Asset pricing Study Guide

📖 Core Concepts Asset Pricing Models – Frameworks that link an asset’s expected return to its exposure to systematic risk factors. Equilibrium vs. Rational Pricing – Equilibrium (e.g., CAPM) derives returns from market‑clearing of supply/demand; Rational (risk‑neutral) pricing uses no‑arbitrage to value derivatives. Systematic Factor – A source of risk that cannot be diversified away (e.g., market index, size, value, momentum). Risk‑Neutral Measure – A probability distribution adjusted so that every asset’s expected return equals the risk‑free rate; used for discounting payoffs. State‑Price Securities (Arrow‑Debreu) – Pure‑security that pays 1 unit only if a particular future state occurs; any security price = weighted sum of state‑prices. Greeks – First‑order sensitivities of an option price to underlying variables (Δ, Γ, Vega, Θ, ρ). --- 📌 Must Remember CAPM: Expected return = risk‑free + β × market risk premium. Fama‑French 3‑factor: Adds size (SMB) and value (HML) to market factor. Carhart 4‑factor: Adds momentum (UMD) to the three F‑F factors. APT: Linear return model with multiple macro‑economic factors, no single market portfolio required. Black‑Scholes: Closed‑form price for European options when the underlying follows a log‑normal process. Black model: Same formula but uses forward prices (for futures/forwards). Garman‑Kohlhagen: Extends Black‑Scholes to FX options; includes domestic & foreign rates. Vasicek: Mean‑reverting short‑rate with constant volatility; can become negative. CIR: Square‑root diffusion → guarantees non‑negative rates. Hull‑White: Time‑varying mean‑reversion level & volatility (extension of Vasicek). Heston: Stochastic variance follows a mean‑reverting square‑root process. Longstaff‑Schwartz: Monte‑Carlo + backward induction for American option valuation. Fundamental Theorem: No‑arbitrage ⇔ existence of a risk‑neutral measure that prices all claims by discounted expectation. --- 🔄 Key Processes Discounted Cash‑Flow Valuation Forecast future cash flows. Discount each flow at the required return from the chosen equilibrium model (captures undiversifiable risk). Sum present values → asset/firm value. Bootstrapping a Yield Curve Start with the shortest‑maturity instrument (e.g., a Treasury bill). Solve for the discount factor that matches its price. Sequentially use longer‑maturity bonds to back out additional discount factors, ensuring the whole curve is arbitrage‑free. Longstaff‑Schwartz Algorithm Simulate many paths of the underlying price. At each exercise date, regress continuation values on basis functions of the state variables. Compare immediate exercise payoff vs. estimated continuation value to decide optimal exercise. Deriving State‑Price Prices from a Model Compute the model‑implied risk‑neutral probability density. Multiply each state’s payoff by the corresponding density and discount at the risk‑free rate → state‑price. --- 🔍 Key Comparisons CAPM vs. APT CAPM: Single market factor, assumes a specific market portfolio. APT: Multiple unspecified macro factors; no need for a market portfolio. Black‑Scholes vs. Black Model Black‑Scholes: Prices options on the spot asset; uses spot price \(S0\). Black: Prices options on forwards/futures; substitutes forward price \(F\). Vasicek vs. CIR Vasicek: Normal diffusion → rates can go negative. CIR: Square‑root diffusion → rates stay non‑negative. Single‑Index vs. Multi‑Factor Single‑Index: Only market index explains returns. Multi‑Factor: Adds size, value, momentum, etc., to capture more variation. Deterministic vs. Stochastic Volatility Deterministic: Volatility is a fixed function of time (e.g., Black‑Scholes). Stochastic: Volatility follows its own random process (e.g., Heston). --- ⚠️ Common Misunderstandings “CAPM predicts actual returns.” → CAPM gives the required return for a given systematic risk, not the realized return. “Black‑Scholes works for American options.” → It only provides