MA2510 - EF3520 Notation Summary Table
Below is an expanded notation guide with a new column providing intuitive, step-by-step explanations and simple examples for each prerequisite’s notation as it appears in Stochastic Calculus for Finance (SCFF). This builds directly on the previous paired template, translating technical notation into practical workflow and real-world intuition.
ALL IN ONE
Expanded Notation Guide with Intuitive SCFF Applications
| Prerequisite | Notation & Definition | Role and Usage in SCFF | Example in SCFF | Step-by-Step Intuitive Explanation & Example | ||
|---|---|---|---|---|---|---|
| Probability Space | $$(\Omega, \mathcal{F}, \mathbb{P})$$: sample space, sigma-algebra, probability measure | Framework for defining all randomness; context for filtrations, martingales, pricing measure ($$\mathbb{Q}$$). | Pricing under risk-neutral measure: $$(\Omega, \mathcal{F}, \mathbb{Q})$$ | Step 1: Imagine a game — $$\Omega$$ is “all possible outcomes,” $$\mathcal{F}$$ defines what we know, $$\mathbb{P}$$/$$\mathbb{Q}$$ is how likely each outcome is.Example: For option pricing, switch from the “real-world” $$\mathbb{P}$$ to the “risk-neutral” $$\mathbb{Q}$$ to compute fair prices. | ||
| Random Variable / Process | $$X: \Omega \to \mathbb{R}$$, $$X_t$$; stochastic process $$(X_t)_{t \geq 0}$$ | Models asset prices, e.g., stock price ($$S_t$$), as random variables indexed by time. | $$S_t$$, $$W_t$$ (Brownian motion) | Step 1: Assign a number to every possible outcome (random variable).Step 2: Allow this number to change over time (stochastic process).Example: $$S_t$$ models how a stock price evolves unpredictably, at every time $$t$$. | ||
| Expectation | $$\mathbb{E}[X]$$; $$\mathbb{E}^{\mathbb{Q}}[X]$$ (under measure $$\mathbb{Q}$$) | Theoretical average ("fair value") under a given probability measure; forms the basis for no-arbitrage pricing in finance. | European call price: $$e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)^{+}]$$ | Step 1: Imagine all future universes and compute the average outcome (“expected value”).Example: Option price = risk-neutral average of payoff across all possible markets, discounted. | ||
| Conditional Expectation | $$\mathbb{E}[X | $$\mathcal{F}_t]$$: expected value given what is known at time $$t$$ | Defines fair prices/martingales, hedging strategies; captures “best estimate” with current info. | Martingale property: $$\mathbb{E}[S_{t+1}$$ | $$\mathcal{F}_t]=S_t$$ | Step 1: “Pause” at time $$t$$, gather all known info ($$\mathcal{F}_t$$).Step 2: Ask—if you could bet on $$X$$ now, what would you expect, knowing only what’s happened up to $$t$$?Example: After seeing prices until today, what’s your best prediction for tomorrow’s price, assuming markets are fair? |
| Distributions | $$F_X(x) = \mathbb{P}(X \leq x)$$; density function for continuous RVs | Models the probability (and shape) of outcomes for asset returns, increments, etc. | Stock returns: log-normal; Brownian motion: $$W_t \sim \mathcal{N}(0, t)$$ | Step 1: Chart how likely any value of $$X$$ is.Example: In Black-Scholes, the distribution of final stock price is log-normal, so all pricing formulas depend on this curve. | ||
| LLN/CLT/Convergence | LLN, CLT formulas; types of convergence | Justifies using averages, Monte Carlo simulations, parameter estimation; proves random walks converge to Brownian motion. | Monte Carlo price estimate: average of thousands of simulated payoffs | Step 1: Repeat an experiment (e.g., simulate asset path) many times.Step 2: By LLN, averages stabilize (estimate price); by CLT, errors are normal.Example: Simulate 10,000 paths → estimated option price. | ||
| Martingale | $$\mathbb{E}[X_{t+1}$$ | $$\mathcal{F}_t]=X_t$$ | Ensures fair prices: discounted portfolios should not offer free money (no-arbitrage). | $$S_t/B_t$$ is a $$\mathbb{Q}$$-martingale | Step 1: If today’s price is “fair,” tomorrow’s expected value (given today’s info) should be today’s price.Example: Under $$\mathbb{Q}$$, the expected value of discount stock price next period = current price, ensuring no free lunch. | |
| Markov Process | $$\mathbb{P}(X_{t+1}\in A$$ | $$\mathcal{F}t) = \mathbb{P}(X\in A$$ | $$(X_t)$$ | Asset price models often “forget” the past; only the present matters for next step. | Black-Scholes model: $$S_t$$ Markov via $$W_t$$ | Step 1: Only current state ($$X_t$$) matters for predicting next step.Example: Future stock price depends solely on its most recent price, not the day before yesterday’s price. |
