The models and articles in this section are valuation-related and deal with topics in finance and economics that happened to interest us at the time. The vast majority of the books and articles that we have come across that deal with mathematical finance are written at the PhD level and include little or nothing in the way of practical examples as to use of the material expounded upon. To rectify this problem we have tried to consistently apply a format where we...

Schurman XGGM Model - Extended Gordon Growth Model in Continuous Time

Schurman CFESC Model - Cash Flow Model for Early Stage Companies

Schurman FLPERP Model - Giving a Perpetuity a Finite Life

Schurman MINTA Model - Minority Interests - Base Case Model

Schurman MINTB Model - Minority Interests - Incorporating a Stochastic Payout Ratio

The Schurman Vector - Modeling Stochastic Paths That Exhibit Mean Reversion

The Pizzeria at the Base of a Volcano

Modeling Fixed Assets, Depreciation Expense and Capital Expenditures Over Time

Butler Pinkerton Total Beta Model - Spreadsheet

Recasting The DDM as a Return Model

Retaining and Reinvesting Free Cash Flow

Adjusting Dividend Yield For Retained Free Cash Flow

Mid-Term Discounting

The Derivation of the CAPM - Two Sources of Uncertainty

Market/Product Diversification and Cost of Capital Implications

Intel - A Case Study

Intel - Historical Share Price Data

Limiting Distribution of a Scaled Symmetric Random Walk

An Introduction to Stochastic Calculus

The Brownian Bridge - Part I: Base Equations

The Brownian Bridge - Part II: Random Path Simulation

The Compensated Poisson Process

A Jump Diffusion Model for Stock Price

A Jump Diffusion Option Pricing Model

Extracting Jump Intensity from the DDM

Combining Diffusion and Jump Size Variances

Valuing a One-Period Call Option Via Partial Differential Equations

Valuing a One-Period Call Option Via Risk-Neutral Probabilities

Deriving the Black-Scholes Equation Via Partial Differential Equations

Deriving the Black-Scholes Equation Via Risk-Neutral Probabilities

Approximating the Black-Scholes Equation Via Finite Differences

Option Value Via A Binomial Tree

Call Option Payoff Distribution

Put Option Payoff Distribution

The Solution to the BSOPM Integral

The Joint Distribution For A Brownian Motion And Its Maximum and Minimum

Path-Dependent Mid-Term Debt Defaults

Pricing A Barrier Call Option

The Debt Leverage Ratio

Estimating a Credit Spread

The Correlated Binomial Distribution - Part I

The Correlated Binomial Distribution - Part II

The Exponential Distribution

The Poisson Distribution

The Gamma Function

The Gamma Distribution

The Beta Function

The Beta Distribution

The Weibull Distribution - Part I

The Weibull Distribution - Part II

The Forward Rate Curve

Bond Price and Yield To Maturity

The Stochastic Discount Rate

Zero Coupon Bond Price Equations

Alternative Zero Coupon Bond Price Equations

The Yield, Forward and Swap Curves

The Coupon Bond Price Equation

Interest Rate Model Calibration to Market

Bond Price Derivatives

Modeling Random Bond Price

Bond Option Price

Bond Price Under The Euler Discretization

Extracting the Risk-Neutral Default Intensity

Solving For The Market Discount Rate of a Risk-Free Bond

Risky Bond Price, Duration and Convexity

Solving For The Market Discount Rate of a Risky Bond

Solving for Debt IRR - Discrete-Time

Solving for Debt IRR - Continuous-Time

Modeling Loan Portfolios

Part II - NIBL growth rate is time-dependent

Part II - The Revised BSOPM - Model Basics

Part III - The Revised BSOPM - Model Calibration

Part IV - The Revised BSOPM - Problem Solution

Part V - The Revised BSOPM - The Greeks

Part VI - The Revised BSOPM - PDE and Proof

Part II - Discount Rate

Part III - Enterprise Value

Solving for Event Probability

Part II - Net Income and Investment

Part III - Enterprise Value

Part IV - Debt Tax Shield Value

Mathematics Supplement

Part II - The Stochastic, Mean-Reverting Short Rate

Part III - Continuous-Time Return Model Including Mean Reversion

Normally-Distributed Portfolio Value

Lognormally-Distributed Asset Values

Lognormally-Distributed Portfolio Value

Basket Options

Reconciling The Change In Portfolio Value Over Time

Eliminating the Arbitrage

Pricing a Credit Default Swap

The Girsanov Multiplier

The Girsanov Multiplier - A Case Study

Continuous-Time Survival Rate Curve

