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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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
Accepted Date and Publication Date:
May 1998Made publicly available
Date Deposited: 06 Apr 2008
Last Modified: 27 Mar 2014 18:34

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