1. Modern Finance Option Pricing Portfolio Selection
Black-Scholes (1973), R. Merton (1973)
Portfolio Selection
single-period models: H. Markowitz (1952)
continuous-time finance: R. Merton (1971)

2. PDE approach PDE models for option pricing
Variational inequality equation
Numerical PDEs

3. PDE models for option pricing
Vanilla options
Barrier options
Asian options
Lookback options
Vasicek model
CIR model

4. Types of equations Elliptic equations Parabolic equations
Hyperbolic equations

5. Initial and boundary conditions
Three ingredients of a PDE model
PDE/solution domain/initial and boundary conditions
Three types of boundary conditions

6. Typical features of PDEs arising from option pricing
Parabolic type, but often degenerate
Unbounded solution domain
Singular in initial value

7. Criteria for boundary conditions
At infinity: no boundary conditions.
Vanilla options: Cauchy problem
Barrier options
Vasicek model
At the degenerate point: Fichera Theorem

8. Interpretation of Fichera Theorem

9. Application of Fichera Theorem
Vanilla options
CIR model
Asian options

10. A special case: lookback options

11. Asymptotic behaviors of solution at infinity and singular points
Vanilla options

12. Well-posed Problem
Existence
Uniqueness
Stability

13. Maximum principle
Heat equation on a bounded domain

14. Comparison principle

15. Uniqueness and stability

16. Applications
Black-Scholes model
Asian options and lookback options