Minimize the Squared Replication Error

Minimize the Expected Squared Replication Error

Reconciling the Change in Asset Value Over Time

Retaining And Reinvesting Dividends

A Case Study

Capitalizing Research And Development Expenditures

Technology Company ROI And NPV

Technology Company Valuation

A Mean-Reverting Cash Flow Model

Valuing The Debt Tax Shield

Solving Univariate Non-Linear Equations - Implied Volatility

Solving Multivariate Non-Linear Equations - Derivation and Application

The Integrating Factor Technique - Part II

Solving a Stochastic Differential Equation

Polylogarithms of Order One

Polylogarithms of Order Two

Polylogarithms of Order Three

Polylogarithms of Order Four

The Fourier Expansion

Case Study: Equation For A Parabola

The Exponential Integral - A Mean-Reverting Revenue Model

The Incomplete Gamma Function - Base Equation for a Mean-Reverting Process

The Incomplete Gamma Function - A Mean-Reverting Revenue Model

The Incomplete Gamma Function - A Mean-Reverting Return Model

Integral Two - Barrier Anti-Derivative

Integral Three - Barrier Probabilities

Integral Four - Barrier Integral Solutions

Integral Five - Expectation of a Diffusion Process

Integral Six - Splitting Expected Share Price Into Component Parts

Estimating Parameters for the Exponential Distribution

The Bivariate Normal Distribution - Correlation

Case Study: Hedging a Loan Guarantee

The Secant Method (Faster)

The Newton-Raphson Method (Fastest)

Univariate Ordinary Least Squares Estimator - Constant is Zero

Multivariate Ordinary Least Squares Estimator

Linear Utility Function

Logrithmic Utility Function

Exponential Utility Function

Quadratic Utility Function

Case Study - Insurance Premiums

Case Study - Investor Risk Aversion Coefficient

The Two-State Model - Option Pricing in Complete Markets

The Two-State Model - Option Pricing in Incomplete Markets

The Utility Option Pricing Model for Incomplete Markets

Common Stock Duration and Convexity

Unlevered Beta

The Mathematics of Diversification

Volatility and Discount Rate

Extracting Individual Asset Return Volatility From An Index

Using a Transition Matrix to Model Events

An Introduction to Finite Difference Methods

Reconciling Cash Flow and Share Price Volatility

Pulling Two Correlated Normally-Distributed Random Variates - Part II: Gaussian Copula (continued)

Pulling Two Correlated Normally-Distributed Random Variates - Part III: Bivariate Normal Distribution

The Cholesky Decomposition - Part I

The Cholesky Decomposition - Part II

Modeling Exponential Arrival Times

Modeling Random Share Price

Integration By Parts - Part II: Weighted Average Revenue Life

The Ratio Test For Convergence - Discrete-Time Case

The Ratio Test For Convergence - Continuous-Time Case

The Taylor Series Expansion

Double Integral of a Minimum Function

Derivatives of the Cumulative Normal Distribution Function

Standardized Normal Random Variates

The Moment Generating Function

The Mean and Variance of a Random Variate Plus/Times a Constant

The Mean and Variance of the Product of Two Normally-Distributed Random Variates

The Mean and Variance of the Exponential of a Normally-Distributed Random Variate

The Mean and Variance of the Sum of Two Normally-Distributed Random Variates

The Mean and Variance of the Product of Two Lognormally-Distributed Random Variates

The Mean and Variance of the Sum of a Normal and Exponentially-Distributed Random Variate

Approximating the Mean and Variance of the Sum of Lognormally-Distributed Random Variates

The Correlation of a Random Variate Plus a Constant with Another Random Variate

Moving the Mean of a Normal Distribution

The Triangular Distribution

The Gaussian Copula

The Gaussian Copula - A Case Study

The Importance Of Modeling Correlation - A Loan Guarantee Problem

A Tale of Two Betas

Auto Lease Residual Risk

1. | Start with a hypothetical problem |

2. | Present and discuss the mathematics needed to solve the problem |

3. | Apply the mathematics to solve the problem |

The valuation models and much of the content are original works and as such are copyrighted. The models and content are free to use but changing the name or expropriating the work as one's own is strictly prohibited. The valuation models available here are fairly simple. We do maintain an entire library of valuation models for banking, tech companies, loan guarantees, complex derivatives, etc.

