Fixed income basic notions and randomization - Статья

FIXED INCOME BASIC NOTIONS and RANDOMIZATION Ilya Gikhman 6077 Ivy Woods Court Mason OH 45040 USA Ph. 513-573-9348 Abstract. In this paper, we outline a randomization of the primary fixed income notions. We present a construction of some stochastic interest rate models. We also consider forward rates which are implied by stochastic bond prices. We highlight to major drawbacks of the commonly used stochastic models. The first drawback is the theoretical possibility that bond admits higher than its face value prior to maturity. The second drawback is related to modeling itself. We model stochastic interest rate in such a way that it would be consistent with deterministic definition of the bond while some popular are not. In the next paper we will pay more attention to FX and a formal definition of the LIBOR rate. Here LIBOR rate is assumed to be known. Key words. Stochastic interest rate, stochastic forward rate, LIBOR. 1. In this section, we present well known basic notions and definitions. Denote B ( t , T ) zero coupon default free bond price at t with expiration at date T and B ( T , T ) = 1. The simple example of the interest rate contract is a government debt security, which for brevity we will call bond. The simple interest rate model of the bond pricing is: B ( t , T ) [ 1 i ( t , T ) ( T - t ) ] = B ( T , T ) Here T - t is taking in 365 ( 360 ) day year format, i.e. (T - t) year fraction = (T - t) days / 365 Thus, simple interest rate over [t , T] period is defined as: i ( t , T ) ( T - t ) = - 1 In this formula the simple interest rate i (t , T) is completely defined by the value B (t , T) at t. Continuous time model of the bond price is governed by the equation: d B ( t , T ) = r ( t , T ) B ( t , T ) d t (1) t I [0 , T] in which we put B (T , T) = 1. Function r (t , T) > 0 in (1) is called continuously compounded interest rate. It is by definition the interest rate over infinitesimal short time period [ t, t , D t ]. In a continuous time the notation r is referred to as to money-market account. Let us briefly recall a discrete scheme bond equation solution construction. This construction can also be used in the stochastic setting. Let: t = t 0 0 Is a known constant that can be interpreted as instantaneous rate at date T which value actually does not known at t. It is common practice to use implied date-t forward rate as the estimate of the forward rate at T. Formally, in continuous time constant ? is the limit of the interest rate over a small period [T , T D] when the length of the period D > 0 tends to 0. We return to this problem in grater details later when we will study this problem in stochastic setting. Now suppose that the coefficient of the above equations (3), (3?) satisfy the conditions: ? ( t , 0 ) = 0 or ? ( t , T , 0 ) = 0 In this case solutions of the equations (3) , (3?) are positive functions that guarantee that: B ( t , T ) ? 1 Let us introduce a forward rate construction that is implied by the bond prices. In this construction we admit that the bond prices or spot rates are observable data. These two alternatives start points are equivalent to each other. We introduce first the forward rate concept. For given time moments 0 ? t ? T ? T h we will distinct at t the rate: r ( T T h , w ) Which will be known T and the date-t forward rate over the future interval [ T , T h ]. Assuming that B ( t , T ) is a given function we define the implied forward rate f ( t , T , T h ) at the moment t over a period [ T , T h ] as a solution of the equation: B ( t , T h ) = B ( t , T ) [ 1 f ( t , T , T h ) h ] - 1 Then: f ( t , T , T h ) h = - 1 = - (4) It follows from that date-t instantaneous h v 0 implied forward rate at T is: f ( t , T ) = - Denote: t = t 0 0 is a scalar Wiener process on this space. Elements ? I W are called in finance market scenarios. Define s-algebra F t , which by definition is the minimal s-algebra generated by the values of Wiener process w ( s ) , 0 ? s ? t .