Financial mathematics Study Guide

📖 Core Concepts Mathematical Finance – Application of math (stochastic calculus, optimization, statistics) to price derivatives and manage risk/portfolios. Two worlds – \(Q\)‑world (risk‑neutral measure): used for pricing; probabilities are “tilted” so discounted asset prices are martingales. \(P\)‑world (real/actuarial measure): used for risk & portfolio management; reflects the true likelihood of future price moves. Fundamental Theorem of Arbitrage‑Free Pricing – A market has no arbitrage iff there exists an equivalent martingale (risk‑neutral) measure \(Q\). Black–Scholes Model – Continuous‑time, log‑normal asset price dynamics under \(Q\) → closed‑form price for European options. Greeks – Partial derivatives of option price w.r.t. model inputs (Δ, Γ, ν, Θ, ρ). Risk Measures – Value‑at‑Risk (VaR) = quantile loss; Expected Shortfall (ES) = average loss beyond VaR. Utility‑Based Portfolio Theory – Maximize expected utility of final wealth; mean‑variance is a special quadratic‑utility case. --- 📌 Must Remember Arbitrage‑Free ⇔ Exists \(Q\) measure (Fundamental Theorem). Risk‑neutral pricing formula: \( \text{Price}0 = e^{-rT}\, \mathbb{E}^Q[\text{Payoff}] \). Black–Scholes PDE: \( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2}+ rS\frac{\partial V}{\partial S} - rV =0\). Black–Scholes closed‑form for a European call: $$ C = S0 N(d1) - K e^{-rT} N(d2) $$ where \(d{1,2}= \frac{\ln(S0/K)+(r\pm\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}\). Delta (Δ) = \( \partial V/\partial S\); Gamma (Γ) = \( \partial^2 V/\partial S^2\); Vega (ν) = \( \partial V/\partial \sigma\); Theta (Θ) = \( \partial V/\partial t\); Rho (ρ) = \( \partial V/\partial r\). VaR (α) = smallest loss \(L\) such that \(P(\text{Loss} > L) \le 1-\alpha\). Expected Shortfall = \( \mathbb{E}[ \text{Loss} \mid \text{Loss} > \text{VaR}\alpha]\). Mean‑variance optimization → solve: minimize \(w^\top \Sigma w\) subject to \(w^\top \mu = \mu{\text{target}}\) and \(\sum wi =1\). --- 🔄 Key Processes Derivatives Pricing under \(Q\) Choose a continuous‑time model (e.g., geometric Brownian motion). Calibrate model parameters (σ, r, etc.) to market‑quoted liquid securities. Solve the pricing PDE or compute the discounted expectation under \(Q\). Apply closed‑form formula (Black–Scholes) or numerical method (finite‑difference, Monte Carlo). Risk‑Neutral Measure Construction Start with real‑world dynamics under \(P\). Perform Girsanov change‑of‑measure to obtain drift = risk‑free rate \(r\). Verify that discounted asset price \(e^{-rt}St\) is a martingale under the new measure. Portfolio Optimization (Mean‑Variance) Estimate expected returns vector \(\mu\) and covariance matrix \(\Sigma\). Solve quadratic program: \(\minw w^\top \Sigma w\) s.t. \(w^\top \mu = \mu^\) and \(\sum wi =1\). Adjust for estimation risk (shrinkage, Bayesian priors). Monte Carlo Pricing (High‑Dimensional) Simulate paths of underlying assets under \(Q\) (or \(P\) for risk metrics). Compute payoff for each path, discount, and average. For American options, use Least‑Squares Monte Carlo: regress continuation values on basis functions to decide early exercise. --- 🔍 Key Comparisons \(Q\) vs. \(P\) measure Purpose: Pricing vs. risk/portfolio decisions. Drift: Risk‑free rate \(r\) vs. real expected return \(\mu\). Use: Closed‑form & risk‑neutral expectations vs. empirical estimation & scenario analysis. Analytic vs. Numerical Pricing Analytic (Black‑Scholes, Black model): Fast, closed‑form, limited to simple payoffs & log‑normal dynamics. Numerical (finite‑difference, Monte Carlo, trees): Handles early exercise, path‑dependence, stochastic volatility; slower, needs convergence checks. Local Volatility vs. Stochastic Volatility Local: σ = σ(S,t) deterministic; calibrated to surface, reproduces market prices but no dynamics for future volatility. Stochastic: σt follows its own stochastic process (e.g., Heston); captures volatility clustering and smile dynamics. VaR vs. Expected Shortfall VaR: Single quantile, not coherent (fails sub‑additivity). ES: Average