**Proprietary Valuation Models:**

Schurman XGGM Model - Extended Gordon Growth Model in Continuous Time

Schurman CFESC Model - Cash Flow Model for Early Stage Companies

Schurman FLPERP Model - Giving a Perpetuity a Finite Life

Schurman MINTA Model - Minority Interests - Base Case Model

Schurman MINTB Model - Minority Interests - Incorporating a Stochastic Payout Ratio

The Schurman Vector - Modeling Stochastic Paths That Exhibit Mean Reversion

The Pizzeria at the Base of a Volcano

Modeling Fixed Assets, Depreciation Expense and Capital Expenditures Over Time

**Commentary on the Valuation Models of Others:**

Butler Pinkerton Total Beta Model - Spreadsheet

**The Dividend Discount Model:**

Recasting The DDM as a Return Model

Retaining and Reinvesting Free Cash Flow

Adjusting Dividend Yield For Retained Free Cash Flow

Mid-Term Discounting

**The Ex-Ante Cost of Capital:**

The Derivation of the CAPM - Two Sources of Uncertainty

Market/Product Diversification and Cost of Capital Implications

**The Ex-Post Cost of Capital:**

Intel - A Case Study

Intel - Historical Share Price Data

**Brownian Motion:**

Limiting Distribution of a Scaled Symmetric Random Walk

An Introduction to Stochastic Calculus

The Brownian Bridge - Part I: Base Equations

The Brownian Bridge - Part II: Random Path Simulation

**Jump Diffusion:**

The Compensated Poisson Process

A Jump Diffusion Model for Stock Price

A Jump Diffusion Option Pricing Model

Extracting Jump Intensity from the DDM

Combining Diffusion and Jump Size Variances

**The Mathematics of Option Valuation:**

Valuing a One-Period Call Option Via Partial Differential Equations

Valuing a One-Period Call Option Via Risk-Neutral Probabilities

Deriving the Black-Scholes Equation Via Partial Differential Equations

Deriving the Black-Scholes Equation Via Risk-Neutral Probabilities

Approximating the Black-Scholes Equation Via Finite Differences

Option Value Via A Binomial Tree

Call Option Payoff Distribution

Put Option Payoff Distribution

The Solution to the BSOPM Integral

**The Mathematics of Barriers:**

The Joint Distribution For A Brownian Motion And Its Maximum and Minimum

Path-Dependent Mid-Term Debt Defaults

Pricing A Barrier Call Option

**Credit Ratings:**

The Debt Leverage Ratio

Estimating a Credit Spread

**Modeling Events:**

The Correlated Binomial Distribution - Part I

The Correlated Binomial Distribution - Part II

The Exponential Distribution

The Poisson Distribution

The Gamma Function

The Gamma Distribution

The Beta Function

The Beta Distribution

The Weibull Distribution - Part I

The Weibull Distribution - Part II

**The Yield Curve:**

The Forward Rate Curve

Bond Price and Yield To Maturity

**Bond Pricing (Vasicek):**

The Stochastic Discount Rate

Zero Coupon Bond Price Equations

Alternative Zero Coupon Bond Price Equations

The Yield, Forward and Swap Curves

The Coupon Bond Price Equation

Interest Rate Model Calibration to Market

Bond Price Derivatives

Modeling Random Bond Price

Bond Option Price

Bond Price Under The Euler Discretization

**Bond Pricing Tools:**

Extracting the Risk-Neutral Default Intensity

**Modeling Bond Prices in Continuous-Time:**

Solving For The Market Discount Rate of a Risk-Free Bond

Risky Bond Price, Duration and Convexity

Solving For The Market Discount Rate of a Risky Bond

**Modeling Debt:**

Solving for Debt IRR - Discrete-Time

Solving for Debt IRR - Continuous-Time

**Modeling Loans:**

Modeling Loan Portfolios

**Valuing Non-Interest-Bearing Liabilities (NIBL):**

Part II - NIBL growth rate is time-dependent

**Loan Guarantees:**

Part II - The Revised BSOPM - Model Basics

Part III - The Revised BSOPM - Model Calibration

Part IV - The Revised BSOPM - Problem Solution

Part V - The Revised BSOPM - The Greeks

Part VI - The Revised BSOPM - PDE and Proof

**Modeling and Pricing for Events:**

Part II - Discount Rate

Part III - Enterprise Value

Solving for Event Probability

**Modeling The Business Cycle:**

Part II - Net Income and Investment

Part III - Enterprise Value

Part IV - Debt Tax Shield Value

Mathematics Supplement

**Return Models and Mean Reversion:**

Part II - The Stochastic, Mean-Reverting Short Rate

Part III - Continuous-Time Return Model Including Mean Reversion

**Portfolio Value Equations:**

Normally-Distributed Portfolio Value

Lognormally-Distributed Asset Values

Lognormally-Distributed Portfolio Value

Basket Options

Reconciling The Change In Portfolio Value Over Time

**Risk-Neutral Pricing:**

Eliminating the Arbitrage

Pricing a Credit Default Swap

The Girsanov Multiplier

The Girsanov Multiplier - A Case Study

**Startups:**

Continuous-Time Survival Rate Curve

**Derivative Pricing in Incomplete Markets:**

Minimize the Squared Replication Error

