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 operators. After the transformation, the solution of the original problemis arrived at by simple algebraic manipulations in thes (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,e, 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, f(t), 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 torthen the present value of a future cash flow,C(t), where,C(t)is a function of time,tis given by:

...... (2)

In the above we have assumed continuous compounding and present value function,p(t)is a function of t. The time is bounded between 0 and some finite quantity,T. 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,tto rate, rand therefore, the present value becomes a function of the rate,r. However, the boundary of the integral is still from 0 to some finite quantity,T.

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 flowP(t)is the Laplace transform of the cash flowC(t).

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 isK(where,K 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.

The Remarkable Power of the Monte Carlo Method

Team Latte
August 7, 2012

Ever since we published the articleThe Essence of Monte Carlo Methodologyon this site we have been inundated with requests to elaborate further on what we explained in that article and give more examples. While we cannot go into very detailed mathematical or technical explanations of the Monte Carlo method on these pages – for that, we would refer our readers to our signature public courseCertificate in Financial Engineering (CFE)– we can certainly try to elucidate a bit more on this remarkable mathematical method and why it is so powerful in solving many of the problems in physics and finance.

In short, Monte Carlo methodology is an extremely powerful method to solve problems – not just stochastic but deterministic problems as well – in physics and finance because of these two mathematical facts:

  1. Every definite integral can be approximated as an average;
  2. Every, complex and intractable, deterministic problem can be expressed as stochastic problem and solved by randomly sampling from a probability distribution.

Well, perhaps we should clarify a bit more on what exactly we mean by a "problem". A "problem" in physics or finance is a dynamical system – where the dependent and the independent variables are related via a function – which is mathematically governed by a differential equation. Be it option pricing, a mechanical pulley, the propagation of wave in a medium, a convertible bond, etc., the dynamics of all these are mathematically governed by a differential equation. And the solution to differential equations is given by integrals.

The real power of Monte Carlo method - or, the Monte Carlo integration methodology – is that it can transform a deterministic problem into a stochastic one by randomizing the independent variable, or the variables,of a given function and then allowing us to sample from a given probability distribution. This means rather than solving the integral analytically – as one would do if one is solving a differential equation analytically – we solve it by transforming the dependent variable of the function into a stochastic (random) variable and then sampling from a probability distribution.

Ordinary differential equations (ODEs), which arise when we have a single dependent variable and a single independent variable related via a function,, are, in general, easier to solve both analytically and numerically. These kinds of problems require evaluating just a single integral. However, when we have a many variable problem, where there is one dependent variable and several independent variables, related via a function of the type,, the problem involves partial differential equations (PDEs) which are, in general, more difficult to solve than ODEs. In such cases, we need to evaluate higher dimensional integrals. This is where a Monte Carlo method becomes truly powerful because solution of the PDEs involving higher dimensional integrals can become very complex.

The option pricing problem is a prime example from finance. Even though the problem is deterministic and can be solved using a differential equation we can transform the valuation problem into an expectation (under some risk-neutral probability measure) and then performing the integration. Also, think of a basket option where the payoff of the option is dependent on a basket of assets, i.e. multiple variables. In such situations if we perform Monte Carlo integration we need to evaluate, say, a triple integral (if there are three assets) whereby we have three random variables to sample from some probability distribution (correlations will give rise to joint distributions). This is still far simpler than solving a partial differential equation – numerically, via lattice methods or otherwise – for the three variables with cross-asset terms.

There are a large number of problems in physics and engineering, such as heat transfer, propagation of waves, etc. that require solution of partial differential equations. And Monte Carlo method can prove to be very useful and efficient numerical method in solving these problems.