# Mathematical models for derivative securities markets

Putyatin, Vladislav Evgenievich
(1998)
Mathematical models for derivative securities markets.
*
University of Southampton, Department of Mathematics,
Doctoral Thesis
*, 108pp.

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## Description/Abstract

The classical Black-Scholes analysis determines a unique, continuous, trading strategy

which allows one to hedge a financial option perfectly and leads to a unique price

for the option. It assumes, however, that there are no transaction costs involved in implementing

this strategy, and the stock market is absolutely liquid. In this work some

new results are obtained to accommodate costs of hedging, which occur in practice,

and market imperfections into the option pricing framework.

In Part One transaction charges are dealt with by means of the mean-variance technique,

originally developed by Markowitz. This approach is based on the minimisation

of the variance of the outcome at expiry subject to spending at most a given initial

endowment. Since "perfect" replication is no longer possible in this case, there will

always be an unavoidable element of risk associated with writing an option. Therefore,

the option price is now not unique. A mean-variance approach makes option pricing

relatively easy and meaningful to an investor, who is supposed to choose a point on

the mean-deviation locus. In the limit of zero transaction costs, the problem naturally

reduces to the Black-Scholes valuation method, unlike alternative approaches based

on the utility-maximisation.

The stochastic optimisation problem obtained is dealt with by means of the stochastic

version of Pontryagin's maximum principle. This technique is believed to be applied to

this kind of problem for the first time. In general the resulting free-boundary problem

has to be solved numerically, but for a small level of proportional transaction costs

an asymptotic solution is possible. Regions of short term and long term dynamics are

identified and the intermediate behaviour is obtained by matching these regions. The

perturbation analysis of the utility-maximisation approach is also revised in this work,

and amendments are obtained.

In addition, the maximum principle is applied to the Portfolio Selection problem of

Markowitz. The dynamical rebalancing technique developed in this work proves more

efficient than the classical static approach, and allows investors to obtain portfolios

with lower levels of risk.

The model presented in Part Two is an attempt to quantify the concept of liquidity

and establish relations between various measures of market performance. Informational

inefficiency is argued to be the main reason for the unavailability of an asset at

its equilibrium price. A mathematical model to describe the asset price behaviour together

with arbitrage considerations enable us to estimate the component of the bid-ask

spread arising from the outstanding information. The impact of the market liquidity

on hedging an option with another option as well as the underlying asset itself is also

examined. Although in the last case uncertainty cannot be completely eliminated from

the hedged portfolio, a unique risk-minimising strategy is found.

Item Type: | Thesis (Doctoral) |
---|---|

Subjects: | H Social Sciences > HG Finance Q Science > QA Mathematics |

Divisions: | University Structure - Pre August 2011 > School of Mathematics |

ePrint ID: | 50648 |

Date Deposited: | 06 Apr 2008 |

Last Modified: | 27 Mar 2014 18:34 |

URI: | http://eprints.soton.ac.uk/id/eprint/50648 |

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