Time Value of Money - Differential Equations

Differential Equations

Ordinary and partial differential equations (ODEs and PDEs) – equations involving derivatives and one (respectively, multiple) variables are ubiquitous in more advanced treatments of financial mathematics. While time value of money can be understood without using the framework of differential equations, the added sophistication sheds additional light on time value, and provides a simple introduction before considering more complicated and less familiar situations. This exposition follows (Carr & Flesaker 2006, pp. 6–7).

The fundamental change that the differential equation perspective brings is that, rather than computing a number (the present value now), one computes a function (the present value now or at any point in future). This function may then be analyzed – how does its value change over time – or compared with other functions.

Formally, the statement that "value decreases over time" is given by defining the linear differential operator as:

This states that values decreases (−) over time (∂_t) at the discount rate (r(t)). Applied to a function it yields:

For an instrument whose payment stream is described by f(t), the value V(t) satisfies the inhomogeneous first-order ODE ("inhomogeneous" is because one has f rather than 0, and "first-order" is because one has first derivatives but no higher derivatives) – this encodes the fact that when any cash flow occurs, the value of the instrument changes by the value of the cash flow (if you receive a $10 coupon, the remaining value decreases by exactly $10).

The standard technique tool in the analysis of ODEs is the use of Green's functions, from which other solutions can be built. In terms of time value of money, the Green's function (for the time value ODE) is the value of a bond paying $1 at a single point in time u – the value of any other stream of cash flows can then be obtained by taking combinations of this basic cash flow. In mathematical terms, this instantaneous cash flow is modeled as a delta function

The Green's function for the value at time t of a $1 cash flow at time u is

where H is the Heaviside step function – the notation "" is to emphasize that u is a parameter (fixed in any instance – the time when the cash flow will occur), while t is a variable (time). In other words, future cash flows are exponentially discounted (exp) by the sum (integral, ) of the future discount rates ( for future, r(v) for discount rates), while past cash flows are worth 0, because they have already occurred. Note that the value at the moment of a cash flow is not well-defined – there is a discontinuity at that point, and one can use a convention (assume cash flows have already occurred, or not already occurred), or simply not define the value at that point.

In case the discount rate is constant, this simplifies to

where is "time remaining until cash flow".

Thus for a stream of cash flows f(u) ending by time T (which can be set to for no time horizon) the value at time t, is given by combining the values of these individual cash flows:

This formalizes time value of money to future values of cash flows with varying discount rates, and is the basis of many formulas in financial mathematics, such as the Black–Scholes formula with varying interest rates.