Mathematical models for derivative securities markets
The classical Black-Scholes analysis determines a unique, continuous, trading strategywhich allows one to hedge a financial option perfectly and leads to a unique pricefor the option. It assumes, however, that there are no transaction costs involved in implementingthis strategy, and the stock market is absolutely liquid. In this work somenew 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 minimisationof the variance of the outcome at expiry subject to spending at most a given initialendowment. Since "perfect" replication is no longer possible in this case, there willalways 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 pricingrelatively easy and meaningful to an investor, who is supposed to choose a point onthe mean-deviation locus. In the limit of zero transaction costs, the problem naturallyreduces to the Black-Scholes valuation method, unlike alternative approaches basedon the utility-maximisation.The stochastic optimisation problem obtained is dealt with by means of the stochasticversion of Pontryagin's maximum principle. This technique is believed to be applied tothis kind of problem for the first time. In general the resulting free-boundary problemhas to be solved numerically, but for a small level of proportional transaction costsan asymptotic solution is possible. Regions of short term and long term dynamics areidentified and the intermediate behaviour is obtained by matching these regions. Theperturbation 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 ofMarkowitz. The dynamical rebalancing technique developed in this work proves moreefficient than the classical static approach, and allows investors to obtain portfolioswith lower levels of risk.The model presented in Part Two is an attempt to quantify the concept of liquidityand establish relations between various measures of market performance. Informationalinefficiency is argued to be the main reason for the unavailability of an asset atits equilibrium price. A mathematical model to describe the asset price behaviour togetherwith arbitrage considerations enable us to estimate the component of the bid-askspread arising from the outstanding information. The impact of the market liquidityon hedging an option with another option as well as the underlying asset itself is alsoexamined. Although in the last case uncertainty cannot be completely eliminated fromthe hedged portfolio, a unique risk-minimising strategy is found.
Year of publication: |
1998-05
|
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Authors: | Putyatin, Vladislav Evgenievich |
Subject: | HG Finance | QA Mathematics |
Saved in:
freely available
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