Martingality plays a crucial role in mathematical finance, in particular arbitrage-freeness of a financial model is guaranteed by the local martingale property of discounted price processes. However, in order to compute prices as conditional expectations the discounted price process has to be a true martingale. If this is not the case, the market and the fundamental (computed) prices deviate, which is interpreted as financial bubble. Moreover, if the discounted price process is a true martingale it can be used to define an equivalent change of measure. Based on general conditions in Kallsen and Shiryaev (2002), we derive explicit sufficient conditions for the true martingality of a wide class of exponentials of semimartingales. Suitably for applications, the conditions are expressed in terms of the semimartingale characteristics. We illustrate their use for stochastic volatility asset price models driven by semimartingales. Finally, we prove the well-definedness of semimartingale Libor models given by a backward construction .
We give a collection of explicit sufficient conditions for the true martingale property of a wide class of exponentials of semimartingales. We express the conditions in terms of semimartingale characteristics. This turns out to be very convenient in financial modeling in general. Especially it allows us to carefully discuss the question of well-definedness of semimartingale Libor models , whose construction crucially relies on a sequence of measure changes .
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Martingality plays a crucial role in mathematical finance, in particular arbitrage-freeness of a financial model is guaranteed by the local martingale property of discounted price processes. However, in order to compute prices as conditional expectations the discounted price process has to be a true martingale. If this is not the case, the market and the fundamental (computed) prices deviate, which is interpreted as financial bubble. Moreover, if the discounted price process is a true martingale it can be used to define an equivalent change of measure. Based on general conditions in Kallsen and Shiryaev (2002), we derive
We give a collection of
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martingality
martingale property
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Suitably for applications, the conditions are expressed
We express the conditions
D
the
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We illustrate their use for stochastic volatility asset price models driven by semimartingales. Finally, we prove the
This turns out to be very convenient in financial modeling in general. Especially it allows us to carefully discuss the question of
R
given by a backward construction
, whose construction crucially relies on a sequence of measure changes