Advanced Stochastic Processes.

Explore 55 formal proofs and analytical renders within the discipline of Advanced Stochastic Processes.

Solving the SDE: Unveiling the Log-Normal Distribution for Geometric Brownian Motion

Exploring the cinematic intuition of Solving the SDE: Unveiling the Log-Normal Distribution for Geometric Brownian Motion.

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Fourier Inversion for Option Pricing: Applying the Heston Characteristic Function

Exploring the cinematic intuition of Fourier Inversion for Option Pricing: Applying the Heston Characteristic Function.

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Advanced

The Martingale Property of the Wiener Process

Exploring the cinematic intuition of The Martingale Property of the Wiener Process.

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Advanced

Itô’s Lemma: The Taylor Expansion for Stochastic Calculus

Exploring the cinematic intuition of Itô’s Lemma: The Taylor Expansion for Stochastic Calculus.

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Advanced

The Girsanov Theorem and the Change of Measure

Exploring the cinematic intuition of The Girsanov Theorem and the Change of Measure.

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Advanced

The Radon-Nikodym Derivative in Risk-Neutral Pricing

Exploring the cinematic intuition of The Radon-Nikodym Derivative in Risk-Neutral Pricing.

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Advanced

The No-Arbitrage Derivation of Put-Call Parity

Exploring the cinematic intuition of The No-Arbitrage Derivation of Put-Call Parity.

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Advanced

The Black-Scholes PDE: A Replication Argument

Exploring the cinematic intuition of The Black-Scholes PDE: A Replication Argument.

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Advanced

Delta-Neutral Hedging: The Dynamics of the Hedge Ratio

Exploring the cinematic intuition of Delta-Neutral Hedging: The Dynamics of the Hedge Ratio.

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Advanced

The Log-Normal Distribution of Geometric Brownian Motion

Exploring the cinematic intuition of The Log-Normal Distribution of Geometric Brownian Motion.

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Advanced

The Ornstein-Uhlenbeck Process: Stationary Distributions

Exploring the cinematic intuition of The Ornstein-Uhlenbeck Process: Stationary Distributions.

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Advanced

The Feller Condition and Positivity in the CIR Model

Exploring the cinematic intuition of The Feller Condition and Positivity in the CIR Model.

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Advanced

Zero-Coupon Bond Pricing in the Vasicek Framework

Exploring the cinematic intuition of Zero-Coupon Bond Pricing in the Vasicek Framework.

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Advanced

The Infinitesimal Generator of a Jump-Diffusion Process

Exploring the cinematic intuition of The Infinitesimal Generator of a Jump-Diffusion Process.

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Advanced

Lévy-Khintchine Representation for Lévy Processes

Exploring the cinematic intuition of Lévy-Khintchine Representation for Lévy Processes.

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Advanced

The Merton Jump-Diffusion Model: Pricing with Discontinuities

Exploring the cinematic intuition of The Merton Jump-Diffusion Model: Pricing with Discontinuities.

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Advanced

Characteristic Functions and the Fourier Inversion Formula

Exploring the cinematic intuition of Characteristic Functions and the Fourier Inversion Formula.

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Advanced

The Heston Model: Solving the Stochastic Volatility SDE

Exploring the cinematic intuition of The Heston Model: Solving the Stochastic Volatility SDE.

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Advanced

The Leverage Effect in Stochastic Volatility Models

Exploring the cinematic intuition of The Leverage Effect in Stochastic Volatility Models.

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Advanced

Cox Processes: The Doubly Stochastic Framework for Credit

Exploring the cinematic intuition of Cox Processes: The Doubly Stochastic Framework for Credit.

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Advanced

Intensity-Based Default Modeling: The Hazard Rate

Exploring the cinematic intuition of Intensity-Based Default Modeling: The Hazard Rate.

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Advanced

Joint Survival Probabilities and Correlation \( \rho \)

Exploring the cinematic intuition of Joint Survival Probabilities and Correlation

\rho

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Recovery Rates in Risky Bond Pricing

Exploring the cinematic intuition of Recovery Rates in Risky Bond Pricing.

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Value-at-Risk (VaR) Analytics for GBM Portfolios

Exploring the cinematic intuition of Value-at-Risk (VaR) Analytics for GBM Portfolios.

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Advanced

Hitting Times and Threshold Probabilities for Brownian Motion

Exploring the cinematic intuition of Hitting Times and Threshold Probabilities for Brownian Motion.

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Advanced

Regime-Switching Dynamics in Electricity Spot Markets

Exploring the cinematic intuition of Regime-Switching Dynamics in Electricity Spot Markets.

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Advanced

Martingale Representation Theorem for Brownian Filtrations

Exploring the cinematic intuition of Martingale Representation Theorem for Brownian Filtrations.

