In 1977, Phelim Boyle pioneered the use of simulation in derivative valuation in his seminal Journal of Financial Economics paper.
[4] This article discusses typical financial problems in which Monte Carlo methods are used.
The Monte Carlo method encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems.
[1] This very general approach is valid in areas such as physics, chemistry, computer science etc.
[1] ("Covering all conceivable real world contingencies in proportion to their likelihood.
Applications: 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).
However, when the number of dimensions (or degrees of freedom) in the problem is large, PDEs and numerical integrals become intractable, and in these cases Monte Carlo methods often give better results.
For more than three or four state variables, formulae such as Black–Scholes (i.e. analytic solutions) do not exist, while other numerical methods such as the Binomial options pricing model and finite difference methods face several difficulties and are not practical.
Monte Carlo methods can deal with derivatives which have path dependent payoffs in a fairly straightforward manner.
On the other hand, Finite Difference (PDE) solvers struggle with path dependence.
This is because, in contrast to a partial differential equation, the Monte Carlo method really only estimates the option value assuming a given starting point and time.
In the Black–Scholes PDE approach these prices are easily obtained, because the simulation runs backwards from the expiry date.
In Monte-Carlo this information is harder to obtain, but it can be done for example using the least squares algorithm of Carriere (see link to original paper)[citation needed] which was made popular a few years later by Longstaff and Schwartz (see link to original paper)[citation needed].
The fundamental theorem of arbitrage-free pricing states that the value of a derivative is equal to the discounted expected value of the derivative payoff where the expectation is taken under the risk-neutral measure [1].
An expectation is, in the language of pure mathematics, simply an integral with respect to the measure.
Today's value of the derivative is found by taking the expectation over all possible samples and discounting at the risk-free rate.
is the discount factor corresponding to the risk-free rate to the final maturity date T years into the future.
Let us suppose that a derivative H pays the average value of S between 0 and T then a sample path
This can be a time-consuming process (an entire Monte Carlo run must be performed for each "bump" or small change in input parameters).
Further, taking numerical derivatives tends to emphasize the error (or noise) in the Monte Carlo value – making it necessary to simulate with a large number of sample paths.
Practitioners regard these points as a key problem with using Monte Carlo methods.
Square root convergence is slow, and so using the naive approach described above requires using a very large number of sample paths (1 million, say, for a typical problem) in order to obtain an accurate result.
Remember that an estimator for the price of a derivative is a random variable, and in the framework of a risk-management activity, uncertainty on the price of a portfolio of derivatives and/or on its risks can lead to suboptimal risk-management decisions.
[26] Not only does this reduce the number of normal samples to be taken to generate N paths, but also, under same conditions, such as negative correlation between two estimates, reduces the variance of the sample paths, improving the accuracy.
Therefore, a standard way of choosing the derivative I consists in choosing a replicating portfolios of options for H. In practice, one will price H without variance reduction, calculate deltas and vegas, and then use a combination of calls and puts that have the same deltas and vegas as control variate.
Importance sampling consists of simulating the Monte Carlo paths using a different probability distribution (also known as a change of measure) that will give more likelihood for the simulated underlier to be located in the area where the derivative's payoff has the most convexity (for example, close to the strike in the case of a simple option).
The simulated payoffs are then not simply averaged as in the case of a simple Monte Carlo, but are first multiplied by the likelihood ratio between the modified probability distribution and the original one (which is obtained by analytical formulas specific for the probability distribution).
When calculating the delta using a Monte Carlo method, the most straightforward way is the black-box technique consisting in doing a Monte Carlo on the original market data and another one on the changed market data, and calculate the risk by doing the difference.
Instead, the importance sampling method consists in doing a Monte Carlo in an arbitrary reference market data (ideally one in which the variance is as low as possible), and calculate the prices using the weight-changing technique described above.