Level 4 — ApplicationThe Greeks

The Greeks

⏱ 60 minutes60 marksprintable — key stays hidden on paper

Level 4 — Application (novel problems, no hints) Time limit: 60 minutes Total marks: 60

Use the Black-Scholes framework where relevant. Assume all options are European on non-dividend-paying stocks unless stated. Present all working.

Question 1 — Delta hedging a portfolio (12 marks)

A trader holds the following position on stock XYZ (spot = $100):

Long 20 call contracts, each contract = 100 shares, call delta = $+0.55$
Short 15 put contracts, put delta = $-0.40$
Short 800 shares of XYZ

(a) Compute the net position delta (in share-equivalents). (4)

(b) State how many shares the trader must buy or sell to become delta-neutral. (2)

(c) The put contracts each have gamma $= 0.03$ . If the stock rises by $2, estimate the new put delta and explain the sign of the change for a short-put holder. (4)

(d) Explain in one or two sentences why delta-neutral is not the same as risk-free. (2)

Question 2 — Gamma, Theta and the P&L balance (12 marks)

An at-the-money long straddle on stock ABC ( $S=50$ ) has the following per-share Greeks: Gamma $= 0.08$ , Theta $= -0.14$ per day (total for the straddle).

(a) Using the gamma-P&L approximation $\text{P\&L}_{\gamma} \approx \tfrac{1}{2}\,\Gamma\,(\Delta S)^2$ , find the daily stock move (up or down) at which the gamma gain exactly offsets one day of theta decay. (5)

(b) Express your answer from (a) as a percentage move of the underlying. (2)

(c) Explain conceptually why long-gamma positions "want" movement while short-gamma positions "want" stillness, linking this to the sign of theta. (3)

(d) The trader realises the stock has been moving about 0.5% per day. Should they hold or close this long straddle, based on the gamma/theta balance? Justify. (2)

Question 3 — Vega, IV crush and an earnings event (14 marks)

A one-week at-the-money call on stock QRS trades at implied volatility of 80% ahead of earnings. The option's Vega is $0.09$ (price change per 1 volatility point, i.e. per 1%). Current option price = $3.20$ .

(a) Immediately after earnings, IV collapses to 35%. Using Vega, estimate the option's price change due to the IV crush alone (assume spot unchanged). (4)

(b) The stock actually gaps up so that the call gains $2.10$ of intrinsic-driven value. Combine this with the IV effect to estimate the net change in the option price. Comment on whether the buyer profited. (4)

(c) Explain why Vega itself is not constant — state two factors (time to expiry and moneyness) that change an option's Vega, and the direction of each effect. (4)

(d) Give one reason a trader might sell options before earnings despite the possibility of a large move. (2)

Question 4 — Volatility skew and Black-Scholes intuition (12 marks)

(a) For equity index options, out-of-the-money puts typically trade at higher implied volatility than out-of-the-money calls. Name this pattern and give the main market reason behind it. (3)

(b) Black-Scholes assumes a single constant volatility. Explain what the existence of a skew implies about the model's assumptions being violated in the real market. (3)

(c) A stock has historical (realised) volatility of 22% but options are priced at an implied volatility of 30%. State what this gap suggests about market expectations, and describe one strategy an "IV is too high" trader could deploy. (4)

(d) Explain why an at-the-money option has the highest Gamma and Vega of the strikes, in intuitive terms. (2)

Question 5 — Managing a live position's Greeks (10 marks)

A market-maker's book currently has: Delta $= +450$ , Gamma $= -120$ , Vega $= +2000$ , Theta $= +85$ /day.

(a) Interpret each Greek: for each, state whether the book benefits from a rise in the underlying, a large move, rising volatility, or the passage of time. (4)

(b) The desk wants to neutralise delta without changing gamma or vega. What single instrument type can achieve this, and why? (3)

(c) The desk is worried about a volatility spike. Given the current book, does a vol spike help or hurt, and what trade reduces this exposure? (3)

Answer keyMark scheme & solutions

Question 1

(a) Convert each to share-equivalent delta (contract = 100 shares):

Long 20 calls: $20 \times 100 \times 0.55 = +1100$
Short 15 puts: short means multiply by $-1$ : $-15 \times 100 \times (-0.40) = +600$
Short 800 shares: $-800$

Net delta $= 1100 + 600 - 800 = +900$ share-equivalents. (4) (1 mark each component sign/magnitude, 1 for total.)

