MA2510 - EF3520 Notation Summary Table

MA2510 - EF3520 Specific learning materials Map

IN GENERAL

Aligning the “cross-fields pre-requisite knowledges” schema with the categorized contents of the “Stochastic calculus for finance” schema.
For each syllabus category, you’ll find:

• Book concept/coverage and justification.

• Stepwise mapping and explanation of how fundamental cross-field knowledge (especially from mathematics, probability, statistics, and finance) is embedded, derived, or needed for mastery of each topic.

• Where appropriate, I unpack the logical flow from foundational theory, definitions, and typical proofs or derivations, showing explicit links.

[Syllabus Category: Stochastic Processes]

Book Concepts & Justification:

• Foundation for all models and results in financial mathematics.
• Includes definitions, properties, types (Markov, martingale), jumps, convergence—all across the key texts.

Cross-fields Pre-requisite Knowledges:

MATHS

• Set Theory & Mathematical Logic: Definition of probability spaces, σ-algebras (collection of events), construction of sample spaces.

• Real Analysis & Measure Theory: Formalizes probability as measure, enables rigorous treatment of convergence (almost surely, in probability, in L^p).

• Proof Theory: Used to derive and prove key properties (e.g., Markov property, independence), via induction, contradiction, or direct arguments.

Pathway to Book Content:

• The formal notion “A stochastic process {X_t; t ∈ T} is a collection of random variables indexed by T over a probability space (\Omega, \mathcal{F}, P)” is built from:

  1. Set Theory: Define sets Ω (sample space), F (event σ-algebra).

  2. Measure Theory: P is a probability measure on F; random variables are measurable mappings Ω→ℝ.

  3. Real Analysis: Defines types of convergence, continuity of paths, regularity.

• Probability Theory: Definitions of stationarity, independence, Markov property, and martingales (conditional expectation—requiring measure-theoretic notions).

• Logic/Proof: All key theorems (e.g., Martingale Convergence, Kolmogorov’s Extension Theorem) are proved using these foundational tools.

[Syllabus Category: Brownian Motion]

Book Concepts & Justification:

• Canonical continuous-time process, modeling random movement (stock price evolution, underlying in Black-Scholes).

• Properties: Continuous paths, independent, stationary increments, Markov, martingale.

Cross-fields Pre-requisite Knowledges:

MATHS & # STATS

• Probability Distributions: Normal (Gaussian) distribution and its properties.

• Real Analysis: Construction of continuous functions, uniform convergence.

• Measure Theory: Kolmogorov’s consistency theorem ensures the existence of Brownian motion.

• Statistical Inference: Understanding normal increments, convergence to normal distribution (e.g., via Central Limit Theorem).

Pathway to Book Content:

• Random walks (Discrete): Defined via binomial distribution.

• Transition: Show convergence by scaling steps/time (Donsker’s theorem, FCLT—functional CLT).

• Proof: Use limit theorems (probability/statistical logic), perform rigorous construction via characteristic functions and measure theory, verify path properties (with real analysis—Kolmogorov’s continuity criterion).

• Martingale property and quadratic variation: Proved with conditional expectation workings from measure-theoretic probability.

[Syllabus Category: Filtration]

Book Concepts & Justification:

• Filtration is the formalization of “information over time”—essential for adaptedness, defining martingales, predictable strategies in trading.

Cross-fields Pre-requisite Knowledges:

MATHS & # STATS

• Set Theory & σ-algebras: Sequence of growing σ-algebras {Ft}{\mathcal{F}_t}{Ft} reflecting information.

• Conditional Probability & Expectation: Key for constructing martingales, adapted processes.

• Real Analysis & Measure Theory: Underpin definition of measurable structures needed for all modern stochastic processes.

Pathway to Book Content:

• Using set theory and logic, define nested family of σ-algebras (Fs⊆Ft\mathcal{F}_s \subseteq \mathcal{F}_tFs⊆Ft, for s<ts < ts<t).

• Measure theory: Guarantees the measurability of adapted processes.