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Duffie-Kan Affine Term Structure Models

Exploring the cinematic intuition of Duffie-Kan Affine Term Structure Models.

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Advanced

The Greeks: Sensitivity Analysis via PDE Differentiation

Exploring the cinematic intuition of The Greeks: Sensitivity Analysis via PDE Differentiation.

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Advanced

Compound Poisson Processes and their Cumulant Generating Functions

Exploring the cinematic intuition of Compound Poisson Processes and their Cumulant Generating Functions.

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Advanced

The Feynman-Kac Theorem: Linking PDEs to Expectations

Exploring the cinematic intuition of The Feynman-Kac Theorem: Linking PDEs to Expectations.

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Advanced

Arbitrage-Free Pricing via Equivalent Martingale Measures

Exploring the cinematic intuition of Arbitrage-Free Pricing via Equivalent Martingale Measures.

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Solving the SDE: Unveiling the Log-Normal Distribution for Geometric Brownian Motion

Ito's Lemma: The Cornerstone of Stochastic Calculus

Girsanov's Theorem: Transforming Measures for Risk-Neutral Valuation

Martingales: The Non-Arbitrage Principle in Discounted Asset Prices

Put-Call Parity: A No-Arbitrage Derivation of Option Relationships

The Black-Scholes PDE: From Assumptions to Closed-Form Solutions

Option Greeks: Calculating and Interpreting Sensitivities for Hedging

Delta-Neutral Hedging: Constructing and Maintaining Riskless Portfolios

The Ornstein-Uhlenbeck Process: Deriving Solution and Mean-Reverting Dynamics

Vasicek Model: Pricing Zero-Coupon Bonds in a Stochastic Interest Rate World

The Cox-Ingersoll-Ross (CIR) Model: Ensuring Positivity and Bond Valuation

Compound Poisson Processes: Building Models for Discrete Jumps

The Lévy-Khintchine Formula: Characterizing Jump-Diffusion Processes

Regime-Switching Models: Unveiling Market Dynamics with Hidden Markov Chains

The Heston Stochastic Volatility Model: Capturing the Leverage Effect

Cox Processes: Doubly Stochastic Intensities in Credit Risk Modeling

Correlated Defaults: Quantifying Portfolio Credit Risk and Joint Survival

Intensity-Based Credit Models: Deriving Survival Probabilities Over Time

Risky Bond Pricing: Integrating Recovery Rates into Valuation Frameworks

Value at Risk (VaR): Analytical Derivation for Log-Normal Returns

Stock Price Thresholds: Probability of Doubling or Loss under GBM

The Infinitesimal Generator: Bridging SDEs and PDEs for Stochastic Processes

The Fundamental Properties of Wiener Processes

Merton's Jump-Diffusion Model: Incorporating Discontinuities in Asset Prices

Fourier Inversion for Option Pricing: Applying the Heston Characteristic Function

The Martingale Property of the Wiener Process

Itô’s Lemma: The Taylor Expansion for Stochastic Calculus

The Girsanov Theorem and the Change of Measure

The Radon-Nikodym Derivative in Risk-Neutral Pricing

The No-Arbitrage Derivation of Put-Call Parity

The Black-Scholes PDE: A Replication Argument

Delta-Neutral Hedging: The Dynamics of the Hedge Ratio

The Log-Normal Distribution of Geometric Brownian Motion

The Ornstein-Uhlenbeck Process: Stationary Distributions

The Feller Condition and Positivity in the CIR Model

Zero-Coupon Bond Pricing in the Vasicek Framework

The Infinitesimal Generator of a Jump-Diffusion Process

Lévy-Khintchine Representation for Lévy Processes

The Merton Jump-Diffusion Model: Pricing with Discontinuities

Characteristic Functions and the Fourier Inversion Formula

The Heston Model: Solving the Stochastic Volatility SDE

The Leverage Effect in Stochastic Volatility Models

Cox Processes: The Doubly Stochastic Framework for Credit

Intensity-Based Default Modeling: The Hazard Rate

Joint Survival Probabilities and Correlation \( \rho \)

Recovery Rates in Risky Bond Pricing

Value-at-Risk (VaR) Analytics for GBM Portfolios

Hitting Times and Threshold Probabilities for Brownian Motion

Regime-Switching Dynamics in Electricity Spot Markets

Martingale Representation Theorem for Brownian Filtrations

Duffie-Kan Affine Term Structure Models

The Greeks: Sensitivity Analysis via PDE Differentiation

Compound Poisson Processes and their Cumulant Generating Functions

The Feynman-Kac Theorem: Linking PDEs to Expectations

Arbitrage-Free Pricing via Equivalent Martingale Measures