Options Basics

Question 1 (12)

(a) Long call payoff at expiry: the holder exercises only if $S_T > K$, gaining $S_T - K$; otherwise lets it expire (payoff 0). $P(S_T) = \max(S_T - K, 0) = \begin{cases} 0 & S_T \le 100 \\ S_T - 100 & S_T > 100 \end{cases}$ (2) Profit subtracts premium $c=4.50$: $\Pi(S_T) = \max(S_T - 100, 0) - 4.50$ (2)

(b) Breakeven: $\Pi = 0 \Rightarrow S_T - 100 - 4.50 = 0 \Rightarrow S_T = 104.50$. (2) Below strike, the right to buy at 100 is worthless because you could buy cheaper in the market, so intrinsic value = 0. (1)

(c) Seller profit = premium received minus payoff obligation: $\Pi_{sell}(S_T) = 4.50 - \max(S_T - 100, 0)$ (2) Max profit = $+4.50$ (when $S_T \le 100$). (1) Max loss = unlimited (as $S_T \to \infty$). (1) It is the mirror: $\Pi_{sell} = -\Pi_{buy}$ (zero-sum, ignoring fees). (1)

Question 2 (10)

(a) Put intrinsic value $= \max(K - S, 0) = \max(250 - 242, 0) = 8$. (2) Time value = premium − intrinsic = $13 - 8 = 5$. (2)

(b) $S=242 < K=250$, so put is ITM by 8 points (right to sell at 250 above market). (2)

(c) At expiry time value $\to 0$; the option is worth only intrinsic since no time remains for further favourable moves. (2) At $S_T=244$: intrinsic $=\max(250-244,0)=6$; profit $= 6 - 13 = -7.00$ (loss of 7.00). (2)

Question 3 (10)

(a) $\text{PCR} = \dfrac{\sum \text{Put OI}}{\sum \text{Call OI}}$. Put OI sum $= 45{,}000+38{,}000+40{,}000+20{,}000+11{,}000 = 154{,}000$. Call OI sum $= 12{,}000+18{,}000+40{,}000+55{,}000+60{,}000 = 185{,}000$. $\text{PCR} = \frac{154{,}000}{185{,}000} \approx 0.832$ (3)

(b) PCR $< 1$ means more call OI than put OI, indicating aggregate bearish/cautious positioning (more call writing/interest above). Standard contrarian view: very low PCR can signal excessive bullishness (overbought); here mildly bearish tilt. (3)

(c) Highest put OI at 19,800 (45,000) → support; highest call OI at 20,200 (60,000) → resistance. (2) Max put OI = heavy put writers who profit if price stays above that strike; they defend it, and put buyers' hedging keeps price from falling below — acting as support. (2)

Question 4 (10)

(a) Using moneyness ($S$ vs $K$): (3, 1 each pair)

• Call: ITM if $S>K$, ATM if $S\approx K$, OTM if $S<K$.
• Put: ITM if $S<K$, ATM if $S\approx K$, OTM if $S>K$.

(b) Exercise is initiated by the option buyer/holder invoking their right. Assignment is imposed on a seller/writer (chosen, often randomly) who must fulfil the obligation — deliver/buy the underlying at strike. (3)

(c) A European call cannot be exercised early — only at expiry. But the trader can sell (close) the option in the market to capture the current value (intrinsic + remaining time value). (2) An American call can be exercised anytime; practically, however, selling is usually better than early exercise because it also captures time value (early exercise forfeits it). (2)

Question 5 (10)

(a) Long put (K=100): $\max(100-S_T,0)-3$. Short call (K=110): $2-\max(S_T-110,0)$. Combined $\Pi(S_T)$:

• $S_T < 100$: $(100-S_T) - 3 + 2 = 99 - S_T$ (2)
• $100 \le S_T \le 110$: $0 - 3 + 2 = -1$ (2)
• $S_T > 110$: $-3 + 2 - (S_T-110) = 109 - S_T$ (2)

(b) Net premium $= -3 + 2 = -1$ (net paid 1). At $S_T=105$ (middle region) $\Pi = -1$. (2)

(c) Max profit occurs as $S_T \to 0$: $99 - 0 = 99$ (bounded by 99 in practice since stock $\ge 0$). Loss unbounded as $S_T \to \infty$ (from short call): region $S_T > 110$, $\Pi = 109 - S_T \to -\infty$. (2)

Question 6 (8)

(a) A buyer pays a premium and can at most lose that premium — payoff is floored at 0, so loss is capped. A naked call seller must deliver at strike regardless of how high price rises; since $S_T$ has no upper bound, the obligation $-(S_T-K)$ grows without limit → unlimited loss. (4)

(b) Time value (theta decay) accelerates near expiry because the probability window for further favourable moves shrinks rapidly and non-linearly. ATM options carry the most time value (greatest uncertainty about final moneyness); deep ITM/OTM options carry little time value. (4)

[
  {"claim":"Long call breakeven at 104.50","code":"K=100; c=4.5; result = (K+c==104.5)"},
  {"claim":"Put profit at S_T=244 is -7","code":"intrinsic=max(250-244,0); profit=intrinsic-13; result=(profit==-7)"},
  {"claim":"PCR approx 0.832","code":"puts=45000+38000+40000+20000+11000; calls=12000+18000+40000+55000+60000; pcr=Rational(puts,calls); result=(abs(float(pcr)-0.832)<0.005)"},
  {"claim":"Straddle-like combo profit at S_T=105 is -1","code":"S=105; longput=max(100-S,0)-3; shortcall=2-max(S-110,0); result=(longput+shortcall==-1)"},
  {"claim":"Combo profit at S_T=115 equals -6","code":"S=115; longput=max(100-S,0)-3; shortcall=2-max(S-110,0); result=(longput+shortcall==-6)"}
]