• Probability/statistics: Adaptedness and conditional expectations (martingales/predictable strategies) are manipulated using these structured filtrations, conditional independence arguments, and classic theorems (tower property for expectation).

[Syllabus Category: Itô Formula (Stochastic Calculus Core)]

Book Concepts & Justification:

• Generalizes chain rule to stochastic (non-differentiable) functions—central to SDEs, asset modeling, Black-Scholes.

Cross-fields Pre-requisite Knowledges:

MATHS, # STATS

• Differential Calculus: Chain rule, Taylor expansions.

• Real Analysis: Rigorous definition of limits, differentiability.

• Measure Theory: Definition of stochastic integrals (Itô integral is defined as L² limit of sums).

• Probability Theory: The distributional properties of Brownian increments (zero mean, variance dt), martingale properties.

Pathway to Book Content:

• Begin with Taylor expansion for smooth functions; see why standard calculus fails for Brownian motion (not differentiable).

• Use probability/statistics: Brownian increments scale as dt\sqrt{dt}dt, and their quadratic variation is dtdtdt, underpinning the “extra term” in Itô formula.

• Prove the formula by approximation via partitions and taking limits in mean-square (measure-theory, real analysis).

• All technical results are derived via precise logical/analytic arguments from these fields.

[Syllabus Category: Two-Instants Model & N-Instants Model]

Book Concepts & Justification:

• Binomial pricing logic, discrete models that build intuition for risk-neutral valuation and hedging.

• Two-instant (single-step), N-instant (multi-step, binomial tree).

Cross-fields Pre-requisite Knowledges:

MATHS, # STATS, # FINANCE

• Discrete Probability: Binomial distribution, calculation of event probabilities.

• Conditional Expectation/statistical inference: Pricing as expectation under risk-neutral measure.

• Linear algebra/Combinatorics: For payoffs, portfolio replication, combinatorial arguments.

• Financial derivatives: Definitions of payoffs, arbitrage, hedging.

Pathway to Book Content:

• Define discrete sample space, analyze outcomes (binomial probabilities, expectation).

• Use logic to argue arbitrage (if prices deviate from expectation, construct “free money” portfolio).

• Move from two-instant to N-instant: Induction and recursion from discrete math.

• Dynamic hedging implemented via stepwise backward induction, justified by properties of expectation (statistics), applied to claims (finance).

[Syllabus Category: Self-Financing Portfolio]

Book Concepts & Justification:

• No infusion/withdrawal of cash—portfolio value changes only from asset price changes; crucial for all pricing.

Cross-fields Pre-requisite Knowledges:

MATHS, # FINANCE

• Basic calculus: Chain rule (discrete and stochastic).

• Linear algebra: Vector representations of portfolios.

• Conditional expectation/probability: Ensures strategies are measurable/predictable.

• Financial derivatives: Understanding of portfolios, hedging, underlying vs. riskless asset.

Pathway to Book Content:

• Define change in portfolio (ΔV) using calculus.

• Use measure-theoretic probability to formalize the “predictability” of strategies (adaptedness to filtration).

• In continuous time, use stochastic calculus (Itô’s lemma, SDEs) to derive differential relationships.

[Syllabus Category: Arbitrage Opportunity]

Book Concepts & Justification:

• Foundational financial concept: possibility of riskless profit with zero net investment.

Cross-fields Pre-requisite Knowledges:

FINANCE, # MATHS, # STATS

• Definitions from finance: Arbitrage, law of one price.

• Linear algebra: Construction of arbitrage portfolios.

• Logic/proof: Proof by contradiction (if arbitrage exists, prices are not consistent), applies arguments from set theory, probability.

Pathway to Book Content:

• Given market prices, use combinations of assets (linear algebra) to illustrate arbitrage.

• Use probability/statistics to show under what measures such an opportunity exists (probability of profit >0).

• Proofs of no-arbitrage ⇒ existence of risk-neutral measure employ logic, probability theory.