| Inference | $$\hat{\mu}, \hat{\sigma}$$: estimators from data | Calibrating model parameters (like volatility) to observed market data; statistical significance/fit. | Estimate volatility $$\sigma$$ for Black-Scholes | Step 1: Collect historic data.Step 2: Use formulas (e.g., sample variance) to estimate drift/volatility for SDE input.Example: Calculate daily return volatility to use in option pricing. | ||
| Set Theory | $$A \in \mathcal{F}$$, unions ($$A \cup B$$), intersections ($$A \cap B$$) | Constructs events and sigma-algebras used throughout models. | Defining “events” in probability space | Step 1: Group or separate outcomes.Example: Define “stock price above $100 AND on a Friday” using set intersection. | ||
| Sigma-Algebra | $$\sigma$$-algebra: $$\mathcal{F}$$ | Encodes what’s “knowable”; vital for adapted processes. | $$\mathcal{F}_t$$: info up to time $$t$$ | Step 1: Formalize what information is available at every moment.Example: The set of all events that have occurred by time $$t$$. | ||
| Measure Theory | Probability measure: $$\mathbb{P}$$, Radon-Nikodym: $$\frac{d\mathbb{Q}}{d\mathbb{P}}$$ | Enables change of measure (e.g., risk-neutral pricing via Girsanov theorem). | Girsanov’s theorem for changing from $$\mathbb{P}$$ to $$\mathbb{Q}$$ | Step 1: “Tilt” the probability to see the world through a new lens (risk-neutral measure).Example: Change real-world probabilities to those required for fair pricing in finance. | ||
| Real Analysis | $$\lim_{n\to\infty}X_n = X$$; limits, continuity, differentiation | Ensures well-behaved processes; required for defining continuous paths, SDE solutions, limits. | Proof that Brownian paths are continuous | Step 1: Use limits to move from discrete to continuous math.Example: Price changes can be made “infinitely small” for calculus. | ||
| Filtration | $${\mathcal{F}t}{t\geq 0}$$: growing family of $$\sigma$$-algebras | Formalizes gradual information flow—a core input to “adaptation” and optional stopping. | Info up to now in pricing/hedging | Step 1: Each $$\mathcal{F}_t$$ tells you everything you know up to time $$t$$.Example: Only use info revealed up to present to make a decision/hedge. | ||
| Stochastic Process | $$(X_t)_{t\geq0}$$, $${S_t}$$, $${W_t}$$ | Framework for all dynamic modeling—tracks how quantities like stock price evolve over time. | $$S_t$$ (stock), $$W_t$$ (Brownian motion) | Step 1: Imagine a movie of a random variable as it moves with time.Example: Price path of a share over a month. | ||
| Brownian Motion | $$W_t$$ or $$B_t$$: continuous, starts at zero, independent increments, normal dist. | Fundamental source of “noise” and uncertainty in asset models. | $$dS_t = \mu S_t dt + \sigma S_t dW_t$$ | Step 1: Model pure “random walk” at the limit—with continuous time and no memory.Example: Core building block of Black-Scholes model for stock prices. | ||
| Stochastic Integral | $$\int_0^t X_s dW_s$$ (Itô integral) | Integrate with respect to random movement (Brownian); needed for solutions in SDEs, hedging replication. | Hedging: $$\int_0^t \phi_s dS_s$$ | Step 1: Generalize traditional integrals to account for randomness (“wiggly” paths).Example: Cost of a dynamic hedging portfolio over time in an uncertain market. | ||
| Itô’s Lemma | $$df(t,X_t) = f_t dt + f_x dX_t + \frac{1}{2} f_{xx} d[X,X]_t$$ (chain rule for stochastic calculus) | Calculate differentials of functions of stochastic processes; forms the backbone for derivations and PDEs used in option pricing. | Derivation of Black-Scholes PDE, $$df(S_t)$$ | Step 1: Extend chain rule to stochastic processes.Example: Find how an option’s price changes as the underlying stock follows a noisy path. | ||
| Quadratic Variation | $$[X]t = \lim{| \mathcal{P} | \to 0} \sum (X_{t_{i+1}}-X_{t_i})^2$$ | Captures “roughness” in sample paths; distinguishes Brownian motion from smooth functions; central to Itô calculus. | $$[W]_t = t$$ | Step 1: Total up all the tiny “wiggles” in a path’s square changes—allows Itô’s lemma to work.Example: For Brownian, total variation grows exactly at rate $$t$$. | ||
| Min/Max Notation | $$a \wedge b = \min(a,b)$$, $$a \vee b = \max(a,b)$$ | Shows up in limits for integration, e.g., covariance of Brownian motion, indicator intervals. | $$\int_0^{t_1\wedge t_2} ds$$ | Step 1: Restrict integration to overlap of intervals.Example: Covariation of two processes up to earliest of two stopping times. |
Key:
- Each “Step-by-Step Intuitive Explanation” walks through what the notation captures and how it guides your thinking, modeling, or calculation in SCFF.
