Stochastic Processes in Financial Modeling
Expert-defined terms from the Certificate in Financial Engineering course at London School of Planning and Management. Free to read, free to share, paired with a professional course.
AR(1) Model – A first‑order autoregressive process where the current valu… #
Related terms: Stationarity, lag operator, time series. The model is X_t = φ X_{t‑1} + ε_t with |φ|<1 for mean‑reversion. In finance, AR(1) describes interest‑rate dynamics or log‑returns over short horizons. Example: modeling daily equity returns where φ≈0.1 captures momentum. Challenges include estimating φ reliably when data are noisy and ensuring residuals ε_t are truly independent.
Brownian Motion – A continuous‑time stochastic process with independent,… #
Related terms: Wiener process, diffusion, martingale. Standard Brownian motion B(t) satisfies B(0)=0, B(t)−B(s)∼N(0, t‑s). It underpins the Black‑Scholes option‑pricing framework, where asset log‑prices follow B(t) with drift. Practical use: Simulating price paths for Monte Carlo valuation. Difficulties arise when real markets display jumps or volatility clustering, violating the Gaussian increment assumption.
Black‑Scholes Model – A closed‑form solution for pricing European options… #
Related terms: Risk‑neutral measure, delta, implied volatility. The price C = S N(d₁) − K e^{‑rT} N(d₂), where d₁, d₂ involve σ, r, T. It provides the “Greeks” for hedging. Example: Computing the fair price of a call on a stock with S=100, K=100, σ=20%, r=5%, T=1 yr. Limitations include inability to capture volatility smile, early exercise features, and discrete dividend effects.
Change‑of‑Measure Technique – A mathematical tool that transforms probabi… #
Related terms: Girsanov theorem, Radon‑Nikodym derivative, martingale pricing. Under Q, discounted asset prices become martingales, simplifying valuation. For instance, applying Girsanov’s theorem to a drifted Brownian motion yields a drift‑free process under Q. The main challenge is verifying the Novikov condition to ensure the measure change is valid, especially in models with stochastic volatility.
Conditional Expectation – The expected value of a random variable given i… #
Related terms: filtration, martingale, law of iterated expectations. In finance, it appears in pricing: the price of a derivative equals the discounted conditional expectation of its payoff under Q. Example: pricing a forward contract as E_Q[S_T|ℱ_0] e^{‑rT}. Calculating conditional expectations analytically can be impossible for complex payoffs, necessitating numerical methods.
Cox‑Ingersoll‑Ross (CIR) Model – A mean‑reverting square‑root diffusion u… #
Related terms: Ornstein‑Uhlenbeck, Feller condition, term structure. The stochastic differential equation d r_t = κ(θ‑r_t)dt + σ√r_t dW_t ensures non‑negative rates if 2κθ ≥ σ² (Feller condition). Practically, the CIR model yields analytical bond prices, aiding curve fitting. Calibration is challenging because the square‑root term makes the likelihood surface flat, and market data may violate the Feller condition.
Diffusion Process – A continuous‑time stochastic process described by a s… #
Related terms: Ito calculus, SDE, generator. General form: DX_t = μ(X_t,t)dt + σ(X_t,t)dW_t. Diffusions model asset prices, volatility, and credit spreads. Example: The Heston model uses a diffusion for both price and variance. Numerical implementation requires discretization schemes (Euler‑Maruyama, Milstein); errors accumulate if step size is too large, especially for stiff drift terms.
Discrete‑Time Markov Chain – A stochastic process where the next state de… #
Related terms: Transition matrix, Chapman‑Kolmogorov equations, absorbing state. In credit risk, a Markov chain models migration among rating categories. For a 3‑state chain (AAA, AA, Default) the transition matrix P gives probabilities of moving between states each year. Calibration uses historical rating transition data. A major difficulty is that real‑world transitions may exhibit path dependence, violating the Markov property.
