Hi!!
I found the Black and Sholes formula for the price of a put option, it
is very similar to the call's formula, only some signs change:
P(S,T) = K.e^(-rT).N(-d2) - S.N(-d1)
See "Black-Scholes - Wikipedia, the free encyclopedia":
http://en.wikipedia.org/wiki/Black-Scholes#The_formula
- Cumulative distribution function:
The cumulative function (cdf) for a given distribution is defined as
the probability that a variable X has a value less than or equal to x,
it is expressed in terms of the distribution's density function, see
for reference:
"Cumulative Distribution Function - 1.3.6.2. Related Distributions":
http://www.itl.nist.gov/div898/handbook/eda/section3/eda362.htm#CDF
The standard normal cdf, conventionally denoted ?, is just the general
cdf evaluated with ? = 0 and ? = 1 , see:
"Normal distribution - Wikipedia, the free encyclopedia":
http://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function
After you get the values d1 and d2, just use a "Standard Normal
Cumulative Distribution Function Table" like the following ones:
"Table of the Standard Normal Cumulative Distribution Function":
http://are.berkeley.edu/~ferre/ztable.pdf
"NORMAL DISTRIBUTION":
http://www.maths.manchester.ac.uk/~cds/internal/tables/normal.pdf
Search strategy:
"Put option" model
"Put option" Black Scholes
normal "cumulative distribution function"
I hope this helps you. Feel free to request for a clarification if you
find something unclear.
Regards,
livioflores-ga |
Request for Answer Clarification by
vaac-ga
on
22 Aug 2006 13:55 PDT
Thank you very much, livioflores-ga, for your comprehensive and well
documented answer. I am sure that a person more versed in math and
statistics than I could derive from this an equation expressing the
premium of a put option in terms of the parameters given below. I am,
unfortunately, not vell verst enough, and cannot use your answer in
its present form.
What I asked for, and what I need is an equation that will give me the
premium of a put option with the stiking price P1, The current price
P2, the time to
expiration (assumed to be long) t, the interest rate i, the average
change in price of similar stocks over a veek v%, the basis of natural
logarithm 2.7183, the assumption that the stock will pay no dividends,
and any other parameter needed, and how to find such a parameter even
approximately.
If the term "cumulative standard normal distribution" appears, as it
does in the call-option calculation, I would appreciate if you could
explain it.
|
Clarification of Answer by
livioflores-ga
on
23 Aug 2006 09:58 PDT
Hi!!
I answered this question in the assumption that you need a formula
similar to the one you find for calls, that is if you have the
Black-Scholes model for puts, like you have for calls, it will be
useful to you; so I thought that the formula I found would be useful
to your purposes.
Note that the only input that is different from yours is regarding the
average change in price of similar stocks over a veek v% (could this
mean week?), for example the Black-Scholes model that you found uses
the Standard Deviation on the stocks returns. Note also that the
average change on prices said nothing about its standard deviation
(average between 999 and 1 is the same that the average between 501
and 499 - that is 500 - but their deviations differes a lot). I found
no model that uses the averages of prices change.
I will continue the research trying to bring you more info that could
lead to a something useful to you, but I guess that you need to modify
the format of the volatility input.
NOTE:
"Volatility most frequently refers to the standard deviation of the
change in value of a financial instrument with a specific time
horizon. It is often used to quantify the risk of the instrument over
that time period. Volatility is typically expressed in annualized
terms, and it may either be an absolute number (5$) or a fraction of
the initial value (5%)."
Volatility - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Volatility
Regards,
livioflores-ga
|
Clarification of Answer by
livioflores-ga
on
26 Aug 2006 07:54 PDT
Hi!!
I am continue working on your question, I hope that by tomorrow night
I can give you an answer that let you calculate the price of put
options including some examples.
Again note that "average change in price of similar stocks" has no
impact on the formula, you must get the volatility, I hope that I can
explain you this term and give you tools to find it.
Regards,
livioflores-ga
|
Clarification of Answer by
livioflores-ga
on
28 Aug 2006 10:55 PDT
Hi!!
Let me start saying what Black-Scholes formula for Put options does:
Given the following inputs:
S ? Stock Price = the current market price of the stock.
T ? Time = time until option expires.
