Monte Carlo and other Simulation Questions

[Soccerboy]

I have trained myself to memorize the following steps in Montecarlo simulation for Pricing derivatives

1. Generate N values from the terminal distribution of Assets
2. Compute Payoff for each value
3. Compute average pay off for all trials
4. Discount all averages at risk free rate

Here is my brief....When it says generate N values, does this mean generate N trees , one one tree with N different Values at each node


Numerical method types include: MonteCarlo, Lattice, Partial differential equations and Closed form equations that are difficult to solve..are these correct ?

In calculating the delta and gamma (Derivative sensitivity measures), we are told that N(d1) is the delta in black scholes and we are also told that Theta(d1)/{S*T^.5)
My question si, what is the difference between N(d1) taken from the Normal Table and Theta(d1), is one the probability and the other the # of standard deviations?
Is S the stock Price and T the time until maturity?

How do Derivatives provide efficient methods to acheive payoffs, not sure I understand why?

EXOGENEOUS factors, say what??..How does this and convexity affect the value of OAS..I thought hard about that one and couldnt make anything of it..see I cannot memorize things without having a slight idea of what is going on

I dont see how the following are "principles" of valuing assets and liabilities: Modelling interest rate behavior, Maping all cash flow of A/L across all paths; adjusting cash flows and discounting using short rates, then find average Present value..I guess Im a little confused as to what they mean by "principles" in this context

Thanks in advance..you are all a gem

[bm1729]

I'll deal with the first two questions only...

When it says generate N values, does this mean generate N trees , one one tree with N different Values at each node
It means generate one tree with N terminal values.

Numerical method types include: MonteCarlo, Lattice, Partial differential equations and Closed form equations that are difficult to solve..are these correct ?
Not quite. Numerical methods are used when it is not possible to obtain a closed-form solution because of path-dependent cash flows.

[Enough Exams Already]

I love the smell of pop quizzes in the morning. It smells like...victory.

1.) When it says "generate N values", it means exactly that. If the derivative you are valuing is path-dependent, that will require generating N trees. But the list you are reciting, IIRC, is for non-path-dependent European options. Hence only the value of the underlying security at maturity is needed.

2.) You are correct.

3.) Firstly, it's a Phi, not a Theta. N(.) is the cdf of a normal distribution, Phi is the pdf of a normal distribution. S is the security price, T is the time to maturity, and you're missing a sigma from the denominator of gamma.

4.) Exogenous factors are things like supply and demandof funds, fiscal and monetary policies insofar as they affect interest rates and inflation, etc.

5.) Could you indentify the context, please? It's hard for me to decipher without a specific reference.

[torrent]

I'll deal with the first two questions only...

When it says generate N values, does this mean generate N trees , one one tree with N different Values at each node
It means generate one tree with N terminal values.

Numerical method types include: MonteCarlo, Lattice, Partial differential equations and Closed form equations that are difficult to solve..are these correct ?
Not quite. Numerical methods are used when it is not possible to obtain a closed-form solution because of path-dependent cash flows.

Monte Carlo is the technique for path dependent cash flows, I don't think "trees" are used at all ("trees" are for the lattice model), unless you are simulating through a lattice (for some American type options).

[Nick Mocciolo]

I don't usually post like this on and ad-hoc basis, but I just wanted to clarify a bit on these issues, since it seems they have created some confusion.

The following methods for pricing derivatives need to be separated from one another, and I hope this high-level outline will help.

1. Closed-form. This method is when we use an analytic formula for pricing a derivative. The best known example is the Black-Scholes formula, but there are also closed form solutions for other types of options.

2. Lattice methods. This is when we build a tree to price the security ... and in doing so we have formulas that tell us where to space the nodes and the values of the stock at each node. They come from the underlying distribution of the stock price (i.e. we "calibrate" the tree so that it replicates the assumed distribution of the stock). That is why for European options the binomial price approaches the B-S price in the limit. This could be a binomial tree (mostly what is covered on C6), but there are other types (e.g. trinomial). The trees may or may not recombine. Tree methods are ideal for the valuation of American style options, for the following reason. To decide whether or not to exercise, we need to compare the xercise value to the value if held. Since we work backwards from right to left in the tree, at any node we always know the value of the option if held as well as the xercise value. Therefore it is easy to take the maximum of these two values. On the contrary, it is very difficult to use this method to value path-dependent options. Why? Because, we start at the end. Therefore at a given node we don't know what path the option has taken ... so how can we calculate its value? You'll notice how annoying it was to calculate the barrier option value in my problem in the CR ... and that only had 8 paths and it was European! Imagine trying that problem using daily time steps for a 5-year option. Now, there are ways to extend tree methods to accomodate these problems (sometimes), but they are definitely beyond the scope of C6.

3. Monte Carlo simulation. This is an entirely different method from lattices. The idea here is that we know the underlying stock's distribution, so we simulate it. Not through a tree, however. We actually generate random numbers and project a future path for the option (we do this many thousands of times). This is different from binomial. In binomial, we basically calculate the value at every node ... in essence we are calculating all of the "paths" through the tree at once. In monte carlo, we calculate only one path at a time. And the path chosen is not a path on any tree ... we do not constrain our values to have to equal the value at some node. The price simply evolves acording to the lognormal distribution (or whatever distribution we assume for the stock). The analogy is, what if you wanted to know the expected value of a roll of a die? You could (in theory) roll the die 1,000,000 times and average the results ... this is MC. (You could also build a six-nomial tree, but this would suck). Eventually the simulation would converge to the actual value of 3.5. Now, MC methods are good for path-dependent but not for American options. Why? Well, at any point in the simulation, I know what path the stock has taken since I simulated it! But, I don't know the value of the option if held, since I have not gotten to the end yet. In other words, MC is forward-looking, but lattice methods are backward looking.

4. Monte Carlo through a tree. This is a hybrid approach. You build the tree. But, instead of starting at the end and calculating the value at every node (which is equivalent to calculating every path through the tree), you instead simulate paths through the tree. I.e., generate a random number ... if < p go up, if >p go down. Do this until you hit the end of the tree, calculate the payoff. Then, repeat this many times, average the payoffs, discount at the risk-free rate. Why do this? Well, suppose you had a non-recombining tree that would take literally years to evaluate. You could instead simulate enough paths through it so that you get close to the answer but with alot less work.

The PDE approach is not covered in C6 really at all so don't worry about knowing alot about it.

I hope this helps.

Best,

Nick

[Soccerboy]

Thanks again All

Torrent

What is the difference between path dependency and trees, aren't trees a form of path dependency?

EEA,

This is a list entitled " Basic Principles for valuation of Assets and Liabilities" from the VISFI Chapter 8

[torrent]

Thanks again All

Torrent

What is the difference between path dependency and trees, aren't trees a form of path dependency?

EEA,

This is a list entitled " Basic Principles for valuation of Assets and Liabilities" from the VISFI Chapter 8

Lattice can be used for valuating path dependent cash flows (but it's not vaery efficient), Nick has an example in his ACTEX review.

Monte Carlo simulation generally does not use a lattice (with the exception of valuation of an American type option which can be exercised at any time).