El voluntariado en la educación superior

Let T^∗> 0be a fixed horizon date for all market activities. By a zero-coupon bond (a discount bond) of maturity T we mean a financial security paying to its holder one unit of cash at a prespecified date T in the future. This means that, by convention, the bond’s principal (known also as face value or nominal value) is one dollar. We assume throughout that bonds are default-free; that is, the possibility of default by the bond’s issuer is excluded. The price of a zero-coupon bond of maturity T at any instant t≤ T will be denoted by B(t, T ); it is thus obvious that B(T, T ) = 1 for any maturity date T ≤ T^∗. Since there are no other payments to the holder, in practice a discount bond sells for less than the principal before maturity – that is, at a discount (hence its name). This is because one could carry cash at virtually no cost, and thus would have no incentive to invest in a discount bond costing more than its face value. We shall usually assume that, for any fixed maturity T ≤ T^∗, the bond price B(·, T ) follows a strictly positive and adapted process on a filtered probability space (Ω, F, P).

117

7.1.1 Term Structure of Interest Rates

Let us consider a zero-coupon bond with a fixed maturity date T ≤ T^∗. The simple rate of return from holding the bond over the time interval [t, T ] equals

1− B(t, T )

B(t, T ) = 1

B(t, T )− 1.

The equivalent rate of return, with continuous compounding, is commonly referred to as a (contin-uously compounded) yield-to-maturity on a bond. Formally, we have the following definition.

Definition 7.1.1 An adapted process Y (t, T ) defined by the formula Y (t, T ) =− 1

T− t ln B(t, T ), ∀ t ∈ [0, T ), (7.1) is called the yield-to-maturity on a zero-coupon bond maturing at time T.

The term structure of interest rates, known also as the yield curve, is the function that relates the yield Y (t, T ) to maturity T. It is obvious that, for arbitrary fixed maturity date T, there is a one-to-one correspondence between the bond price process B(t, T ) and its yield-to-maturity process Y (t, T ). Given the yield-to-maturity process Y (t, T ), the corresponding bond price process B(t, T ) is uniquely determined by the formula¹

B(t, T ) = e−Y (t,T )(T −t), ∀ t ∈ [0, T ]. (7.2) The discount function relates the discount bond price B(t, T ) to maturity T. At the theoretical level, the initial term structure of interest rates may be represented either by the family of current bond prices B(0, T ), or by the initial yield curve Y (0, T ), as

B(0, T ) = e^{−Y (0,T )T}, ∀ T ∈ [0, T^∗]. (7.3) In practice, the term structure of interest rates is derived from the prices of several actively traded interest rate instruments, such as Treasury bills, Treasury bonds, swaps and futures. Note that the yield curve at any given day is determined exclusively by market prices quoted on that day.

The shape of an historically observed yield curve varies over time; the observed yield curve may be upward sloping, flat, descending, or humped. There is also strong empirical evidence that the movements of yields of different maturities are not perfectly correlated. Also, the short-term interest rates fluctuate more than long-term rates; this may be partially explained by the typical shape of the term structure of yield volatilities, which is downward sloping. These features mean that the construction of a reliable model for stochastic behavior of the term structure of interest rates is a task of considerable complexity.

7.1.2 Forward Interest Rates

Let f (t, T ) be the forward interest rate at date t≤ T for instantaneous risk-free borrowing or lending at date T. Intuitively, f (t, T ) should be interpreted as the interest rate over the infinitesimal time interval [T, T + dT ] as seen from time t. As such, f (t, T ) will be referred to as the instantaneous, continuously compounded forward rate, or shortly, instantaneous forward rate. In contrast to bond prices, the concept of an instantaneous forward rate is a mathematical idealization rather than a quantity observable in practice. As we shall see in what follows, however, one of the popular approaches to the bond price modelling is based on the exogenous specification of a family f (t, T ), t≤ T ≤ T^∗, of instantaneous forward interest rates. Given such a family f (t, T ), the bond prices are then defined by setting

B(t, T ) = exp

−

 _T

t

f (t, u) du

, ∀ t ∈ [0, T ]. (7.4)

1We assume that the limit of Y (t, T ), as t tends to T, exists.

7.1. ZERO-COUPON BONDS 119 On the other hand, if the family of bond prices B(t, T ) is sufficiently smooth with respect to maturity T, the implied instantaneous forward interest rate f (t, T ) is given by the formula

f (t, T ) =−∂ ln B(t, T )

∂T (7.5)

which, indeed, can be seen as the formal definition of the instantaneous forward rate f (t, T ).

Alternatively, the instantaneous forward rate can be seen as a limit case of a forward rate f (t, T, U ) which prevails at time t for riskless borrowing or lending over the future time interval [T, U ]. The rate f (t, T, U ) is in turn tied to the zero-coupon bond price by means of the formula

B(t, U )

B(t, T ) = e−f(t,T,U)(U−T ), ∀ t ≤ T ≤ U, (7.6) or equivalently

f (t, T, U ) = ln B(t, T )− ln B(t, U)

U− T . (7.7)

Observe that Y (t, T ) = f (t, t, T ), as expected – indeed, investing at time t in T -maturity bonds is clearly equivalent to lending money over the time interval [t, T ]. On the other hand, under suitable technical assumptions, the convergence f (t, T ) = lim_U↓Tf (t, T, U ) holds for every t ≤ T. For convenience, we focus on interest rates that are subject to continuous compounding. In practice, interest rates are quoted on an actuarial basis, rather than as continuously compounded rates. For instance, the actuarial rate (or effective rate) a(t, T ) at time t for maturity T (i.e., over the time interval [t, T ]) is given by the following relationship

(1 + a(t, T ))^T−t= ef (t,t,T )(T−t)= e^{Y (t,T )(T−t)}, ∀ t ≤ T.

This means, of course, that the bond price B(t, T ) equals B(t, T ) = 1

(1 + a(t, T ))^T−t, ∀ t ≤ T.

Similarly, the forward actuarial rate a(t, T, U ) prevailing at time t over the future time period [T, U ] is set to satisfy

(1 + a(t, T, U ))^U−T = exp

f (t, T, U )(U− T )

= B(t, T )/B(t, U ).

7.1.3 Short-term Interest Rate

Most traditional stochastic interest rate models are based on the exogenous specification of a short-term rate of interest. We write rt to denote the instantaneous interest rate (also referred to as a short-term interest rate, or spot interest rate ) for risk-free borrowing or lending prevailing at time t over the infinitesimal time interval [t, t + dt]. In a stochastic setup, the short-term interest rate is modelled as an adapted process, say r, defined on a filtered probability space (Ω, F, P) for some T^∗> 0. We assume throughout that r is a stochastic process with almost all sample paths integrable on [0, T^∗] with respect to the Lebesgue measure. We may then introduce an adapted process B of finite variation and with continuous sample paths, given by the formula

B_t= exp

 ^t

0

r_udu

, ∀ t ∈ [0, T^∗]. (7.8)

Equivalently, for almost all ω∈ Ω, the function Bt= Bt(ω) solves the differential equation dBt= rtBtdt, with the conventional initial condition B0 = 1. In financial interpretation, B represents the price process of a risk-free security which continuously compounds in value at the rate r. The process B is referred to as an accumulation factor or a savings account in what follows. Intuitively, Btrepresents the amount of cash accumulated up to time t by starting with one unit of cash at time 0, and continually rolling over a bond with infinitesimal time to maturity.