Options out-of-the-money

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. 

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. 

Overall, we find a difference between the simulated and approximated swaption prices of up to 1.3% for out-of-the-money options (see figure ( 5.3.SI)). Nevertheless, based on our results of the last section, where the lEE performed up to 10 times more accurate than the corresponding MC simulation study, we expect that the difference between the simulated and the approximated option prices is mainly linked to the impreciseness of the MC approach (figure (5.3.SI)). The lEE approach performs very efficient and accurate, even if we compute the price of a 1x20 swaption (see table 5.1). Again, all 21 single probabilities TlJ iEEm for i = 0,. ..,20 are in between the 97.5% confidence coming from the simulation approach. ... 

Another finding of special interest is the significant impact of the dependency structure. The price of an out-of-the-money option, given a completely uncorrelated RF (7 = 00) is about 60 times as high as the price of the perfectly correlated counterpart (7 = 0). 

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 correlation effect is not completely symmetric. For out-of-the-money options we find that p = -1 leads to an increase in the option price of about 25%, whereas the converse correlation imphes a decrease of about 27%. 

The volatility figure used in a B-S computation is constant and derived mathematically, assuming that asset prices move according to a geometric Brownian motion. In reality, however, asset prices that are either very high or very low do not move in this way. Rather, as a price rises, its volatility increases, and as it falls, its variability decreases. As a result, the B-S model tends to undervalue out-of-the-money options and overvalue those that are deeply in the money. 

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. 

Moreover, the volatility input has a different effect depending on if the option is in or out of the money. In fact, the value of the convertible bond is more sensitive to changes in volatility when the option is in the money (price of the tmderlying asset... 

For readers who are already familiar with option theory, this characteristic can be restated as follows When the coupon rate for the issue is below the market yield, the embedded call option is said to be out-of-the-money. When the coupon rate for the issue is above the market yield, the embedded call option is said to be in-the-the money. ... 

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. 

The time value of an option is the amount by which the option value exceeds the intrinsic value. Because of the risk they are taking on, illustrated in the payoff profiles above, option writers almost always demand premiums that are higher than the contracts intrinsic value. The value of an option that is out of the money is composed entirely of time value. Time value reflects the potential for an option to move into, or more deeply into, the money before expiry. It diminishes up to the option s expiry date, when it becomes zero. The price of an option on expiry is composed solely of intrinsic value. FIGURE 8.4 lists basic option market terminology. 

In pricing an option that expires in the future, however, the relevant factor is not historical but future volatility, which, by definition, cannot be measured directly. Market makers get around this problem by reversing the process that derives option prices from volatility and other parameters. Given an option price, they calculate the implied volatility. The implied volatilities of options that are either deeply in or deeply out of the money tend to be high. 

According to (8.12), only two outcomes are possible at maturity either the option is in the money and the holder earns Sr - X, or it is out of the money and expires worthless. Modifying (8.12) to incorporate probability... 

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... 

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. 

In exotic credit options, one or more parameters differ from the vanilla norm. A barrier credit option, for example, specifies a credit event that would trigger the option or inactivate it. A digital credit option has a binary payout if it is at or out of the money at expiration, it pays zero otherwise, it pays a fixed amount, no matter how far in the money it is. 

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