Quant Interview FAQ — Derivatives

Each question below includes a Short Answer, a concrete Example, and a Detailed Explanation with quant-level depth, including formulas, edge cases, and practical caveats.

Full topic regarding derivatives is covered in the Derivatives section.

1) Forward vs. Futures

Short Answer
Forward = OTC, single settlement at $T$, bilateral credit risk; Futures = exchange-traded, standardized, daily mark-to-market with margin and clearing.

Example
Crude oil June forward struck at $K$ vs. NYMEX CL June futures: same notionally, but futures P&L is realized daily; the forward’s P&L is realized once at expiry.

Detailed Explanation

• Pricing: With deterministic rates, $F_0^{fut} = F_0^{fwd} = S_0 e^{(r-q)T}$. With stochastic rates and $\mathrm{corr}(S,r)\neq 0$, daily settlement creates a convexity bias: $\mathbb{E}[F^{fut}] \gtrless F^{fwd}$ depending on sign of correlation.
• Credit & Funding: Forwards embed bilateral CVA/DVA, CSA terms (thresholds, MTA, collateral rate), and potentially different discounting curves (OIS vs. legacy IBOR). Futures replace bilateral credit risk with CCP exposure and margin liquidity risk.
• Operations: Futures require intraday margin; forwards require CSA collateral management and closeout mechanics (ISDA).
• Hedging Impact: Futures P&L realized early changes effective reinvestment rate and can create basis vs a forward hedge.

2) No-Arbitrage Pricing

Short Answer
Portfolios with identical future cash flows must have the same price today.

Example
Equity forward parity: $F_0=S_0 e^{(r-q)T}$. If market quotes $F_0^\star > S_0 e^{(r-q)T}$, do cash-and-carry (borrow cash, buy spot, short forward) to lock risk-free profit.

Detailed Explanation

• Cash-and-carry: Profit at $T$ equals $F_0^\star - S_0 e^{(r-q)T}$.
• Reverse carry: If $F_0^\star < S_0 e^{(r-q)T}$, short spot, invest proceeds, long forward; profit $= S_0 e^{(r-q)T}-F_0^\star$.
• Carry components: $q$ can be dividends (equity), foreign rate (FX: $r_d-r_f$), or convenience yield vs storage (commodities).
• Real-world frictions: Bid–ask, short-borrow fees, taxes, discrete dividends, and collateral rates shrink or flip apparent arbitrages.

3) Risk-Neutral Valuation

Short Answer
Under $\mathbb{Q}$, discounted prices are martingales; price = discounted expectation of payoff under $\mathbb{Q}$.

Example
European option: $V_0=e^{-rT}\mathbb{E}^{\mathbb{Q}}[f(S_T)]$, with $dS_t=(r-q)S_t\,dt+\sigma S_t\,dW_t^{\mathbb{Q}}$.

Detailed Explanation

• Change of Measure: Girsanov transforms drift $\mu\to r-q$ to remove risk premia from pricing; risk is handled via replication.
• Completeness: If the market is complete (e.g., BSM), replication is unique ⇒ a unique $\mathbb{Q}$. In incomplete markets (jumps, stoch-vol), additional criteria (e.g., minimal martingale measure) pick a $\mathbb{Q}$.
• Numeraire: Pricing invariance across numeraires (bank account, $T$-bond, annuity) leads to forward measures simplifying some products (e.g., caplets).

4) BSM PDE (Derivation)

Short Answer
Delta-hedge $V(S,t)$ to eliminate diffusion; the residual riskless portfolio must earn $r$ ⇒ \(V_t+\tfrac12\sigma^2 S^2 V_{SS}+rS V_S-rV=0.\)

Example
For a call with $V(T,S)=\max(S-K,0)$, solving the PDE yields the Black–Scholes formula.

