Type of Credit: Required
Credit(s)
Number of Students
This course will be focused on several main topics covering the application of quantitative and numerical methods in Finance.
能力項目說明
The lectures will be oriented in three main categories: the underlying linear algebraic structure leading to dynamic asset pricing; the treatment of convergence especially Cauchy sequences and its various implications including fixed point theorems; and the application of ordinary differential equations to financial and economic problems.
| 教學週次Course Week | 彈性補充教學週次Flexible Supplemental Instruction Week | 彈性補充教學類別Flexible Supplemental Instruction Type |
|---|---|---|
With limited time, a selection of topics from the following shall be taught:
1 Replication/attainability, and market completeness. A linear algebraic interpretation. System of equations, solvability and invertibility and uniquess.
2 Finte one-period market model. Vector spaces. Subspaces. Linear independence and dependence. Redundant and non-redundant assets. Spanning set and basis concepts. Dimensionality and bases. Rank-Nullity theorem.
3 Characterizing no-arbitrage. Farka’s lemma. The existence of a state vector. Separting of hyperplanes and Reitz representation. Linearity and strict-positivity of a pricing
system.
4 Types of Arbitrage opportunities. Dominating trading strategies. Law of One Price. APT leading to the concepts of asset pricing based on: (i), Arrow Debreu state prices; (ii), artificial/risk-neutral probabilities; (iii), pricing kernel/stochastic discount factor. The choice of numeraire and the concept of forward measure.
5 Self-financing as an intertemporal budget constraint. Equivalent statements of self-financing. The need for properly defined stochastic integrals.
6 The problematic riskless hedge of Black-Scholes(1973). Instantaneous investment gain, and instantaneous portfolio value process. The need for a measure transform.
7 Tools for analyzing convergence. Convergence of sequences. Manipulation of limits. Subsequnces. Axiom of bound. Monotonicity and boundedness.
8 Monotonic subsequences. Divergence and subsequences. Bolzano-Weierstrass Theorem. General principle for convergence. Cauchy sequences and the concepts of completeness and closure.
9 Space of random variables. Inner product spaces. Hibert space and orthogonal projection. Conditional expectations and martingales.
10 Contractive mapping and Banach’s Fixed Point Theorem. Generation of Cauchy sequences and convergence of Fixed Point Iterations.
11 Linear first order differential equations. Existence and Uniquness theorems. Separable equations. Homogeneous equations.
12 Exact equations, sufficient and necessary condition for exactness. Integrating factors.
13 Autonomous equations, equilibrium solution, stability, criteria for stability.
14 Principle of Superposition. Wronskian. Fundamental set of solutions. Linear independence. Equivalent conditions for fundamental solutions.
15 General solutions for non-homogeneous equations with constant coefficients. Method of undetermined coefficients and variation of parameters.
16 Improper integrals, definition of Laplace transform. Some basic theorems. Inverse Laplace transform, one-to-one correspondence between function and its Laplace transform.
Mid-term Examination 50%
Final Examination 50%