K ? Option Strike Price or exercise price.
r ? risk free rate (yield of appropriate US govt. treasury).
s ? volatility of returns (How much has the stock fluctuated in the
past?) expressed in terms of the standard deviation of such
fluctuation.
You will get the following output:
Theoretical Put Option Premium or Put Option Price.
According to your request you have:
"the stiking price P1, The current price P2, the time to expiration
(assumed to be long) t, the interest rate i, the average change in
price of similar stocks over a veek v%, the basis of natural logarithm
2.7183, the assumption that the stock will pay no dividends"
The last assumption give us the chance to use the Black-Scholes model
since one of its assumptions is that stock pays no dividends during
the option's life.
-P1 is the strike price (K in the answer's formula)
-P2 is the stock price (S in the answer's formula)
-t is T in the answer's formula
-i is r in the answer's formula; remember that it is the risk free
rate, you can use for this value the 10-YEAR TREASURY NOTE (^TNX), the
current value is 4.81%. For updates see its Yahoo! Finance page
(search for Index Value):
http://finance.yahoo.com/q?s=%5ETNX
-Regarding the average change in price of similar stocks, you can use
it as an estimated volatility, but later I will give you some clues on
how to get a more accurate value.
The first thing you must do is to calculate d1 and d2 to two decimal
places using the following formulas:
ln (S / K) + (r + (s^2)/2) * T
d1 = --------------------------------------
s * sqrt(T)
d2 = d1 - s * sqrt(T)
Recall that:
S = current stock price
N = cumulative standard normal distribution
T = time until option expiration (in years)
r = risk-free interest rate
K = option strike price
e = the constant 2.7183..
s = standard deviation of stock returns
ln() = natural logarithm of the argument
sqrt() = square root of the argument
^ means exponentiation (i.e., 2^3 = 8)
Suposse you get:
d1 = 0.43
d2 = 0.18
The next step is to find the value of N(-d1) and N(-d2). Use normal
distribution tables to find them, I suggest you this one:
"Table of the Standard Normal Cumulative Distribution Function":
http://are.berkeley.edu/~ferre/ztable.pdf
N(-d1):
Go to the table and find the -0.4s row on the left; the first item of
this row is N(-0.40), the second is N(-0.41), etc.; so you must use
the fourth value that is N(-0.43)=0.3336
Using the same method you will find that N(-d2)=0.4286
Now plug these values together with the other data you have into the
Black- Scholes formula, use the calculator and you will get the Put
Option Premium.
I found a page that gives a complete example on how to use the
Black-Scholes model for a call option, but for the put option premium
uses the "Put?Call Parity" Theorem:
C + PV(K) = P + S
where:
C = the current market value of the call;
PV(K) = the present value of the strike price K, discounted from the
expiration date at a suitable risk free rate;
P = the current market value of the put;
S = the current market value of the stock.
So if you have the call premium you can get the put premium by:
P = C + PV(K) - S
Note that PV(K) = K.e^(-rT); then, if you have the call premium the put premium is:
P = C + K.e^(-rT) - S
See:
"Put-Call Parity - riskglossary.com":
http://www.riskglossary.com/link/put_call_parity.htm
The page with the example is:
"Black scholes model" by Patrick Lynch:
http://www.accaglobal.com/publications/studentaccountant/456433
Regarding volatility you can find how to estimate it in an easy way here:
"CRB Option Implied Volatility and Futures Historical Volatility":
http://www.crbtrader.com/support/options.asp
You can also use the CBOE.com IV Index:
http://www.cboe.com/tradtool/IVolService8.aspx
Just input the stock symbol and click GO button, for example current
3M Company's volatilities are (plug the MMM symbol at the box):
Historical = 13.88% (10 days) / 11.09% (20 days) / 19.21% (30 days)
Implied = 16.63% (Call) / 16.48% (Put)
You can use an online calculator:
"CBOE - Theoretical Listed Option Price Calculator":
http://www.cboe.com/LearnCenter/OptionCalculator.aspx
For additional references see:
"Introduction to Derivatives ? Options":
http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt
"Equity Options":
http://www.m-x.ca/f_publications_en/en.guide.options.pdf
I hope this clarify the answer. Feel free to continue using the
clarification feature if you still need it.
