Options Strategies

Question 1 (10)

(a) Net debit $=6.20-2.40=\mathbf{3.80}$. (1)

Payoff (excluding debit first, then subtract):

• $S_T<100$: both calls expire worthless → gross $0$; net $=-3.80$. (1)
• $100\le S_T\le110$: long call ITM, short OTM → gross $=(S_T-100)$; net $=(S_T-100)-3.80$. (1)
• $S_T>110$: gross $=(S_T-100)-(S_T-110)=10$; net $=10-3.80=6.20$. (1)

(b)

• Max profit $=$ spread width $-$ debit $=10-3.80=\mathbf{6.20}$. (1)
• Max loss $=$ debit $=\mathbf{3.80}$. (1)
• Breakeven: $(S_T-100)-3.80=0\Rightarrow S_T=\mathbf{103.80}$. (1)

(c) The bull call spread is a moderately bullish play; selling the higher-strike call reduces the debit (from 6.20 to 3.80), lowering breakeven and cost, at the price of capping upside. Preferable when IV is elevated (long option overpriced, short leg recovers premium) or when the target price is bounded near 110. (3 — view 1, IV/cost 1, cap trade-off 1)

Question 2 (10)

(a) Position = long stock @50 + long 52 put @1.50.

• For $S_T<52$: put pays $52-S_T$; stock worth $S_T$ → floor value $=52$. Net P/L $=(S_T-50)+(52-S_T)-1.50=\mathbf{0.50}$. (1)
• For $S_T\ge52$: put worthless; P/L $=(S_T-50)-1.50=S_T-51.50$. (1)
• Breakeven: $S_T=\mathbf{51.50}$. (1)
• Floor position value $=52$; guaranteed minimum P/L $=+0.50$ (locked-in gain above cost). (1)

(b) Add short 60 call @1.20.

• Net protection cost $=1.50-1.20=\mathbf{0.30}$. (1)
• Floor value still $52$ (put); floor P/L $=(52-50)-0.30=\mathbf{1.70}$. (1)
• Capped upside: above 60 call caps stock at 60 → max value 60; max P/L $=(60-50)-0.30=\mathbf{9.70}$. (1)
• Between 52 and 60 P/L $=(S_T-50)-0.30$. (1)

(c) High IV inflates the call premium you sell, so the collar can be built at near-zero or credit cost while the expensive put is largely paid for by the rich short call — cheap downside insurance financed by overpriced upside. (2)

Question 3 (12)

(a) Net credit $=(2.10-1.10)+(2.30-1.30)=1.00+1.00=\mathbf{2.00}$. (1)

• Max profit $=$ net credit $=\mathbf{2.00}$ (S between 190 and 210). (1)
• Put spread width $=190-185=5$; call spread width $=215-210=5$. (1)
• Max loss $=$ width $-$ credit $=5-2.00=\mathbf{3.00}$. (1)
• Lower breakeven $=190-2.00=\mathbf{188.00}$. (1)
• Upper breakeven $=210+2.00=\mathbf{212.00}$. (1)

(b)

import numpy as np
def iron_condor_payoff(S):
    S = np.asarray(S, dtype=float)
    credit = 2.00
    long_put  = np.maximum(185 - S, 0)   # bought
    short_put = np.maximum(190 - S, 0)   # sold
    short_call= np.maximum(S - 210, 0)   # sold
    long_call = np.maximum(S - 215, 0)   # bought
    return credit + long_put - short_put - short_call + long_call

(structure 2, correct signs 1, credit added 1)

(c) Deploy when the view is neutral/range-bound and IV is high (rich premium to sell) with an expectation of IV contraction — defined risk on both wings. (2)

Question 4 (9)

(a) Total cost $=4.80+4.20=\mathbf{9.00}$. (1)

• Payoff: for $S_T<150$: $(150-S_T)-9$; for $S_T>150$: $(S_T-150)-9$; i.e. $|S_T-150|-9$. (1)
• Upper BE $=150+9=\mathbf{159}$. (1)
• Lower BE $=150-9=\mathbf{141}$. (1)

(b) Minimum move $=9$ points $=9/150=\mathbf{6\%}$ of strike. (2)

(c) The straddle's cost scales directly with implied volatility — high IV means an expensive straddle and a wider required move to profit. Around earnings, IV is elevated (priced-in expected move); after the event IV collapses (IV crush), so even a real price move may not cover the premium paid, producing a loss. (3)

Question 5 (10)

(a) Net cost $=5.00-2(2.00)=5.00-4.00=\mathbf{1.00}$ debit. (1) Payoff (gross by region):

• $S_T<100$: all worthless → net $-1.00$. (1)
• $100\le S_T\le110$: long call $(S_T-100)$; short calls OTM → net $(S_T-100)-1.00$. (1)
• $S_T>110$: $(S_T-100)-2(S_T-110)= -S_T+120$; net $(120-S_T)-1.00=119-S_T$. (1)

(b)

• Max profit at $S_T=110$: $(110-100)-1.00=\mathbf{9.00}$. (1)
• Upside breakeven: $119-S_T=0\Rightarrow S_T=\mathbf{119}$. (1)
• Above 119 P/L turns negative and decreases $1$-for-$1$ with $S_T$ because of the net one extra (naked) short call above 110 → unlimited upside risk. (2)

(c) Suitable for a moderately bullish view targeting the short strike (~110) with expected IV contraction; the extra sold call finances the long call but exposes you if the stock rockets past 119. (2)

Question 6 (9)

(a) Bull put (short put) spread (or naked/cash-secured put) — bullish + collect premium in high IV; defined risk, profits from time decay and drop in IV. (3)

(b) Iron condor (or short strangle if undefined risk acceptable) — neutral, range-bound, sell rich premium in high IV, benefit from IV crush. (3)

(c) Long straddle / long strangle — direction-agnostic, needs a big move, cheap to buy while IV is low (buy vol before it rises). (3)

[
  {"claim":"Bull call spread BE = 103.80, max profit 6.20, max loss 3.80","code":"debit=6.20-2.40; be=100+debit; mp=10-debit; ml=debit; result=(abs(be-103.80)<1e-9) and (abs(mp-6.20)<1e-9) and (abs(ml-3.80)<1e-9)"},
  {"claim":"Iron condor credit 2.00, max loss 3.00, BEs 188 and 212","code":"credit=(2.10-1.10)+(2.30-1.30); ml=5-credit; lbe=190-credit; ube=210+credit; result=(abs(credit-2)<1e-9) and (abs(ml-3)<1e-9) and (abs(lbe-188)<1e-9) and (abs(ube-212)<1e-9)"},
  {"claim":"Long straddle cost 9, BEs 141/159, required move 6%","code":"cost=4.80+4.20; lbe=150-cost; ube=150+cost; move=cost/150*100; result=(abs(cost-9)<1e-9) and (abs(lbe-141)<1e-9) and (abs(ube-159)<1e-9) and (abs(move-6)<1e-9)"},
  {"claim":"1x2 ratio: net debit 1, max profit 9 at 110, upside BE 119","code":"cost=5-2*2; mp=(110-100)-cost; be=119; check=(119-be); result=(abs(cost-1)<1e-9) and (abs(mp-9)<1e-9) and (abs(check)<1e-9)"}
]