Risk Latte - Laplace Transforms and the Time Value of Money - I

Laplace Transforms and the Time Value of Money - I

Team Latte
December 25, 2010

Recently, we asked a CFE Quiz on our facebook page regarding the application of Laplace transform in quantitative finance and we received a large number of emails from readers and CFE students who attempted the quiz. The question was: What is the simplest and yet, the most profound application of Laplace Transform in Finance? Explain very briefly.

The simplest and yet, the most profound application of Laplace transform is to estimate the time value of money.

Laplace Transforms are used to convert time domain relationships to a set of equations expressed in terms of the Laplace operator . After the transformation, the solution of the original problemis arrived at by simple algebraic manipulations in the (Laplace) domain rather than the timedomain.

But why do we need to do such a transformation? The quick answer is to simply mathematical calculations. It is a bit like why we use logarithms in mathematical calculations. Logarithms simply the math considerably and makes a problem more tractable. When we take logarithms we transform numbers to the power of 10 or some other base, say, , which becomes natural logarithm. What we achieve by this is to transform mathematical manipulations and divisions to simple additions and subtractions. Similarly, Laplace transforms can be applied to linear, differential equations in a way that the differential equations are transformed into simple algebraic equations which can be solved easily.

The Laplace Transform of a time variable function,, is defined as:

   ...............(1)

Now think of the present value problem in Finance. One of the most fundamental and elementary problems in finance is to estimate the present value of a future cash flow. If the discount rate (interest rate) is constant and equal to then the present value of a future cash flow, , where, is a function of time, is given by:

   ...............(2)

In the above we have assumed continuous compounding and present value function, is a function of t. The time is bounded between 0 and some finite quantity, . In the limiting case, the summation is replaced by an integral and the above present value equation can be expressed as:

   ...............(3)

Due to the presence of the integral, the domain of the computation changes from time, to rate, and therefore, the present value becomes a function of the rate, . However, the boundary of the integral is still from 0 to some finite quantity, .

Now, if we change the upper bound to infinity, i.e. , then the definite integral will become:

   ...............(4)

Note, that equation (4) is now an exact replica of equation (1), where the “” (rate) domain acts like the Laplace domain, “”. In fact, we can write equation (4) as:

   ...............(4)

Therefore, the present value of a future cash flow is the Laplace transform of the cash flow .

Of course, one needs to be cognizant of the fact that the bounds of the integral above are from 0 to infinity. In other words, the Laplace transform equation, (3), exactly translates into the discrete time present value equation (2) only when we are considering a very long period of time.

If the cash flow is $1 then by equation (4), and if the cash flow is (where, is constant) then equation (4) will yield . Actually, this can be very easily verified using an Excel spreadsheet. Choose any interest rate, say, 5% and choose a cash flow equal to $100. Then, over say, a 100 year period (100 years is long enough to be the real life equivalent of infinity) the present value of all the cash flows (summed up over 100 years) would be equal to $1,985. Using the Laplace transform result, you’d get $2,000.

In the next article, we will show how Laplace transforms can be useful to deduce present value rules. Not many analytic solutions exist for present value problems but thanks to Laplace transforms we can deduce some of the closed form solutions quite easily.


Reference:
   - Process Dynamics: Laplace Transforms by M Tham, Department of Chemical and
     Process Engineering, University of Newcastle upon Tyne
   - Laplace Transforms as Present Value Rules – A Note by Stephen A. Buser, The Ohio
     State University.

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