C(omp)++ | A Portal for the Computational Finance Community: Joerg Kienitz

In this blog we look into Las Vegas methods. American / Bermudan options exhibit early exercise opportunities. Thus the option holder does not have to wait until maturity to execute his right to exercise the option. The question arising here is the time point when it is optimal to exercise. The difference between the two types is that an American option can be exercised at any time until maturity whereas a Bermudan option has specified dates at which exercise is possible. The American option can be seen as a limiting case of a Bermudan. We actually only consider Bermudan types in Monte-Carlo.

One of the most well known and widely applied methods is the Longstaff-Schwarz regression method. Here an early exercise policy is determined using linear regression.

The price of an American / Bermudan option is given by

(1)

where U denotes the discounted expected payoff at time Joerglasvegaspostfig2final_3

Since in the above mentioned method we derive one of many exercise policies and formula (1) states that to actually price the contract we need the optimal policy we only compute a lower bound on the price.

Let us briefly review the method. We denote bythe possible exercise dates.

Firstly, you have to generate M₁ paths corresponding to your underlying dynamic. Second you apply regression on the possible exercise dates as follows:
Let Joerglasvegaspostfig4 denote the discounted price at time Joerglasvegaspostfig5 as above and denote by Joerglasvegaspostfig6_2 regression functions, e.g. polynomials. We suggest to compute

(2)

Joerglasvegaspostfig7

This task essentially involves matrix operations as inversion and multiplication. Working backwards through the exercise dates leads to regression coefficients at those points.

Now, run another simulation to generate M₂ paths. Together with the results in the first phase, the regression phase, we can determine the continuation value Joerglasvegaspostfig8 using Joerglasvegaspostfig9_3

This could be seen as the expected price under the assumption not to exercise immediately. An exercise policy for each path can be derived by comparing the continuation value CV to the value computed due to the given payoff at current time point. This phase we call the pricing phase. The obtained result determines a vector of size M₂ with values in the set Joerglasvegaspostfig10 . The values determine the exercise policy corresponding to the paths. The symbol Joerglasvegaspostfig11 suggests that we do not exercise until maturity and the option expires worthless. Now computing the resulting values with respect to the exercise policy and averaging gives the desired value of the American / Bermudan option.

Of course this method incorporates errors such as

Estimation of continuation value
In general smooth parameterisation functions used in the regression phase
Too few paths might be considered
Foresight bias

In future blogs we will explore some more details of the Longstaff-Schwarz method as well as review some other methods to price American / Bermudan options. Until then, I would be very pleased if anybody would like to share experiences on the pricing of early exercise features using Monte-Carlo, e.g. what is your favourite method? In which context do you apply such methods? Fixed-Income? Equity?

Joerg

Our quantitative finance team is involved in many parts of the banks business. We are not specialised to work in certain areas and markets like most quant teams in the city. We have to deal with a number of issues like portfolio optimisation, dynamic portfolio trading strategies or derivatives pricing. Furthermore, we have to consider the broad range of asset classes.

Because there is a limited amount of manpower available we look for methods that can be applied to a large class of problems. To this end we decided to implement some robust, flexible and widely applicable tools. One of those tools is the Monte Carlo method.

The Monte Carlo method can be applied to a wide range of problems in finance. We use it for examining dynamic trading strategies like CPPI, computing VaR and derivatives pricing.
Often Monte Carlo simulation is the only chance to get results. This is the case if you work on high dimensional, path-dependent problems. Monte Carlo simulation uses ideas from probability theory as well as calculus but also from number theory. There are many sources of input for new ideas and new methods. It is a very active field for researchers.

The method has – of course - some drawbacks. Some of the main criticisms put forward are that the procedure is time consuming when implemented on a computer and it is often hard to compute the Greeks and prices for American / Bermudan options. But latest research shows that difficulties can be overcome in applications. For example Giles and Glasserman developed new methods for computing Greeks or Schoenmakers tackled the problem for computing prices for Bermudan options in high dimensional Libor Market models.

Together with Daniel Duffy we work on a generic and robust method to implement the Monte Carlo method such that it can easily be extended and customised to fit your needs. This is hard work but we hope to come up with a nice setup.

The method of choice is programming in C++ using Microsoft Excel as a frontend. Excel is used for supplying data, outputting data and graphical display it. All the numerical stuff is done in C++. This gives us the opportunity to work object oriented and get speedy methods.

The complexity of the models used for derivatives pricing and portfolio optimisation does certainly increase. Complex stochastic models are put forward to capture the features observed in the markets. The Gaussian paradigm does not hold any more!

Therefore: Monte Carlo now! Not later!

This is my answer to the initially posed question. I suggest to set up a robust and flexible Monte Carlo engine. Then, you are able to tackle a lot of problems in finance appropriately.

Any comments?!

Cheers, Joerg