Market Participants

Question 1

(a) Revenue rate $g(\delta) = 2\delta A e^{-k\delta}$. $g'(\delta) = 2A e^{-k\delta} + 2\delta A(-k)e^{-k\delta} = 2A e^{-k\delta}(1 - k\delta)$. (2) Set $g'=0$: since $2Ae^{-k\delta}>0$, need $1-k\delta=0 \Rightarrow \boxed{\delta^\* = 1/k}$. (2) Second derivative / sign test: $g'>0$ for $\delta<1/k$ and $g'<0$ for $\delta>1/k$, so it's a maximum. (Or $g''(\delta^\*)=2A e^{-k\delta}(-k)(1-k\delta) -2Ake^{-k\delta}\big|_{\delta=1/k} = -2Ake^{-1}<0$.) (2)

(b) $R(\delta)=2\delta A e^{-k\delta} - c\delta^{-1}$. $R'(\delta) = 2Ae^{-k\delta}(1-k\delta) + c\delta^{-2}$. (3) FOC $R'(\delta)=0$: $2Ae^{-k\delta}(1-k\delta) + \frac{c}{\delta^2}=0.$ This mixes an exponential term $e^{-k\delta}$ with a rational/polynomial term $c/\delta^2$; such an equation cannot be inverted using elementary algebra (it is transcendental — no finite combination of algebraic operations isolates $\delta$). Hence numerical root-finding is required. (3)

(c) Let $f(\delta)=R'(\delta)=2Ae^{-k\delta}(1-k\delta)+c\delta^{-2}$. $f'(\delta)=2A\big[-ke^{-k\delta}(1-k\delta) + e^{-k\delta}(-k)\big] -2c\delta^{-3}$ $=2Ae^{-k\delta}\big[-k(1-k\delta)-k\big] -2c\delta^{-3} = 2Ae^{-k\delta}(k^2\delta -2k) - 2c\delta^{-3}$. (2)

import math
A, k, c = 100, 2, 5
def f(d):  return 2*A*math.exp(-k*d)*(1-k*d) + c/d**2
def fp(d): return 2*A*math.exp(-k*d)*(k*k*d - 2*k) - 2*c/d**3
d = 0.6                     # start near 1/k = 0.5
for _ in range(100):
    step = f(d)/fp(d)
    d -= step
    if abs(step) < 1e-10:  # stopping criterion
        break
print(round(d,5))          # ~0.63988

Root $\delta^\* \approx 0.640$. (3) Economic meaning: the profit-maximising half-spread once inventory-carrying cost is included is wider than the pure-revenue optimum $1/k=0.5$ — the MM widens quotes to compensate for inventory risk. (1)

(d) Market makers post two-sided quotes continuously, guaranteeing that a counterparty exists at all times → they supply immediacy/liquidity and earn the spread as compensation for risk. A broker merely routes/agency-executes client orders (no obligation to quote, earns commission, takes no principal risk). A prop firm trades its own capital for directional/statistical profit and has no obligation to provide continuous liquidity. (Any 2 correct distinctions.) (4)

Question 2

(a) Model $r_i=\alpha x_i+\varepsilon_i$ with $x_i=F_i-D_i$. Minimise $S(\alpha)=\sum_i (r_i-\alpha x_i)^2$. $\dfrac{dS}{d\alpha}=-2\sum x_i(r_i-\alpha x_i)=0$. (3) $\Rightarrow \sum x_i r_i = \alpha\sum x_i^2 \Rightarrow \boxed{\hat\alpha=\dfrac{\sum x_i r_i}{\sum x_i^2}}$. (2) Second derivative $2\sum x_i^2>0 \Rightarrow$ minimum. (1)

(b) $\sum x_i r_i = 500(0.6)+(-300)(-0.3)+800(1.0)=300+90+800=1190$. (2) $\sum x_i^2 = 500^2+300^2+800^2 = 250000+90000+640000=980000$. (2) $\hat\alpha = 1190/980000 = 0.001214...\approx \boxed{1.214\times10^{-3}\ \%/\text{₹cr}}$. (2)

(c) Sequence (T+1):

• Day T (trade day): order matched on the exchange order book; a contract note is generated. The clearing corporation performs novation — it becomes buyer to every seller and seller to every buyer, and computes net obligations (funds & securities). (2)
• Day T+1 (settlement day): buyer's funds are debited and paid to the clearing corp; seller's securities are debited from their demat account at the depository (NSDL or CDSL) and, via the clearing corp, credited to the buyer's demat account the same day (T+1). (3)
• Roles: Clearing corporation = guarantees settlement, manages risk/margins, nets obligations. NSDL/CDSL = hold securities in electronic (dematerialised) form and effect the book-entry transfer of ownership. (1)
• Why novation: by interposing itself as central counterparty, the clearing corp removes counterparty default risk — neither party depends on the other's solvency; the CCP guarantees the trade, enabling anonymous trading and settlement finality. (2)

Question 3

(a) (8) — 2 marks each rule, correct "2 of 3" logic.

def is_eligible(paid_up, market_cap, public_pct, net_worth_list):
    c1 = paid_up >= 10                              # ₹10 cr
    c2 = market_cap >= 25                           # ₹25 cr
    c3 = public_pct >= 25                           # 25% float
    positive_years = sum(1 for nw in net_worth_list if nw > 0)
    c4 = positive_years >= 2                        # 2 of last 3
    return c1 and c2 and c3 and c4

(b) Trace with (12, 40, 22, [−2,5,6]):

• c1: 12 ≥ 10 → True (1)
• c2: 40 ≥ 25 → True (1)
• c3: 22 ≥ 25 → False ← binding constraint (float too low) (1)
• c4: positive years = 2 (5,6) ≥ 2 → True. Verdict: NOT eligible. Binding constraint = public shareholding of 22% < 25% minimum. (1)

(c) (6)

• SEBI (statutory regulator under SEBI Act 1992) frames listing regulations (LODR) and mandates continuous disclosure: periodic financials, material-event disclosure, insider-trading rules, corporate-governance norms. It can suspend/delist non-compliant firms. (3)
• Minimum-public-shareholding (e.g., 25%) ensures a genuine free float → better price discovery, liquidity, and prevents promoters from cornering the stock. Retail investors are price-takers with limited information and cannot negotiate access; adequate float and mandatory disclosure protect them from manipulation and illiquidity. Institutional investors have research teams, direct access, and bargaining power, so they are less dependent on these protections — hence the rule primarily shields the retail segment. (3)

[
  {"claim":"Q1a optimal half-spread delta* = 1/k for k=2 gives 0.5","code":"k=2; delta=1/k; result = (delta==Rational(1,2))"},
  {"claim":"Q1c Newton root of R'(delta) approx 0.63988","code":"import mpmath as mp; A,k,c=100,2,5; f=lambda d: 2*A*mp.e**(-k*d)*(1-k*d)+c/d**2; root=mp.findroot(f,0.6); result = abs(float(root)-0.63988)<1e-3"},
  {"claim":"Q2b alpha_hat = 1190/980000","code":"num=500*Rational(6,10)+(-300)*Rational(-3,10)+800*1; den=500**2+300**2+800**2; ah=num/den; result = (ah==Rational(1190,980000)) and abs(float(ah)-0.001214285)<1e-6"},
  {"claim":"Q3b firm NOT eligible due to public_pct 22<25","code":"paid,mc,pub,nw=12,40,22,[-2,5,6]; c1=paid>=10;c2=mc>=25;c3=pub>=25;c4=sum(1 for x in nw if x>0)>=2; result = (not (c1 and c2 and c3 and c4)) and (c3==False)"}
]