Approximation of Girsanov’s measure with logarithmic returns in the case of heavy-tailed distributions

The article is devoted to further development of the topic of applying the extended Girsanov principle for heavy-tailed distributions. The extended Girsanov principle involves finding the conditional mathematical expectation of the ratio of prices of underlying assets of option contracts at the current time to prices of underlying assets at the previous time. To do this, it is necessary to choose an appropriate model that will best describe the dynamics of this price ratio. The linear return or logarithmic return is typically used as the modeling object. In the case of linear returns, the risk-neutral dynamics for the heavy-tailed distribution (Su Johnson) was obtained in [1]. In [2], the effectiveness of the found martingale measure was demonstrated in the example of option contract valuation. However, there may be a need to use logarithmic returns, which have several useful properties (non-negativity of underlying asset prices, symmetry with respect to price increases and decreases). This article derives the martingale measure for the case where the approximation of logarithmic returns is considered. In the simplest case, the approximation coincides with linear returns, and as the degree of approximation increases, it approaches logarithmic returns.