Doob‑Meyer Decomposition – The representation of a sub‑martingale as the… #
Related terms: Compensator, optional decomposition, semimartingale. In pricing, the decomposition isolates the “risk‑neutral” martingale part from the “drift” that reflects risk premia. For a counting process N_t with intensity λ_t, N_t = M_t + ∫₀^t λ_s ds. Understanding the compensator is essential for intensity‑based credit models. Computing the decomposition analytically is often infeasible, requiring estimation of the intensity from market data.
Empirical Characteristic Function – The Fourier transform of the empirica… #
Related terms: Fourier inversion, moment generating function, estimation via CF. In stochastic volatility models, the true characteristic function is known analytically; matching it to the empirical counterpart yields efficient estimators. For example, calibrating the Heston model by minimizing the distance between model and empirical characteristic functions. The method is sensitive to sample size and may suffer from numerical instability at high frequencies.
Exact Simulation – A technique that generates sample paths of a stochasti… #
Related terms: Acceptance‑rejection, Brownian bridge, Lévy area. For the Ornstein‑Uhlenbeck process, one can sample directly from its Gaussian transition law. Exact simulation eliminates bias in Monte Carlo pricing of path‑dependent options. However, exact methods are limited to a few tractable processes; for most SDEs one must resort to approximate schemes.
Feynman‑Kac Formula – A link between certain partial differential equatio… #
Related terms: Backward Kolmogorov equation, PDE pricing, probabilistic representation. The price V(t,x) of a contingent claim satisfies ∂V/∂t + ℒV = rV with terminal condition payoff, where ℒ is the generator of the underlying diffusion. Equivalently, V(t,x)=E_Q[e^{‑r(T‑t)} g(X_T) | X_t=x]. This formula justifies Monte Carlo valuation as an alternative to solving PDEs. Applying it requires that the coefficients satisfy regularity conditions; otherwise the representation may break down.
Geometric Brownian Motion (GBM) – A stochastic process where the logarith… #
Related terms: Log‑normal distribution, Black‑Scholes, drift‑diffusion. DS_t = μS_t dt + σS_t dW_t. It models equity prices in the classic Black‑Scholes framework. Example: Simulating a stock price path with μ=5%, σ=20% over one year. GBM fails to capture empirical features such as heavy tails and volatility clustering, prompting extensions like stochastic volatility or jump‑diffusion models.
Girsanov Theorem – A result that describes how the dynamics of a Brownian… #
Related terms: Change of measure, Radon‑Nikodym density, drift adjustment. If under P, dW_t = dW_t^P, then under Q defined by dQ/dP = ℰ(θ·W)_T, the process W_t^Q = W_t^P – ∫₀^t θ_s ds becomes a Brownian motion. This is pivotal for moving from the physical to the risk‑neutral world. The theorem’s applicability requires Novikov’s condition; violating it leads to non‑equivalent measures and invalid pricing.
Heston Stochastic Volatility Model – A two‑factor diffusion where the ass… #
Related terms: Variance swap, characteristic function, volatility smile. DS_t = μS_t dt + √v_t S_t dW_t^S, dv_t = κ(θ‑v_t)dt + ξ√v_t dW_t^v, corr(dW_t^S, dW_t^v)=ρ. The model reproduces observed implied‑volatility smiles and allows semi‑closed‑form option pricing via Fourier inversion. Calibration involves solving a non‑linear optimization over (κ,θ,ξ,ρ, v₀). Challenges include parameter identifiability and ensuring the Feller condition for v_t to stay positive.
Ito’s Lemma – The stochastic calculus analogue of the chain rule, providi… #
Related terms: Stochastic differential, drift, diffusion term. If X_t follows dX_t = μ dt + σ dW_t and f(t,X_t) is twice differentiable, then df = (∂f/∂t + μ∂f/∂x + ½σ²∂²f/∂x²)dt + σ∂f/∂x dW_t. Ito’s lemma converts a non‑linear payoff into an SDE, essential for deriving hedging ratios. Misapplication (e.G., Ignoring the second‑order term) leads to pricing errors, especially for options with high curvature.
