Investments, tenth edition

 Example  21.8 

Speculating on Mispriced Options 

 Notice in Example 21.8 that the profit is not exactly independent of the stock price. This 

is because as the stock price changes, so do the deltas used to calculate the hedge ratio. The 

hedge ratio in principle would need to be continually adjusted as deltas evolve. The sensi-

tivity of the delta to the stock price is called the    gamma    of the option. Option gammas are 

analogous to bond convexity. In both cases, the curvature of the value function means that 

hedge ratios or durations change with market conditions, making rebalancing a necessary 

part of hedging strategies. 

 Suppose you bet on volatility by purchasing calls instead of puts. How would you 

hedge your exposure to stock-price fluctuations? What is the hedge ratio? 

 CONCEPT CHECK 

21.9 

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P A R T   V I

  Options, Futures, and Other Derivatives

 A variant of the strategy in Example 21.8 involves cross-option speculation. Suppose 

you observe a 45-day expiration call option on IBM with strike price 95 selling at a price 

consistent with a volatility of  s 5  33% while another 45-day call with strike price 90 has 

an implied volatility of only 27%. Because the underlying asset and expiration date are 

identical, you conclude that the call with the higher implied volatility is relatively over-

priced. To exploit the mispricing, you might buy the cheap calls (with strike price 90 and 

implied volatility of 27%) and write the expensive calls (with strike price 95 and implied 

volatility 33%). If the risk-free rate is 4% and IBM is selling at $90 per share, the calls 

purchased will be priced at $3.6202 and the calls written will be priced at $2.3735. 

 Despite the fact that you are long one call and short another, your exposure to IBM 

stock-price uncertainty will not be hedged using this strategy. This is because calls with 

different strike prices have different sensitivities to the price of the underlying asset. The 

lower-strike-price call has a higher delta and therefore greater exposure to the price of 

IBM. If you take an equal number of positions in these two options, you will inadvertently 

establish a bullish position in IBM, as the calls you purchase have higher deltas than the 

calls you write. In fact, you may recall from Chapter 20 that this portfolio (long call with 

low exercise price and short call with high exercise price) is called a  bullish spread.

 

To establish a hedged position, we can use the hedge ratio approach as follows. 

Consider the 95-strike-price options you write as the asset that hedges your exposure to the 

90-strike-price options you purchase. Then the hedge ratio is

 H 5

Change in value of 90-strike-price call for $1 change in IBM

Change in value of 95-strike-price call for $1 change in IBM

 5

Delta of 90-strike-price call

Delta of 95-strike-price call

. 1

You need to write  more  than one call with the higher strike price to hedge the purchase of 

each call with the lower strike price. Because the prices of higher-strike-price calls are less 

sensitive to IBM prices, more of them are required to offset the exposure.