overall volatility of the market, and for reasons specific to the asset itself When deriving implied volatility from exchange-traded options, market makers compute more than one value, because different options on the same asset will imply different volatilities depending on how close to at the money the option is. The price of an at-the-money option is more sensitive to volatility than that of a deeply in- or out-of-the-money one. 

At-the-money options have the greatest time value in-the-money contracts have more time value than out-of-the-money ones. These relationships reflect the risk the different options pose to the market makers that write them. Out-of-the-money call options, for instance, have the lowest... 

At-the-money options—which constitute the majority of OTC contracts—are the riskiest to write. They have 50—50 chances of being exercised, so deciding whether or not to hedge them is less straightforward than with other options. It is this uncertainty about hedging that makes them so risky. Accordingly, at-the-money options have the highest time values. 

As in chapter (3), where we approximated the density function of a and lognormal-distributed random variable, we again focus in our analysis on all parameters K, such that T1 [AT] > 10 holds. This implies that we are able to analyze the accuracy of this method, even for far out-of-the-money options. 

The convertible bond will be more sensitive with the change of volatility when the option is at the money, or the share price is closed to the conversion price. 

Moneyness—Is the option worth exercising If so, it is said to be in-the-money (ITM). Our call option struck at 98 would be in-the-money if the underlying bond was trading above 98. If the bond were trading below 98, the call would instead be out-of-the-money (OTM). Finally, if the current price of the underlying asset was the same as the strike price, 98 in this example, the option would be at-the-money (ATM). Premium—The amount paid by the buyer of an option is called the premium. This is normally paid up-front. 

An option that has intrinsic value is in the money. One with no intrinsic value is out of the money. An option whose strike price is equal to the underlying s current price is at the money. This term is normally used only when the option is first traded. 

At-the-money An option for which the strike price is identical to the underlying asset price... 

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. 

After a demonstration at the Hanford Site C Reactor in 1998, the DOE estimated that it would cost approximately 50,000 to remediate the 1956 contaminated lead bricks on site. Costs would range from 0.96 per pound if the bricks were presurveyed for contamination levels to 0.99 per pound if the bricks were not presurveyed. The presurveying option is less expensive because not all of the bricks would require decontamination. These estimates do not include money earned from the salvage value of the bricks (D198327, pp.l6, 17). The DOE notes that TechXtract was not cost effective at Hanford due to the cheap costs of landfill disposal at the facility (D222719, p. 6). 

For instance, at maturity, the stock price can have a maximum value of 11.51 and a minimum value of 0.35, according to the assumed volatility. Therefore, at higher node, the value of option will be equal to 8.91 as the difference between the stock price of 11.51 and conversion price or strike price of 2.6. In contrast, at lower node, because the stock price of 0.35 is lower than conversion price of 2.6, the option value will be equal to 0. Particular situation is in the middle of the binomial tree in which in the upstate the stock price is 2.84 and downstate is 1.41, meaning that in the first case the option is in the money, while in the second case is out of the money. 

Conversely, at the lowest node, the hedge ratio is 0 because the option is out of money or 0. This means that in the first case the bond trade like the equity, while in the second case like a conventional bond. Therefore when the share price increases the delta approaches unity, implying that the option is deeply in the money. In contrast, when the share price is low relative to the conversion price, the sensitivity of the convertible and therefore of the embedded option is low. 

Determine the Value of a Callable Bond Since the option is held by the issuer, the option element decreases the value of the bond. Therefore, the value of a callable bond is found as an option-free bond less the option element according to Formula (11.3). For the hypothetical bond, the price is 106.13-2.31 or 103.82. This is shown in Figure 11.11. The binomial tree shows that at maturity the option free and callable bond have the same price, or 100. Before the maturity, if the interest rates go down, the callable bond s values are less than an option-free bond, and in particular when the embedded option is deeply in the money, the callable values equal the strike price according to the caU schedule. Conversely, when the interest rates go up, the option free and callable bonds have the same price. 

The net cost of this strategy is 0.82 million, which is the maximum amount the investor can lose if he or she is wrong and bond prices fall. If bond prices stay the same, the net loss is only 0.34 million, as the option bought is in-the-money, and will expire with some intrinsic value that will be returned to the investor. If bond prices rise up to 1.50%, the investor can enjoy the benefit until the 118.00 strike is reached and profits are capped at around 1.2 million. 

A DNT option is a path-dependent digital option. The digital characteristic means that payment at maturity is not dependent on how much the option is in-the-money instead, the payout, if made, is fixed at the outset. The path-dependent characteristic means that market rates throughout the lifetime of the option are monitored, not just on the maturity date. If, at any time, the nnderlying market rate touches (trades at or through) the barrier levels, this triggers the payout at maturity. 

This may be a relatively large sum to put aside all at once, especially for a first-year engineering student. A more realistic option would be to put aside some money each year. Then the question becomes, how much money do you need to put aside every year for the next five years at the given interest rate to have that 2000 available to you at the end of the fifth year To answer this question, we need to develop the formula that deals with a series of pajmients or series of deposits. This situation is discussed next. 

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