Discount bond options

Discount bond options are not very liquid, but they form an elementary component for pricing other options. For example, a floating-rate cap can be decomposed into a portfolio of European puts on discount bonds. Similarly with the European option contingent on the bank account we can price European options contingent to discount bonds. 

One notable exception from this general class is the CIR model. There is a closed-formula for this case too. Following Clewlow and Strickland the price at time 0 of a European pure discount bond option is... 

The transforms 0t(z) and S, (z), by itself can be seen as a modified characteristic function. Unfortunately, there exists no closed-form of the transform S,(z), meaning that the standard Fourier inversion techniques can be applied only for the computation of options on discount bonds. On the other hand, the transform S,(n) can he used to compute the n-th moments of the underlying random variable V (7b, 7 ). Then, by plugging the moments (cumulants) in the lEE scheme the price of an option on coupon hearing bond can be computed, even in a multi-factor framework. 

In this section, we start from a simple multi-factor HJM term structure model and derive the drift term of the forward rate dynamics required to obtain an arbitrage-free model framework (see HJM [35]). Furthermore, we derive the equivalence between the HJM-firamework and a corresponding extended short rate model. Then, by applying our option pricing technique (see chapter (2)) we are able derive the well known closed-form solution for the price of an option on a discount bond (e.g. caplet or floorlet). 

Now, by applying the Fourier inversion technique we derive the well known formula for the price of an option on a discount bond. Therefore, we first compute the exponential affine solution of the transform... 

Fig. 7.1 Relative deviation between the USV and the average variance model computing an option on a discount bond for 0 = vo...

This is the same model as that proposed by Rendleman and Bartter. This model is the only log-normal, single-factor model that leads to closed formulae for pure discount bonds. Nonetheless there is no closed formula for a European option on a pure discount bond. 

This model is similar to Dothan model being a log-normal short rate model. This model, however, does not lead to explicit formulae for pure discount bonds or for options contingent on them. In addition this is an... 

Bonds are traded generally over the counter. Futures contracts on bonds may be more liquid and may remove some of the modelling difficulties generated by the known value at maturity of the bonds. Hedging may be more efficient in this context using the futures contracts on pure discount bonds (provided they are liquid) rather than the bonds themselves. Chen provides closed-form solutions for futures and European futures options on pure discount bonds, under the Vasicek model. 

Ren-Raw Chen, Exact Solutions for Futures and European Futures Options on Pure Discount Bonds, Journal of Financial and Quantitative Analysis 27, no. 1 (March 1992), pp. 97-107. 

There is a similar put-call parity for European options contingent on a discount bond. If Pp(0, t) Tq-,k he price at r = 0 of a European put option on the discount bond with maturity T, then for B(0) = 1,... 

Initially the first formulas on pricing options on pure discount bonds used the Vasicek model for the term structure of interest rates. Thus, given that r follows equation (18.6), the price of a European call option with maturity Tq with exercise price fC on a discount bond maturing at T(Tq < T) is... 

Taking the same example as that developed to demonstrate the Vasicek model earlier, we now price the 3-year European call option on a 10-year pure discount bond using the CIR model for the short interest rates. Recall that face value is 1 and exercise price K is equal to 0.5. As in the example with the Vasicek model, we consider that o = 2% and tq = 3.75%. The CIR model overcomes the problem of negative interest rates (acknowledged as a problem for the Vasicek model) as long as 2a > o. This is true, for example, if we take a = 0.0189 and P = 0.24. Feeding this information into the above formulae is relatively tedious. A spreadsheet application is provided by Jackson and Staunton, After some work we get that the price of the call is... 

When short rates are modelled with single-factor models, Jamshidian proved that an option on a coupon bond can be priced by valuing a portfolio of options on discount bonds. This approach does not work in multifactor models as proved by El Karoui and Rochet." ... 

Under the one-factor HJM model corresponding to the Ho-Lee model, a European option on a coupon bond can be valued as a portfolio of options contingent on zero discount bonds with maturities Ti,T2,...,Tm- Let Tq be the maturity of such a European option. 

As explained in the introduction, the value of a convertible bond is the sum of two main components, the option-free bond and a call option on underlying security. The value of the option-free bond, or bond floor, is determined as the sum of future payments (coupon and principal at maturity). Therefore, the bond component is influenced by three main parameters, that is the maturity, the coupon percentage on par value and the yield to maturity (discount rate). Differently, the value of a call option can be found mainly through two option pricing models, Black Scholes model and binomial tree model. 

Craisider a hypothetical situation. Assume that an option-free bond paying a semi-annual coupon 5.5% on par value, with a maturity of 5 years and discount rate of 8.04% (EUR 5-year swap rate of 1.04% plus credit spread of 700 basis points). Therefore, the valuation of a conventional bond is performed as follows (Figure 9.4). 

Exhibit 3.1 depicts this inverse relationship between an option-free bond s price and its discount rate (i.e., required yield). There are two things to infer from the price/discount rate relationship depicted in the exhibit. First, the relationship is downward sloping. This is simply the inverse relationship between present values and discount rates at work. Second, the relationship is represented as a curve rather than a straight line. In fact, the shape of the curve in Exhibit 3.1 is referred to as convex. By convex, it simply means the curve is bowed in relative to the origin. This second observation raises two questions about the convex or curved shape of the price/discount rate relationship. First, why is it curved Second, what is the import of the curvature ... 

Price/Discount Rate Relationship for an Option-Free Bond... 

There are valuation models that can be used to value bonds with embedded options. These models take into account how changes in yield will affect the expected cash flows. Thus, when V and V+ are the values produced from these valuation models, the resulting duration takes into account both the discounting at different interest rates and how the expected cash flows may change. When duration is calculated in this manner, it is referred to as effective duration or option-adjusted duration or OAS duration. Below we explain how effective duration is calculated based on the lattice model and the Monte Carlo model. 

Solving equation (11.6) gives p = 0.5926 and - p - 0.4074. These are the two probabilities for which the probability-weighted average, or expected, value of the bond discounts to the true market price. These risk-neutral probabilities can be used to derive a probability-weighted expected value for the option in figure 11.5 at point 1, which can be discounted at the six-month rate to give the option s price at point 0. The process is shown in (11.8). 

The market quotes bonds with embedded options in terms of yield spreads. A cheap bond trades at a high spread, a dear one at a low spread. The usual convention is to quote the spread between the redemption yield of the bond being analyzed and that of a government bond having an equivalent maturity. This is not an accurate measure of the actual difference in value between the two bonds, however. The reason is that, as explained in chapter 1, the redemption yield computation unrealistically discounts all a bond s cash flows at a single rate. 

All bond instruments are characterized by the promise to pay a stream of future cash flows. The term structure of interest rates and associated discount function is crucial to the valuation of any debt security and underpins any valuation framework. Armed with the term structure, we can value any bond, assuming it is liquid and default-free, by breaking it down into a set of cash flows and valuing each cash flow with the appropriate discount factor. Further characteristics of any bond, such as an element of default risk or embedded option, are valued incrementally over its discounted cash flow valuation. 

The convertible price accounts for both the conventional bond element and the embedded option element. If we assume the share price in period t< is 97.01, then in period 6o the share can assume only one of two possible values, 106.25 or 92.24 (see Figure 13.6). In these cases, the value of the call option Ch and Cl will be equal to the higher of the bonds conversion value or its redemption value, which is 106.25 if there is a rise in the price of the underlying or 102.50 if there is a fall in the price of the underlying. This is the range of possible final values for the bond however, we require the current (present) value, so we discount this at the appropriate rate. 

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