A trader holds a call option with delta 0.4 and constructs a delta-neutral hedge. Market implied volatility then spikes upward by 10 percentage points. What is the effect on the position?
ANo effect — the position is delta-neutral, so all market moves are hedged
BThe position loses value because delta-neutral positions are inherently short volatility
CVega causes the option to gain value; the delta-neutral hedge eliminates only directional exposure, not volatility exposure
DThe hedge must be incorrect because delta-neutral positions cannot experience vega effects
Delta-neutral hedging eliminates only directional risk (first-order sensitivity to the underlying price). It does not eliminate vega exposure. When implied volatility spikes, all long options gain value through vega, regardless of whether they are delta-hedged. This is why professional options traders manage Greeks separately: being delta-neutral says nothing about your vega, gamma, or theta profile. A position can be delta-neutral but carry enormous volatility risk.
Question 2 Multiple Choice
An at-the-money call option has positive gamma. If the underlying stock rises significantly, what happens to the option's delta?
ADelta stays near 0.5 because ATM options always have delta 0.5
BDelta increases toward 1.0 as the option moves into-the-money — gamma has caused the position to become more directionally sensitive
CDelta decreases toward 0 because a rising stock price reduces the option's time value
DGamma and delta are independent; delta does not change when the stock moves
Gamma is the rate of change of delta. For a long call starting at-the-money (delta ≈ 0.5), a large upward move takes the option deep in-the-money and delta rises toward 1.0. This convexity is the key value of gamma: as the stock rises you become more long; as it falls you become less long. This self-reinforcing behavior in favorable directions is what you're paying for through negative theta.
Question 3 True / False
A long call option and a long put option on the same stock with the same strike and expiration have opposite signs of vega.
TTrue
FFalse
Answer: False
Both long calls and long puts have positive vega. This surprises students who assume that since calls and puts move in opposite directions with the stock, their volatility sensitivity must also be opposite. But higher volatility benefits the holder of any long option — more volatility means a greater chance of a large move, which helps a call (if the stock goes up) or a put (if it goes down). Only short option positions have negative vega, regardless of call or put.
Question 4 True / False
A position that is simultaneously long gamma and long theta is the standard profile of a long option position.
TTrue
FFalse
Answer: False
Long options have positive gamma but negative theta — this is the fundamental tradeoff in options. You pay for convexity through time decay. Gamma and theta are always in opposition for a simple long or short option: positive gamma comes with negative theta; negative gamma comes with positive theta. There is no free lunch — if you want convexity, you pay daily rent through time decay.
Question 5 Short Answer
Explain the relationship between gamma and theta for a long option position, and what this reveals about what you're actually paying for when you buy an option.
Think about your answer, then reveal below.
Model answer: Gamma and theta are opposite sides of the same tradeoff. Positive gamma (convexity) means the position benefits from large moves in either direction — as the stock rises, delta increases and you get more long; as it falls, delta decreases and you get less long. This self-adjusting property is valuable. Negative theta means the option loses value every day as expiration approaches. When you buy an option, you are paying for the right to this convexity. Theta represents the daily 'rent' on that optionality — if the stock doesn't move enough to offset the time decay, the option expires worthless.
The gamma-theta tradeoff is the core economic logic of options pricing. The daily theta payment is the fair price for the gamma benefit given current volatility expectations. If realized volatility turns out higher than implied, long gamma positions profit; if lower, they lose. Buying options is essentially a bet that realized volatility will exceed implied volatility.