Practice Quiz 4 Solutions — Option Pricing & the Greeks
Setup: Stock at $100. 1-year European call with strike $100 priced at $12. Risk-free rate 5%, volatility 30%.
(a) Using put-call parity, what is the price of the put?
Put-call parity: $C - P = S - K e^{-rT}$ (or using discrete discounting: $C - P = S - K/(1+r)$).
Using discrete discounting:
\[P = C - S + \frac{K}{1+r} = 12 - 100 + \frac{100}{1.05} = 12 - 100 + 95.24 = \$7.24\]The put is worth approximately $7.24.
(b) Direction of call price changes:
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Volatility rises from 30% to 40%: Call price increases. Higher volatility increases the probability of large upside moves, which benefits the call holder (downside is bounded at zero).
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Stock price drops from $100 to $90: Call price decreases. The call is now out of the money — lower stock price means lower probability of finishing in the money.
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Time to expiration shortens from 1 year to 6 months: Call price decreases. Less time means less opportunity for the stock to move favorably — the option’s time value decays.
(c) Is higher volatility bad for the holder of a long call?
No — higher volatility is good for the holder of a long call. The call’s downside is limited to the premium paid, but its upside is unlimited. Greater volatility increases the chance of large favorable moves without increasing the maximum loss, so the option becomes more valuable (positive vega).