Numerical solution for derivative models using finite difference methods and how this can be used with Monte Carlo simulation
(2019) In Master's Theses in Mathematical Sciences FMSM01 20191Mathematical Statistics
- Abstract
- Derivative models often come in the form of stochastic differential equations.
From these equations a partial differential equation (PDE) can be derived.
By discretizing the PDE the numerical solution is obtained on a form where
the value of the derivative can be seen as a probabilistic weighting of future
values. These probabilities can be used to simulate trajectories of the under-
lying assets. This connection between the finite difference scheme and the
simulation is rather unique for this pricing method and can be very useful.
The probability weights can be forced to have a perfect probability interpre-
tation in some cases, meaning they are positive and less than one, but in
other cases we will end up with negative weights... (More) - Derivative models often come in the form of stochastic differential equations.
From these equations a partial differential equation (PDE) can be derived.
By discretizing the PDE the numerical solution is obtained on a form where
the value of the derivative can be seen as a probabilistic weighting of future
values. These probabilities can be used to simulate trajectories of the under-
lying assets. This connection between the finite difference scheme and the
simulation is rather unique for this pricing method and can be very useful.
The probability weights can be forced to have a perfect probability interpre-
tation in some cases, meaning they are positive and less than one, but in
other cases we will end up with negative weights meaning we somehow have
to simulate using negative probabilities. This paper presents how to price
and simulate options with these methods in a few different situations and
how to solve some of the problems that may come up. (Less) - Popular Abstract
- Simulating option prices using negative probabilities
Being able to determine a fair price for an option is really useful
in finance but its not very simple. In many situations it is also of
interest to simulate the underlying asset that gives the option its
value. Finding a connection between these subjects is therefore of
great interest.
A common way of pricing options is to solve a partial differential equation
that must be satisfied. With our way of solving this equation the solution
will have a form where the value of the option is a weighting of future option
values. These weights can be seen as probabilities which means we can use
them to simulate the underlying asset forward in time, at least if the weights
are positive... (More) - Simulating option prices using negative probabilities
Being able to determine a fair price for an option is really useful
in finance but its not very simple. In many situations it is also of
interest to simulate the underlying asset that gives the option its
value. Finding a connection between these subjects is therefore of
great interest.
A common way of pricing options is to solve a partial differential equation
that must be satisfied. With our way of solving this equation the solution
will have a form where the value of the option is a weighting of future option
values. These weights can be seen as probabilities which means we can use
them to simulate the underlying asset forward in time, at least if the weights
are positive and sum to one (as probabilities should do). With this method
we can easily generate a large number of samples of the future value of the
underlying asset. These samples can then by used to estimate the option
value and thus also the fair price of the option. But even more importantly
we can use the samples for risk analysis of the option. Unfortunately we can
only guarantee that all weights have a perfect probability interpretation in
some very specific situations. In most cases some weights will be negative
meaning we somehow have to simulate using negative probabilities. This
may seem like an impossible task but there is in fact a way to make this
work. However the risk analysis works best if we don’t have any negative
probabilities. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/8986416
- author
- Hallabro, Marcus
- supervisor
- organization
- course
- FMSM01 20191
- year
- 2019
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Finite Difference Method, Option Pricing, Feynman-Kac Rep- resentation, Monte Carlo Simulation, Negative Probabilities.
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUTFMS-3375-2019
- ISSN
- 1404-6342
- other publication id
- 2019:E41
- language
- English
- id
- 8986416
- date added to LUP
- 2019-06-20 10:23:32
- date last changed
- 2024-09-30 12:39:55
@misc{8986416,
abstract = {{Derivative models often come in the form of stochastic differential equations.
From these equations a partial differential equation (PDE) can be derived.
By discretizing the PDE the numerical solution is obtained on a form where
the value of the derivative can be seen as a probabilistic weighting of future
values. These probabilities can be used to simulate trajectories of the under-
lying assets. This connection between the finite difference scheme and the
simulation is rather unique for this pricing method and can be very useful.
The probability weights can be forced to have a perfect probability interpre-
tation in some cases, meaning they are positive and less than one, but in
other cases we will end up with negative weights meaning we somehow have
to simulate using negative probabilities. This paper presents how to price
and simulate options with these methods in a few different situations and
how to solve some of the problems that may come up.}},
author = {{Hallabro, Marcus}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{Numerical solution for derivative models using finite difference methods and how this can be used with Monte Carlo simulation}},
year = {{2019}},
}