Within this XL Service we enables the evaluation of standard exotic options (Asian, Lookback, Barrier, Parisian, Parasian) in accordance with the Black-Scholes model using Monte Carlo and Finite Differencing pricing techniques.
Here we consider a number of standard Exotic and Vanilla Options
contracts which are evaluation in accordance with the Black-Scholes
model using either Monte Carlo and/or Finite Differencing PDE
pricing techniques.
In particular, we offer the following contract pricing models:
Asian Options - Evaluate the present value using Monte Carlo or
Finite Differencing pricing techniques.
Lookback Options - Evaluate the present value using Monte Carlo
or Finite Differencing pricing techniques.
Barrier Options - Evaluate the present value using a Monte Carlo
approach.
Parisian Options - Evaluate the present value using a Monte Carlo
approach.
Parasian Options - Evaluate the present value using a Monte Carlo
approach.
European (Vanilla) Options - Evaluate the present value using a
Monte Carlo or Finite Differencing PDE pricing techniques
Binary Options (in particular, cash-or-nothing Binary Option) -
Evaluate the present value using Monte Carlo or Finite Differencing
PDE pricing techniques.
Remark: Though the pricing and evaluation of the Greeks for
Vanilla and (cash-or-nothing) Binary options exhibit analytic solutions,
we have decided to include evaluation methods using Monte Carlo and Finite
Differencing pricing techniques so that our implementations can be tested
in these cases against known analytic formulae. These analytic formulae
have been implemented within the Options XL Service.
Within the below sections we describe details concerning the provided
methods and guidelines in there use.
Types of Parameters
For each of the option pricing evaluation methods you will need to give
various parameters in order to define and then price the option contract
being considered. Each of these parameters will provide one of the following
two types of information:
Financial Information - Parameters which provide financial information
will contain information about either the contracts definition or the underlying
market variables on which the contract depends.
Pricing Technique Information - Parameters which vary the dynamics
of the pricing approach used. In the case of Monte Carlo the corresponding
parameters will allow you to set the following:
Number of time steps - Now to simulate each Random Walk we divide the time
interval into a finite number of sub-intervals. We then generate the Random
Walk by using a `random step' on each of these intervals one at a time, from
the initial point until the maturity. This parameter specifies the number of
intervals (and hence number of steps) which are used in order to generate
each one of these approximate Random Walks. Please see the PDF documentation
for further details concern exactly how the procedure works.
Number of Random Walks - The total number of Random Walks used in the
Monte Carlo Simulation. As the number of Random walks increases the accuracy
In the case of Finite Differencing Pricing Techniques the corresponding
parameters will allow you to set the following:
Algorithm Type - We allow you to select between the Explicit, Fully Implicit
or Cranck-Nicholson finite differencing PDE algorithms. In most instances the
Cranck-Nicholson should be the preferred algorithm. For further details concerning
the nature of these algorithms please see the PDF documentation.
Number of Time Steps - The number of time steps used in the generation of the
finite differencing grid.
Number of Path Dependent Variable steps (for path dependent options) - The number
of path dependent variable steps used in the generation of the finite differencing grid.
Maximum underlying price (variable) - The maximum value of the underlying price
(or variable) represented within the finite differencing grid.
Maximum value of the path dependent variable (for path dependent options) - The
maximum value of the path dependent variable which is represented within the finite
differencing grid.
Tolerance - This parameter is only used in the case of American options and refers
to the internal tolerance of the error which is used during the computation. The
smaller the value (in principle) the greater the resulting accuracy (and a longer
computation time). A typical value for this parameter is 0.00000001.
The financial parameters and the pricing algorithm parameters have the following
significant difference:
The financial situation is given by the contract definition and the
market variables whereas the pricing method parameters are user defined.
So, for a given contact and market conditions the financial parameters are either
deduced from the definition of the contract under consideration or are read/evaluated
from observed market variables. The model parameters such as the number of Random Walks
in the case of Monte Carlo simulation or the maximum underlying asset value in the
case of finite differencing, however must to set by the user. Therefore, in order to
set these model parameters the user will need to use there judgment in order to select
suitable choices. These choices will effect the efficiency (i.e. accuracy and time)
of a pricing model applied to the evaluation of a given option contract.
Are general advice when using these methods within an application is to vary
each one of the parameters in turn until you have the right balance of efficiency
for your intended purposes.