[Syllabus Category: Risk-Neutral Measure]

Book Concepts & Justification:

• Change of probability measure under which discounted asset prices are martingales.

• Foundation for fair pricing via expectation of discounted payoff.

Cross-fields Pre-requisite Knowledges:

MATHS, # STATS

• Probability theory: Definition of measure, conditional expectation (martingales), Radon-Nikodym derivative, Girsanov’s theorem.

• Measure theory: Change of measure; absolute continuity.

• Financial derivatives: Discounting, no-arbitrage price formulation.

Pathway to Book Content:

• Martingales (measure-theoretic probability): Proofs use filtration, measure changes, stochastic integration.

• Girsanov’s theorem (proof and application): Deep use of measure theory, logic.

• Pricing formulæ: Derived as mathematical expectation under new (risk-neutral) measure, grounded in statistics and financial logic (no-arbitrage principle).

[Syllabus Category: Market Completeness]

Book Concepts & Justification:

• Uniqueness of risk-neutral measure equivalent to ability to replicate all claims.

• Linked to martingale representation theorem.

Cross-fields Pre-requisite Knowledges:

MATHS, # FINANCE

• Linear algebra: Uniqueness of solutions in systems (replication strategies).

• Probability/statistics: Representation of martingales, completeness.

• Financial derivatives: Hedging, replication.

Pathway to Book Content:

• Show via linear algebra/statistics that each claim can be written as a sum of elementary assets (basis).

• Prove with probability/martingale theory: Uniqueness of measure ⇔ completeness.

[Syllabus Category: Partial Differential Equations (PDEs)]

Book Concepts & Justification:

• Analytical machinery for pricing derivatives in continuous time (Black-Scholes, Feynman-Kac, complex options).

Cross-fields Pre-requisite Knowledges:

MATHS

• ODEs, PDEs: Existence, uniqueness, method of characteristics.

• Multi-variable calculus, differential operators.

• Real analysis: Continuity, differentiability, solutions in function spaces.

• Probability theory: Feynman-Kac (linking SDEs to PDEs).

Pathway to Book Content:

• Begin with stochastic model (SDE via Itô calculus).

• Apply Itô formula to price function—derive diffusive term, drift, time decay.

• Collect PDE for option price (e.g., Black-Scholes PDE) using calculus and analysis.

• Connect probabilistic solution (Feynman-Kac) to PDEs.

[Syllabus Category: Black-Scholes Formula]

Book Concepts & Justification:

• Explicit formula for European options, the apex of modern financial mathematics.

Cross-fields Pre-requisite Knowledges:

MATHS, # STATS, # FINANCE

• Multi-variable calculus: Derivation, integration.

• Probability theory: Expectation, distributions (normal).

• PDEs: Solution of Black-Scholes PDE.

• Financial derivatives: European options definitions/payoff structure.

• Statistical inference: Use of cumulative normal distribution (Φ) in formula.

Pathway to Book Content:

• Derive model assumptions and SDE for asset price (Itô calculus).

• Apply risk-neutral valuation: express price as expectation of discounted payoff (integral).

• Solve PDE (or close SDE solution) analytically, requiring all mathematical/statistical machinery—a synthesis of the entire chain of knowledge.

Summary Table: Schematic Crossfields Pre-requisite Flow for Each Major Category

This structure exposes the precise intellectual plumbing: each financial or calculus tool is rigorously constructed and justified from deep mathematical, financial, and statistical foundations, following an ordered development that mirrors the logic of both modern textbooks and professional practice in quantitative finance.

From Book: Stoc Cal for FINANCE 1

Certainly! Below is a comprehensive, ranked enumeration of the prerequisite knowledge for "Stochastic calculus for finance," organized by the three pivots: #STATS and Probability theory, #MATHS, and #Finance. For each, I explain the foundational definitions, key theorems, and how each area contributes to the main categories and results in stochastic calculus for finance, with explicit links to the stepwise development of the subject.

