Сандық Монте-Карло әдістері корпоративтік және математикалық қаржыда

Probabilistic measurement methods
Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions (sources of uncertainty) of the problem increase. Monte Carlo methods were first introduced to finance in 1964 by David B. Hertz through his Harvard Business Review article, discussing their application in Corporate Finance. In 1977, Phelim Boyle pioneered the use of simulation in derivative valuation in his seminal Journal of Financial Economics paper. This article discusses typical financial problems in which Monte Carlo methods are used. It also touches on the use of so called "quasi random" methods such as the use of Sobol sequences.

It is also natural to use a control variate. Let us suppose that we wish to obtain the Monte Carlo value of a derivative H, but know the value analytically of a similar derivative I. Then H* = (Value of H according to Monte Carlo) + B*[(Value of I analytically) − (Value of I according to same Monte Carlo paths)] is a better estimate, where B is covar(H,I)/var(H). The intuition behind that technique, when applied to derivatives, is the following: note that the source of the variance of a derivative will be directly dependent on the risks (e. g. delta, vega) of this derivative. This is because any error on, say, the estimator for the forward value of an underlier, will generate a corresponding error depending on the delta of the derivative with respect to this forward value. The simplest example to demonstrate this consists in comparing the error when pricing an at the money call and an at the money straddle (i. e. call+put), which has a much lower delta. 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). This will ensure that paths whose probability have been arbitrarily enhanced by the change of probability distribution are weighted with a low weight (this is how the variance gets reduced). This technique can be particularly useful when calculating risks on a derivative. 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. This results in a risk that will be much more stable than the one obtained through the black box approach.

Instead of generating sample paths randomly, it is possible to systematically (and in fact completely deterministically, despite the "quasi random" in the name) select points in a probability spaces so as to optimally "fill up" the space. The selection of points is a low discrepancy sequence such as a Sobol sequence. Taking averages of derivative payoffs at points in a low discrepancy sequence is often more efficient than taking averages of payoffs at random points.

Boyle, P., Broadie, M. and Glasserman, P. Monte Carlo Methods for Security Pricing. Journal of Economic Dynamics and Control, Volume 21, Issues 8 9, Pages 1267 1321
Rubinstein, Samorodnitsky, Shaked. Antithetic Variates, Multivariate Dependence and Simulation of Stochastic Systems. Management Science, Vol. 31, No. 1, Jan 1985, pages 66–67