|
PureBytes Links
Trading Reference Links
|
Jesus,
Michael what are you doing at AIS Futures Management, LLC ? I don't want
to offend you or anyone else - really, believe me - but I hope you are not
responsible in any way to trade customer funds (based on your knowledge).
My impression is that you are lacking almost completely the theory of
Derivatives Pricing. (quote: OptionStation says B-S is used for this and
F.Blacks Model is used for that....) In order to use financial software
appropriately I strongly suggest you should take a course on the subject
(see book recommendation). Keying numbers into pre-coded formulas not
knowing what those formulas are actually calculating could lead to a total
desaster!
A very good book on the subject is John C. Hull's "Options, Futures, and other
Derivatives". It is used by a lot of Universities as the 'standard work'
as well as practioners. It covers all aspects of Derivatives pricing
including an introduction to exotic options. Both Models you stated as
well as others are covered.
Now a non-complete , little introduction to both Models:
a) Both Models were originally developped to price EUROPEAN TYPE OPTIONS.
B-S for Options on Stocks paying no dividends, F.Blacks for Futures
Options.
European type Options can only be excercised at Expiration date. Most
traded Futures Options are on the other hand AMERICAN TYPE (can be
excercised at any time). Unfortunately NO analytic formulas are
available for valuing American Futures Options - analytic
approximations as well as numeric procedures do however exist.
(Sorry since I do not have/use OptionStation I cannot tell you what
formulas they are using)
b) the formulas:
B-S Option values(EUROPEAN) for a stock paying no dividends:
Call = S*N(d1) - E*exp(-rt)*N(d2)
Put = E*exp(-rt)*N(-d2) - S*N(-d1)
d1 = [ln (S/E) + (r + s~²/2)*t]/[s~* SquareRoot of t]
d2 = d1 - s~* SquareRoot of t
where S is the Spot Price of the stock, E is the Excercise Price,N is
the Cumulative Normal Distribution, t is the Time to Expiration, s~ is
the Volatility and r is the continuously compounded risk-free rate of
interest.
B-S Op. values (EUROPEAN) for a stock with a continuous Dividend yield
Call = S*exp(-qt)*N(d1) - E*exp(-rt)*N(d2)
Put = E*exp(-rt)*N(-d2) - S*exp(-qt)*N(-d1)
d1 = [ln(S/E) + (r - q + s~²/2)*t]/[s~ * SquareRoot of t]
d2 = d1 - s~ * SquareRoot of t
where q is the dividend yield (percentage number)
Fisher Black for Commodity Futures Options (EUROPEAN STYLE):
Call = S*exp(-rt)*N(d1) - E*exp(-rt)*N(d2)
Put = E*exp(-rt)*N(-d2) - S*exp(-rt)*N(-d1)
d1 = [ln(S/E) + s~²/2*t]/[s~ * SquareRoot of t]
d2 = d1 - s~ * SquareRoot of t
c) Fisher Black developped his model in the late 70's. He wrote an article
about it in the Journal of Financial Economics named 'The Pricing of
Commodity Contracts'. You could try to find it in your local library.
His model is based on the assumption that the price of a futures
contract follows what is called Geometric Brownian Motion:
dF = µF dt + s~ dz.
µ is the expected growth rate , s~ is the volatility and dz is a so
called Wiener Process. (A Wiener process is a special version of a
Markov stochastic process. Models of Stock Price behaviour are usually
described in terms of Wiener processes. It was brought over from
Physics where it was used to describe the path/motion of a particle
that is subject to a large number of shocks->the so called Brownian
Motion) It can be shown that this assumption leads to the futures
price being treated in the same way as a security providing a continous
dividend yield equal to r. Compare the formulas above: For simple
calculation purposes you need to remember only the second one. Why?
Because if you have a stock that pays no dividends you set q=0. This
leads to the whole term exp(-qt) being 1. So you get the first formula
set. (in d1 and d2 r-q resembles r - 0 or just r). For the F.Black
formula you set q = r. This leads to the formula stated above: note in
d1 and d2 you set r - q = r -r so it shortens the formulas for d1 and
d2 as indicated above.
Hope this gives a first impression. I do recommend however to buy the book
and work it through. You will have a much better understanding of
Derivatives later on. PS: Be careful with the formulas - I checked them
once but I keyed them in quite fast so I might have overlooked something.
Best regards and successful studying,
Tom
|