Detailed Explanation

• Itô’s Lemma: $dV=V_t dt+V_S dS+\tfrac12 V_{SS}(dS)^2$. With $dS=\mu S dt+\sigma S dW$.
• Hedge: Take $\Delta=V_S$, portfolio $\Pi=V-\Delta S$ eliminates $dW$.
• No-arb: $d\Pi=r\Pi dt$ ⇒ PDE above with appropriate boundary (terminal payoff), and conditions at $S\to 0,\infty$.
• Dividends: With continuous yield $q$, PDE becomes $V_t+\tfrac12\sigma^2 S^2 V_{SS}+(r-q)S V_S-rV=0$.

5) Meaning of $N(d_1)$ and $N(d_2)$

Short Answer
$N(d_2)$ ≈ risk-neutral probability of finishing ITM; $N(d_1)$ relates to delta/expected exercise under $\mathbb{Q}$.

Example
Call: $C=S_0 e^{-qT}N(d_1)-K e^{-rT}N(d_2)$. If $N(d_2)=0.60$, there’s a 60% $\mathbb{Q}$-chance of $S_T>K$.

Detailed Explanation

• Term 1: $S_0 e^{-qT}N(d_1)$ = PV of expected asset delivered at exercise (adjusted for $q$).
• Term 2: $K e^{-rT}N(d_2)$ = PV of exercise price paid, weighted by exercise probability.
• Delta: $\Delta_{call}=e^{-qT}N(d_1)$, connecting $N(d_1)$ to hedging ratio.

6) Volatility’s Impact

Short Answer
Higher $\sigma$ raises option value due to convexity (Jensen’s inequality).

Example
ATM call $S_0=K=100$, $T=1$, $r=q=0$: $C(\sigma=10\%)\approx 3.99$ vs $C(\sigma=30\%)\approx 11.92$ (BSM).

Detailed Explanation

• Convex Payoff: Upside unbounded, downside floored at 0 ⇒ dispersion benefits long options.
• Vega: $\nu=S_0 e^{-qT}\phi(d_1)\sqrt{T}>0$. Peaks near ATM and for longer $T$.
• Skews: Market smiles imply state-dependent effective volatility, making sensitivity path-dependent in practice.

7) Delta Hedging

Short Answer
Hold $\Delta=\partial V/\partial S$ shares against an option to neutralize small $S$ moves.

Example
Short 100 calls with $\Delta=0.55$ ⇒ buy 55 shares to be delta-neutral initially; rebalance as $\Delta$ changes.

Detailed Explanation

• Gamma–Theta: Frequent rebalancing needed if $\Gamma$ large; long gamma gains from realized variance but pays theta (time decay).
• Discrete Hedging Error: Hedging discretely produces residual P&L $\approx \tfrac12\Gamma (\Delta S)^2-\Theta \Delta t$ plus transaction costs.
• Smile Dynamics: “Sticky-delta” vs “sticky-strike” conventions materially affect hedge slippage.

8) Put–Call Parity

Short Answer
$C-P=S_0 e^{-qT}-K e^{-rT}$ for European options with same $(K,T)$.

Example
$S_0=100$, $r=5\%$, $q=2\%$, $T=1$. If $C=9.0$, parity implies $P=9.0-100e^{-0.02}+100e^{-0.05}\approx 6.4$.

Detailed Explanation

• Replication: Long call + short put = synthetic forward $S_T-K$.
• Uses: Build synthetics (e.g., covered call ↔ short put), detect data inconsistencies, infer missing quotes.
• Edge Cases: Early-exercise (American) parity becomes an inequality; discrete dividends must be PV-adjusted.

9) Volatility Smile/Skew

Short Answer
Implied vol varies with strike/maturity due to non-Gaussian returns and supply/demand.

Example
Equities: OTM puts rich (downward skew); FX: more symmetric smiles with risk-reversal asymmetry.

Detailed Explanation

• Drivers: Leverage effect ($\rho_{S,\sigma}<0$), crash risk premia, hedging pressure, jumps/stoch-vol.
• Modeling: SVI per maturity, SABR/Heston dynamics, local vol for exact fit vs. dynamics realism trade-off.
• Arb-free: Enforce butterfly (convexity in $K$) and calendar (monotonic in $T$) constraints.

10) Vega

Short Answer
Sensitivity to volatility: $\nu=S_0 e^{-qT}\phi(d_1)\sqrt{T}$.

