Level 1 — RecognitionThe Greeks

The Greeks

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Chapter: 5.3 The Greeks Level: 1 — Recognition Time Limit: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. Delta measures the sensitivity of an option's price to a change in:

(a) time to expiry
(b) the underlying asset's price
(c) volatility
(d) the risk-free interest rate

Q2. A long call option has a Delta that ranges between:

(a) $-1$ and $0$
(b) $0$ and $1$
(c) $-1$ and $1$
(d) $0$ and $\infty$

Q3. Gamma is largest for options that are:

(a) deep in-the-money
(b) deep out-of-the-money
(c) at-the-money near expiry
(d) far from expiry

Q4. Theta for a typical long option position is usually:

(a) positive
(b) negative
(c) zero
(d) undefined

Q5. Vega measures sensitivity to a change in:

(a) implied volatility
(b) underlying price
(c) interest rate
(d) dividends

Q6. Rho has the greatest impact on:

(a) short-dated options
(b) at-the-money options only
(c) long-dated options
(d) options on non-dividend stocks only

Q7. "IV crush" typically occurs:

(a) the moment volatility rises before earnings
(b) immediately after a scheduled event resolves uncertainty
(c) only when a stock is delisted
(d) when interest rates fall

Q8. Implied volatility is derived from:

(a) past price movements of the underlying
(b) the current market price of the option
(c) the company's earnings history
(d) the risk-free rate alone

Q9. A volatility "smile" refers to a pattern where implied volatility is:

(a) constant across all strikes
(b) higher for strikes far from the money on both sides
(c) lower for out-of-the-money puts
(d) always downward sloping with strike

Q10. In the Black-Scholes model, an increase in volatility, all else equal, will make a call option:

(a) cheaper
(b) more expensive
(c) unchanged
(d) worthless

Section B — Matching (1 mark each, 5 marks)

Q11. Match each Greek (i–v) to what it measures (A–E).

Greek		Measures
(i) Delta		(A) rate of change of Delta w.r.t. underlying
(ii) Gamma		(B) sensitivity to interest rate
(iii) Theta		(C) sensitivity to underlying price
(iv) Vega		(D) sensitivity to volatility
(v) Rho		(E) sensitivity to time decay

Write your answers as (i)–?, (ii)–?, etc.

Section C — True/False WITH Justification (2 marks each: 1 for T/F, 1 for reason)

Q12. A delta-neutral position has zero directional exposure to small moves in the underlying.

Q13. Historical volatility and implied volatility are always equal.

Q14. Theta decay accelerates as an at-the-money option approaches expiry.

Q15. Buying an option just before an earnings announcement is a reliable profit strategy because IV is high.

Q16. Gamma is the same for a long call and a long put with the same strike and expiry.

Q17. To hedge the directional risk of a long call, a trader can short shares of the underlying.

Q18. A higher interest rate increases the value of a European put option (Rho of a put is positive).

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (b). Delta is $\partial V/\partial S$ , sensitivity to underlying price. (1)

Q2 — (b). A long call's delta runs from $0$ (deep OTM) to $1$ (deep ITM). (1)

Q3 — (c). Gamma peaks for ATM options near expiry, where delta changes most rapidly around the strike. (1)

Q4 — (b). Long options lose value as time passes (time decay), so Theta is negative. (1)

Q5 — (a). Vega is $\partial V/\partial\sigma$ , sensitivity to implied volatility. (1)

Q6 — (c). Rho grows with time to expiry because discounting of the strike matters more over longer horizons. (1)

Q7 — (b). After a scheduled event (e.g., earnings), uncertainty resolves and implied volatility collapses. (1)

Q8 — (b). IV is the volatility input that makes the model price match the observed market option price. (1)

Q9 — (b). A smile shows elevated IV on both OTM wings relative to ATM. (1)

Q10 — (b). More volatility raises option value (both calls and puts) in Black-Scholes. (1)