#STATS and Probability theory

1. Probability Spaces and Random Variables

• Axiomatic Definition:
  A probability space is a triple $(\mathrm{\Omega },\mathcal{F},P)$ where $\mathrm{\Omega }$ is the sample space, $\mathcal{F}$ is a σ-algebra of events, and $P$ is a probability measure with $P(\mathrm{\Omega })=1$.
• Random Variable:
  A measurable function $X:\mathrm{\Omega }\to \mathbb{R}$.
• Contribution:
  Forms the basis for defining stochastic processes, filtrations, and all probabilistic modeling in finance.

2. Distributions and Expectation

• Discrete and Continuous Distributions:
  Binomial, normal, Poisson, etc.
  $E[X]=\sum xP(X=x)$ (discrete), $E[X]=\int x{f}_X(x)dx$ (continuous).
• Contribution:
  Used in modeling asset returns, option payoffs, and in the binomial and Black-Scholes models.

3. Conditional Probability and Expectation

• Definition:
  $P(A|B)=\frac{P(A\cap B)}{P(B)}$;
  $E[X|\mathcal{G}]$ is the conditional expectation given a sub-σ-algebra $\mathcal{G}$.
• Contribution:
  Essential for martingales, filtrations, and risk-neutral pricing.

4. Law of Large Numbers (LLN) and Central Limit Theorem (CLT)

• LLN:
  $\frac{1}{n}\sum _{i=1}^n{X}_i\to E[X]$ almost surely as $n\to \mathrm{\infty }$.
• CLT:
  $\frac{1}{\sqrt{n}}\sum _{i=1}^n(X_i-E[X])\to N(0,{\sigma }^2)$ in distribution.
• Contribution:
  Justifies convergence of binomial models to Brownian motion; underpins the use of normal distributions in continuous models.

5. Martingales and Markov Processes

• Martingale:
  $E[X_{t+1}|{\mathcal{F}}_t]=X_t$.
• Markov Property:
  $P(X_{t+1}\in A|{\mathcal{F}}_t)=P(X_{t+1}\in A|X_t)$.
• Contribution:
  Central to risk-neutral pricing, hedging, and the structure of asset price models.

6. Useful Theorems

• Chebyshev’s Inequality:
  $P(|X-E[X]|\ge k\sigma )\le 1/k^2$.
• Doob’s Martingale Convergence Theorem:
  If $X_t$ is a bounded martingale, $X_t$ converges almost surely.
• Contribution:
  Used in proofs of convergence, pricing, and hedging arguments.

7. Joint, Marginal, and Conditional Distributions

• Definition:
  $f_{X,Y}(x,y)$, $f_X(x)=\int f_{X,Y}(x,y)dy$, $f_{X|Y}(x|y)=f_{X,Y}(x,y)/f_Y(y)$.
• Contribution:
  Required for multi-asset models, correlation, and copula modeling.

8. Statistical Inference and Estimation

• Parameter Estimation:
  MLE, method of moments.
• Hypothesis Testing:
  $H_0$ vs $H_1$, p-values, confidence intervals.
• Contribution:
  Used in model calibration, volatility estimation, and risk management.

9. Regression and Time Series

• Linear Regression:
  $Y={\beta }_0+{\beta }_{1}X+ϵ$.
• Time Series Models:
  AR, MA, ARMA, GARCH.
• Contribution:
  Empirical modeling of asset returns, volatility, and risk factors.

#MATHS

1. Set Theory and Mathematical Logic

• Axiomatic Foundation:
  Sets, functions, relations, logical connectives, quantifiers.
• σ-algebra:
  Collection of subsets closed under countable unions and complements.
• Contribution:
  Underpins the construction of probability spaces, filtrations, and measurable functions.

2. Real Analysis and Measure Theory

• Measure:
  A function $\mu$ assigning non-negative values to sets, with countable additivity.
• Lebesgue Integral:
  $\int fd\mu$ for measurable $f$.
• Convergence Theorems:
  Monotone Convergence, Dominated Convergence.
• Contribution:
  Rigorous foundation for probability, expectation, and stochastic integration.

