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.
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.) In terms of financial theory, this, essentially, is an application of risk neutral valuation; see also risk neutrality.
Applications:
- In Corporate Finance, project finance 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. Note that whereas equity options are more commonly valued using other pricing models such as lattice based models, for path dependent exotic derivatives – such as Asian options – simulation is the valuation method most commonly employed; 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, swaptions, and convertible bonds. As for equity, for path dependent interest rate derivatives – such as CMOs – simulation is the primary technique employed; (Note 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. Here analysts may apply Principal component analysis, where through dimensionality reduction, a limited set of factors may be simulated instead of each of the individual sources of uncertainty.
- A similar approach is used in calculating value at risk, or "VaR", an estimate of how much a position, "desk", or other area might lose with a given probability (or confidence level) and in a set time period. A typical application of VaR is in investment banking, where the bank holds economic “risk capital” corresponding to the estimated number; see . VaR is also used in portfolio risk management, where, as above, simulation allows the fund manager to estimate losses at a given horizon and confidence level, and to then hedge as / if appropriate.
- Post crisis, banks will make various “valuation adjustments” - collectively XVA - when assessing the value of derivative contracts that they have entered into. The purpose of these is twofold: primarily to hedge for possible losses due to the other parties' failures to pay amounts due on the derivative contracts (credit valuation adjustment); but also to determine (and hedge) the amount of capital required under the bank capital adequacy rules. These are calculated under a simulation framework as the risk-neutral expectation value of the possible loss or other impact. See .
- Structurers use simulation to estimate the likely payout - and possibility of losses - of their bespoke structured note or other structured product, typically comprising several component securities.
- 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.
- Discrete event simulation can be used in evaluating a proposed capital investment's impact on existing operations. Here, a "current state" model is constructed. Once operating correctly, having been tested and validated against historical data, the simulation is altered to reflect the proposed capital investment. This "future state" model is then used to assess the investment, by evaluating the improvement in performance (i.e. return) relative to the cost (via histogram as above); it may also be used in stress testing the design. See .
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). 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.
Control variate method
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
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.
Quasi-random (low-discrepancy) methods
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.
Notes
- Frequently it is more practical to take expectations under different measures, however these are still fundamentally integrals, and so the same approach can be applied.
- More general processes, such as Lévy processes, are also sometimes used. These may also be simulated.
See also
- Quasi-Monte Carlo methods in finance
- Monte Carlo method
- Historical simulation (finance)
- Stock market simulator
- Real options valuation