Options in-the-money

Now, comparing the non-differentiable RF model with the integrated T-differentiable counterpart, we find the similar partially offsetting volatility effects. Interestingly, now the price of an in-the-money option given anon-differentiable RF is lower that the price of the T-differentiable counterpart and vice verca (see figure (6.4)). Note that the two different RF models coincide for 7 = 0 and 7= < . 

FIGURE 9.15 The value of a convertible bond - Volatility sensitivity for in the money options. 

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. ... 

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%. 

In other words, the option is valuable when the principal is lower than 100. When the inflation rate drops and the economy is in a hypothetical deflation scenario, the option is in the money. This increases the value of an inflation-linked bond. Note that the probability of a deflation scenario depends on the market sentiment and implied inflation rate between conventional and... 

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... 

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. 

Determine the Value of a Putable Bond As exposed in Formula (11.4), the value of a putable bond is the sum of an option-free bond and an embedded put optimi. Therefore, conversely to a callable bond, the embedded option increases the value of the bond. When the option is deeply in the money, the bond matches the values defined in the put schedule. When the option has no value, option free and putable bonds have the same price. The value of our hypothetical putable bond is 106.13 + 0.33 or 106.45. This is illustrated in Figure 11.14. 

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. 

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. 

An interest rate cap is effectively a call on interest rates. If interest rates exceed the strike rate, the cap expires in-the-money. Just as a put option... 

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. 

Convexity is a positive number for most normal bonds. However, for bonds with embedded call options such as mortgage-backed securities, it is always negative. Intuitively, it is obvious that if interest rates fall, the bond prices rise and the option to call the bond turns in the money and is often exercised, which shortens the duration of the bond and hence the rate of change of duration with respect to change in yields is negative. 

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. 

In-the-money The term for an option that has intrinsic value... 

The strike price. Since the deeper in the money the option, the more likely it is to be exercised, the difference between the strike and the underlying asset s price when the option is struck influences the size of the premium. 

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... 

Pricing an option therefore requires knowing the value of both />, the probability that the option will expire in the money, and E[Sr 5t- > A] — X, its expected payoff should this happen. In calculating/, the probability function is modeled. This requires assuming that asset prices follow a stochastic process. 

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