Jump‑Diffusion Model – An extension of diffusion processes that incorpora… #
Related terms: Merton model, Lévy process, jump intensity. DS_t = μS_t dt + σS_t dW_t + (J‑1)S_t dN_t, where N_t is a Poisson process with intensity λ and J is the random jump multiplier. This captures sudden price moves and fat‑tailed return distributions. Example: Pricing a barrier option where jumps can cause knock‑out events. Calibration is difficult because jump parameters (λ, jump size distribution) are weakly identified from daily data.
Kullback‑Leibler Divergence – A measure of the distance between two proba… #
Related terms: Relative entropy, likelihood, information criteria. For distributions P and Q, D_{KL}(P‖Q)=∫ p(x) log(p(x)/q(x))dx. In finance, minimizing D_{KL} between market‑implied and model‑generated return distributions yields parameters that best fit observed option prices. It is asymmetric; the choice of direction matters for risk‑adjusted pricing. Computing the integral may require numerical quadrature when analytic forms are unavailable.
Long‑Memory Process – A stochastic process whose autocorrelation function… #
Related terms: Fractional Brownian motion, Hurst exponent, ARFIMA. In volatility modeling, long‑memory implies that shocks persist, affecting option pricing and risk management. Example: Using an ARFIMA(0,d,0) model for realized variance where d≈0.4 Captures the slow decay. Estimating the memory parameter d is sensitive to sample size and can be biased by structural breaks.
Markov Property – The characteristic that the future evolution of a proce… #
Related terms: Memoryless, transition kernel, strong Markov property. Mathematically, P(X_{t+s}∈A | ℱ_t) = P(X_{t+s}∈A | X_t). This property simplifies pricing by allowing use of transition densities. Many models (e.g., GBM, OU) are Markovian, while stochastic volatility models with hidden factors may violate it from the perspective of observable variables. Violations complicate the use of PDE methods, often requiring filtering techniques.
Monte Carlo Simulation – A numerical method that estimates expectations b… #
Related terms: Variance reduction, pseudo‑random numbers, quasi‑Monte Carlo. To price a European call, simulate N paths of S_T, compute payoff max(S_T‑K,0), discount and average. Monte Carlo handles high‑dimensional problems and path‑dependent options (e.G., Asian options). Main challenges are slow convergence (rate 1/√N) and the need for variance‑reduction techniques such as antithetic variates, control variates, or low‑discrepancy sequences.
Ornstein‑Uhlenbeck (OU) Process – A mean‑reverting Gaussian diffusion oft… #
Related terms: Vasicek model, stationary distribution, speed of reversion. DX_t = κ(θ‑X_t)dt + σ dW_t. The stationary distribution is N(θ, σ²/(2κ)). Example: Modeling natural gas spot price where κ controls how quickly deviations from the long‑run mean θ disappear. OU assumes normality, which may underestimate extreme moves; extensions include adding jumps or stochastic volatility.
Partial Differential Equation (PDE) Approach – Solving the pricing proble… #
Related terms: Finite difference, Crank‑Nicolson, boundary conditions. For a European option under GBM, the Black‑Scholes PDE ∂V/∂t + ½σ²S²∂²V/∂S² + rS∂V/∂S – rV = 0 must be solved with terminal payoff V(T,S)=max(S‑K,0). Numerical schemes discretize time and asset space, offering fast valuations for vanilla products. Difficulties include handling free‑boundary problems for American options and ensuring stability for highly non‑linear PDEs.
Path‑Dependent Option – A derivative whose payoff depends on the entire t… #
Related terms: Asian option, lookback option, barrier option. For an Asian call, payoff = max( (1/n)∑_{i=1}^n S_{t_i} – K, 0). Pricing requires averaging across simulated paths, making Monte Carlo the preferred method. Analytical solutions exist only for a few special cases. Computational cost grows with the number of monitoring dates, and variance reduction becomes essential.
