What to Trade

Question 1 (10)

(a) Buy at ask $M+\tfrac{s}{2}$, sell at bid $M-\tfrac{s}{2}$. Spread cost per round trip $=s$ on a value of $M$. Brokerage charged both sides $=2bM$. Round-trip cost fraction: $C=\frac{s}{M}+2b$

• $s/M$ = spread as fraction of mid (implicit cost). (2)
• $2b$ = explicit brokerage both legs. (2)

(b) Stock A: $s=4$, $C_A=\frac{4}{200}+2(0.0003)=0.02+0.0006=0.0206=2.06\%$. (2) Stock B: $s=1$, $C_B=\frac{1}{200}+0.0006=0.005+0.0006=0.0056=0.56\%$. (2)

(c) Choose the lower-cost, tighter-spread stock (B) for scalping since frequent trades make spread the dominant cost; edge per trade is small so cost must be minimized. (2)

Question 2 (12)

(a) Total free-float mcap $=\sum p_i q_i$. Raw sum changes with corporate actions, so we normalise by divisor $D$: $\text{Index}=\frac{\sum_i p_i q_i}{D}$ $D$ is set so the index equals a chosen base value at the base date; adjusting $D$ on splits/additions keeps the index continuous. (4)

(b) Current mcap $=500(100)+300(200)+800(50)=50000+60000+40000=150000$. Index base $1000$: $D=\frac{150000}{1000}=150$. (2) New mcap with X=550: $550(100)+60000+40000=55000+100000=155000$. New index $=155000/150=1033.33$. (3)

(c) Bank Nifty has fewer constituents (~12) and heavy concentration in top private banks; large weighted names move it more, so idiosyncratic bank moves aren't diluted → higher volatility than the broader 50-stock Nifty. (3)

Question 3 (12)

(a) Long call payoff at expiry $=\max(S_T-K,0)-c$. Below $K$ lose premium $c$; above $K$ gains linearly. (2.5) Short put payoff $=p-\max(K-S_T,0)$. Keep premium $p$ above $K$; lose linearly below. (2.5)

(b) Long future payoff $=S_T-22000$. Long put payoff $=\max(22000-S_T,0)-150$. Combined: $\Pi=(S_T-22000)+\max(22000-S_T,0)-150$

• If $S_T<22000$: $(S_T-22000)+(22000-S_T)-150=-150$.
• If $S_T\ge22000$: $(S_T-22000)+0-150=S_T-22150$. Max loss $=150$ (floor). (5)

(c) This is a protective put (synthetic long call). View: bullish but wanting downside protection. (2)

Question 4 (12)

(a) $\rho=\dfrac{\sum(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2}\sqrt{\sum(y_i-\bar y)^2}}$; numerator = covariance (co-movement), denominator = product of std devs (normalises to $[-1,1]$). (3)

(b) $\bar P=\frac{2-1+3+0}{4}=1$, $\bar Q=\frac{1-2+2-1}{4}=0$. Deviations P: $1,-2,2,-1$; Q: $1,-2,2,-1$. Products: $1,4,4,1$ → $\sum=10$. $\sum(P-\bar P)^2=1+4+4+1=10$; $\sum(Q-\bar Q)^2=1+4+4+1=10$. $\rho=\frac{10}{\sqrt{10}\sqrt{10}}=\frac{10}{10}=1.0$. (6)

(c) Two highly correlated stocks are effectively one bet; taking full positions in both doubles risk without diversifying. Trader sizes them as one exposure or picks the stronger only. (3)

Question 5 (8)

def screen(stocks):
    out = []
    for s in stocks:
        if (s['avg_volume'] > 1_000_000 and      # liquidity  (2)
            s['atr_pct'] > 1.5 and                # volatility (1)
            s['rel_strength'] > 1.0 and           # leadership (1)
            s['price'] < 3000):                   # affordability(1)
            out.append(s)
    out.sort(key=lambda s: s['rel_strength'], reverse=True)  # sort (2)
    return out                                    # return (1)

Correct filter logic (5) + sort desc (2) + return (1).

Question 6 (6)

(a) $RS=\dfrac{P_{stock}}{P_{benchmark}}$ (or ratio of normalised returns). $RS>1$ (or rising) means the stock outperforms the benchmark → leadership. A rising RS line = accumulating relative strength. (3)

(b) Stock return $=20\%$, benchmark $=10\%$. RS of returns $=\frac{1.20}{1.10}=1.0909$; stock outperforms benchmark → relative leader. (3)

[
  {"claim":"Round-trip cost Stock A = 2.06%","code":"s=4; M=200; b=0.0003; C=s/M+2*b; result=abs(C-0.0206)<1e-9"},
  {"claim":"Round-trip cost Stock B = 0.56%","code":"s=1; M=200; b=0.0003; C=s/M+2*b; result=abs(C-0.0056)<1e-9"},
  {"claim":"Index divisor 150 and new value 1033.33","code":"D=150000/1000; newidx=155000/D; result=(D==150) and abs(newidx-1033.3333333)<1e-3"},
  {"claim":"Correlation PQ = 1.0","code":"P=[2,-1,3,0]; Q=[1,-2,2,-1]; mP=sum(P)/4; mQ=sum(Q)/4; cov=sum((p-mP)*(q-mQ) for p,q in zip(P,Q)); sp=sum((p-mP)**2 for p in P); sq=sum((q-mQ)**2 for q in Q); rho=cov/(sp*sq)**Rational(1,2); result=simplify(rho-1)==0"},
  {"claim":"RS ratio of returns = 1.0909","code":"rs=Rational(120,100)/Rational(110,100); result=abs(float(rs)-1.0909090909)<1e-6"},
  {"claim":"Protective put max loss = 150","code":"import numpy as np; loss=min((St-22000)+max(22000-St,0)-150 for St in range(20000,24001,50)); result=loss==-150"}
]