- “Example” cell ties each concept to a tangible, real-world, or computational operation in financial mathematics.
This detailed breakdown ensures that for every symbol and prerequisite, you see both how it is defined, where it is used, and—crucially—how to intuitively interpret and apply it in real modeling and pricing situations in Stochastic Calculus for Finance.
SEPERATE Notation Guide and Concept Connections
Probability Theory (#STATS and Probability)
| Prerequisite | Notation & Definition | Role and Usage in SCFF | Example in SCFF | ||
|---|---|---|---|---|---|
| Probability Space | $$ (\Omega, \mathcal{F}, \mathbb{P}) $$: sample space, sigma-algebra, probability measure[1][2] | Foundation for stochastic processes; context for all randomness, filtrations, martingales, Brownian motion | Pricing models start with a fixed $$\Omega, \mathbb{P}$$, e.g., risk-neutral measure | ||
| Random Variable | $$ X: \Omega \to \mathbb{R} $$, $$ X_t $$ (process at time t)[1] | Underpins stochastic processes (indexed collection, i.e., $${X_t}_{t \geq 0}$$) | $$ S_t $$: asset price as random variable indexed by time | ||
| Expectation | $$ \mathbb{E}[X] $$, $$ \mathbb{E}^\mathbb{Q}[X] $$ (under measure $$\mathbb{Q}$$)[1] | Pricing (risk-neutral expectation), properties of martingales ($$\mathbb{E}[X_{t+1}]$$ | \mathcal{F}_t]=X_t$$), integral part of Itô lemma | Option prices: $$ \mathbb{E}^\mathbb{Q}[h(S_T)] $$ | |
| Conditional Expectation | $$ \mathbb{E}[X]$$ | $$\mathcal{F}_t] $$[1] | Defines martingales; crucial for “memoryless” property in finance, dynamic programming, filtering | Hedging basis, martingale representation theorem | |
| Distributions | $$ P_X $$, cdfs: $$ F_X(x) = \mathbb{P}(X \leq x) $$[1][2] | Modeling underlying asset returns, increments of Brownian motion ($$\sim \mathcal{N}(0, t)$$) | Black-Scholes: lognormal returns | ||
| LLN/CLT/Convergence | LLN: $$\lim_{n \to \infty} \bar{X}_n = \mu$$, CLT: $$(\sum X_i - n\mu)/\sqrt{n\sigma^2} \to N(0,1)$$[1] | Justifies model calibration, convergence to Brownian motion, statistical estimation of drift/volatility parameters (large data) | Monte Carlo simulation for price convergence | ||
| Martingale | $$ \mathbb{E}[X_{t+1}$$ | $$\mathcal{F}_t]=X_t $$ (martingale property)[3] | No-arbitrage, risk-neutral valuation, pricing via martingale methods, constructing measure changes | $$ S_t/B_t $$ is a martingale under $$\mathbb{Q}$$ | |
| Markov Process | $$ \mathbb{P}(X_{t+1} \in A$$ | $$\mathcal{F}t) = \mathbb{P}(X \in A$$ | $$(X_t) $$[1] | Used in Black-Scholes model, filtering, reduced-form modeling | Brownian motion, Ornstein-Uhlenbeck |
| Inference | Notation varies: estimators ($$\hat\mu$$), hypothesis testing | Calibrating models to market data: volatility, drift, residual analysis | Calibrating $$\sigma$$ in Black-Scholes |
Mathematical Foundations (#MATHS)
| Prerequisite | Notation & Definition | Role and Usage in SCFF | Example in SCFF |
|---|---|---|---|
| Set Theory | Elements: $$A \in \mathcal{F}$$, unions ($$A \cup B$$), intersections ($$A \cap B$$), complements ($$A^c$$) [4] | Definition of events, sigma-algebras $$\mathcal{F}$$, filtrations | Construction of probability space, event analysis |
| Sigma-Algebra | $$\sigma$$-algebra: $$\mathcal{F}$$, sets closed under complements and countable unions[2] | Models available information (filtration), builds up progressive “revealing” in processes | $$\mathcal{F}_t$$ = info up to time t in pricing |
| Measure Theory | Probability measure: $$\mathbb{P}$$, Radon-Nikodym derivative ($$\frac{d\mathbb{Q}}{d\mathbb{P}}$$) [2][5] | Risk-neutral measure change, Girsanov theorem, integration of stochastic processes | Equivalent martingale measures |
