Wednesday, 7 October 2009

Black and Scholes Volatility

Just occasionally I throw in a bit of finance into these pages to keep my more technically inclined readers happy. This offering is to help clear up the mystery of how to estimate volatility into the option pricing model.

The Black, Scholes and Merton model for pricing options assume that the underlying asset (share price, rate of exchange, NPV) is continuously traded and follows a random process through time along a rising trend of know drift and constant variance. This price process is known as geometric brownian motion and it results in a log-normal distribution of values. The random process is assumed to be continuous and if we take price changes over (theoretically, infinitely) short periods of time the distribution of price changes is normal.

So in order to estimate volatility from past data we need a statistically significant number of price readings (I use 101 daily price observations) and then take a daily return (P1/P-1) -1 on each. If the natural log of this number is taken we have an approximation of the continuously generated daily return. The standard deviation of these 100 returns is the crude historical volatility. A better measure can be developed whereby the volatilities are weighted using a sum of the day's digits approach and a weighted standard deviation calculated. This weighted volatility appears in most cases to give better than 2% accuracy on most actively traded near the money options. More sophisticated techniques are available (GARCH 2 for example) but do not in my experience give notably better results.

10 comments:

Anonymous said...

hi Bob what does project duration means ?

Anonymous said...

yeah.. plz makes a short note on duration method... i've read 3 books in finance but haven't found anything bout this method.. I wiki and google it also found nothing...

Anonymous said...

I can bet he wouldn't make a release on duration method.As the only way to learn this is from his textbook.He dont want to devalue his book.so you can either to purchase his book or ignore it.ACCA recently appoint a new examer.We can do reasonable assumption the project duration method wouldn't appear in DEC paper.

Professor Bob Ryan said...

Please read the notice on the blog with respect to all queries relating to P4 and what might or not be in the examination.

Anonymous said...

Dear Prof,

Actually we ask for duration method to be posted on the blog for our knowledge and working purpose. If u dont mind, please write a short note in this method. I am currently working and studying ACCA at the same time, i hope you understand my situation, if i still study full time for this paper, i might read your recommended text. As for your info, i read a lot of text and i don't take an easy approach to sit on the next p4 exam. So, if you could help by posting a duration method, its definition and concept, it would be appreciated. Thanks

confused-student said...

i think project duration method relates to this paper:
A Comparison of Different Project Duration Forecasting Methods using Earned Value Metrics by S. VANDEVOORDE and M. VANHOUCKE
Earned value project management is a well-known management system that integrates cost, schedule and technical performance. It allows the calculation of cost and schedule variances and performance indices and forecasts of project cost and schedule duration. The earned value method provides early indications of project performance to highlight the need for eventual corrective action. Earned value management was originally developed for cost management and has not widely been used for forecasting project duration. However, recent research trends show an increase of interest to use performance indicators for predicting total project duration

confused-student said...

or is it the same as MODIFIED DURATION of bonds and estimating the change in price (NPV) based on change in yield (discount rate)?
I am confused, plz help me out, prof Bob, as I live in a remote country w/o any ability to purchase your book

registoni said...

in june 2009 exam the question asked for calculation of duration, which the notion taken from bond pricing and applied to project appraisal.
Duration of a bond is the weighted average of the maturity of individual cash flows, with the weights being proportional
to their present values. Duration measures the average time taken by a bond, on a discounted basis, to pay back the original investment.

I think everyone was confused and thought about discounted payback period, but duration is a bit different.
You can read about duration method in any book on fixed-income securities, not only in prof Bob's books :)

Anonymous said...

With all due respect, can i know a lil bit on a duration method for my working purpose as a finance staff.

I am not the ACCA since u cant give any comment on readings, exam and anythings about the its paper.

registoni said...

i read about duration method in HULL's Options, Futures, and Other Derivatives (7ed) but I suspect this method is explained in any serious book on fixed-income securities (bonds) or financial markets.