Q1. Monte Carlo vs Binomial Tree - when shall you use one or the other?

Binomial Tree

- The great virtue of a binomial tree --> easy to do early exercise.

- Rate of convergence for continuous payoffs is 1/n, where n = number of steps.

- Number of computations increases as n2 --> rate of convergence is t-1/2  (t =  computational time)

- For higher dimensional trees, convergence will be slower.

- Path-dependent is not natural in trees but can be dealt with by using an auxiliary variable which increases the dimension by 1.

Monte Carlo

Great virtues:

- Convergence is order t-1/2 in all dimensions

- Path-dependence is easy

Great downside:

- early excercise is hard.

- slowness of convergence in low dimensions

--> although no slower than a binomial tree

--> there are other faster lattice methods in low dimensions.

The convergence speed and early exercise can be coped with, but it requires much work.

As a rule of thumb:

- use binomial trees for low-dimensional problems involving early exercise

- use MC for high-dimensional problems involving path-dependence.

Q2.

Q3. Explain the Longstaff-Schwartz algorithm for pricing an early exercisable option with MC?

As we saw, it is difficult to use MC for products with early exercise features, such as American options.

The Longstaff-Schwartz method is one way to handle early exercise using MC.

When pricing an early exercisable product --> need to know at a given time if it will be worth more by exercising or holding.

Continuation value (CV) = value of holding onto the option  --> Longstaff-Schwartz method estimates the CV via regression.

A regression needs to be carried out against some basis functions.

Choosing good basis functions that will estimate the continuation value accurately is an art.

The Longstaff-Schwartz algorithm is:

1) Run MC paths and store the exercise value divided by the numeraire at each possible exercise date for each path.

We also need to store the value of the basis functions at the same points.

2) With all this data we now look at the second last exercise date.

We perform a regression (least-squares) of the value of the last exercise date against the basis functions.

This gives coefficients, , for the basis functions.

3) Using these coefficients we can calculate the CV at the second last exercise time for each path.

4) We now replace the deflated exercise value stored initially with the continuation value, if the CV is greater.

5) Continue back through the exercise dates until reaching time 0.

The (biased) value of the product will be the average of the time zero values multiplied by the numeraire value.

Q4. In the pricing of options, why doesn't it matter if the stock price exhibits mean reversion?

A possible mean reversion process for a stock is: .

These are the real world dynamics of stock process, but to price an option we are interested in the risk-neutral dynamics.

When we move to the risk-neutral dynamics the drift of the stock becomes rdt.

Therefore, the real world drift, be it mean reverting or not, does not change the price of the option.

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