Program

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Module Six

Advanced Topics

The lognormal random walk and the Black-Scholes model have been very successful in practice. Yet there is plenty of room for improvement. The benefits of new models will be discussed from theoretical, practical and commercial viewpoints. When pricing complex products it is necessary to be able to correctly value vanilla products. Modern models adopt frameworks that ensure that basic products are perfectly calibrated initially.

The models derived in earlier parts of the course are only as good as the solution. Increasingly often the problems must be solved numerically. We explain the main numerical methods, and their practical implementation.

 

Exotic Options

•    Implicit finite-difference methods including Crank-Nicolson
•    Douglas schemes
•    Richardson extrapolation
•    American-style exercise and exotic options
•    About the explicit finite-difference method for two-factor models
•    About the ADI and Hopscotch methods

 

Further Finite Difference Methods

•    The relationship between implied volatility and actual volatility in a deterministic world
•    The difference between ‘random’ and ‘uncertain’
•    How to price contracts when volatility, interest rate and dividend are uncertain
•    Non-linear pricing equations
•    Optimal static hedging with traded options
•    How non-linear equations make a mockery of calibration

 

'Advanced' Volatility Modelling in Complete Markets

•    Further Monte Carlo Methods
•    The connection to statistics
•    The Feynman-Kac theorem
•    The basic Monte Carlo algorithm, standard error and uniform variates
•    Non-uniform variates, efficiency ratio and yield
•    Co-dependence
•    Wiener path construction; Poisson path construction
•    Numerical integration of stochastic differential equations
•    Variance reduction techniques
•    Sensitivity calculations
•    Weighted Monte Carlo

 

Further Monte Carlo

•    The Poisson process for modeling jumps
•    Hedging in the presence of jumps
•    How to price derivatives when the path of the underlying can be discontinuous
•    Modeling volatility as a stochastic variable
•    How to price contracts when volatility is stochastic
•    The market price of volatility risk

 

Correlation and State Dependance

•    The meaning of correlation, in particular linear correlation
•    Understand when to use and not to use linear correlation
•    Analyzing correlation sensitivity and state dependence in finite
•    Difference, examples using simple equity exotic and structured products
•    Correlation sensitivity and state dependence in credit risk modeling
•    An uncertain correlation model for Mezzanine tranche.

 

Incomplete Markets: Jump Diffusion and Stochastic Volatility

•    The Poisson process for modeling jumps
•    Hedging in the presence of jumps
•    How to price derivatives when the path of the underlying can
be discontinuous
•     Modeling volatility as a stochastic variable
•    How to price contracts when volatility is stochastic
•    The market price of volatility risks

 

Martingales and PDEs: Which, When and Why Part II

•    Quantitative finance, computational finance, financial mathematics or mathematical finance?
•    Who does what in QuantLand?
•    The link between PDEs and probabilities
•    Problems, methods and models
•    Managing model risk