Poisson Process – A counting process with independent, exponentially dist… #
Related terms: Jump intensity, compound Poisson, Lévy process. N(t) counts the number of events up to time t; P[N(t)=k]=e^{‑λt}(λt)^k/k!. In credit risk, default arrivals are often modeled as a Poisson process. In jump‑diffusion models, the Poisson component drives discrete price jumps. Estimating λ from sparse data can be unstable, especially when events are rare.
Quadratic Variation – The limit of the sum of squared increments of a sto… #
Related terms: Itô integral, semimartingale, path regularity. For Brownian motion B(t), [B]_t = t. Quadratic variation differentiates between processes of finite variation (e.G., Deterministic functions) and those with stochastic volatility. In finance, it is the theoretical foundation of realized variance estimators. High‑frequency data suffer from microstructure noise, biasing the observed quadratic variation.
Risk‑Neutral Measure – An equivalent probability measure under which disc… #
Related terms: Q‑measure, equivalent martingale measure, numéraire. Changing from the physical measure P to Q involves adjusting drifts while keeping volatilities unchanged. Under Q, the expected return of any tradable asset equals the risk‑free rate r. Computing Q requires the market price of risk λ(t) and often Girsanov’s theorem. In incomplete markets, multiple risk‑neutral measures exist, leading to price bounds rather than unique values.
Sharpe Ratio – A performance metric that compares excess return to volati… #
Related terms: Risk‑adjusted return, Sortino ratio, information ratio. In portfolio optimization, maximizing the Sharpe ratio leads to the tangency portfolio on the efficient frontier. Example: A fund with annual return 12%, risk‑free rate 2%, and volatility 10% has Sharpe = (12‑2)/10 = 1.0. The ratio assumes returns are normally distributed; heavy tails or skewness can mislead investors, prompting use of alternative risk measures.
Stochastic Differential Equation (SDE) – An equation describing the dynam… #
Related terms: Drift, diffusion, Ito calculus. Solving SDEs yields the transition law of the underlying variable, essential for pricing derivatives. Analytical solutions exist for only a few SDEs (e.G., GBM, OU). Numerical solvers (Euler‑Maruyama, Milstein) approximate paths but introduce discretization bias, especially when σ depends on X_t.
Term Structure Model – A framework that describes the evolution of intere… #
Related terms: Yield curve, Heath‑Jarrow‑Morton (HJM), affine term structure. Models can be short‑rate based (e.G., Vasicek, CIR) or forward‑rate based (HJM). They produce bond price formulas P(t,T)=E_Q[e^{‑∫_t^T r_s ds}|ℱ_t]. Calibration uses market data such as zero‑coupon yields or swap rates. High‑dimensional HJM models may suffer from over‑parameterization and instability in calibration.
Variance Gamma (VG) Process – A pure‑jump Lévy process obtained by time‑c… #
Related terms: Lévy process, subordinator, characteristic function. X_t = θ G_t + σ W_{G_t}, where G_t ~ Gamma(t/ν, ν). VG captures skewness and kurtosis without stochastic volatility. Option pricing can be performed via Fourier inversion using its known characteristic function. Parameter estimation is sensitive to the choice of ν; small ν leads to frequent small jumps, while large ν produces rare large jumps.
Volatility Surface – A three‑dimensional representation of implied volati… #
Related terms: Smile, skew, interpolation. The surface reveals market expectations of future volatility; for equity options, it typically shows a “smile” where out‑of‑the‑money strikes have higher implied vol. Practitioners fit parametric models (e.G., SABR) or use spline interpolation to obtain smooth surfaces for pricing exotic options. Maintaining arbitrage‑free conditions (no calendar or butterfly arbitrage) during fitting is a key challenge.
