Pricing of zero-coupon bond options

Starting from the risk-neutral bond price dynamics (5.4), we derive the well known closed-form solution for the price of a zero-coupon bond option. Thus, as shown in section (2.1) the price of a call option on a discount bond is given by 

the abiUty to derive a closed-form of t z) crucially depends on the ability to find a solution for the transform lj(z). 

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In chapter (2), we derive a unified framework for the computation of the price of an option on a zero-coupon bond and a coupon bond by applying the well known Fourier inversion scheme. Therefore, we introduce the transform t (z), which later on can be seen as a characteristic function. In case of zero-coupon bond options we are able to find a closed-form solution for the transform t z) and apply standard Fourier inversion techniques. Unfortunately, assuming a multi-factor framework there exists no closed-form solution of the characteristic function Et z) given a coupon bond option. Hence, in this case Fourier inversion techniques fail. 

The zero-correlation (y = ) price of a coupon bond option with a moneyness 1.14 is about 1.7 times as high as the corresponding option price obtained by a perfect correlation stnjcture (y = 0). The corresponding zero-coupon bond option price is about 60 times as high as its perfect correlation equivalent. 

In this thesis we derived new methods for the pricing of fixed income derivatives, especially for zero-coupon bond options (caps/floor) and coupon bond options (swaptions). These options are the most widely traded interest rate derivatives. In general caps/floors can be seen as a portfolio of zero-coupon bond options, whereas a swaption effectively equals an option on a coupon bond (see chapter (2)). The market of these LIBOR-based interest rate derivatives is tremendous (more than 10 trillion USD in notional value) and therefore accurate and efficient pricing methods are of enormous practical importance. 

The price process under the new measure Tq, either is used to derive the formula for the zero-coupon bond option (see section (5.2.1)), the characteristic function in (5.2.2), or finally to compute the moments of the underlying random variable (section (5.3.3) and (5.3.4)). 

Following the last section, we introduce the FRFT technique to derive the price of a zero-coupon bond option. In doing so, we are able to compare the option price coming from the FRFT approach with the appropriate closed-form solution (5.27). 

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... 

Following chapter (5.2) we obtain the price of a zero-coupon bond option by computing the risk-neutral probabilities... 

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. 

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... 

Thus, we end up with the well known Black and Scholes -like formula for the price of a European call option on a zero-coupon bond... 

We first set the scene by introducing the interest-rate market. The price of a zero-coupon bond of maturity T at time t is denoted by P(t, T) so that its price at time 0 is denoted by / (O, T). The process followed by the bond price is a stochastic one and therefore can be modelled equally, options that have been written on the bond can be hedged by it. If market interest rates are constant, the price of the bond at time t is given by This enables us to state that given... 

An example will provide an idea of how a variation of one of the models proposed by Hull and White described above by the first of equation (18.12) models can be nsed to price an option on a zero-coupon bond. If the assumptions are made that both P, the reversion rate, and o, the volatility, are constant then the model can be restated as... 

We will use the Vasicek model for pricing a 3-year European call option on a 10-year zero-coupon bond with face value 1 and exercise price K equal to 0.5. As in Jackson and Staunton, we use for the parameters of this model the values estimated by Chan, Karolyi, Longstaff, and Sanders for US 1-month Treasury bill yield from 1964 to 1989. Thus a = 0.0154, p = 0.1779, and o = 2%. In addition, the value of the short... 

EXHIBIT 18.6 Calculations of Elements for Pricing an European Call Option on a Zero-Coupon Bond when Short Rates are Following the Vasicek Model... 

The value of the call option is 0.335 that is the sum of all zero-coupon bond call option prices in the last column. 

P t, T) = the price at time r of a zero-coupon bond maturing at time T f = the forward price of the underlying asset with maturity T f = the forward price at time t X = the strike price of the option N = normal distribution... 

The theoretical price of a call option written on a zero-coupon bond is calculated using equation (8.28). 

For an underlying coupon-paying bond, the equation must be modified by reducing P by the present value of all coupons paid during the life of the option. This reflects the fact that prices of call options on couponpaying bonds are often lower than those of similar options on zero-coupon bonds because the coupon payments make holding the bonds themselves more attractive than holding options on them. 

Assume now that the one-year zero-coupon bond in the example has a call option written on it that matures in six months (at period 1) and has a strike price of 97.40. FIGURE 11.5 is the binomial tree for this option, based on the binomial lattice for the one-year bond in figure 11.4. The figure shows that at period 1, if the six-month rate is 5.50 percent, the call option has no value, because the bond s price is below the strike price. If, on the other hand, the six-month rate is at the lower level, the option has a value of 97.5562 - 94.40, or 0.1562. 

What is the value of this option at point 0 Option pricing theory states that to calculate this, you must compute the value of a replicating porlfolio. In this case, the replicating portfolio would consist of six-month and one-year zero-coupon bonds whose combined value at period 1 will be zero if the six-month rate rises to 5.50 percent and 0.1562 if the rate at that time is 5.01 percent. It is the return that is being replicated. These conditions are stated formally in equations (11.4) and (11.5), respectively. 

Solving the two equations gives Cj = —65.3566 and C2 = 67.1539. This means that to construct the replicating portfolio, you must purchase 67.15 of one-year zero-coupon bonds and sell short 65-36 of the six-month zero-coupon bond. The reason for constructing the portfolio, however, was to price the option. The portfolio and the option have equal values. The portfolio value is known it is the price of the six-month bond at period 0 multiplied by Cj, plus the price of the one-year bond multiplied by Cj, or... 

Thus far our coverage of valuation has been on fixed-rate coupon bonds. In this section we look at how to value credit-risky floaters. We begin our valuation discussion with the simplest possible case—a default risk-free floater with no embedded options. Suppose the floater pays cash flows quarterly and the coupon formula is 3-month LIBOR flat (i.e., the quoted margin is zero). The coupon reset and payment dates are assumed to coincide. Under these idealized circumstances, the floater s price will always equal par on the coupon reset dates. This result holds because the floater s new coupon rate is always reset to reflect the current market rate (e.g., 3-month LIBOR). Accordingly, on each coupon reset date, any change in interest rates (via the reference rate) is also reflected in the size of the floater s coupon payment. 

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