Zero-coupon bonds

Cox, Ingersoll and Ross [22] and Jamshidian [42] demonstrate that closed-form solutions for zero-coupon bond options can be derived for single-factor square root and Gaussian models. More generally, Duffie, Pan and Singleton... 

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. 

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... 

The change in measure amounts to a change in numeraire by using a zero-coupon bond P t, Ti) with a specified maturity. Mathematically, the change of measure modifies the probability such that the expectation of the product can be computed as the product of the expectations under the new measure. 

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)). 

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... 

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... 

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... 

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 continuously compounded gross redemption yield at time ton a default-free zero-coupon bond that pays 1 at maturity date 7 is x. We assume that the movement in X is described by... 

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... 

Of course, interest rates are not constant but Equation (3.1) is valuable as it is used later in constructing a model. By using Equation (3.1), we are able to produce a yield curve, given a set of zero-coupon bond prices. For modelling purposes, we require a definition of the short rate, or the current interest rate for borrowing a sum of money that is paid back a very short period later (in fact, almost instantaneously). This is the rate payable at time t for repayment at time t+M where Af is an incremental passage of time. This is given by... 

This is convenient because this means that the price at time t of a zero-coupon bond maturing at T is given by Equation (3.7), and forward rates can be calculated from the current term structure or vice versa. 

In this section, we describe the relationship between the price of a zero-coupon bond and spot and forward rates. We assume a risk-free zero-coupon bond of nominal value 1, priced at time t and maturing at time T. We also assume a money market bank account of initial value P t, T) invested at time t. The money market account is denoted M. The price of the bond at time t is denoted P t, T) and if today is time 0 (so that t > 0), then the bmid price today is unknown and a random factor (similar to a future interest rate). The bond price can be related to the spot rate or forward rate that is in force at time t. 

Consider the scenario below, used to derive the risk-free zero-coupon bond price. ... 

The continuously compounded constant spot rate is r as before. An investor has a choice of purchasing the zero-coupon bond at price P(t, T), which will return the sum of 1 at time T, or of investing this same amount of cash in the money market account, and this sum would have grown to 1 at time T. We know that the value of the money market accoxmt is given by Me If M must have a... 

The zero-coupon bond price may also be given in terms of the spot rate r(t, T), as shown in Equation (3.18). From our earlier analysis, we know that... 

Equation (3.22) describes the bond price as a function of the spot rate only, as opposed to the multiple processes that apply for aU the forward rates from t to T. As the bond has a nominal value of 1, the value given by Equation (3.22) is the discount factor for that term the range of zero-coupon bond prices would give us the discount function. 

A short-rate model can be used to derive a complete term structure. We can illustrate this by showing how the model can be used to price discount bonds of any maturity. The derivation is not shown here. Let P t, T) be the price of a risk-free zero-coupon bond at time t maturing at time T that has a maturity value of 1. This price is a random process, although we know that the price at time T will be 1. Assume that an investor holds this bond, which has been financed by borrowing funds of value C,. Therefore, at any time t the value of the short cash position must be C,= —P(t, T) otherwise, there would be an arbitrage position. The value of the short cash position is growing at a rate dictated by the short-term risk-free rate r, and this rate is given by... 

Therefore, once we have a full description of the random behaviour of the short-rate r, we can calculate the price and yield of any zero-coupon bond at any time, by calculating this expected value. The implication is clear specifying the process r t) determines the behaviour of the entire term structure so, if we wish to build a term structure model, we need only (under these assumptions) specify the process for r t). 

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