Level 2 — RecallThe Greeks

The Greeks

⏱ 30 minutes40 marksprintable — key stays hidden on paper

Subject: Stock-Market | Chapter: The Greeks Difficulty Level: 2 (Recall — definitions, standard problems, short derivations) Time Limit: 30 minutes Total Marks: 40

Instructions

Answer all questions. Show working where calculation is required. Use $...$ for any mathematical expressions.

Q1. Define Delta and state its typical numeric range for (a) a call option and (b) a put option. (4 marks)

Q2. A trader holds 5 call option contracts (each contract = 100 shares) with a Delta of $0.40$ per option. Calculate the total position delta (in equivalent shares) and state the directional exposure. (4 marks)

Q3. Define Gamma and explain in one sentence why an at-the-money option near expiry has high Gamma. (4 marks)

Q4. An option currently has Delta $= 0.50$ and Gamma $= 0.08$ . If the underlying rises by $\$ 2$, estimate the new Delta using a first-order (linear) approximation. (4 marks)

Q5. Define Theta. State whether Theta is typically positive or negative for a long option holder, and what it represents in dollar terms per day. (4 marks)

Q6. A long call has a Theta of $-0.05$ (per day, per share). For 3 contracts (100 shares each), estimate the dollar value lost to time decay over 2 days, assuming all else constant. (4 marks)

Q7. Define Vega. If an option has a Vega of $0.12$ and implied volatility rises from $20\%$ to $25\%$ , estimate the change in option price per share. (4 marks)

Q8. Distinguish between implied volatility and historical volatility in one or two sentences each. (4 marks)

Q9. Explain the phenomenon of IV crush and identify one typical event that causes it. (4 marks)

Q10. Briefly explain what the volatility skew (or smile) describes, and name the Greek that measures interest-rate sensitivity. (4 marks)

End of Paper

Answer keyMark scheme & solutions

Q1. (4 marks)

Delta = the rate of change of an option's price with respect to a $\$ 1 $change in the underlying price;$ \Delta = \partial V/\partial S$. (2 marks)
(a) Call: $0$ to $+1$ . (1 mark)
(b) Put: $-1$ to $0$ . (1 mark) Why: Delta measures directional (first-order price) exposure; calls gain as price rises (positive), puts gain as price falls (negative).

Q2. (4 marks)

Shares per position $= 5 \times 100 = 500$ shares. (1 mark)
Total delta $= 500 \times 0.40 = 200$ delta-equivalent shares. (2 marks)
Directional exposure: bullish/long — equivalent to being long 200 shares. (1 mark)

Q3. (4 marks)

Gamma = rate of change of Delta with respect to a $\$ 1 $move in the underlying;$ \Gamma = \partial \Delta/\partial S = \partial^2 V/\partial S^2$. (3 marks)
ATM near expiry: small price moves flip the option between ITM/OTM, so Delta changes rapidly → high Gamma. (1 mark)

Q4. (4 marks)

New Delta $\approx \Delta_{old} + \Gamma \times \Delta S$ . (1 mark)
$= 0.50 + 0.08 \times 2$ (1 mark)
$= 0.50 + 0.16 = 0.66$ . (2 marks)

Q5. (4 marks)

Theta = rate of change of an option's price with respect to the passage of time (time decay); $\Theta = \partial V/\partial t$ . (2 marks)
For a long option holder: typically negative. (1 mark)
Represents the dollar value the option loses per day, all else equal. (1 mark)

Q6. (4 marks)

Shares $= 3 \times 100 = 300$ . (1 mark)
Daily decay $= 300 \times 0.05 = \$ 15$ per day. (1 mark)
Over 2 days $= \$ 15 \times 2 = $30$ lost. (2 marks)

Q7. (4 marks)

Vega = change in option price for a $1$ percentage-point ( $1\%$ ) change in implied volatility. (2 marks)
IV change $= 25\% - 20\% = 5$ points. (1 mark)
Price change $= 0.12 \times 5 = \$ 0.60$ per share. (1 mark)

Q8. (4 marks)

Historical (realized) volatility: measured from past actual price movements of the underlying (backward-looking). (2 marks)
Implied volatility: the volatility derived from the current market price of the option via a pricing model (forward-looking, market's expectation). (2 marks)

Q9. (4 marks)

IV crush = a sudden, sharp drop in implied volatility (and hence option premiums) after an anticipated event resolves the uncertainty. (2 marks)
Typical event: earnings announcement (also: FDA decision, product launch, macro data release). (2 marks)

Q10. (4 marks)

Volatility skew/smile: describes how implied volatility varies across different strike prices (for a given expiry), rather than being constant. (2 marks)
Interest-rate-sensitivity Greek: Rho. (2 marks)

[
  {"claim":"Q2 total position delta = 200 shares", "code":"result = (5*100*0.40 == 200)"},
  {"claim":"Q4 new delta approx 0.66", "code":"result = abs((0.50 + 0.08*2) - 0.66) < 1e-9"},
  {"claim":"Q6 time decay over 2 days = $30", "code":"result = (3*100*0.05*2 == 30)"},
  {"claim":"Q7 price change from vega = $0.60", "code":"result = abs(0.12*(25-20) - 0.60) < 1e-9"}
]