Best regards,
livioflores-ga
|
Request for Answer Clarification by
vaac-ga
on
29 Aug 2006 17:00 PDT
Thank you, Livioflore-Ga for yours excellent and well documented
answer. I checked all the URL's and they all lead to useful option
information except the one next to last
"http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt"
which gives a grey screen with only "retry" and "cancel" on it and
which cannot be removed from the computer except through Ctrl-Alt-Del.
Could you please check if this url's spelling is right or the
dfficulties are due to the inadvecacy of winows98?
I understand now that the excessive time consumption necessary for
getting the closing of the previous 20 days in order to calculate
volatility -- cannot be avoided. Not, unless a macro can be composed
which will do this Automativally .I would not know how to do this or
if it can be done.
Regards
vaac-ga
|
Clarification of Answer by
livioflores-ga
on
29 Aug 2006 20:58 PDT
Hi!!
Thank you for the good comments!!
Regarding the document after the link:
http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt
It is a MS PowerPoint presentation and you need to have it installed
in your computer, it is included with the MS Office package.
If you do not have it installed you can download for free the MS PowerPoint Viewer:
"Download details: PowerPoint Viewer 2003":
http://www.microsoft.com/downloads/details.aspx?FamilyID=428d5727-43ab-4f24-90b7-a94784af71a4&displaylang=en
If you have PowerPoint installed or after install the Viewer you are
continue having troubles with the link do the following:
- Open this page with Microsoft's Internet Explorer
- Do a right click on the link
- Select the option Save link as... (or a similar option like Save Target As)
- Save the file and then open it with PowerPoint or the Viewer.
Regarding the Volatility calculation I do not know how to make a
proper macro for this but I think that an Excel spreadsheet where you
input the 21 days data is easy to do, just input the Close Prices of
the stock for the last 21 days at column A (A1 to A21). Start for the
most recent day, that is at A1 set the close price for today's close,
A2 for yesterday´s close, etc.
Then at column B do the "today's close / previous close" (that is B1:
=A1/A2, B2: =A2/A3, ... ,B20: =A20/A21).
Then at column C do the log calculation (C1: =LN(B1), etc.).
At column D set the squares of the natural logs (D1: =C1*C1 ,etc.).
At cell C21 set the sum of the natural logs over the past 20 days,
that is C21: =SUM(C1:C20) .
At cell D21 set the sum of the squares of the natural logs over the
past 20 days, that is D21: =SUM(D1:D20) .
At cell C22 divide the sum of the natural logs by 20, C22: =C21/20 .
At cell D22 divide the sum of the squares of the natural logs by 20, D22: =D21/20 .
At C23: =D22-(C22*C22)
At A24 do the last calculation that will give you the 20-day historic
volatility for today:
A24: =SQRT(C23)*SQRT(252)*100
The only work you must do is to input the Close Prices of the stock
for the last 21 days at column A; the rest will be solved
automatically.
Please check if this spreadsheet guide is right according to the
guidelines given at:
"CRB Option Implied Volatility and Futures Historical Volatility":
http://www.crbtrader.com/support/options.asp
I hope this helps you.
Regards,
livioflores-ga
|
Request for Answer Clarification by
vaac-ga
on
30 Aug 2006 17:13 PDT
Thank you very much for your answer and please excuse my ignorance. I
have downloaded "Power point viewer" but still cannot get
"http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt"
You say "open this page with microsoft...", which page is "this page"
and how do you open it? On which link shall I do a right click?
Thanks in advance
Regards
vaac-ga
|
Clarification of Answer by
livioflores-ga
on
30 Aug 2006 20:28 PDT
Hi again!!
Do not worry, that part of my clarification is a little ambiguous.
When I said open this page I am refering to THIS page:
http://answers.google.com/answers/threadview?id=757943
Just open it with with Microsoft's Internet Explorer;
- Do a right click on the following link:
http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt
- Select the option Save link as... (or a similar option like Save Target As)
- Save the file and then open it with PowerPoint or the Viewer.
I hope you can open the file after this clarification.
Best regards,
livioflores-ga
|
Clarification of Answer by
livioflores-ga
on
30 Aug 2006 20:32 PDT
Hi!!