exact prices for European‑style options; early exercise features require other methods (e.g., binomial trees, Longstaff‑Schwartz). “CIR guarantees positive rates for all parameters.” → Positivity holds only if the Feller condition \(2\kappa\theta \ge \sigma^2\) is satisfied. “Higher‑order Greeks are unimportant.” → For large moves or long‑dated options, Gamma, Vanna, and Vomma can dominate risk. “Bootstrapping yields a unique curve.” → Different interpolation choices (linear, cubic spline) produce slightly different curves; the underlying principle is the same. --- 🧠 Mental Models / Intuition Factor Loading = Exposure: Think of each factor like a “lever” you pull; the larger the loading, the more the asset’s return moves with that lever. Risk‑Neutral World = “Fair‑Game” Casino: All assets earn the risk‑free rate; the only difference is the probability weighting, not the payout. Mean‑Reversion = “Elastic Band”: The short rate is pulled back toward a long‑run average (the band’s equilibrium point). Stronger pull → faster return to mean. Stochastic Volatility = “Weather”: The “temperature” (volatility) itself fluctuates randomly, affecting the price of “clothes” (options). --- 🚩 Exceptions & Edge Cases Negative Rates: Vasicek permits them; CIR does not (unless the Feller condition fails). Zero‑Coupon vs. Coupon Bonds: Bootstrapping formulas differ; zero‑coupon bonds give direct discount factors. FX Options: Garman‑Kohlhagen requires both domestic (\(rd\)) and foreign (\(rf\)) risk‑free rates; ignoring \(rf\) leads to mispricing. Momentum Factor: Only meaningful when the data window captures persistent return trends; may disappear in short‑term samples. --- 📍 When to Use Which | Situation | Recommended Model / Method | |-----------|----------------------------| | Single‑factor market risk | CAPM (or single‑index) | | Evidence of size/value effects | Fama‑French 3‑factor | | Momentum observed | Carhart 4‑factor | | Multiple macro‑economic drivers | APT | | European option on non‑dividend stock | Black‑Scholes | | Option on futures/forward | Black model | | FX option | Garman‑Kohlhagen | | Need for stochastic volatility | Heston | | Short‑rate modeling with possible negative rates | Vasicek | | Short‑rate modeling that must stay positive | CIR (check Feller condition) | | Term‑structure fitting with a tree | Black‑Derman‑Toy | | American option valuation | Longstaff‑Schwartz (Monte‑Carlo) or binomial tree | | Building a risk‑free discount curve | Bootstrapping from liquid bonds | | Multi‑curve environment (e.g., OIS discounting) | Multi‑curve bootstrapping framework | | Deriving state‑price securities | Use risk‑neutral densities from any calibrated model (e.g., HJM, Heston). | --- 👀 Patterns to Recognize “+β·(Market‑Premium)” pattern → CAPM‑type return formula. Linear combination of factors → APT or multi‑factor models. Mean‑reverting drift term \(\kappa(\theta - rt)\) → Vasicek, Hull‑White, Heston variance process. Square‑root diffusion \(\sqrt{rt}\) → CIR, Heston variance. Log‑normal dynamics ⇒ closed‑form Black‑Scholes price (European options). Forward price substitution ⇒ Black model (options on futures). Two‑rate terms (domestic & foreign) ⇒ Garman‑Kohlhagen (FX). --- 🗂️ Exam Traps Confusing “required return” with “actual return” in CAPM questions. Choosing Black‑Scholes for American options → will be marked wrong. Applying CIR without checking the Feller condition → may lead to impossible negative variances in calculations. Assuming all multi‑factor models are “APT”. APT is a specific linear‑factor framework; Fama‑French and Carhart are empirical factor models. Neglecting foreign interest rate in FX option pricing → Garman‑Kohlhagen vs. Black‑Scholes mix‑up. Mixing risk‑neutral discount rate (risk‑free) with required return (CAPM rate) when discounting cash flows. Treating the Greeks as independent – remember they are interrelated; e.g., Δ and Γ are linked via the curvature of the price surface. ---