White Noise – A stochastic process with zero mean, constant variance, and… #
Related terms: Gaussian noise, innovation, discrete‑time shock. In discrete models, ε_t ~ N(0,σ²) represents white noise. In continuous time, dW_t/dt is formally white noise. It provides the randomness in diffusion processes. Real financial data exhibit autocorrelation and heteroskedasticity, so white‑noise assumptions may be violated, prompting the use of GARCH‑type structures for residuals.
Yield Curve – A plot of interest rates (or yields) against different matu… #
Related terms: Spot rate, forward rate, bootstrapping. The curve reflects market expectations of future rates and risk premia. In term‑structure modeling, the curve is used to calibrate short‑rate models or to construct discount factors via bootstrapping. Sudden shifts (e.G., “Twist” or “steepening”) present challenges for risk management, as they affect the valuation of cash‑flow‑sensitive securities.
Zero‑Coupon Bond – A bond that pays no periodic coupons and matures at a… #
Related terms: Discount factor, spot rate, pure discount bond. Price at time t: P(t,T)=E_Q[e^{‑∫_t^T r_s ds}|ℱ_t]. Zero‑coupon bonds are building blocks for constructing the term structure and for replicating cash flows. In models with stochastic rates, analytical bond prices may be unavailable, requiring numerical integration or Monte Carlo. Accurate pricing demands careful handling of the integral of the short rate.
Affine Jump‑Diffusion – A class of models where drift, diffusion, and jum… #
Related terms: Duffie‑Pan‑Singleton, exponential affine, tractability. General form: DX_t = (b + β X_t)dt + Σ√X_t dW_t + dJ_t, with jump intensity λ = λ₀ + λ₁ X_t. Affine structure yields exponential‑affine solutions for bond and option prices, facilitating calibration. However, ensuring positivity of state variables and satisfying the Feller condition become more intricate when jumps are present.
Barndorff‑Nielsen and Shephard (BNS) Model – A stochastic volatility mode… #
Related terms: Volatility clustering, Lévy process, integrated variance. Dv_t = −λ v_t dt + dZ_t, where Z_t is a compound Poisson or Gamma process. The model captures sudden spikes in volatility and heavy‑tailed returns. Option pricing uses Fourier methods thanks to known characteristic functions. Calibration can be computationally intensive due to the need to estimate jump‑size distribution parameters.
Calibration – The process of adjusting model parameters so that model out… #
Related terms: Optimization, loss function, over‑fitting. Typical objective: Minimize Σ_i w_i (Market_i – Model_i(θ))², where θ are parameters. Calibration may use gradient‑based algorithms (Levenberg‑Marquardt) or global heuristics (genetic algorithms). Over‑fitting occurs when the model captures noise rather than underlying dynamics, leading to poor out‑of‑sample performance. Regularization techniques and cross‑validation help mitigate this risk.
Dirichlet Boundary Condition – A type of boundary condition specifying th… #
Related terms: Neumann condition, finite difference, domain truncation. For a barrier option, the price V(t,S) = 0 when S reaches the barrier level, representing a Dirichlet condition. Implementing it in finite‑difference grids requires setting the grid points at the barrier to zero each time step. Incorrect handling can produce spurious oscillations or violate the option’s payoff structure.
Ergodicity – A property indicating that time averages converge to ensembl… #
Related terms: Stationary distribution, mixing, long‑run mean. In a mean‑reverting process like OU, ergodicity ensures the process will spend proportionate time near its long‑run mean θ. For risk management, ergodicity justifies using historical averages to estimate future risk measures. Certain models with explosive drift (e.G., Geometric Brownian motion with positive drift) are non‑ergodic, limiting the reliability of long‑term forecasts.
Feller Condition – A criterion ensuring that the square‑root diffusion in… #
Related terms: CIR process, non‑negativity, boundary behavior. Condition: 2Κθ ≥ σ². When satisfied, the origin is unattainable, preventing negative interest rates. Violations lead to the possibility of hitting zero, which may be acceptable for some interest‑rate environments but complicates analytical bond pricing. Adjusting parameters to satisfy the condition may reduce model flexibility during calibration.
