monte carlo simulation vs finite difference method

R

rkdlbh

hello all

Probably a very basic question for most but for someone just starting to learn about options, has proven quite problematic!

Currently working on a project regarding these two option price predicting techniques. really struggling to work out how they each work. I kind of understand that Monte Carlo is about expectations. You run a bunch of simulations using a RNG, transform that random number into an outcome for your simulation...

but really struggling on how finite difference method works. please could someone just provide me with a basic overview on how this work and which method is better to use? thanks in advance
 
Don't shoo away a potential topic that could up our readership. I have met a ton of rocket scientists on this board alone. Most are just too quiet.

Here is the old college try:

Monte Carlo simulation is an approximation via use of random numbers. It is commonly used by the Statistical branch of the mathematical sciences.

Here is a simple example of using Monte Carlo to calculate Pi. It is usually introduced in beginning comp sci. It has an interesting history , too.

https://learntofish.wordpress.com/2010/10/13/calculating-pi-with-the-monte-carlo-method/

In the Applied Maths branch, you have Finite Difference methods which is a numerical method (an efficient technique unlike the way you solve by hand) of solving partial differential equations (PDE). That's part of Calculus for you heathens.

For FDM, I leave you to research the use on wikipedia. As I don't know anything more about those topics :p
 
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In a word, finite differences is about estimating derivatives. Monte Carlo is about estimating integrals. The relationship is the relationship between derivative and integrals in the form of the

Fundamental Theorem of Calculus:

ftoc.gif


The second fundamental theorem of calculus holds for
Inline6.gif
a continuous function on an open interval
Inline7.gif
and
Inline8.gif
any point in
Inline9.gif
, and states that if
Inline10.gif
is defined by

ne2.gif


and now you see how MC and FDs are related by using this equation (integrals and derivatives are mathematical inverses of each other):

ne3.gif


Sometimes it is better to approach a problem through derivatives, sometimes through integrals. The FTOC and SFTOC tells you that you can recover the solution from either end. But just like when you are taking a trip, one way to the destination might be considerably harder than another - not just computationally, but theoretically.
 
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rkdlbh said:
hello all

Probably a very basic question for most but for someone just starting to learn about options, has proven quite problematic!

Currently working on a project regarding these two option price predicting techniques. really struggling to work out how they each work. I kind of understand that Monte Carlo is about expectations. You run a bunch of simulations using a RNG, transform that random number into an outcome for your simulation...

but really struggling on how finite difference method works. please could someone just provide me with a basic overview on how this work and which method is better to use? thanks in advance
More...


Oops. I re-read your post. This isn't one of those "Why is the sky blue?" questions. Your really do need the guys at Wilmott or nuc phynance. o_O
 
Sorry if I'm being repetitive (I didn't read the full thread chain)

Finite differences and Monte Carlo Methods are not mutually exclusive. They can and SHOULD be used jointly.

Finite differences are used to to solve systems of partial differential equations (PDEs) numerically. If those PDEs contain any sort of stochastic component (noise) , than one should use Monte Carlo Simulation to vary the stochastic components of those PDEs and see how it affects the solution/trajectories the PDEs take. If you are doing options pricing using Black-Scholes, than you want to solve the Black Scholes equation (a PDE) and your monte carlo variables could be the stocks price, the interest rate, the drift rate, the standard deviation of returns, etc... Performing the Monte Carlo simulation will allow you to establish the Expected Value of the contract at some future date, as well as the distribution of that contracts price given stochastic variation of the parameters of the Black-Scholes model
 
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