For some reason the link to the PowerPoint presentation was mispelled
in my last clarification. Please save the file after the following
link (332Kb):
http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt
Good luck!!
|
Request for Answer Clarification by
vaac-ga
on
31 Aug 2006 16:50 PDT
Sorry, but I still cannot get the url
"http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt". Where
is this link "http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt"
or
"http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt" (which
you claim are misspelled but which are identical) on which
I am supposed ti right click?
Thanks and regards
vaac-ga
|
Clarification of Answer by
livioflores-ga
on
31 Aug 2006 20:16 PDT
Hi!!
One of the links that I posted (the one I claimed is misspelled) has a
minus sign (-) added and it does not work, the correct link is:
http://www.usc.uwo.ca/clubs/investment/downloads/options.ppt
Here is a cached version of the file (without images and graphs):
http://72.14.207.104/search?q=cache:ayhOYtMSZmAJ:www.usc.uwo.ca/clubs/investment/downloads/options.ppt+&hl=en&ct=clnk&cd=3
Regards,
livioflores-ga
|
Request for Answer Clarification by
vaac-ga
on
11 Sep 2006 15:36 PDT
Thank you for the cached version of the file (without images and graphs):
http://72.14.207.104/search?q=cache:ayhOYtMSZmAJ:www.usc.uwo.ca/clubs/investment/downloads/options.ppt+&hl=en&ct=clnk&cd=3
It works!!
You have spent much,much time with me to get me an extremely
comprehensive and extremely well documented answer. I did not yet get
a chance to put it to use except for calculating volatility for a
stock of symbol holz, and trying out Schwab's "option calculator"
I have calculated the volatility according to your instructions and
tried to calculate the Jan 2009 40-striking price for a put for a
stock symbol HOLX selling at 42.84 on Sept 1, 06. Using Charles
Schwabs "opion calculator" putting in 4.81 for interest rate 13.7 for
volatility and 0 for dividend (the stock does not pay any). The model
used, Black & Sholl, gave me a price of 0.88 while the bid and ask are
7.90 and 8.8. The answer for the call, 8.03, for which I did not look
up the bid and ask is probably the right order of magnitude.
I cannot expect you to explain why Schwabs calculaor does not work,
Thank yuo very much
vaac-ga
|
Clarification of Answer by
livioflores-ga
on
11 Sep 2006 23:05 PDT
Hi!!
Thank you for the good rating and the generous tip.
Regarding the calculator I do not know the Schwab's "option
calculator" but I think that you are using a very low value for the
current volatility.
According to "IVolatility.com - Services & Tools -> Knowledge Base ->
Education -> Calculators Help", the more proper value for this input
is the last night's implied Volatility for the stock.
http://www.ivolatility.com/info/calchelp.html
Recall that the calculator associated to IVolatility.com is "CBOE -
Theoretical Listed Option Price Calculator":
http://www.cboe.com/LearnCenter/OptionCalculator.aspx
At the CBOE.com IV Index
(http://www.cboe.com/tradtool/IVolService8.aspx) page I searched for
the HOLX symbol and found a current implied volatility for puts of
47.23% .
Using the above value on the CBOE calculator I got for an American
Option a price of $8.421 and for an European option a price of $7.878
.
I used the following calculators to recheck the option price and got
the results listed below:
"Robert's Online Option Pricer!":
$8.2565
http://www.intrepid.com/~robertl/option-pricer1/option-pricer.cgi
"BIG APPLET -- OPTION PRICING":
$7.75
http://www.margrabe.com/OptionPricing.html
"DUKE UNIVERSITY -- Option Pricer":
$8.412
http://www.duke.edu/~charvey/OptionPricer/op.htm
As you can see the above values are close to the bid and ask values that you have.
I hope that you find this helpful.
Regards,
livioflores-ga
|
Request for Answer Clarification by
vaac-ga
on
12 Sep 2006 17:18 PDT
Many thanks for the un-hoped for clarification of option calculators.
I did know that there are several ones on the internet and free
reagards
vaac-ga
|
Clarification of Answer by
livioflores-ga
on
12 Sep 2006 20:26 PDT
You are welcome.
Best regards,
livioflores-ga
|
,
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