Example
$S_0=K=100$, $T=1$, $r=q=0$, $\sigma=20\%$ ⇒ $\nu\approx 39.9$ per unit vol (i.e., 0.399 per 1% vol point).

Detailed Explanation

• Term Structure: Per-maturity vega buckets; vega not fungible across $T$.
• Smile: Skew vega (dV/d skew) and curvature vega (vomma) matter for surface moves.
• Hedging: Use options near ATM and close $T$ to neutralize efficiently.

11) Gamma

Short Answer
Curvature w.r.t. $S$: $\Gamma=e^{-qT}\phi(d_1)/(S_0\sigma\sqrt{T})$.

Example
Near-ATM, short-dated options have large $\Gamma$ (sensitive delta).

Detailed Explanation

• Risk/Reward: Long gamma benefits from realized volatility; short gamma earns theta but is exposed to large moves.
• Inventory: Market makers run gamma targets and rebalance based on liquidity/vol.

12) Theta

Short Answer
Time decay: typically negative for long options, positive for short.

Example
Short-dated ATM options can lose value rapidly into expiry (theta acceleration).

Detailed Explanation

• Components: “Carry” from discounting and from expected drift under $\mathbb{Q}$; discrete dividends can flip signs around ex-dates.
• Trade Design: Structures like calendars exploit theta/vega interplay.

13) Cost-of-Carry Forward Pricing

Short Answer
$F_0=S_0 e^{(r-q+c-\delta)T}$ with storage cost $c$ and convenience yield $\delta$.

Example
Gold with $r=4\%$, storage $c=1\%$, $\delta=0$ ⇒ $F_0=S_0 e^{0.05T}$.

Detailed Explanation

• FX: $F_0=S_0 e^{(r_d-r_f)T}$.
• Commodities: Scarcity $\Rightarrow \delta>0$ (backwardation).
• Curve: Forward curve encodes expectations + risk premia + inventory/flow constraints.

14) Futures Convexity Adjustment

Short Answer
With stochastic rates, futures ≠ forwards due to daily settlement.

Example
Eurodollar futures convexity vs FRA often approximated by $\tfrac12\sigma_r^2 T_1 T_2$ (order-of-magnitude guidance).

Detailed Explanation

• Mechanism: Covariance of daily gains with discounting shifts fair futures price.
• Sign: If underlying positively co-moves with rates, long futures benefit ⇒ futures > forward.

15) Greeks of Digital Options

Short Answer
Extremely sharp near strike: large gamma/vega, unstable delta.

Example
Cash-or-nothing call price $= e^{-rT}N(d_2)$; $\Delta= e^{-rT}\phi(d_2)/(S_0\sigma\sqrt{T})$.

Detailed Explanation

• Hedging: Use tight call spreads to approximate a digital and smooth greeks.
• Risk: Jump/announcement risk is acute due to step payoff.

16) Asian Options

Short Answer
Payoff depends on average; lower variance ⇒ cheaper than vanilla.

Example
Arithmetic Asian call: $(\bar S - K)^+$ with $\bar S=\frac1n\sum S_{t_i}$.

Detailed Explanation

• Pricing: Geometric Asians have closed forms; arithmetic often via MC or analytic approximations (Turnbull–Wakeman).
• Greeks: Pathwise estimators preferred; bridge corrections reduce bias.

17) Barrier Options

Short Answer
Activation/extinction based on path crossing; many closed forms via reflection.

Example
Down-and-out call = vanilla call − down-and-in call.

Detailed Explanation

• Monitoring: Continuous vs discrete matters (discrete cheaper knock-out); Brownian bridge improves MC accuracy.
• Greeks: Discontinuous near barrier (kinks); hedging requires careful sizing and possibly semi-static portfolios.

18) Stochastic Volatility (Heston)

Short Answer
Volatility follows a mean-reverting square-root process; semi-closed forms via characteristic functions.

Example
$dv=\kappa(\theta-v)dt+\eta\sqrt{v}\,dW^v$, $d\langle W^S,W^v\rangle=\rho\,dt$.

