Questions: Option Greeks: Delta, Gamma, Vega, and Theta

Question 1

A trader holds 100 call options with delta = 0.4 and hedges by selling 40 shares (delta-neutral). The stock then rises $5. Is the portfolio still delta-neutral?

• A. Yes — delta-hedging permanently eliminates all price exposure
• B. No — as the stock rose, the option's delta increased (due to gamma), so the original hedge is now stale
• C. No — the hedge should have used put options rather than short shares
• D. Yes — delta-hedging is exact for moves of any size

Answer: (B)

Delta-hedging is only instantaneously correct for infinitesimally small moves. Gamma measures how quickly delta changes with stock price — for a call, as the stock rises toward and past the strike, delta increases toward 1. After a $5 rise, the option's delta might be 0.55, meaning the hedge (still 40 shares) is now too small to neutralize the exposure. To remain delta-neutral, the trader must buy more shares. This continuous rebalancing requirement is the cost of being short gamma.

Question 2

An options trader says: 'I'm selling volatility.' Which position does this most accurately describe?

• A. Buying call options on a stock expected to have high realized volatility
• B. Selling a straddle (a call and a put at the same strike) to collect premium while expecting the stock to stay near its current price
• C. Buying put options as a hedge against a market decline
• D. Shorting shares to profit from an expected price drop

Answer: (B)

Selling volatility means taking positions that profit from low realized volatility — typically selling options and collecting premium (short vega). A straddle sale profits if the stock stays near the strike; both the call and put expire worthless and the seller keeps the premium. The key reframe is that options markets are volatility markets: buying options goes long vega and long gamma; selling options goes short vega and short gamma. Options A, C, and D are directional price bets, not volatility positions.

Question 3

An option holder benefits from time passing — theta is positive for long option positions because expiration reduces uncertainty.

• T. True
• F. False

Answer: False

Theta is almost always negative for option holders. As expiration approaches with no change in stock price or volatility, the option loses time value — the probability of a large favorable move decreases, so the time premium erodes. This decay accelerates sharply near expiration for at-the-money options. Option sellers (short positions) have positive theta, collecting that daily decay. The holder paid for time value; it erodes against them each day.

Question 4

A delta-neutral portfolio has zero exposure to stock price changes, regardless of how large those changes are.

• T. True
• F. False

Answer: False

Delta-neutrality holds only instantaneously, for infinitesimally small price changes. For any finite move, gamma matters — the portfolio's delta changes as the stock moves, so directional exposure develops. To hedge against larger moves, a trader would need to be gamma-neutral as well (typically achieved by holding options with offsetting gammas). Even then, even higher-order Greeks become relevant for very large moves. Delta-hedging is a first-order, local approximation that requires continuous rebalancing to remain accurate.

Question 5

Explain the gamma-theta tradeoff in options trading. Why can't a trader simultaneously be long gamma and collect positive theta?

Model answer: Gamma and theta are intrinsically linked: being long gamma means owning options, which means paying time premium — negative theta. Each day, the option's time value erodes (theta drain). Conversely, selling options gives positive theta (collecting daily decay) but leaves you short gamma and vulnerable to large moves. The tradeoff is: you pay daily (theta) for the right to profit from large moves (gamma), or you collect daily (theta) by accepting the risk of large moves. You cannot have both.

This tradeoff is the central risk-reward tension in options trading. A long gamma position profits when realized volatility exceeds implied volatility — when the stock actually moves more than the market priced in. A short gamma position profits from calm markets. The daily theta paid or collected is the market's 'fair' price for the gamma exposure. Managing this — judging whether actual volatility will beat or underperform implied volatility — is the core skill of options market-making and is why options are fundamentally instruments for trading volatility, not just direction.