MAFS525 – Computational Methods for Pricing Structured Products
Course objective
This course introduces the computational techniques in pricing of structured products.
First we discuss the three common classes of numerical methods: lattice tree methods,
finite difference methods and Monte Carlo simulation methods. These computational
techniques are then applied to pricing structured products that are commonly traded in the
financial markets. The classes of structured products include: accumulators, convertible
bonds, variance products, equity-linked variable annuities, etc.
Instructor
Professor Kwok Yue-Kuen, Office: Room 3445; Office phone: 2358-7418
E-mail: maykwok@ust.hk
Textbooks/ references
 Yue Kuen KWOK, Mathematical Models of Financial Derivatives, second
edition, Springer Verlag, 2008 (Chapter 6).
 Domingo TAVELLA and Curt RANDALL, Pricing Financial Instruments, John
Wiley & Sons, Inc., 2000.
 Rudiger SEYDEL, Tools for computational finance, second edition, Springer
Verlag (2003).
 Claudio ALBANESE and Giuseppe CAMPOLIETI, Advanced derivatives pricing
and risk management, Academic Press, 2006.
 Daniel J. DUFFY, Finite difference methods in financial engineering, John Wiley
and Sons, 2006.
Course content
1. Lattice tree methods
1.1 Binomial option pricing models revisited
 Risk neutral valuation principle
 Multi-period extension
 Dynamic programming procedure
 Estimating delta and Greek letters
 Discrete dividend models
 Pricing of path dependent derivatives
1.2 Trinomial schemes
 Discounted expectation approach
 Multi-state extension – Kitchken-Kamrad’s approach
1.3 Forward shooting grid algorithms (strongly path dependent options)
 Cumulative Parisian feature
 Call options with strike reset feature
 Floating strike arithmetic averaging call
 Accumulators
2. Finite difference methods
2.1 Discretization of the Black-Scholes equation
 Explicit schemes
 Implicit Crank-Nicolson scheme
2.2 Pricing of American options
 Front fixing method
 Point relaxation method
2.3 Numerical approximation of auxiliary conditions
 Barrier options
 Lookback options
 Interest rate models
2.4 Properties of numerical solutions
 Truncation errors and order of convergence
 Numerical stability
 Spurious oscillations
2.5 Jump-diffusion models
 Poisson process as counting process
 Partial integral-differential equations
 Numerical schemes
2.6 Multi-state models
 Alternating-Direction-Implicit methods
 Factorization of operators
 Hopscotch method
 Fractional steps
 Treatment of boundary conditions
3. Monte Carlo simulation
3.1 General formulation of Monte Carlo procedure
 Expected value and variance of estimate
 Multistate extension – correlated random samples
 Computational efficiencies
 Calculating the greeks
3.2 Variance reduction techniques
 Antithetic variates method
 Control variate method
3.3 Valuation of American options
 Method of parametrization of the early exercise boundary
 Linear regression method via basis functions
4. Structured products
4.1 Participating life insurance policies
 Product nature and pricing model
 Finite difference scheme
 Mortality risk
4.2 Variance swaps
 Product nature
 Replication by options, forward and log contract
 Mathematical formulation of pricing equations
 Finite difference solution approach
 Numerical results and pricing behavior
4.3 Convertible bonds
 Product nature
 Lattice tree algorithm
 Finite difference scheme
 Anatomy of embedded features
4.4 Hull-White interest rate models
 Analytic procedure of fitting term structure of bond prices
 Calibration of interest rate trees against market discount curves
 Extension to other interest rate models
 Pricing of interest rate products using the calibrated interest rate trees
4.5 Mortgage backed securities
 Product nature
 Finite difference schemes
 Anatomy of embedded features
Grading policy
Two sets of homework
2 x 5% = 10%
 Lattice tree methods (due on June 26)
 Finite difference methods and Monte Carlo simulation
(due on Aug 2)
Two computer assignments
 Reverse convertibles (due on July 26)
20%
 Accumulators (due on Aug 14)
20%
Two 60-minute tests
July 26
August 14
25%
25%
_____________
Total = 100%