GARCH Model – Generalized Autoregressive Conditional Heteroskedasticity,… #
Related terms: ARCH, volatility clustering, conditional variance. Standard GARCH(1,1): Σ_t² = ω + α ε_{t‑1}² + β σ_{t‑1}². It captures volatility persistence observed in asset returns. Example: Estimating daily equity volatility using maximum likelihood. While useful for risk forecasting, GARCH assumes conditional normality; heavy‑tailed innovations require extensions like EGARCH or GJR‑GARCH.
Hedging Ratio (Delta) – The sensitivity of an option price to a small cha… #
Related terms: Greeks, risk management, replicating portfolio. Delta indicates the number of units of the underlying needed to offset price risk. For a call option, Δ∈(0,1). Traders continuously adjust delta to maintain a neutral position, a process called delta‑hedging. In practice, discrete rebalancing introduces hedging error, especially when the underlying exhibits jumps or stochastic volatility.
Itô Integral – The stochastic integral ∫₀^t σ_s dW_s, defined as the limi… #
Related terms: Stratonovich integral, semimartingale, quadratic variation. The Itô integral is central to SDE solutions and obeys Itô’s isometry: E[(∫₀^t σ_s dW_s)²] = E[∫₀^t σ_s² ds]. It differs from the Stratonovich integral, which follows the usual chain rule. Misusing the integral type leads to incorrect drift terms in model derivations.
Jump‑to‑Default Model – A reduced‑form credit model where default occurs… #
Related terms: Intensity‑based, reduced‑form, hazard rate. The survival probability up to time T is S(T)=e^{‑∫₀^T λ(s) ds}. Credit default swaps (CDS) are priced using this survival function. Calibration involves matching market CDS spreads to implied λ. Time‑varying intensity adds realism but introduces identification problems, as λ and recovery rates can trade off in fitting the same spreads.
Kramers‑Moyal Expansion – A series expansion relating the evolution of pr… #
Related terms: Fokker‑Planck equation, master equation, transition rates. Truncating after the second term yields the Fokker‑Planck (diffusion) equation, justifying the use of SDEs for continuous dynamics. Higher‑order terms capture jump behavior. In practice, estimating higher‑order coefficients from data is noisy, limiting the expansion’s empirical use.
Levy Process – A stochastic process with stationary, independent incremen… #
Related terms: Brownian motion, Poisson jump, infinite activity. Characteristic function φ(u)=exp[t ψ(u)], where ψ is the Lévy exponent. Models like Variance Gamma, Normal Inverse Gaussian, and CGMY belong to this class. Lévy processes provide flexible return distributions with skewness and heavy tails. Calibration often requires Fourier methods; ensuring numerical stability of the exponent ψ can be challenging.
Martingale – A stochastic process whose conditional expectation equals it… #
E., E[X_t|ℱ_s]=X_s for s≤t. Related terms: fair game, no‑arbitrage, local martingale. Under the risk‑neutral measure, discounted asset prices form martingales, guaranteeing the absence of arbitrage. In practice, verifying martingale property helps confirm model consistency. Certain models produce strict local martingales (e.g., certain stochastic volatility specifications) leading to price bubbles and non‑standard option pricing behavior.
Markov Chain Monte Carlo (MCMC) – A class of algorithms that generate sam… #
Related terms: Metropolis‑Hastings, Gibbs sampler, posterior sampling. In Bayesian calibration of stochastic‑process models, MCMC draws from the posterior of parameters given market data. Example: Estimating Heston parameters by sampling from their joint posterior. Convergence diagnostics (trace plots, Gelman‑Rubin statistic) are essential; poor mixing can result in biased parameter estimates and excessive computation time.
