Options strike price

The simplest hedge against a drop in bond prices is to buy put options on the bond. Let s first consider using the June 2003 put option struck at 116.50, which are priced at 1.26, or 627,480 for 498 contracts. Exhibit 17.20 shows the payoff profile for the bond, the put option, and the combination. The kink in the option payoff profile at an option strike price of 116.50 corresponds to a price of around 105.75 in the bond being hedged. 

P = the bond s value at the specified node S = the call option strike price, determined by the call schedule... 

As noted in Chapter 8, the value of an option is a function of five factors The price of the underlying asset The options strike price The options time to expiry The volatility of the underlying assets price returns The risk-free interest rate applicable to the life of the option... 

Before-tax profit increases to 3707 if the average annual price appreciation doubles to 14%. Anyone holding this kind of option stands only to profit, not to lose. If the stock fails to rise beyond the strike price, the options expire unused. 

The owner of a swaption with strike price K maturing at time 7b, has the right to enter at time To the underlying forward swap settled in arreas. A swaption may also be seen as an option on a coupon bearing bond (see e.g. Musiela and Rutkowski [61]). 

In the following, we derive a theoretical pricing framework for the computation of options on bond applying standard Fourier inversion techniques. Starting with a plain vanilla European option on a zero-coupon bond with the strike price K, maturity T of the underlying bond and exercise date To of the option, we have... 

This implies that the payoff of a caplet clet t, Tq, 7i) = leti To) is equivalent to a put option on a zero-coupon bond P t,T) with face value = 1 + ACR and a strike price AT = 1. Therefore, we obtain the date-t price of a caplet... 

In section (5.2.1), we have derived the closed-form solution for the price of a zero-coupon bond option. Then, later on in section (5.2.2) we introduced the FRFT-technique and showed that this method works excellent for a wide range of strike prices by solving the Fourier inversion numerically. Now, we show that also the IFF is an efficient and accurate method to compute the single exercise probabilities (see section (2.2)) for a G 0,1 via... 

Putting the strike price equal to the promised payment F, when the firm value is greater than that amount, the value of the option is 0 and the debt holder will receive the face value conversely, the value of debt at maturity is lower than face value and the option has value. In the second case, the bondholders receive a payment value equal to the firm value and shareholders get nothing. 

Therefore, at maturity T, the value of a call option is determined as the relationship between the stock price St of the underlying asset and the strike price X as follows (Equation 9.8) ... 

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. 

Determine the Value of an Embedded Call Option After determining the value of an option-free bond, we calculate the value of the option element. On the maturity, the value of the option is 0 because the bond s ex coupon value is 100 and equal to the strike price. In other nodes, the option has value if the strike price is less than bond s price. The strike price for each node is shown in Table 11.3. Consider also that the value of the option decreases as the bond approaches maturity due to the decreasing probability to redeem the bond. Figure 11.10 shows the value of a call option. The holder of the option has substantially the choice to exercise the option or wait a further period. Therefore, the value of the option if exercised is given by (11.7) ... 

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. 

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 an Embedded Put Option Conversely to a callable bond, the embedded option of a putable bond is a put option. Therefore, the value is estimated as the maximum between 0 and the difference between the strike price and bond s price. The strike price is defined according to the put schedule, while the bond s price is the value of the option-free bond at each node as shown in Figure 11.9. The value at maturity of a putable option if exercised is given by Formula (11.10) ... 

Strike Price (or Exercise Price)— The price, fixed at the outset, at which the underlying asset may be bought (for a call) or sold (for a put). In the previous example, the option was a call struck at 98, giving the holder the right to buy the bond at this 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. 

Intrinsic Value—This is the value that would be realised if the option were exercised right now at prevailing market prices, provided that exercise was worthwhile. For example, a call option struck at 98 would have an intrinsic value of 3 if the underlying bond were trading at 101. The put option at the same strike price would have an intrinsic value of 0, however, not -3, as it would not be worth exercising. 

Each of these is an option on the underlying futures. If an investor exercises one call option contract, he or she will be assigned a long position of one contract in the underlying future, at the strike price. Equally, after exercising one put option contract, an investor will be assigned a... 

The longer the option s time to expiry, the more valuable is the option. In all cases, the options expiring in May are more valuable than those expiring in April with the same strike price and, similarly, June options are more valuable still. This is sensible when one thinks of the risk experienced by the option seller. The further the expiry date, the greater the uncertainty as to where the price of the underlying bond will be in the future, and therefore the more valuable the right to execute a trade in the future at a price fixed at the outset. 

The STIR option—being a call in this example— will only be worth exercising if the futures price rises above the strike price of 97.750. If, for example, the option is exercised when the June 2003 contract is trading at 98.000, the investor s margin account would be credited with 625 upon exercise. 

The exhibit shows the option premiums quoted to the nearest 0.005 and strike prices separated by 0.125 intervals as required by the contract specification. The underlying June 2003 future settled at 97.745 that day (implying a 3-month interest rate in June of 2.255%), so the table highlights the 97.750 ATM options. 

The most commonly used volatility strategy is to buy a straddle, which involves buying both a call and a put option at the same strike price. At maturity, this will give a V-shaped profile. However, straddles are normally short-term strategies, and are seldom held until maturity. For this reason, it is more relevant to look at the straddle profile in the short-term, well before the maturity of the options. Exhibit 17.25 shows the curved profiles of the straddle components—long a call and long a put option—when the straddle is created. The combination of the two gives the straddle a U-shaped profile at the outset as the chart illustrates. 

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... 

We now revisit the earlier Vasicek example for short interest rates to consider the case where the underlying bond pays an annual coupon at a 5% rate (p = 0.05), all the other characteristics remain as before. In order to calculate the call price of the coupon-bond European option first we need to calculate the interest rate such that the present value at the maturity of the option of all later cash flows on the bond equals the strike price. This is done by trial and error using equation (18.48) and the value we get here is = 22.30%. Next, we map the strike price into a series of strike prices via equation (18.50) that are then associated with coupon payments considered as zero-coupon bonds and calculate the value of the European call options contingent on those zero-coupon bonds as in the above example. The calculations are described in Exhibit 18.7. 

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