Monte Carlo Simulation

michaelcwman · June 13th, 2003, 5:24 pm

Is MC Sim basically a tool used to test a new deriv instrument by simulating the various scenarios the deriv can encounter? Is this form of simulation comparable to test-driving the deriv? Are the scenarios in which the deriv put through analysed to price the derivative?

JabairuStork · June 13th, 2003, 5:50 pm

I guess the best answer is not really. You are asking some complex questions based on a lot of assumptions, so it is hard to give a direct answer without unpacking all those assumptions.Monte carlo, in the most basic sense, is just drawing random numbers according to a set of rules that simulate the outcome space of a probabilistic experiment. Evaluation of the outcomes of that experiment will give you a distribution that converges to the true distribution, given that everything is correctly specified (that is a big caveat.)MC is typically not the first choice for valuing derivatives because it is slow to run enough trials to get reliable results. However, for many instruments that have complex payoff functions, path dependence, or other difficulties, MC is the best (sometimes only) way to value them.I'm not sure what you mean by "test driving" the derivative.Also, MC is not scenario based. Running scenarios is sort of a degenerate form of quasi-MC, in which the simulated behavior of the underlying state variables is not random. Scenario analysis is closer to determinstic, in terms of the values taken on by the state variables, and is a very crude yet very fast way to describe a broad region in which the price of the derivative can take values.Does this help at all?

michaelcwman · June 13th, 2003, 6:09 pm

Right, so if I wanted to know all the possible paths that a derivative could take I would use deterministic analysis, and MC is different from this because it is about generating random numbers (according to specified rules). The two are therefore different concepts, right?I guess the thrust of my question was if a particular complex derivative has been structured to meet a particular problem, how would this derivative be evaluated, and subsequently priced? With evaluation I mean whether the deriv does its job...I thought pricing would also come into this because the more random paths that a derivative can take the more expensive it would be cuz the deriv seller is taking on more risk...

chiral3 · June 14th, 2003, 1:14 am

Monte Carlo is very, very old. Before computers the idea existed and was used (for problems with fast convergence ) Here is a modern overview

spacemonkey · June 14th, 2003, 1:15 am

On one level, the monte-carlo method is just scenario analysis. To solve a problem with some random element, simulate it a million times and see what happens. However, although this might be useful for risk analysis or insurance, this is not the way that MC is used for derivative pricing. It turns out that the problem of pricing a derivative boils down to calculating the average value of the payoff from the derivative. This average is not the real world average, but the average in an artificial world in which stock prices are as likely to move up as move down. Sometimes this average can be calculated exactly using various techniques, but often it is much too complicated and has to be calculated approximately by computer. The method of last resort is to simply generate a possible future path of the stock price at random assuming this artificial behaviour, calculate what the payoff would be for this future and then calculate the average payoff over millions of possible futures. This is the monte-carlo technique. The important difference in this application is that the simulated behaviour is not supposed to be realistic. There are many other problems in different areas of science and mathematics in which a non-random problem has a random analogue which can then be solved approximately by simulating the random problem and looking at the statistics of the result.

michaelcwman · June 14th, 2003, 1:41 am

I am getting a little confused now, because the last two comments affirmed what I previously thought but that was said to be wrong by one of the earlier comments...so from what I understand now is that MC sim is a technique used to simulate various scenarios to simulate the real world to the best of its ability, and the premise of MC simulation is that the real world, or whatever is simulated, is 'random'...this is a fundamental rule of quant finance, esp. in relation to stock market, as it subscribes to the randomness of the market and not that the market moves in any systematic way i.e. the way traders see it. Yet MC is not realistic because the world is not perfectly random, and this is one of its weaknesses, is this a correct inference from what has been posted?That aside, MC sim is used to price deriv by finding the average of a million possible hypothetical pay-offs and then use some model to compute the price to be required for a seller of such a derivative...