Mean‑Reversion Speed (κ) – The parameter governing how quickly a mean‑rev… #
Related terms: Ornstein‑Uhlenbeck, speed of adjustment, half‑life. In the OU SDE, κ>0 determines the exponential decay rate of deviations: X_t−θ decays as e^{‑κt}. A higher κ yields a shorter half‑life (ln 2/κ). In interest‑rate modeling, κ influences the curvature of the yield curve. Estimating κ is sensitive to measurement error; overly high values may over‑fit short‑term fluctuations.
Monte‑Carlo Greeks – Sensitivities of option prices computed via Monte Ca… #
Related terms: Bump‑and‑revalue, adjoint algorithmic differentiation, finite differences. Delta can be estimated by differentiating the payoff along each simulated path (pathwise method) when the payoff is smooth. For discontinuous payoffs (e.G., Digital options), the likelihood‑ratio method is preferred. Monte‑Carlo Greeks suffer from higher variance than price estimates, requiring variance‑reduction techniques and large sample sizes.
Numerical Stability – The property of an algorithm to produce bounded err… #
Related terms: Convergence, discretization error, CFL condition. In finite‑difference PDE solvers, the Courant‑Friedrichs‑Lewy (CFL) condition ensures stability: Δt ≤ (Δx)²/(2σ²) for explicit schemes. Implicit schemes (e.G., Crank‑Nicolson) are unconditionally stable but may introduce oscillations if not properly damped. In SDE simulation, the Milstein scheme offers higher order accuracy but can become unstable for stiff drift terms unless adaptive step sizes are used.
Option Pricing Kernel – Also called the state‑price density, it links pay… #
Related terms: Stochastic discount factor, Radon‑Nikodym derivative, pricing measure. For any payoff h(S_T), price = E_Q[π_T h(S_T)], where π_T = e^{‑∫₀^T r_s ds} dQ/dP. The kernel encapsulates market risk preferences. In practice, extracting the kernel from option prices (e.G., Via Breeden‑Litzenberger) yields the risk‑neutral density. Ill‑posedness can cause instability, requiring smoothing or regularization techniques.
Partial Correlation – The correlation between two variables after removin… #
Related terms: Conditional correlation, graphical models, precision matrix. In multivariate stochastic modeling, partial correlation helps identify direct dependencies among factors, useful for building parsimonious covariance structures. Estimating it from high‑dimensional data may suffer from over‑fitting; shrinkage estimators improve robustness.
Quadratic Hedging – A hedging approach that minimizes the expected square… #
Related terms: Mean‑variance hedging, L² minimization, risk‑minimization. Given a contingent claim H, the optimal hedge h* solves min_h E[(H – h·ΔS)²]. In incomplete markets, quadratic hedging provides a tractable solution when exact replication is impossible. The resulting hedge depends on the covariance structure of asset returns; estimation error can degrade performance, especially for assets with low liquidity.
Risk‑Adjusted Return – A performance measure that accounts for the amount… #
Related terms: Reward‑to‑risk, performance attribution, downside risk. Risk‑adjusted metrics guide portfolio selection by penalizing volatility or downside moves. For assets with asymmetric return distributions, downside‑focused measures (e.G., Sortino) provide a more accurate picture. However, these metrics rely on reliable estimates of risk, which can be unstable during market stress.
SABR Model – Stochastic Alpha Beta Rho model, an arbitrage‑free approach… #
Related terms: Implied volatility, smile dynamics, normal SABR. Dynamics: DF_t = σ_t F_t^β dW_t, dσ_t = ν σ_t dZ_t, corr(dW_t,dZ_t)=ρ. Closed‑form approximations (Hagan et al.) Provide implied volatility as a function of strike and maturity. Calibration targets market smiles across tenors. The model can produce negative rates when β=0 (normal SABR), but for β=1 (log‑normal) it may violate positivity under extreme volatility shocks.
Stochastic Discount Factor (SDF) – A random variable that discounts futur… #
Related terms: Pricing kernel, state‑price density, risk‑neutral measure.