3. Linear Algebra

• Vector Spaces, Matrices:
  Linear systems, eigenvalues, diagonalization.
• Contribution:
  Used in multi-asset models, solving for hedging portfolios, and covariance matrices.

4. Multi-variable Calculus and ODE/PDE

• Partial Derivatives, Chain Rule:
  $\frac{\partial f}{\partial x}$, $\frac{d}{dt}f(X(t))$.
• Ordinary and Partial Differential Equations:
  $u_t+\frac{1}{2}{\sigma }^2{x}^2{u}_{xx}+rx{u}_x-ru=0$ (Black-Scholes PDE).
• Contribution:
  Derivation and solution of the Black-Scholes PDE, Feynman-Kac formula, and pricing PDEs for exotic options.

5. Combinatorics

• Counting Principles:
  Binomial coefficients, permutations, combinations.
• Contribution:
  Binomial model, path counting in discrete models.

6. Stochastic Analysis

• Stochastic Integral (Itô Integral):
  ${\int }_0^{t}f(s)dB(s)$.
• Quadratic Variation:
  $[B](t)=t$ for Brownian motion.
• Itô’s Lemma:
  $df(X_t)=f^{\prime }(X_t)d{X}_t+\frac{1}{2}{f}^{″}(X_t)d[X{]}_t$.
• Contribution:
  Foundation for stochastic calculus, SDEs, and option pricing.

7. Moment Generating Functions

• Definition:
  $M_X(t)=E[e^{tX}]$.
• Contribution:
  Used in proofs of CLT, distributional convergence, and pricing formulas.

8. Proof Theory

• Direct, Contradiction, Induction:
  Used in all rigorous derivations and theorem proofs.
• Contribution:
  Ensures logical consistency and rigor in all results.

#Finance

1. Financial Derivatives

• Definition:
  Contracts whose value depends on underlying assets (stocks, bonds, indices).
• Types:
  European/American options, forwards, futures, swaps, exotics.
• Payoff Functions:
  $C=(S_T-K{)}^{+}$ for a call option.
• Contribution:
  The objects being priced and hedged in all models.

2. Arbitrage and Law of One Price

• Arbitrage:
  Riskless profit with zero net investment.
• Law of One Price:
  Identical payoffs must have identical prices.
• Contribution:
  Motivates risk-neutral pricing and market completeness.

3. Hedging and Self-financing Portfolios

• Replication:
  Constructing a portfolio to match a derivative’s payoff.
• Self-financing:
  No cash inflows/outflows except from asset price changes.
• Contribution:
  Central to pricing and risk management.

4. Risk-neutral Valuation

• Risk-neutral Measure ($\mathbb{Q}$):
  $E^{\mathbb{Q}}[e^{-rT}f(S_T)]$ is the price of a claim $f(S_T)$.
• Contribution:
  The core pricing principle in modern finance.

5. Market Completeness

• Definition:
  Every contingent claim can be replicated.
• Contribution:
  Ensures unique pricing and hedging strategies.

6. Econometrics and Time Series

• Model Calibration:
  Estimation of volatility, drift, and other parameters from data.
• Time Series Analysis:
  Modeling and forecasting asset returns and volatility.
• Contribution:
  Empirical foundation for model selection and risk management.

Stepwise Illustration: How Each Pivot Contributes to "Stochastic Calculus for Finance"

Example: Derivation of the Black-Scholes Formula

Step 1: Model Asset Price as a Stochastic Process

• #STATS/Probability: Define $S_t$ as a stochastic process, typically geometric Brownian motion.
• #MATHS: Use measure theory to define the probability space; real analysis for path properties.
• #Finance: $S_t$ is the price of the underlying asset.

Step 2: Construct a Self-financing Portfolio

• #Finance: Portfolio consists of ${\mathrm{\Delta }}_t$ shares of stock and $B_t$ in the money market.
• #MATHS: Linear algebra for portfolio value; calculus for changes in value.

