Algorithms for Worst-Case Design and Applications to Risk Management

By Berç Rustem; Melendres Howe | Go to book overview

Chapter 8
A continuous minimax strategy for options hedging

In this chapter, we consider how minimax can provide a robust hedging strategy for written call options. The contingent nature of the liability behind an option makes it important to address the management of this kind of liability. We formulate a minimax hedging strategy that minimizes the effect of a predefined worst-case scenario, mainly in terms of bounds on the UNDERlying source of uncertainty, that is, the future price of the asset that underlies the option. We then identify variants, including multiperiod strategies, and discuss their performance relative to a standard strategy referred to as delta hedging. We also look into an alternative formulation of the minimax strategy using an evaluation of asset returns via the Capital Asset Pricing Model, or CAPM, and discuss its performance. We present the application of minimax to bond options and discuss the complexity involved in such an application. Finally, we include numerical results from an algorithmic point of view to demonstrate the performance of the minimax algorithm when applied to the hedging problem.


1 INTRODUCTION

We present an application of continuous minimax within the context of options hedging. An option is a contract that entitles the holder to buy or sell a specific number of shares of a given stock, at or within a certain period of time, for an agreed price1. The problem of hedging the risk of an option is mainly confined to the selling of the option where the seller incurs a liability contingent on the asset underlying the option. Because of the contingent nature of the liability, the seller of the option has to adjust her expectation of the magnitude of the liability, and in some cases the timing as well, and prepare her position such that she minimizes the potential negative impact to her of such a liability. Whereas the selling of an option is risky, with a potential loss of possibly unlimited magnitude, the buying of an option is mainly regarded as nonrisky in the sense that the buyer is acquiring an insurance policy for which the buyer pays a price to have the opportunity to exercise the option and benefit from it. The only risk to the buyer is the

____________________
1
The price that a buyer of an option pays is called the premium.

-179-

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Algorithms for Worst-Case Design and Applications to Risk Management
Table of contents

Table of contents

  • Title Page *
  • Algorithms for Worst-Case Design and Applications to Risk Management *
  • Contents vii
  • Preface xiii
  • Chapter 1 - Introduction to Minimax 1
  • References 17
  • Comments and Notes *
  • Chapter 2 - A Survey of Continuous Minimax Algorithms 23
  • References *
  • Comments and Notes *
  • Chapter 3 - Algorithms for Computing Saddle Points 37
  • References *
  • Comments and Notes *
  • Chapter 4 - A Quasi-Newton Algorithm for Continuous Minimax 63
  • References *
  • Appendix A - Implementation Issues *
  • Appendix B - Motivation for the Search Direction D̄ *
  • Comments and Notes *
  • Chapter 5 - Numerical Experiments with Continuous Minimax Algorithms 93
  • References 119
  • Chapter 6 - Minimax as a Robust Strategy for Discrete Rival Scenarios 121
  • References *
  • Chapter 7 - Discrete Minimax Algorithm for Equality and Inequality Constrained Models 139
  • References *
  • Chapter 8 - A Continuous Minimax Strategy for Options Hedging 179
  • References *
  • Appendix A - Weighting Hedge Recommendations, Variant B* *
  • Appendix B - Numerical Examples 237
  • Comments and Notes 244
  • Chapter 9 - Minimax and Asset Allocation Problems 247
  • References *
  • Comments and Notes *
  • Chapter 10 - Asset/liability Management under Uncertainty 291
  • References *
  • Comments and Notes *
  • Chapter 11 - Robust Currency Management 341
  • References *
  • Appendix - Currency Forecasting *
  • Comments and Notes *
  • Index 381
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