Question

Suppose that call options on ExxonMobil stock with time to expiration 6 months and strike price $97 are selling at an implied volatility of 29%. ExxonMobil stock currently is $97 per share, and the risk-free rate is 6%. If you believe the true volatility of the stock is 31%.

a. If you believe the true volatility of the stock is 31%, would you want to buy or sell call options?

b. Now you need to hedge your option position against changes in the stock price. How many shares of stock will you hold for each option contract purchased or sold? (Round your answer to 4 decimal places.)

Answer 1

(a.) Options Prices and Implied Volatility are directly proportional. In other words, as the prices of the options tends to increase, the implied volatility too increases and vice-versa.

Keeping all other factors unchanged, if the implied volatility is lesser than the actual or true volatility, say in this case; (29% < 31%), then there is a high chance that the implied volatility will tend to match up with the true volatility in the near future of expiration.

Call Options are such which are bought when the traders speculates that the underlying future prices of the security will rise higher. Therefore, in this scenario it will be advisable to buy the call options since it is expected to rise, with the rise in the implied volatility.

(b) No. of shares of the stock to be hold for each option contract are calculated as follows:

N (d1) where;

N (*) = Cumulative Normal Probability &

    d1 = In (S / K) + [(r + 0.5×σ²) × t] ÷ (σ √t)

        = In (97 / 97) + [(0.06 + 0.5×0.31²) × 0.5] ÷ (0.31×√0.5)

        = (0 + 0.054025) ÷ (0.219203102)

        = 0.246460928

∴ N (0.246460928) = 0.597337

                                = 0.5974 (approx)

∴ No. of shares of the stock to be hold for each option contract are

                                = 0.5974 (approx)

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