| Real Analysis | Limits, continuity, convergence: $$\lim_{n \to \infty} X_n = X$$[5] | Convergence of random variables and processes, continuous-time models | Quadratic variation, SDE solution existence |
| Filtration | $${\mathcal{F}t}{t\geq 0}$$: non-decreasing family of $$\sigma$$-algebras[6][7] | Models information flow, required for adaptation, optional stopping theorem | $$\mathcal{F}_t$$: info up to t, “natural” filtration |
| Stochastic Process | $${X_t: t \geq 0}$$, $$(X_t)_{t\geq 0}$$[8] | Price paths, index to time; core framework for modeling asset evolution | $$S_t, W_t$$ : stock price, Brownian motion |
| Brownian Motion | $$W_t$$ or $$B_t$$: continuous path, independent increments, $$W_0=0$$[9][10] | Canonical model for asset returns, foundation for SDEs and option pricing | $$dW_t$$ in Black-Scholes, geometric BM |
| Stochastic Integral | $$\int_0^t X_s dW_s$$, “Itô integral”[11][12] | Integration against Brownian motion, Itô’s lemma | Hedging strategies, SDE solutions |
| Itô’s Lemma (Formula) | For $$X_t$$ solving $$dX_t = \mu_t dt + \sigma_t dW_t$$: | Chain rule for stochastic functions, option pricing PDEs | Derivation of Black-Scholes PDE |
| $$ df(t, X_t) = f_t dt + f_x dX_t + \frac{1}{2} f_{xx} d[X,X]_t $$[13] | |||
| Quadratic Variation | $$[X]t = \lim{| \mathcal{P}| \to 0} \sum (X_{t_{i+1}} - X_{t_i})^2$$ [11] | Captures “roughness” of paths, key in SDEs and in Itô’s formula | $$[W]_t = t$$ for Brownian motion |
| Min/Max Notation | $$a \wedge b = \min(a, b)$$, $$a \vee b = \max(a, b)$$[14] | Appears as limits in integrals, e.g., quadratic covariation | $$\int_0^{t_1 \wedge t_2} \ldots ds$$ |
Direct Connection: Prerequisite to SCFF Usage
Below is a stepwise connection showing how each prerequisite appears and evolves in SCFF notation and application.
-
Probability Space $$(\Omega, \mathcal{F}, \mathbb{P})$$:
- All financial models start here. When using risk-neutral pricing, $$\mathbb{Q}$$ replaces $$\mathbb{P}$$.
-
Random Variable / Stochastic Process ($$X_t$$):
- Notation $${S_t}$$, $${W_t}$$, $$dX_t$$.
- Models the asset's price, the underlying noise, or state variable.
-
Filtration ($${\mathcal{F}_t}$$):
- Specifies information available at each moment, essential to define adapted processes.
-
Expectation ($$\mathbb{E}[,\cdot,]$$), Conditional Expectation ($$\mathbb{E}[,\cdot,|\mathcal{F}_t]$$):
- Valuation formulas, martingale properties, hedging analysis.
-
Martingale:
- Price processes discounted at the risk-free rate ($$e^{-rt}S_t$$) must be martingales under $$\mathbb{Q}$$.
-
Brownian Motion ($$W_t$$), Stochastic Differential Equations (SDE):
- The fundamental noise process in finance: $$dS_t = \mu S_t dt + \sigma S_t dW_t$$.
- Used to model random movements of asset prices.
-
Itô's Lemma and Stochastic Integral:
- Provides differential calculus for functions of stochastic processes—basis for option pricing PDEs.
-
Quadratic Variation ($$[X]_t$$):
- Used in Itô's lemma and SDE computation. For Brownian motion, $$[W]_t = t$$.
-
Law of Large Numbers/CLT/Convergence:
- Justifies Monte Carlo simulation and estimation; forms the theoretical backbone for stochastic integration limits.
-
Measure and Integration:
- Integral notation throughout ($$\int$$, $$\int_0^T$$), required for pricing expectations and SDEs.
Conclusion
Stochastic Calculus for Finance notates and operationalizes each of the statistical and mathematical prerequisites with systematic symbols and formulations. Understanding the foundational notation in probability (#STATS) and mathematics (#MATHS) is essential—their usage is explicit in every aspect of modeling, hedging, pricing, and theoretical finance[11][1][14][2][6][9][13][5][8][3].
This mapping ensures you can follow any derivation or financial modeling step by systematically referencing back to core theory, notational discipline, and mathematical rigor.