Detailed Explanation

• Calibration: Fit to smile surface across $K,T$ by minimizing price or IV errors.
• Dynamics: Negative $\rho$ creates equity-type skew; mean-reversion sets term structure.
• Greeks: Additional vanna, volga exposures; hedging needs both underlyings and volatility instruments.

19) SABR

Short Answer
Rates/FX model producing analytic IV approximations with parameters controlling level ($\alpha$), skew ($\rho$), and curvature ($\nu$); $\beta$ sets log-normal vs normal.

Example
FX often uses $\beta\approx 1$; rates sometimes $\beta<1$ for low-rate environments.

Detailed Explanation

• Hagan Formula: Widely used closed-form IV; care with extreme $K,F$ and very short $T$.
• Calibration: ATM volatility pins $\alpha$; risk-reversal pins $\rho$; butterfly pins $\nu$.

20) Local Volatility (Dupire)

Short Answer
Deterministic $\sigma_{loc}(t,S)$ reproducing the entire vanilla surface exactly.

Example
Dupire formula: \(\sigma_{loc}^2(t,K)=\frac{\partial_T C + qC - rK\partial_K C}{\tfrac12 K^2 \partial_{KK} C}\Big|_{T=t}.\)

Detailed Explanation

• Use: Good for barrier/exotics when exact vanilla fit is mandated.
• Limit: Unrealistic dynamics (sticky-strike), may mis-hedge under surface moves; numerically sensitive to noisy $\partial_{KK} C$.

21) Monte Carlo vs PDE

Short Answer
MC handles high-dimensional/path-dependent payoffs; PDE efficient in low dimensions and for early exercise.

Example
American put via finite-difference (PDE with free boundary) vs Bermudan via LSMC.

Detailed Explanation

• MC: Error $\mathcal{O}(1/\sqrt{M})$; QMC lowers effective variance; Greeks via pathwise/LR estimators.
• PDE: Fast & accurate in 1–2D with well-posed boundaries; tricky beyond 2D or with complex path terms.

22) Gamma–Theta Tradeoff

Short Answer
Long gamma benefits from movement but pays theta; short gamma earns theta but is hurt by movement.

Example
Long straddle: positive gamma/vega, negative theta; P&L thrives on realized vol exceeding implied.

Detailed Explanation

• P&L Attribution: $\Delta \text{P\&L}\approx \Delta\,\Delta S+\tfrac12\Gamma(\Delta S)^2+\nu\,\Delta\sigma+\Theta\,\Delta t$.
• Strategy: Market makers run near-neutral delta and target gamma/theta depending on vol views.

23) Forward-Start Options

Short Answer
Strike set at future date; prices depend on time to maturity after start.

Example
At $t_1$, strike $K=S_{t_1}$; payoff at $T$: $(S_T-S_{t_1})^+$.

Detailed Explanation

• Valuation: Under BSM, reduces to vanilla with maturity $T-t_1$ and ATM at $t_1$.
• Use: Equity comp, forward vol trades; greeks tied to forward measure over $[t_1,T]$.

24) Variance Swaps

Short Answer
Exchange realized variance for fixed variance strike; priced via strip of OTM options.

Example
Payoff $= N\left(\sigma_{\text{real}}^2 - K_{\text{var}}\right)$ with realized variance from high-frequency returns.

Detailed Explanation

• Replication: $K_{\text{var}}=\dfrac{2 e^{rT}}{T}\int_0^\infty \frac{P(K)-C(K)}{K^2}\,dK$ (OTM strip).
• Risks: Vol-of-vol, jumps, discretization. Corridors and gamma swaps extend concept.

25) Portfolio Greeks

Short Answer
Aggregate by summation across positions (linearity).

Example
$\Delta_{book}=\sum_i \Delta_i Q_i$, $\nu_{book}=\sum_i \nu_i Q_i$.

Detailed Explanation

• Hierarchy: Position → strategy → book; limits set per Greek and scenario.
• Surface Risk: Include $\partial\sigma/\partial K$ and $\partial\sigma/\partial T$ (skew/term risk).
• Stress: Nonlinear interactions under jumps/liquidity shocks require scenario P&L beyond first/second order.