loss beyond VaR, coherent risk measure. --- ⚠️ Common Misunderstandings “Risk‑neutral = real world” – No, \(Q\) is a pricing construct; actual expected returns differ. “Black‑Scholes works for all options” – Only for European, frictionless, log‑normal assets; fails for American, path‑dependent, or stochastic volatility products. “Higher Δ always means better hedge” – Delta hedging ignores higher‑order risks (Gamma, Vega); large Gamma can cause hedging error. “VaR tells you the worst loss” – VaR only gives a threshold; extreme tail losses can be far larger. “Mean‑variance optimal portfolios are always optimal under any utility” – Only true for quadratic utility or normally distributed returns. --- 🧠 Mental Models / Intuition Martingale ↔ “Fair Game” – Under \(Q\), the expected discounted price of any tradable asset equals its current price; think of a fair coin toss where the expected future wealth is unchanged. Risk‑Neutral Pricing = “Pricing as if investors are indifferent to risk” – The market price is the expected payoff discounted at the risk‑free rate, regardless of actual risk preferences. Greeks as “sensitivities” – Imagine moving a knob (price, vol, time, rate) slightly; the Greek tells you how much the option price knob‑turn moves. Stochastic Volatility = “Volatility that itself wiggles” – Not just a static knob, but a second random process that adds extra uncertainty. --- 🚩 Exceptions & Edge Cases Jump‑diffusion / Lévy processes – Black‑Scholes assumes continuous paths; heavy‑tailed or jump models (Mandelbrot’s α‑stable) capture sudden moves. Negative interest rates – Classical Black‑Scholes formula still works mathematically, but interpretation of forward price and discounting may need adjustment. Discrete dividend payments – Adjust underlying price or use a dividend‑adjusted forward; standard BS assumes no dividends. American options on dividend‑paying stocks – Early exercise may be optimal; tree or LSMC required. --- 📍 When to Use Which Closed‑form Black‑Scholes → European vanilla on non‑dividend stocks, log‑normal assumption acceptable, quick “back‑of‑the‑envelope”. Black model → Interest‑rate caps/floors, swaptions (underlying forward rates). Binomial/Trinomial tree → American options, discrete dividend effects, intuitive visual of early exercise. Finite‑difference PDE → When a PDE has known boundary conditions but no closed form (e.g., barrier options). Monte Carlo → High‑dimensional problems, path‑dependent payoffs, stochastic volatility, or when Greeks via bump‑and‑reprice are acceptable. Least‑Squares Monte Carlo → American options with many risk factors. Mean‑variance optimization → When returns are approximately normal or utility is quadratic. Utility‑based (concave) optimization → Non‑normal returns, investors with varying risk aversion. --- 👀 Patterns to Recognize Log‑normal payoff → Black‑Scholes applicable (European call/put). Early‑exercise feature + dividend → Look for tree or LSMC rather than BS. Smile/skew in implied vol surface → Indicates need for local or stochastic volatility model. Heavy tails in return data → Consider Lévy/α‑stable or Student‑t distributions, not Gaussian. Risk‑neutral drift = r → Whenever you see a PDE or expectation under \(Q\), replace drift with risk‑free rate. --- 🗂️ Exam Traps “Use Black‑Scholes for American options” – BS only gives a lower bound; exam will often present an American put with dividends to catch this. Mixing up Δ and Γ – Δ is first‑order; Γ is curvature. A question may ask which Greek measures convexity – answer Γ, not Δ. VaR vs. ES confusion – If the question asks for a coherent risk measure, pick ES; VaR is not coherent. Assuming σ is constant – In questions about “volatility smile”, the constant‑σ assumption is a trap; answer should mention local/stochastic volatility. Forgetting the discount factor – Pricing formula must include \(e^{-rT}\); forgetting it leads to a price that is too high. Misidentifying the measure – If the problem states “under the real‑world probability”, you must use \(P\) (e.g., for expected portfolio return), not \(Q\). ---