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)

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

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

Types of equations Elliptic equations Parabolic equations Hyperbolic equations

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

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

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

Interpretation of Fichera Theorem

Application of Fichera Theorem Vanilla options CIR model Asian options

A special case: lookback options

Asymptotic behaviors of solution at infinity and singular points Vanilla options

Well-posed Problem Existence Uniqueness Stability

Maximum principle Heat equation on a bounded domain

Comparison principle

Uniqueness and stability

Applications Black-Scholes model Asian options and lookback options