Minimize the Expected Squared Replication Error

**Modeling Dividends and Other Distributions:**

Reconciling the Change in Asset Value Over Time

Retaining And Reinvesting Dividends

A Case Study

**Technology Companies:**

Capitalizing Research And Development Expenditures

Technology Company ROI And NPV

Technology Company Valuation

**The Schurman Parabola:**

A Mean-Reverting Cash Flow Model

Valuing The Debt Tax Shield

**The Newton-Raphson Method of Solving Nonlinear Equations:**

Solving Univariate Non-Linear Equations - Implied Volatility

Solving Multivariate Non-Linear Equations - Derivation and Application

**The Secant Method of Solving Nonlinear Equations:**

**Solving Differential Equations in Finance and Economics:**

The Integrating Factor Technique - Part II

Solving a Stochastic Differential Equation

**Mathematical Series:**

Polylogarithms of Order One

Polylogarithms of Order Two

Polylogarithms of Order Three

Polylogarithms of Order Four

**The Fourier Series:**

The Fourier Expansion

Case Study: Equation For A Parabola

**The Exponential Integral:**

The Exponential Integral - A Mean-Reverting Revenue Model

**The Incomplete Gamma Function:**

The Incomplete Gamma Function - Base Equation for a Mean-Reverting Process

The Incomplete Gamma Function - A Mean-Reverting Revenue Model

The Incomplete Gamma Function - A Mean-Reverting Return Model

**Integral Solutions:**

Integral Two - Barrier Anti-Derivative

Integral Three - Barrier Probabilities

Integral Four - Barrier Integral Solutions

Integral Five - Expectation of a Diffusion Process

Integral Six - Splitting Expected Share Price Into Component Parts

**Maximum Likelihood Estimation:**

Estimating Parameters for the Exponential Distribution

**Multivariate Distributions:**

The Bivariate Normal Distribution - Correlation

Case Study: Hedging a Loan Guarantee

**Solving for Internal Rate of Return:**

The Secant Method (Faster)

The Newton-Raphson Method (Fastest)

**Ordinary Lease Squares Estimation (i.e. Linear Regression):**

Univariate Ordinary Least Squares Estimator - Constant is Zero

Multivariate Ordinary Least Squares Estimator

**Utility Functions:**

Linear Utility Function

Logrithmic Utility Function

Exponential Utility Function

Quadratic Utility Function

Case Study - Insurance Premiums

Case Study - Investor Risk Aversion Coefficient

**Utility Option Pricing Model (UOPM) [Proprietary]**

The Two-State Model - Option Pricing in Complete Markets

The Two-State Model - Option Pricing in Incomplete Markets

The Utility Option Pricing Model for Incomplete Markets

**The Practitioners Toolbox:**

Common Stock Duration and Convexity

Unlevered Beta

The Mathematics of Diversification

Volatility and Discount Rate

Extracting Individual Asset Return Volatility From An Index

Using a Transition Matrix to Model Events

An Introduction to Finite Difference Methods

Reconciling Cash Flow and Share Price Volatility

**The Monte Carlo Toolbox:**

Pulling Two Correlated Normally-Distributed Random Variates - Part II: Gaussian Copula (continued)

Pulling Two Correlated Normally-Distributed Random Variates - Part III: Bivariate Normal Distribution

The Cholesky Decomposition - Part I

The Cholesky Decomposition - Part II

Modeling Exponential Arrival Times

**Rolling Our Own Probability Distribution:**

Modeling Random Share Price

**Some Fun Math Stuff:**

Integration By Parts - Part II: Weighted Average Revenue Life

The Ratio Test For Convergence - Discrete-Time Case

The Ratio Test For Convergence - Continuous-Time Case

The Taylor Series Expansion

Double Integral of a Minimum Function

**Probability and Statistics:**

Derivatives of the Cumulative Normal Distribution Function

Standardized Normal Random Variates

The Moment Generating Function

The Mean and Variance of a Random Variate Plus/Times a Constant

The Mean and Variance of the Product of Two Normally-Distributed Random Variates

The Mean and Variance of the Exponential of a Normally-Distributed Random Variate

The Mean and Variance of the Sum of Two Normally-Distributed Random Variates

The Mean and Variance of the Product of Two Lognormally-Distributed Random Variates

The Mean and Variance of the Sum of a Normal and Exponentially-Distributed Random Variate

Approximating the Mean and Variance of the Sum of Lognormally-Distributed Random Variates

The Correlation of a Random Variate Plus a Constant with Another Random Variate

Moving the Mean of a Normal Distribution

The Triangular Distribution

The Gaussian Copula

The Gaussian Copula - A Case Study

The Importance Of Modeling Correlation - A Loan Guarantee Problem

**Published Articles:**

A Tale of Two Betas

Auto Lease Residual Risk