A basic stochastic model of market behaviour of asset prices is the
Ito process, a generalized Wiener process. The price x depends on the
time t in the following way:
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(1)
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the function a is called the drift rate, and b the variance rate. Z
refers to a Brownian motion or Wiener process. This is a
process in which the distribution of x changes randomly, independently
of its history, in such a way that the variance increases by a unit
amount over a unit time interval
If C is the price of an asset, and you have an option to purchase
that asset at a price K (the strike price) at a future time T, then the
value of that option at time T is clearly known; it is
The reason is that if the price S at that time exceeds the agreed
strike price K, then you have a profit S-K, while if it does not, the
option is unexercised. For a put option, the position is reversed:
As usual, we neglect brokerage costs and assume various degrees of
perfection about the market. For more information on these, the user
should consult the various references listed below. The kind of option
we are now discussing is called a European call option. A put option
would be an option to sell at an agreed price, rather than buy. The term
European refers to the fact that the option is an option to purchase on
a specified future date. If the agreement had been that the asset could
have been purchased at any time up until that date, it would be called
an American option.
At time t earlier than T, the option will have a value which depends
on the current price S and the time t. It will not, given the assumption
of independence in the Wiener process, depend on the price history.
There is a mathematical result, Ito's lemma, which relates the rates of
change of such a derived quantity; it is called Ito's lemma, and when
applied in this case gives the Black-Scholes equations:
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(2)
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Here we are using a more conventional financial notation. r is
sometimes called the riskless interest rate, but it can also be regarded
as an average rate of rise of comparable securities. s
is the volatility, or variance rate, now assumed constant.
When would you use Fastflo to solve an option pricing
problem?
The conventional Black-Scholes equations have analytic solutions, and
evaluation of these will always be faster and more accurate than using
pde methods. Where using FEM becomes attractive is when there are
special conditions on price limits (barriers), for example. These
complicate analytic solutions greatly, but do not add to the difficulty
of a pde solution. This is true even if, for example, the conditions
relate to the price of another security, also modelled by a stochastic
process. Sometimes the volatility is also allowed to be stochastic, and
becomes a second independent variable. This is also easily handled in Fastflo.
A major advantage for a practitioner in using Fastflo is that
the language used in the scripts make it easy to adapt to novel
conditions that may be proposed.
Fastflo's
accuracy
In Black-Scholes modelling, as in fluid mechanics, numerical
difficulties are most frequent when the diffusivity, or volatility, is
small. In a recent paper, Zvan,
Forsyth and Vetzal [1] looked at finite differences applied to a
simple Black-Scholes problem with low volatility, and studies the
spurious oscillations in the resulting solution. They were able to show
that using van Leer flux limitation methods that the oscillations could
indeed be very much reduced.
We checked this using Fastflo and found that, firstly, the
oscillations were much less than with the finite difference methods, and
the solutions apparently a lot more accurate. They were not, however, as
accurate as the van Leer -based approximations. We then tried removing
the advection, using a moving frame of reference. This is easy to do in Fastflo
even if continuous adaption to the boundaries is required. In the simple
case of a vanilla European option, that is not a problem; just solving
in discounted currency is sufficient. That is adequate to remove almost
all the errors in the cases examined. The results are included in a Case
Study.
A two dimensional problem
We have looked at Asian options, Garch (1,1)
models, stochastic volatility models and others. They are all readily
described in a Fastflo environment. Details will appear soon.
References
[1]
R. Zvan, P. Forsyth and K. Vetzal. Robust
numerical methods for PDE models of Asian options. J. Computational
Finance, 1:39-78, 1998
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