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 Subject: Option Value Category: Business and Money > Finance Asked by: rr64519-ga List Price: \$25.00 Posted: 28 Aug 2005 20:32 PDT Expires: 27 Sep 2005 20:32 PDT Question ID: 561624
 ```1. What?s the value of a \$100 call option. (European Option) 2. What?s the value of a \$100 put option. (European Option) If: -Option exercise price = \$ 25 -Standard deviation of the continuously compounded return of the stock = .8 -Time period = 2 years -Current stock price = \$ 20 -Interest rate = 5% Please use Black-Scholes Model and show work.``` Request for Question Clarification by elmarto-ga on 29 Aug 2005 06:59 PDT ```Hello, Could you please clarify what do you mean by "a \$100 call option"? Does it mean that you bought it for \$100 (not that this would have any effect on its value)? Best regards, elmarto``` Clarification of Question by rr64519-ga on 29 Aug 2005 07:42 PDT `Typo. It should be a \$20 call and put option, which matches the stock price.` Clarification of Question by rr64519-ga on 29 Aug 2005 07:46 PDT `I am incorrect. Let me keep looking for clarification.` Clarification of Question by rr64519-ga on 29 Aug 2005 14:26 PDT `Just ignore the \$100 for both. I should have never put them in there.` Clarification of Question by rr64519-ga on 30 Aug 2005 16:43 PDT `Any update?` Request for Question Clarification by elmarto-ga on 30 Aug 2005 18:07 PDT ```Just one more clarification: the Black-Scholes model results in a formula that shows the value of a put or call option given the parameters you've supplied (exercise price, std. dev., etc). Do you need me to explain how the Black-Scholes model is derived (which would be a bit lengthy) or just show you the formula, plug your parameters, and give you the value of the options? Best regards, elmarto``` Clarification of Question by rr64519-ga on 31 Aug 2005 07:12 PDT `No need for a derivation. I know how ugly that is :). Just plug and play is fine.` Request for Question Clarification by elmarto-ga on 31 Aug 2005 07:31 PDT ```Hello, Sorry, I forgot to ask just one more thing in my previous request. Is the standard deviation you provide annualized? Monthly? In any case it seems quite high at 0.8 (80%). Could you confirm that this number is right, and whether it is annual, semi-annual, etc? Thank you very much, elmarto``` Clarification of Question by rr64519-ga on 31 Aug 2005 09:14 PDT ```80% is correct. What would it normally be and how will the 80% effect the numbers. I will crunch the numbers once you are done to see as well.```
 ```Hello! At last, here's the answer :) First of all, I'll assume that the standard deviation you provide is for annual returns. That said, you can find the Black-Scholes formula at the following site: Wikipedia http://en.wikipedia.org/wiki/Black-Scholes Let's first compute "d1" and "d2" as described in that link. Using the same notation, you have: S = 20 K = 25 T = 2 r = 0.05 sigma = 0.8 Plugging these value into the formula gives: d1 = 0.4568408747... d2 = -0.6745299752... Now, the formula for the options value makes use of the standard normal cummulative distribution function (cdf). Since there is no explicit formula for this function, I used the online calculator at the following site: Normal Calculator http://cnx.rice.edu/content/m11328/latest/ [make sure you select "Area left of" in this calculator] Finally, plugging d1 and d2 in the calculator, and using these results in the call value formula, we get that the theoretical value of the call option , according to this model, is: C = 7.86 Likewise, the theoretical value of the put option is: P = 10.48 If you want to take a shortcut, you can just use the follwing Java-based online European option value calculator Option pricing http://www.margrabe.com/OptionPricing.html If you plug the parameters of your problem, you'll get the same results as with the formula (probably a bit different because I did some rounding). Regarding the 80%, forget what I said in the clarification request about it being quite high. There are lots of stocks with volatility even higher than 80% annualy. Google search terms black scholes formula ://www.google.com.ar/search?hl=es&q=black+scholes+formula&meta= option calculator black scholes ://www.google.com.ar/search?hl=es&q=option+calculator+black+scholes&meta= I hope this helps! If you have any questions regarding my answer, please don't hesitate to request a clarification. Otherwise I await your rating and final comments. Best wishes! elmarto``` Request for Answer Clarification by rr64519-ga on 01 Sep 2005 13:48 PDT `Thx. I am reviewing it now.` Request for Answer Clarification by rr64519-ga on 01 Sep 2005 14:58 PDT ```OK. I am plugging the numbers into the formula for d1 and dont get the same answer you do. Can you tell me where I am messing up, see below: ln(20/25) = .2231 (.05 + 1/(2*.8*.8))2 = 1.66 sqrt(.8*.8*2) = 1.13 I come up with d1 = 1.88/1.13 = 1.66 ???``` Clarification of Answer by elmarto-ga on 01 Sep 2005 16:00 PDT ```Hello! There are some mistakes in your calculations: ln(20/25) = -0.2231 (not 0.2231) (r + (sigma^2)/2)*T = (0.05 + 0.64/2)*2 = 0.74 (not 1.66) sigma*sqrt(T) = 0.8*sqrt(2) = 1.1313... (this one is correct) Therefore, d1 = (-0.2231 + 0.74)/1.1313 = 0.45... Please let me know if you have any other doubt regarding my answer. Best wishes! elmarto```