(b) To be delta-neutral, offset $+900$ → sell 900 shares. (2)

(c) New put delta $\approx -0.40 + \Gamma \cdot \Delta S = -0.40 + 0.03\times 2 = -0.40 + 0.06 = -0.34$ . (2) Put delta rises toward zero as spot rises (puts become more OTM). For a short-put holder the position delta ( $+0.40$ per share held short) becomes less positive — the short put's directional exposure shrinks. (2)

(d) Delta-neutral removes only first-order directional risk. The position still carries gamma, theta, vega and rho — e.g. a large move or vol change changes P&L. (2)

Question 2

(a) Set gamma gain = theta cost: $\tfrac12 \Gamma (\Delta S)^2 = |\Theta|$ $\tfrac12 (0.08)(\Delta S)^2 = 0.14 \Rightarrow (\Delta S)^2 = \frac{0.14}{0.04}=3.5$ $\Delta S = \sqrt{3.5} \approx 1.87.$ So a move of about ±$1.87 breaks even. (5)

(b) $\frac{1.87}{50} = 0.0374 \approx 3.74\%$ . (2)

(c) Long gamma → curvature is positive, so any move (up or down) creates a positive convexity gain, but you pay for it via negative theta (long options decay). Short gamma is the mirror: you collect theta but lose on movement. The theta sign is always opposite to the gamma sign — you get paid to take the risk you dislike. (3)

(d) Realised move ≈ $0.5\% \times 50 = 0.25$ , far below the $1.87 break-even. Gamma gains ($ \tfrac12(0.08)(0.25^2)\approx0.0025 $) are tiny vs$ 0.14$ theta. Close the straddle — realised vol is too low to justify holding. (2)

Question 3

(a) IV drop $= 80 - 35 = 45$ points. Price change $\approx \text{Vega}\times \Delta \sigma = 0.09 \times (-45) = -4.05$ . But the option is only worth $3.20$ ; the linear estimate over a huge move overshoots — price can't go below intrinsic/zero. Estimate from Vega alone $\approx -\$ 4.05$ (capped at losing the full premium). (4) (3 for calculation, 1 for noting the cap/limitation.)

(b) Net $\approx +2.10 \text{ (intrinsic)} - 4.05 \text{ (IV)} = -1.95$ (or, capped, roughly the intrinsic $2.10$ minus the extrinsic destroyed). The stock moved up but the IV crush removed most/all extrinsic value; net the buyer likely lost money despite being directionally right — classic IV-crush trap. (4)

(c) Vega falls as time to expiry shrinks (less time = less volatility exposure), and Vega is highest at-the-money and falls for deep ITM/OTM strikes. (4) (2 each.)

(d) Selling captures the elevated pre-earnings IV; if the move is smaller than the IV implies, the IV crush hands the seller profit (theta + vol collapse). (2)

Question 4

(a) Volatility skew (put skew / "smirk"). Reason: demand for downside protection (crash/tail hedging) and the leverage effect — markets fall faster than they rise — bid up OTM put IV. (3)

(b) A single constant vol would produce a flat IV across strikes. The skew shows real return distributions have fat tails and negative skewness, violating Black-Scholes' lognormal/constant-vol assumption. (3)

(c) Implied (30%) > historical (22%): the market expects more future volatility than recently realised (or a risk premium). An "IV too high" trader could sell options / short a straddle or strangle (delta-hedged), profiting if realised vol stays near 22%. (4)

(d) ATM options have the most uncertainty about finishing ITM vs OTM, so their delta is most sensitive to spot (high Gamma) and price most sensitive to vol (high Vega). Deep ITM/OTM outcomes are nearly settled, so both Greeks decay away. (2)

Question 5

(a)

Delta $+450$ : benefits from underlying rising.
Gamma $-120$ : hurt by large moves (benefits from stillness).
Vega $+2000$ : benefits from rising volatility.
Theta $+85$ : benefits from passage of time (consistent with short gamma). (4)

(b) Trade the underlying stock (or futures) — it has delta but zero gamma and zero vega, so it shifts delta without disturbing the other Greeks. (3)

(c) Book is long vega (+2000), so a vol spike helps the book (would gain). If instead they wish to reduce vega exposure, they would sell options (long-dated/ATM) to cut vega. (Accept: spike helps; to reduce vega sell options.) (3)

[
  {"claim":"Q1a net delta = +900","code":"calls=20*100*0.55; puts=-15*100*(-0.40); shares=-800; result=(calls+puts+shares==900)"},
  {"claim":"Q1c new put delta = -0.34","code":"nd=-0.40+0.03*2; result=abs(nd-(-0.34))<1e-9"},
  {"claim":"Q2a breakeven move sqrt(3.5) approx 1.871","code":"dS=sqrt(Rational(14,100)/(Rational(1,2)*Rational(8,100))); result=abs(float(dS)-1.8708286)<1e-4"},
  {"claim":"Q2b percentage move approx 3.74%","code":"pct=sqrt(3.5)/50*100; result=abs(float(pct)-3.7416573)<1e-3"},
  {"claim":"Q3a IV crush price change = -4.05","code":"chg=0.09*(35-80); result=abs(chg-(-4.05))<1e-9"},
  {"claim":"Q3b net change = -1.95","code":"net=2.10+0.09*(35-80); result=abs(net-(-1.95))<1e-9"}
]