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# P1.T4.817. Option delta (Hull Ch.19)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning objectives: Describe and assess the risks associated with naked and covered option positions. Explain how naked and covered option positions generate a stop loss trading strategy. Compute the delta of an option.

Questions:

817.1. As a market maker, Toughgreen Financial has written (aka, taken a short position in) a single contract for 100 call options on a non-dividend-paying stock whose volatility is 36.0% per annum when the riskless rate is 3.0%. Their strike price is $100.00 but the options are underwater because the current stock price is$90.00. The option term is six months. At the current stock price, each option has a value of $5.88 and each option's percentage delta, Δ = +0.410. From Toughgreen's perspective, if the stock price increases by +$3.00 to $93.00, which is nearest to the impact on the position's value? a. A loss of few dollars less than$8.00; ie, 100 * [$5.88 * ($3.00/$90.00) * 0.410] minus (-) gamma adjustment b. A loss of few dollars more than$8.00; ie, 100 * [$5.88 * ($3.00/$90.00) * 0.410] plus (+) gamma adjustment c. A loss of few dollars less than$123.00; ie, 100 * ($3.00 * 0.410) minus (-) gamma adjustment d. A loss of few dollars more than$123.00; ie, 100 * ($3.00 * 0.410) plus (+) gamma adjustment 817.2. A non-dividend-paying stock has a volatility, σ, equal to 35.0% and a current price of$60.00 while the riskfree rate is 4.0%. Which is nearest to the percentage delta, Δ(p), of a six-month put option with an exercise price of $75.00? a. -2.2127 b. -0.7580 c. -0.2420 d. +0.2429 817.3. Greendex Financial LLC is a market maker who, at the request of its client, has written out-of-the-money (OTM) call options on the stock of Industrial Automation, Inc. Greendex wrote 100 contracts, and the size of each contract is 100 options with nine month maturities. Because S(0) =$20.00, K = \$25.00, σ = 40.0%, Rf = 4.0% and T = 0.75 years, the percentage delta Δ(c) of each call option is +0.350.

Greendex immediately seeks to neutralize its delta exposure (i.e., achieve a position delta equal to zero) by trading put options on the same underlying stock, Industrial Automation, Inc. Compared to the call options, these put options will have an identical strike price and maturity. Which of the following trades is advisable to neutralize its delta, and after having neutralized delta, what is the nature of Greendex's retained gamma exposure?

a. Write about 54 put option contracts (100 put options per contract) but Acme is exposed to wild movements in the stock price
b. Write about 54 put option contracts (100 put options per contract) but Acme is exposed to a virtually constant stock price
c. Write about 185 put option contracts (100 put options per contract) but Acme is exposed to wild movements in the stock price
d. Write about 185 put option contracts (100 put options per contract) but Acme is exposed to a virtually constant stock price