Patrik · June 14th, 2003, 7:33 am

To be brief and inprecise (about one use of MC in pricing):You have an option with a pathdependent payoff, let's say the the payoff is a constant X minus the averageof the underlying during some period . In order to price this via MC you need to specify a model for how the underlying behaves. This model is assumed to have stochastic elements in finance. With this model specified you can simulate one possible outcome of the future by drwaing random numbers that represents the stochastic elements in the model of the underlying. This enables you to calculate the payoff of one possible outcome according to your model. In MC you (often) switch to what's called a risk-neutral worldwhich means that all instruments evolves at a risk-free rate of interest.The idea then is to simulate very many such outcomes and calculate the payoff for each outcome. Then you approximate the expected payoff for this derivative with the average of those payoffs you calculated (which idealy would have been infinitely many). The expected payoff dicounted at a risk-free rate is then your estimation of the price of the derivative.Hope this practical example cleared some things up for your.

michaelcwman · June 14th, 2003, 11:34 am

yes thanx very much Patrik...this is a very clear example, and now I can see that MC is used in conjunction with a model that simulates the behaviour of the underlying...I guess that model is separately constructed for every derivative in order to reflect the underlying asset. In any even, it is interesting that you tied into the concept of discounting the pay-off in order to come up with a present value of that pay-off to price that derivative...sometimes you come across these concepts, and it isn't until people put it into different contexts that you finally see how the concept really works...thanx! I believe someone in this thread mentioned that MC sim has its limitations but strictly in terms of generating the random numbers based to simulate behaviour of the underlying, how effective is MC? If MC is not used for this particular purpose, what other random-number generating technique is most frequently used instead by deriv modellers?

Patrik · June 17th, 2003, 9:44 pm

Now when you have got the basic idea of the technique it is probably best to readabout it in hull or wilmott - one can say a whole lot about the questions you justasked and I think it's easier to have a look in a book.

rhmari · September 2nd, 2005, 11:36 am

hi guys,does anybody has a spreadsheet excel for pricing a vanilla option using monte carlo with the VBA codethanks for your help,

mj · September 2nd, 2005, 3:35 pm

my take on Monte Carlo is that it's a method of carrying out numerical integration. and for high-d problems is the fastest one.

xmulh2 · September 2nd, 2005, 4:14 pm

QuoteOriginally posted by: mjmy take on Monte Carlo is that it's a method of carrying out numerical integration. and for high-d problems is the fastest one.for numerical integration, you can use some good softwares, such as Matlab. I mainly use the MC to price the path dependent derivatives.

spacemonkey · September 2nd, 2005, 8:44 pm

QuoteOriginally posted by: xmulh2QuoteOriginally posted by: mjmy take on Monte Carlo is that it's a method of carrying out numerical integration. and for high-d problems is the fastest one.for numerical integration, you can use some good softwares, such as Matlab. I mainly use the MC to price the path dependent derivatives.Pricing path-dependent options is just a very high dimensional integral.

mensa0 · September 3rd, 2005, 3:25 am

This may be a bit OT, but I have used MC for possible cash flows in capital budgeting analysis and then used these to compute a distribution of NPVs. The cash flows in each trial are not random, however. For example, suppose the first year's cash flow is (randomly) large. I'd say that the t=2,...N) cash flows would tend to be above average as well (product is "hot!".) The opposite would be true for a low (random) cash flow at t=1 in another trial (product's a "dud".)This is the same idea for a price series with drift. Purely random MC runs will not give a true picture of the distribution of outcomes. Now, what is the drift? I have built in skewness in the MC runs as well, based on how large the initial cash flow estimate is from my expectation.Lots of interesting issues here!!Mike

Robske24 · September 5th, 2005, 12:27 pm

MC simulation will not give you a range of possible paths a derivative can take. MC simulation is basically just another way to model data and estimate parameters. In a statistical sense what you are doing is taking what is called a prior distribution and then running a whole lot of simulations which will converge to the true values or a "chain". The average of all these simulations is taken. What you are left with are the parameter estimates for your model and a posterior distribution. MC is quite complex, and is generally only useful in more complex modelling situations. http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtmlThis website has a great program designed specifically for monte carlo simulation.