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Q: Put option evaluation ( Answered ,   0 Comments ) Question
 Subject: Put option evaluation Category: Business and Money > Accounting Asked by: vaac-ga List Price: \$10.00 Posted: 20 Aug 2006 16:08 PDT Expires: 19 Sep 2006 16:08 PDT Question ID: 757943
 ```I wonder if anybody can show me how to calculate 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 and how to find such a parameter. The internet site http://bradley.bradley.edu/~arr/bsm/pg04.html gives you the Black and Sholes model for calls but I cannot find anything similar for puts. In this call calculation I do not quite understand what "n=cumulative standard normal distribution" is. So if you have a model for puts and this term is there, please explain it``` Subject: Re: Fundamentals of Corporate Finance:The Time Value of Money Answered By: livioflores-ga on 20 Aug 2006 18:50 PDT Rated: ```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```
 vaac-ga rated this answer: and gave an additional tip of: \$3.00 `Extraordinary comprehensive and extremely thorougly well documented answer!`  