University Finance Seminar - Lent Term, 2000

Friday, 3 March 2000

Held jointly by the Faculties of Mathematics and of Economics and
Politics
and the Judge Institute of Management Studies in the University of
Cambridge.


Professor Terry Lyons and Dr William Shaw will be speaking on Implied
Volatility between 4.15 and 6.15 pm in Lecture Theatre 1 at the Judge
Institute of Management Studies in the University of Cambridge.

All welcome

PROGRAMME

4.00 pm Tea

4.15 pm Professor Terry Lyons, Imperial College & Oxford University
Volatility of implied volatility

It seems common sense that the optimal portfolio designed to hedge an
exotic
derivative contract should contain freely traded call options as well as
the
underlying stock and the numeraire. However, standard stochastic
volatility
models are not complete, and so do not in general give hedging
information.

We point out that there are quite reasonable models for the "volatility
of
implied volatility" and that such models are complete providing they
satisfy
a consistency condition. Experimental data does not contradict, and in
some
parts confirms this approach. We hope that in the long term it will lead
to
effective approaches to protecting portfolios from volatility risk.

5.15 pm Dr William Shaw, Nomura, London and Balliol College, Oxford
Implied volatility and numerical instability
In any modelling process the calculation of an observable effect is a
mapping
from the space of parameters associated with the theory to the space of
observable parameters. The form of the mapping may range from an
explicit
formula through to an intensive numerical calculation. The inference of
theoretical parameters from observations represents an inversion of such
a
mapping and it is necessary to be careful to establish when the
inversion
represents a process that is both well-defined and stable.

The inverse function theorem is a critical element of the inversion
process
when the mapping is non-linear.

In option pricing an inversion of common interest is the computation of
implied volatility from market price data. This talk will explore the
consequences of the failure and near-failure of the inverse function
theorem
as applied to volatility for some simple options of interest. I will
argue
that except in very limited circumstances, the implied volatility may
well be
meaningless.


6.15 pm Drinks
Please refer any queries to 01223 339641 or by fax on 01223 339652

Mary Jane Jerkins
PA to Professor M A H Dempster
Judge Institute of Management Studies
Trumpington Road
Cambridge CB2 1AG
Tel: 01223 339641
Fax: 01223 339652

-------------------------------
NEW BOOK


Option Valuation under Stochastic Volatility : with Mathematica Code
by Alan L. Lewis /  Published February 1, 2000


Book Description

This book provides an advanced treatment of option pricing for traders,
money
managers, and researchers. Providing largely original research not
available
elsewhere, it covers the latest generation of option models where both
the
stock price and its volatility follow diffusion processes. These new
models
help explain important features of real-world option pricing, including
the
"volatility smile" pattern. The book includes Mathematica code and 37
illustrations.

Table of Contents

  Preface
Historical Volatility of the S&P 500 Index
1. Introduction and Summary of Results
    Summary of Results
    The Hedging Argument of Black and Scholes
    The Drift Cancellation and Option Sensitivities
    The Hedging Argument under Stochastic Volatility
    The Martingale Approach
    App. 1.1 Parameter Estimators for the GARCH Diffusion Model
    App. 1.2  Solutions to PDEs

2. The Fundamental Transform
    Assumptions
    The Transform-based Solution
    Some Models with Closed-form Solutions
    Analytic Characteristic Functions
    A Bond Price Analogy and Option Price Bound
    App. 2.1 Recovery of the Black and Scholes Solution
    App. 2.2 Mathematica Code for Chapter 2
   App. 2.3 General Properties of Option Prices

3. The Volatility of Volatility Series Expansion
    Assumptions
    General Steps in the  expansion
    The Two Series for a Parameterized Model
    App. 3.1  Details of the Volatility of Volatility Expansion

4. Mixing Solutions and Applications
    The Basic Mixing Solution
    Connection between Mixing Densities and the Fundamental Transform
    A Monte Carlo Application
    Arbitrary Payoff Functions
    A More General Model without Correlation

5. The Smile
     Introduction and Summary of Results
    The Symmetric Case
    The Correlated Case
    Deducing the Risk-adjusted Volatility Process from Option Prices
    App. 5.1 Calculating Volatility Moments
    App. 5.2 Working with Differential Operators in  Mathematica
    App. 5.3  Additional Mathematica Code for Chapter 5
    App. 5.4  Calculating with the Mixing Theorem

6. The Term Structure of Implied Volatility
    Deterministic Volatility
    Deterministic Volatility II: a Transform Perspective
    Stochastic Volatility≈The Eigenvalue Connection
    Example I: The Square Root Model
    Example II: The 3/2 Model
    Example III: The GARCH Diffusion Model
    A Variational Principle Method
    A Differential Equation (Dsolve) Method
    App. 6.1 Mathematica Code for Chapter 6

7. Utility-based Equilibrium Models
    A Representative Agent Economy
    Examples
    The Pure Investment Problem with a Distant Planning Horizon
    Preference Adjustments to the Volatility of Volatility Series
Expansion
    The Effect of Risk Attitudes on Option Prices

8. Duality and Changes of Numeraire
    Put-Call Duality
    Introduction to the Change of Numeraire
    Mathematics of the Change of Numeraire
    Implications for the Term Structure

9. Volatility Explosions and the
    Failure of the Martingale Pricing Formula
    Introduction
   The Feller Boundary Classifications
   Volatility Explosions I
   Volatility Explosions II. Failure of the Martingale Pricing Formula
   When Martingale Pricing Fails: Generalized Pricing Formulas
   Generalized Pricing Formulas and the Transform-based Solutions
   Generalized Pricing Formulas. Example I: the 3/2 Model
   Generalized Pricing Formulas. Example II: the CEV Model

10. Option Prices at Large Volatility
      Introduction
      Asymptotica for the Fundamental Transform

11. Solutions to Models
      The Square Root Model
      The 3/2 Model
      Geometric Brownian Motion

References
Index
Frequent Notations and Abbreviations

About the Author

Alan Lewis has been active in option valuation and financial research
for
over 20 years. He served as the Director of Research, Chief Investment
Officer, and President of the mutual fund family for a money manager
specializing in derivative securities. He has published articles in many
of
the leading financial journals including: The Journal of Business, The
Journal of Finance, The Financial Analysts Journal, and Mathematical
Finance.
He received a Ph.D. in physics from the University of California at
Berkeley
and a B.S. from Caltech.

---------------

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