Step 3: Apply Itô’s Lemma

• #MATHS: Itô’s formula (stochastic calculus) to $V(t,S_t)$, the value of the option.
• #STATS/Probability: Martingale property under risk-neutral measure.

Step 4: Derive the Black-Scholes PDE

• #MATHS: Collect terms to form the PDE: $V_t+rS{V}_S+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}-rV=0$.
• #Finance: Boundary/terminal conditions from option payoff.

Step 5: Solve the PDE

• #MATHS: Use analytical or numerical methods (PDE theory, real analysis).
• #STATS/Probability: Solution involves the cumulative normal distribution.

Step 6: Interpret and Use the Formula

• #Finance: Price and hedge options, compute Greeks, manage risk.
• #Econometrics: Calibrate model parameters from market data.

Summary Table: Cross-field Prerequisites and Their Roles

In summary:

• #STATS and Probability theory provides the language and theorems for randomness, expectation, and convergence.
• #MATHS supplies the rigorous structure, logic, and analytical tools for modeling, integration, and solving equations.
• #Finance gives the economic context, practical motivation, and application for pricing, hedging, and managing risk.

All three are essential, and their interplay is what enables the full development and application of stochastic calculus for finance.

From Book Stoc Cal for Finance 2

Certainly! Below is a structured enumeration of the prerequisite knowledges for "Stochastic calculus for finance," organized by the three pivots: #STATS and Probability theory, #MATHS, and #Finance. For each, I provide a ranked order, explain the axiomatic definitions and key theorems, and show how each subject area contributes to the pivotal parts of "Stochastic calculus for finance" (SCFF). Where appropriate, I include step-by-step derivations and equations.

#STATS and Probability theory

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1. Probability Spaces and Random Variables

Axiomatic Definition:
A probability space is a triple $(\mathrm{\Omega },\mathcal{F},P)$ where:

• $\mathrm{\Omega }$: sample space (set of all possible outcomes)

• $\mathcal{F}$: σ-algebra of events (subsets of $\mathrm{\Omega }$)

• $P$: probability measure ($P: \mathcal{F} \to $, countably additive, $P(\mathrm{\Omega })=1$)

A random variable $X:\mathrm{\Omega }\to \mathbb{R}$ is measurable: $X\le x\in \mathcal{F}$ for all $x$.

Contribution to SCFF:

• Foundation for all stochastic processes, Brownian motion, and martingales.

• Underpins the definition of filtrations, conditional expectations, and measure changes.

2. Distributions and Expectation

Theory:

• Discrete: $P(X=x_i)=p_i$, $\sum p_i=1$

• Continuous: $P(a<X<b)={\int }_a^b{f}_X(x),dx$, where $f_X$ is the pdf.

• Expectation: $E[X]=\sum x_i{p}_i$ (discrete), $E[X]=\int x{f}_X(x),dx$ (continuous)

Contribution to SCFF:

• Used in pricing (expected discounted payoff), risk-neutral valuation, and in defining martingales.

3. Conditional Probability and Expectation

Axiomatic Definition:

• $P(A|B)=\frac{P(A\cap B)}{P(B)}$

• $E[X|\mathcal{G}]$: conditional expectation given σ-algebra $\mathcal{G}$, defined as the $\mathcal{G}$-measurable random variable $Y$ such that for all $A\in \mathcal{G}$, ${\int }_{A}Y,dP={\int }_{A}X,dP$.

Contribution to SCFF:

• Central to the definition of martingales, filtrations, and risk-neutral pricing.

• Used in the Feynman-Kac theorem and in the derivation of the Black-Scholes PDE.

4. Key Distributions and Theorems

Distributions:

• Binomial, Poisson, Normal, Exponential, Gamma, Beta, etc.

• Central Limit Theorem (CLT): Sums of i.i.d. random variables converge in distribution to a normal distribution.

Theorems:

• Law of Large Numbers (LLN): Sample averages converge to the expected value.

