Mon Feb 18 18:36:26 EST 2008

> Are we supposed to compute the implied volatilities for each data point?

  Yes, using a model other than vanilla Black-Scholes, or an approximation 
algorithm other than the Newton-Raphson / bisection methods used to 
compute the numbers in the data set.  


> We haven't really studied anything other than Black-Scholes.  
> Can the implied volatility be computed for other options pricing models?

  Yes.  Let s be an estimate of the future volatility of the underlying.
Then the theoretical option price C is a function of s and other relevant
inputs, e.g. C = f(s, ...).  Now assume an inverse function (of f) 
s' = g(C', ...).  Then s' is the volatility implied by a market price C'.
You can compute the implied volatility of an options price under any 
pricing model by inverting the pricing model f().  One reason we think 
this is an interesting problem is because inverting f() is seldom possible 
in a clean, closed-form way.

  One reason that there is a "smile" in Black-Scholes models has to 
do with the "fat tails" we discussed: equities do not follow a stationary
log-normal process.  In a perfect option pricing model
and an efficient options market, the implied volatility of the various
market prices for options at different strike prices and expiration dates
should be constant.  After all, your assumptions about the volatility of 
the underlying security should not change just because you are pricing a
different option.  Therefore, variation in the volatility implied by
the various options' market prices indicates either that the model is 
inaccurate, or the options are mispriced.  It's widely accepted that
Black-Scholes is but a useful approximation.

  If you use another options pricing model, you will get a different 
smile than comes up in a Black-Scholes model -- maybe even a smirk, 
a scowl or a grimace!   What you want is unemotional accuracy:   :^|

  If you prefer to use Black-Scholes based methods instead of studying 
and inverting a model, this paper describes and cites various methods 
for approximating the Black-Scholes implied volatility:
http://www.eecs.harvard.edu/~parkes/cs286r/spring08/reading3/chambers.pdf


> My understanding of the volatility smile is that it is a plot of implied 
> volatilities versus the strike price for options with the same maturity 
> date. Is that correct?

  Yes.


> If so, this leads us to the assumption that there could be one smile per 
> maturity date per asset. In this case, are we to pick some maturity 
> dates and use those for analysis?

  Yes, that is correct.  The data set contains only "near" maturity dates. 
For current date T, the next expiration date D1 > T always appears, 
(quotations do not appear when D1=T), and following month's expiration 
date D2 begins appearing on the Monday 25 days before D2 (or the next 
business day).

  If you want to look at how the smile changes as time to maturity 
decreases (e.g.


> In the updated assignment description, it says "our data set does not 
> contain sufficient future prices to consider time to expiration". What 
> exactly does this mean?

  It meant that we did not have sufficient future expiration months to 
study the volatility term structure of the implied volatility surface
by looking at several different expiration months.  This was badly
worded; of course you may need to know days to maturity to compute
the implied volatility.  I have corrected this.


>       We have chosen a subset of stocks we are going to analyze and assumed 
> they form an equally weighted index. We can consider options for each stock 
> individually, or synthesize an option on the index. When we compute the 
> implied volatility, are we doing this individually for each stock or only for 
> the synthesized option?
>
>     We are worried because the amount of data is an order of magnitude 
> higher if we compute for each stock individually.

   It's unclear to me how you would answer the questions about market 
prices if you synthesized an option, because you would not be able to know 
what the market prices were.  One goal of the assignment is to learn about 
any inconsistencies between market prices of options for the same 
security.

   If you need to reduce the size of your data corpus, may I suggest using 
a smaller number of equities in your index?



Mon Feb 18 19:04:02 EST 2008

> We are assuming that the volatility surface is convex. Hence, we define a 
> 'bump' in the volatility surface as a point where the surface becomes 
> concave. That is: a point that has higher implied volatility than any of its 
> neighbors.

  Well, the extrema are expected have higher IV than their neighbors.
Moerover, your definition is too strong: consider a surface like this:

                                                            
 *                     *                                    
                                                            
   *                                                        
                    *                                       
                                                            
      *                                                     
         ^                                                  
               *                                            
                                                            
           x                                                
                                                            

Point ^ is not higher than (or even equal to) its neighbors; nonetheless
it doesn't fit the smile very well: one example is that the curve between 
defined by ^ and its neighbors is concave down.

While local concavity is one way of identifying a "bump", be careful:
point x could also be the culprit in the above graph!  You can use any 
metric you wish for deciding when something is "out of line", but be
prepared to justify it.