Monte Carlo Methods in Finance - Overview

Overview

The Monte Carlo Method encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems. Essentially, the Monte Carlo method solves a problem by directly simulating the underlying (physical) process and then calculating the (average) result of the process. This very general approach is valid in areas such as physics, chemistry, computer science etc.

In finance, the Monte Carlo method is used to simulate the various sources of uncertainty that affect the value of the instrument, portfolio or investment in question, and to then calculate a representative value given these possible values of the underlying inputs. ("Covering all conceivable real world contingencies in proportion to their likelihood." ) In terms of financial theory, this, essentially, is an application of risk neutral valuation; see also risk neutrality.

Some examples:

• In Corporate Finance, project finance and real options analysis, Monte Carlo Methods are used by financial analysts who wish to construct "stochastic" or probabilistic financial models as opposed to the traditional static and deterministic models. Here, in order to analyze the characteristics of a project’s net present value (NPV), the cash flow components that are (heavily ) impacted by uncertainty are modeled, incorporating any correlation between these, mathematically reflecting their "random characteristics". Then, these results are combined in a histogram of NPV (i.e. the project’s probability distribution), and the average NPV of the potential investment - as well as its volatility and other sensitivities - is observed. This distribution allows, for example, for an estimate of the probability that the project has a net present value greater than zero (or any other value). See further under Corporate finance.

• In valuing an option on equity, the simulation generates several thousand possible (but random) price paths for the underlying share, with the associated exercise value (i.e. "payoff") of the option for each path. These payoffs are then averaged and discounted to today, and this result is the value of the option today; see Monte Carlo methods for option pricing for discussion as to further - and more complex - option modelling.

• To value fixed income instruments and interest rate derivatives the underlying source of uncertainty which is simulated is the short rate - the annualized interest rate at which an entity can borrow money for a given period of time; see Short-rate model. For example for bonds, and bond options, under each possible evolution of interest rates we observe a different yield curve and a different resultant bond price. To determine the bond value, these bond prices are then averaged; to value the bond option, as for equity options, the corresponding exercise values are averaged and present valued. A similar approach is used in valuing swaps and swaptions. (Note that whereas these options are more commonly valued using lattice based models, for path dependent interest rate derivatives - such as CMOs - simulation is the primary technique employed.; note also that "to create realistic interest rate simulations" Multi-factor short-rate models are sometimes employed.)

• Monte Carlo Methods are used for portfolio evaluation. Here, for each sample, the correlated behaviour of the factors impacting the component instruments is simulated over time, the resultant value of each instrument is calculated, and the portfolio value is then observed. As for corporate finance, above, the various portfolio values are then combined in a histogram, and the statistical characteristics of the portfolio are observed, and the portfolio assessed as required. A similar approach is used in calculating value at risk.

• Monte Carlo Methods are used for personal financial planning. For instance, by simulating the overall market, the chances of a 401(k) allowing for retirement on a target income can be calculated. As appropriate, the worker in question can then take greater risks with the retirement portfolio or start saving more money.

Although Monte Carlo methods provide flexibility, and can handle multiple sources of uncertainty, the use of these techniques is nevertheless not always appropriate. In general, simulation methods are preferred to other valuation techniques only when there are several state variables (i.e. several sources of uncertainty). These techniques are also of limited use in valuing American style derivatives. See below.