Preface Mathematical finance theory has been established as one sector of finance theory. This theory is based on probability theory and many probabilists have contributed to this field. The Black-Scholes model is a typical model for the complete market. This model is an outstanding model convenient to analyze, but it is well-known that in the real world the completeness of the market is not usually satisfied. The distribution of the log return of an asset usually has a fat tail and asymmetry, and the market is usually incomplete. Therefore we need other models for the incomplete market. The geometric Lévy process (GLP) model is one of the most important models for the incomplete market. This model is able to possess a fat tail property, an asymmetric distribution and smile/smirk properties of implied volatility. An incomplete market has many martingale measures (or risk-neutral measures) by the second fundamental theorem of mathematical finance. So we have to select a suitable martingale measure among them in order to discuss option pricing based on arbitrage theory. Many kinds of martingale measures have been proposed for this. Among them the minimal entropy martingale measure (MEMM) is the most important candidate. Among the many GLP models, the importance of the geometric stable process (GSP) model is recognized by Fama [36] and Mandelbrot [77]. Since then many researchers have studied this model. The suitable martingale measure for the GSP model had not been evident for many years, but since the existence of the MEMM for GSP model was proved and an explicit form of it was obtained, the GSP model can be applied to a wide class of problems concerning option pricing.
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