Path-Dependent Monte Carlo

[unleash]

Greetings!! My first post but been reading for months

I'm having a little bit of trouble understanding when you use MC to price a path-dependent derivative...

My understanding of monte carlo is that you're computing an integral which, when valuing derivatives, means you're calculating the expectation which is used with the risk-neutral formula to compute a derivative's price.

What is the difference between MC and FDM? When do you use which?

I know you can use MC to traverse a tree, so am I correct in seeing MC as a method of running through a tree (i.e. simulating a path through the tree) or is MC completely separate to a tree structure?
When valuing path-dependent derivatives, a barrier option for instance, you have to simulate the payoff along the path and check if it's hit the barrier. That's a tree structure isn't it? Is the MC part is the simulation of the path?

I've seen it said that MC is used for "multiple sources of uncertainty or with complicated features that make them difficult to value through straightforward BS or lattice-style computation"... but I'm unsure about how you can value path-dependent derivative without it being in the context of a lattice structure?

I guess what I'm really asking:
a) what is the difference between MC, FDM and tree methods?
b) Are there any situations when you HAVE to use MC? (particularly in the context of path-dependent payoffs and where does dimensionality come in?)

Really appreciate your help

 

[Lun]

FDM is for solving PDE. With a PDE, if you can't solve it analytically, you need to solve it numerically, FDM is one of the numercial methods.

Under many cases, you develop a model with SDE. By Ito lemma, the SDE is transformed to PDE. Black Scholes model is such an example in which you can solve analytically, no FDM is needed. For other pricing models, the PDE cannot be solved analytically, then you need FDM.

 

[unleash]

would an example be where you assume GBM but have a lookback option where the boundary condition means you can't get a solution to simulate the terminal distribution?

Would be great if you could give an example of a situation when you'd use FDM compared to MC

And maybe an example where you HAVE to use MC

Much appreciated :D

 

[Lun]

If you can transform it to PDE, then FDM is the tool to solve PDE numercially.

PDE is an equation which you need to solve.
MC is a simulation.
One is an equation, one is a simulation. It's very fundamental, don't mix them.

 

[unleash]

examples would really help (i'm an undergrad noob)

 

[Lun]

let's put the simplest example, an European option. Don't think about path-dependent options first, it just makes your mind more complicated. Go to understand what basically MC & FDM are first.

Option price = Expectation(future payoff) x discounted factor

If you can express the payoff in terms of GBM, then
MC = numerically imagine many possible scenarios to calculate the expectation
or
MC = simulate many possible scenarios to calculate the expectation numerically

For BS again, if you're with SDE, you turn it into PDE by Ito lemma. Then, you can solve the PDE analytically. Alternatively, you can solve the PDE by FDM.

 

[unleash]

ah ok that makes sense

so would you resort to using MC when you can't turn the SDE into a PDE because of boundary conditions? I'm still having difficulty understanding when you'd HAVE to use MC. I know you only use MC when there's no analytical solution or high dimensionality. When would you HAVE to use MC?

 

[Lun]

I would say, we just choose the appropriated one from all methods. We won't say when have to use which.

What's more important is that you know how to use these methods, and choose the appropriated one under a given situation.

 

[unleash]

I understand your reluctance to make a broad generalisation on where to use one versus another but having tried to understand the differences between the methods, I still find myself unable to know when specifically you would resort to MC (I know it's not particularly efficient except in high dimensions) so any help, if you could, would be great in understanding the differences between them in the context of path-dependent pricing

really appreciate your time to reply! :p

 

[bob]

ah ok that makes sense

so would you resort to using MC when you can't turn the SDE into a PDE because of boundary conditions? I'm still having difficulty understanding when you'd HAVE to use MC. I know you only use MC when there's no analytical solution or high dimensionality. When would you HAVE to use MC?

Well, pretty much any case someone posts in answer to this question is practically begging for someone else to come in with a link to some paper somewhere showing there's a PDE method that's appropriate. So I won't claim that you HAVE to use MC for the following deal I came across recently; I will merely say that I would. (Note these are not the actual T&C--just something to give the flavor of what it is like.)

The deal defines a basket of 12 stocks and is 5 years long. The deal is broken down into 5 one-year periods. At inception of the deal, each of the stocks is assigned a lower barrier level equal to 85% of its spot price at the time the deal is struck.

In each of the first 4 periods, a breach occurs if at least one of the 12 stocks touches its lower barrier level. In the 5th period, a breach occurs if at least 4 of the 12 stocks touch their respective lower barrier levels.

At the end of each period, the lower barrier level of the worst-performing stock in that period (that is, the one with the greatest percent decline since the beginning of the period) is reset to 85% of its closing price at the end of the period.

The payoff of the deal occurs at the end of year 5 and is equal to:
$1M * N, where N is the largest-numbered period in which no breach occurred. If a breach occurred in all 5 periods, then the payoff is zero.

(As an editorial note, it is very hard to argue with people who claim derivatives are gambling rather than risk management when things like this come across your desk.)

 

[iHateVariance]

it probably was a cost efficient solution for a corporate client. right Bob?

 

[Daniel Duffy]

examples would really help (i'm an undergrad noob)

For MC, see http://en.wikipedia.org/wiki/Monte_Carlo_methods_for_option_pricing

For Lattices, see http://en.wikipedia.org/wiki/Binomial_options_pricing_model

On PDE/FDM, here are some articles and schemes from my own threads.

http://www.datasimfinancial.com/forum/viewforum.php?f=24&sid=8c9556af65af41b12ded1a09aab8a221

In general, trying to find exact solutions for PDE (especially nonlinear PDE) is almost impossible but in some cases it is possible by clever change of variables.

 

[karafrylee]

I know you can use MC to traverse a tree, so am I correct in seeing MC as a method of running through a tree (i.e. simulating a path through the tree) or is MC completely separate to a tree structure?

Would like to know the answer to that too if anyone can help. I'd also like to know if its any good combining CRR binomial lattice model with MC?

 

[quotes]

I bet Monte-Carlo has far little possible paths than a binomial tree. Say, 2^100 paths only need 100 time points in a tree.
Monte-Carlo has a finer ending-time distribution, but too little paths have insufficient representation in a path dependent model.
Therefore, I prefer binomial model to Monte-Carlo in path dependent models.
American options are better priced in a tree as well.
Of course, any rebuttal to the passage above is welcome.