Estimate the historical standard deviation of GOOG*. Compare the implied standard deviation with the historical standard deviation.

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Answer the following questions given the following call
option prices on Google (GOOG) and on Apple (APPL). Note that these are actual
option prices on 2/21/13 and these contracts have 60 days till expiration. The
2-month T-bill rate is about 4.75%. Show all work.

Part One

1. Estimate the theoretical option values for the call on
GOOG with K =800 and for the call on APPL with K = 450 using the
Black-Scholes-Merton and Binomial Models program (available under doc sharing).
You can also use the following website to calculate the option prices and
implied volatility 

www.option-price.com

2. When estimating
the option values assume various standard deviations of returns of 10%, 15%,
20%, … up to 100% or until you find the theoretical option value is close to
the actual one.

3. Draw a graph showing the relationship between standard
deviations and option values.

4. Based on the
graph, what does the actual option value imply about the expected future
standard deviation (volatility)? Which option has higher implied volatility and
is it surprising?

Part Two

1. Estimate the historical standard deviation of GOOG*.

2. Compare the implied standard deviation with the
historical standard deviation.

3. What can you infer from the difference, if any, between
the two numbers?

Part three

1. Compute the implied volatility for all options.

2. Do you have the same implied volatility for the two
options on the same underlying? If not (in which case it is referred to as
volatility smile), what might be able to explain the differences? (Hint: refer
to the chapter on volatility smile).