Quantitative Finance & Risk Management

QuantFin (Quantitative Finance) integrates ML, stat mod, stoch calc, and num meth to model fin mkts, price deriv, and manage risk. Core pillars: asset pric, port opt, risk mgt, algo trad. Asset pric relies on no-arb arg, deriv pric via BS PDE or martingale meth. BS Model (Black-Scholes) assumes log-normal SDE: dS = μSdt + σSdW; soln: S(t)=S(0)exp[(μ−½σ²)t + σW(t)]. Risk-neut val: deriv price = E^Q[e^−rT payoff], Q=risk-neut meas. Ext: local vol (Dupire), stoch vol (Heston: dV=κ(θ−V)dt+ξ√V dZ), jump-diff (Merton). Implied vol surf reveals skew/smirk; vol arb exploits mispricing. Port opt: Markowitz mean-var (MVP) → quadratic prog: min w'Σw s.t. w'μ=μ₀, w'1=1. Ext: Black-Litterman (incorp investor views), robust opt (handle est uncer), risk-parity (risk contrib equal). RiskMgt: VaR (Value at Risk) = max loss at α% conf over Δt; est via hist sim, param (Normal/GED), or MC. Crit: VaR not subadditive → CVaR (cond VaR) preferred; coherent risk meas satisfy mono, trans inv, pos homog, subadd, tail sens. Basel III: Tier 1 cap, LCRR, NSFR, stress test req. Credit risk: PD (def prob), LGD (loss given def), EAD (exp exposure). Models: KMV (dist to def), Jarrow-Turnbull (reduced form), Merton (struct). CDS pric ≈ PD × (1−R). Mkt microstr: order book dyn, slippage, impact mod (Almgren-Chriss). HFT uses low-lat infra, pred mod, latency arb. ML appl: pric (LSTM vol forec), risk (anom detec), port (NN alloc), alpha gen (NLP news senti). GARCH-family (GARCH, EGARCH, TGARCH) mod vol clus and lev eff. Backtest val strat: avoid data-snooping, inc trans cost, slippage, regime shift. Pitfalls: overfit (esp ML), model risk (mis-spec, par drift), illiq ass (neglect bid-ask, depth), tail risk underest (Normal assump), black swan events. SoA: XVA (CVA, DVA, FVA, MVA, KVA) adjust deriv val for counterparty risk, fund cost, cap. CVA = EPE × (1−R) × PD; EPE=exp pos expos. FVA reflects fund imbal. Deep hedging (Buehler et al.) uses RL to learn opt hedge pol under non-ideal mkt. Quantum fin: QML for port opt (QAOA), MC accel (amplif), option pric (HHL). Data: high-freq tick, TAQ, options chain, macro ind. Tools: Python (NumPy, pandas, PyTorch), R, C++, KDB+. Reg: MiFID II, Dodd-Frank, EMIR. Stress test: hist (2008, 2020), hyp, reverse. Liquidity risk: fund vs mkt liq; LCR (high-qual liq asset / net cash outflow) ≥100%. Op risk: fraud, sys fail, model error; mod via LDA (loss dist appr). Climate risk: TCFD, phys & trans risk, scen anal. ESG integ: screen, best-in-class, impact invest. Key metrics: Sharpe (excess ret / vol), Sortino (↓vol), Calmar (ann ret / max draw), IR (α / σα). Model val: resid diag, out-of-samp perf, sens anal, stress test. Corr dyn: DCC-GARCH, copulas (t, Clayton, Gumbel) for tail dep. Crisis mod: regime-switch (MS-GARCH), EVT for tail index est. Ethical issues: front-run, data privacy, algo bias. Future: real-time risk, explainable AI, gen AI for strat synth, green fin quant.