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...
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Start with a hypothetical problem |
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Present and discuss the mathematics needed to solve the problem |
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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 RO1C Model - Discounts for Reliance on One Customer
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 - PDF
Butler Pinkerton Total Beta Model - Spreadsheet
The Dividend Discount Model:
The Dividend Discount Model (DDM) Derivation
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 - A Single Source of Uncertainty
The Derivation of the CAPM - Two Sources of Uncertainty
Market/Product Diversification and Cost of Capital Implications
The Ex-Post Cost of Capital:
The Actual and Adjusted Share Closing Price
Intel - A Case Study
Intel - Historical Share Price Data
Brownian Motion:
The Scaled Random Walk
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 Poisson Process
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:
An Introduction to the Valuation Process
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 Reflection Principle For A Brownian Motion
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:
Default Rates, Recovery Rates, and Credit Spreads
The Debt Leverage Ratio
Estimating a Credit Spread
Modeling Events:
The Binomial Distribution
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 Spot Rate Curve
The Forward Rate Curve
Bond Price and Yield To Maturity
Bond Pricing (Vasicek):
The Short Rate Process
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:
Bond Duration and Convexity
Extracting the Risk-Neutral Default Intensity
Modeling Bond Prices in Continuous-Time:
Risk-Free Bond Price, Duration and Convexity
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:
Debt Amortization - Discrete-Time
Solving for Debt IRR - Discrete-Time
Solving for Debt IRR - Continuous-Time
Perpetual Debt Valuation - Non Mean-Reverting
Perpetual Debt Valuation - Mean-Reverting
Modeling Loans:
Modeling Static Pools
Modeling Loan Portfolios
Valuing Non-Interest-Bearing Liabilities (NIBL):
An Introduction to the Process
NIBL growth rate is constant
NIBL growth rate is time-dependent
Loan Guarantees:
The Two State Guarantee Model
The Revised BSOPM - Model Basics
The Revised BSOPM - Model Calibration
The Revised BSOPM - Problem Solution
The Revised BSOPM - The Greeks
The Revised BSOPM - PDE and Proof
Modeling and Pricing for Events:
Annualized Cash Flow
Discount Rate
Enterprise Value
Solving for Event Probability
Modeling The Business Cycle:
Cyclical Revenue
Net Income and Investment
Enterprise Value
Debt Tax Shield Value
Mathematics Supplement
Return Models and Mean Reversion:
Discrete-Time Return Model Excluding Mean Reversion
The Stochastic, Mean-Reverting Short Rate
Continuous-Time Return Model Including Mean Reversion
Return model valuation multiples - Non-Banks
Return model valuation multiples - Banks
Portfolio Value Equations:
Normally-Distributed Asset Values
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:
Finding the Arbitrage
Eliminating the Arbitrage
Pricing a Credit Default Swap
The Girsanov Multiplier
The Girsanov Multiplier - A Case Study
Startups:
Discrete-Time Survival Rate Curve
Continuous-Time Survival Rate Curve
Derivative Pricing in Incomplete Markets:
Pricing Redundant Assets in a Complete Market
Minimize the Squared Replication Error
Minimize the Expected Squared Replication Error
Modeling Dividends and Other Distributions:
The Effect Of Dividends On Future Asset Price
Reconciling the Change in Asset Value Over Time
Retaining And Reinvesting Dividends
A Case Study
Technology Companies:
Modeling Technology Product Revenue
Capitalizing Research And Development Expenditures
Technology Company ROI And NPV
Technology Company Valuation
The Schurman Parabola:
A Mean-Reverting Revenue Model
A Mean-Reverting Cash Flow Model
Valuing The Debt Tax Shield
The Newton-Raphson Method of Solving Nonlinear Equations:
Solving Univariate Non-Linear Equations - Derivation and Application
Solving Univariate Non-Linear Equations - Implied Volatility
Solving Multivariate Non-Linear Equations - Derivation and Application
The Secant Method of Solving Nonlinear Equations:
Solving Univariate Non-Linear Equations - Derivation and Application
Solving Differential Equations in Finance and Economics:
The Integrating Factor Technique - Part I
The Integrating Factor Technique - Part II
Solving a Stochastic Differential Equation
Mathematical Series:
Polylogarithms of Order Zero
Polylogarithms of Order One
Polylogarithms of Order Two
Polylogarithms of Order Three
Polylogarithms of Order Four
The Fourier Series:
Derivatives and Integrals Of Trigonometric Functions
The Fourier Expansion
Case Study: Equation For A Parabola
The Exponential Integral and Incomplete Gamma Function:
The Exponential Integral - Derivation and Solution
The Exponential Integral - A Mean-Reverting Revenue Model
The Incomplete Gamma Function - Derivation and Solution
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
Mathematical Supplement - Excel VBA Toolbox
Integral Solutions:
Integral One - Option Valuation Integrals
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 Normal Distribution
Estimating Parameters for the Exponential Distribution
Multivariate Distributions:
The Bivariate Normal Distribution - No Correlation
The Bivariate Normal Distribution - Correlation
Case Study: Hedging a Loan Guarantee
Solving for Internal Rate of Return:
The Bisection Method (Slowest)
The Secant Method (Faster)
The Newton-Raphson Method (Fastest)
Ordinary Lease Squares Estimation (i.e. Linear Regression):
Univariate Ordinary Least Squares Estimator - Constant is Non-Zero
Univariate Ordinary Least Squares Estimator - Constant is Zero
Multivariate Ordinary Least Squares Estimator
Utility Functions:
Introduction to Utility Functions
Linear Utility Function
Logrithmic Utility Function
Exponential Utility Function
Quadratic Utility Function
Case Study - Insurance Premiums
Case Study - Investor Risk Aversion Coefficient
Case Study - Utility-Derived Discount Rates
Utility Option Pricing Model (UOPM) [Proprietary]
Introduction - The Two-State Asset Pricing Model
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:
Valuation in Discrete and Continuous Time
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 I: Gaussian Copula
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:
The Parabolic Probability Distribution
Modeling Random Share Price
Some Fun Math Stuff:
Integration By Parts: The Mathematics And A Simple Example
Integration By Parts: Weighted Average Revenue Life
Integration By Parts: Antiderivative Of A Natural Log
The Ratio Test For Convergence - Discrete-Time Case
The Ratio Test For Convergence - Continuous-Time Case
The Taylor Series Expansion
An Application of L'Hopital's Rule
Double Integral of a Minimum Function
Probability and Statistics:
The Calculus of the Normal Distribution
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:
Valuing Loan Guarantees
A Tale of Two Betas
Auto Lease Residual Risk