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# L1.T4.9. Gamma-neutral option positions

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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AIMs: Explain how to implement and maintain a gamma‐ neutral position. Discuss the relationship between delta, theta, and gamma. Describe how hedging activities take place in practice, and discuss how scenario analysis can be used to formulate expected gains and losses with option positions. Describe how portfolio insurance can be created through option instruments and stock index futures.

Questions:

9.1. A delta-neutral option portfolio has a position gamma of +300. If call options have a (percentage) delta of 0.58 and gamma of 0.120, what trades will neutralize the delta and gamma of the portfolio?
a. Long 1,500 put options and sell 950 shares
b. Short 1,500 put options and buy 950 shares
c. Long 2,500 call options and sell 1,450 shares
d. Short 2,500 call options and buy 1,450 shares

9.2. A market maker writes (sells) a contract of 100 call options, where the percentage (per option) delta of the call options is 0.60 and the gamma is 0.080. The market maker wants to neutralize both the delta and gamma of this position (delta-gamma-neutral) with two additional trades: the underlying shares; and put options on the stock with percentage delta of -0.40 and gamma of 0.020. What are the trades?
b. Sell 200 put options and sell 80 shares
d. Sell 400 put options and sell 220 shares

9.3. Hull (equation 17.4) shows that the relationship between theta, delta and gamma is given by: theta + (Rf * S * delta) + (0.5*variance(S)*S^2*gamma) = Rf*Value(option portfolio), where (Rf) is the riskfree rate and (S) is the stock price. The price of a one-year European call option with a strike price of $100 is$13.75 when the stock price is also \$100. The volatility is 30.0% and the riskfree (Rf) rate is 4.0% per annum. The option's (percentage) delta is 0.612 and gamma is 0.0128. What is the option's theta?
a. -5.333
b. -7.658
c. -9.112
d. -11.115

9.4. A delta-neutral option portfolio has a large and positive position theta. Which of the following trades is most likely to neutralize the portfolio's gamma?
b. Write put options
c. Sell shares
d. None of the above: theta must be negative!

9.5. A delta-neutral portfolio of options is net short and the options are, on average, at-the-money (ATM) with near-term maturities. Which of the following is most likely true about the portfolio's theta?
a. Large and negative
b. Small and negative
c. Small and positive
d. Large and positive