• Chebyshev’s Inequality: $P(|X-E[X]|\ge ϵ)\le \frac{\mathrm{Var}(X)}{{ϵ}^2}$

• Martingale Convergence Theorem: If $M_n$ is a martingale and bounded in $L^1$, then $M_n$ converges almost surely.

Contribution to SCFF:

• CLT: Justifies the convergence of binomial models to Brownian motion.

• Poisson: Foundation for jump processes.

• Martingale theorems: Underpin risk-neutral pricing and hedging.

5. Stochastic Processes

Definition:
A stochastic process is a collection ${X_t}_{t\in T}$ of random variables indexed by time.

Types:

• Markov process: $P(X_{t+1}|X_t,X_{t-1},\dots ,X_0)=P(X_{t+1}|X_t)$

• Martingale: $E[X_{t+1}|{\mathcal{F}}_t]=X_t$

• Stationary, independent increments (e.g., Brownian motion, Poisson process)

Contribution to SCFF:

• All asset price models are stochastic processes.

• Markov and martingale properties are essential for pricing and hedging.

6. Statistical Inference and Time Series

Theory:

• Estimation, hypothesis testing, regression, time series analysis.

Contribution to SCFF:

• Used in model calibration, parameter estimation, and empirical finance.

#MATHS

1. Set Theory and Mathematical Logic

Axiomatic Definition:

• Sets, subsets, unions, intersections, complements.

• Logic: Proofs, quantifiers, logical connectives.

Contribution to SCFF:

• Foundation for probability spaces, σ-algebras, filtrations, and rigorous proofs.

2. Measure Theory and Real Analysis

Axiomatic Definition:

• Measure: A function $\mu$ on a σ-algebra with countable additivity.

• Lebesgue integral: $\int f,d\mu$, generalizes Riemann integral.

• Convergence notions: almost surely, in probability, in $L^p$.

Contribution to SCFF:

• Probability as measure, random variables as measurable functions.

• Rigorous definition of expectation, conditional expectation, and convergence.

• Underpins the construction of Brownian motion and stochastic integrals.

3. Linear Algebra

Theory:

• Vectors, matrices, systems of equations, eigenvalues/eigenvectors.

Contribution to SCFF:

• Used in multi-asset models, solving for hedging portfolios, and market completeness (uniqueness of risk-neutral measure).

4. Combinatorics

Theory:

• Counting, binomial coefficients, permutations, combinations.

Contribution to SCFF:

• Binomial model construction, calculation of probabilities in discrete models.

5. Multi-variable Calculus, ODEs, and PDEs

Axiomatic Definition:

• Differentiation and integration in multiple variables.

• ODE: $y^{\prime }=f(y,t)$

• PDE: $u_t=\frac{1}{2}{\sigma }^2{u}_{xx}+rx{u}_x-ru$ (Black-Scholes PDE)

Contribution to SCFF:

• Itô’s formula (multi-variable chain rule for stochastic processes).

• Black-Scholes PDE and Feynman-Kac theorem.

• Solution of pricing equations for derivatives.

6. Stochastic Analysis

Theory:

• Stochastic integrals (Itô integral), quadratic variation, stochastic differential equations (SDEs).

Contribution to SCFF:

• Foundation for Itô calculus, modeling of asset prices, derivation of hedging strategies.

#Finance

1. Financial Derivatives

Definition:

• Contracts whose value depends on underlying assets (stocks, bonds, etc.).

• Types: European/American options, forwards, futures, swaps, exotics.

Contribution to SCFF:

• The objects being priced and hedged using stochastic calculus.

• Payoff functions provide boundary/terminal conditions for PDEs.

2. Arbitrage and Hedging

Theory:

• Arbitrage: Riskless profit opportunities.

• Hedging: Constructing portfolios to offset risk.

Contribution to SCFF:

• Fundamental Theorems of Asset Pricing: No-arbitrage $⟺$ existence of risk-neutral measure; completeness $⟺$ unique risk-neutral measure.

• Self-financing portfolios and replication strategies.

3. Risk-neutral Valuation

Theory:
Pricing by expected value under risk-neutral measure: