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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.
OPTION STRIKE EXP VOL LAST
GOOG 800 APR 378 28.20
S=795.53 690 APR 53 101.57
APPL 450 APR 530 18.55
S=446.06 480 APR 856 7.81
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).

*: How to estimate the historical standard deviation:
  1. You need to obtain historical stock prices. I recommend using Yahoo Finance http://finance.yahoo.com/ (enter symbol)--chart--(daily or weekly) historical quotes
  2. Calculate stock return
    Return on day T+1 = (stock price on day T+1 - stock price on day T)/ stock price on day T
    Rt+1 = (St+1 – St) / St
    • For simplicity we can ignore dividends, since GOOG pays no dividend.
    • A minimum of 30 returns is required—I am not requiring an exact time period or the length of time period, so that your answers will be slightly different and answers identical to those of other students might receive point deductions.
  3. The standard deviation (SD) of Stock Returns can then be estimated using EXCEL function STDEV.S(range of return data). Further, the SD needs to be annual standard deviation. If you use daily returns, the SD you obtain is a daily one; you need to transform it to Annual SD = Daily SD * (365)0.5.  For weekly data, Annual SD = Weekly SD * (52)0.5

Question Set #27

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