Section B

Q11 (5 marks, 1 each):

(i) Delta – C
(ii) Gamma – A
(iii) Theta – E
(iv) Vega – D
(v) Rho – B

Section C (2 marks each: 1 T/F, 1 justification)

Q12 — TRUE. (1) Delta neutral means net delta ≈ 0, so first-order (small) moves in the underlying produce no P&L; residual exposure is via gamma for larger moves. (1)

Q13 — FALSE. (1) HV is computed from past realized returns; IV is forward-looking from market prices. They differ and reflect different information/expectations. (1)

Q14 — TRUE. (1) Time value of an ATM option decays roughly with $\sqrt{T}$ , so the rate of decay steepens sharply as $T\to 0$ . (1)

Q15 — FALSE. (1) High IV means the option is expensive; after the event IV crush lowers the option's value, so a directional move may be needed just to break even. Not reliable. (1)

Q16 — TRUE. (1) For the same strike/expiry, a call and put have identical gamma (from put-call parity, the linear stock/bond terms drop out under differentiation twice). (1)

Q17 — TRUE. (1) A long call has positive delta; shorting shares (negative delta) offsets it to reduce directional exposure (delta hedging). (1)

Q18 — FALSE. (1) Higher rates decrease a European put's value; Rho of a put is negative (calls have positive Rho). (1)

[
  {"claim":"Long call delta from BS is between 0 and 1 (Q2)", "code":"from sympy import symbols, exp, sqrt, log, erf; S,K,r,sig,T=symbols('S K r sig T', positive=True); d1=(log(S/K)+(r+sig**2/2)*T)/(sig*sqrt(T)); N=lambda x:(1+erf(x/sqrt(2)))/2; delta=N(d1).subs({S:100,K:100,r:0.05,sig:0.2,T:1}); val=float(delta); result = (0 < val < 1)"},
  {"claim":"Call vega positive: price increases with volatility (Q10)", "code":"from sympy import symbols, exp, sqrt, log, erf; S,K,r,sig,T=symbols('S K r sig T', positive=True); d1=(log(S/K)+(r+sig**2/2)*T)/(sig*sqrt(T)); d2=d1-sig*sqrt(T); N=lambda x:(1+erf(x/sqrt(2)))/2; C=S*N(d1)-K*exp(-r*T)*N(d2); vals={S:100,K:100,r:0.05,T:1}; c_lo=float(C.subs({**vals,sig:0.2})); c_hi=float(C.subs({**vals,sig:0.3})); result = (c_hi > c_lo)"},
  {"claim":"Gamma of call equals gamma of put for same strike/expiry (Q16)", "code":"from sympy import symbols, exp, sqrt, log, erf, diff, pi, simplify; S,K,r,sig,T=symbols('S K r sig T', positive=True); d1=(log(S/K)+(r+sig**2/2)*T)/(sig*sqrt(T)); d2=d1-sig*sqrt(T); N=lambda x:(1+erf(x/sqrt(2)))/2; C=S*N(d1)-K*exp(-r*T)*N(d2); P=K*exp(-r*T)*N(-d2)-S*N(-d1); gc=diff(C,S,2); gp=diff(P,S,2); sub={S:100,K:100,r:0.05,sig:0.2,T:1}; result = abs(float(gc.subs(sub))-float(gp.subs(sub))) < 1e-9"},
  {"claim":"European put Rho is negative (Q18)", "code":"from sympy import symbols, exp, sqrt, log, erf, diff; S,K,r,sig,T=symbols('S K r sig T', positive=True); d1=(log(S/K)+(r+sig**2/2)*T)/(sig*sqrt(T)); d2=d1-sig*sqrt(T); N=lambda x:(1+erf(x/sqrt(2)))/2; P=K*exp(-r*T)*N(-d2)-S*N(-d1); rho=diff(P,r); val=float(rho.subs({S:100,K:100,r:0.05,sig:0.2,T:1})); result = (val < 0)"}
]