Τιμολόγηση αριθμητικών ασιατικών δικαιωμάτων ευρωπαϊκού-τύπου
The aim of the paper is to develop and compare bounds on the pricing formulas for European type discrete Asian options with fixed and floating strike. Due to the dependence structure between the prices of the underlying asset, no simple exact pricing formula exists, not even in a Black-Scholes setting. In the recent literature, several approximations and bounds for the price of this type of option are proposed. In our approach we use a general technique for deriving upper (and lower) bounds for stop-loss premiums of sums of dependent random variables, as explained in Kaas et al. and Dhaene et al., and additionally, the ideas of Rogers and Shi, of Nielsen and Sandmann and of M. Vanmaele et al. The lower bound approximates very accurate the real value of the Asian option. However, the comonotonic upper bounds perform rather badly for some strike prices. Therefore, we construct sharper upper bounds based upon the traditional comonotonic bounds. Making use of the ideas of Rogers and Shi (1995), the first upper bound is obtained as the comonotonic lower bound plus an error term. Next this bound is refined by making the error term dependent on the strike. Further, we study the case that the Asian option can be decomposed into two parts. One part which can be evaluated exactly and another part to which comonotonic bounds are applied. This so-called partially exact/comonotonic upper bound consists of an exact part of the option price and some improved comonotonic upper bound for the remaining part. One of the aims of this paper is to identify the currently best lower and upper bounds.We will show that the lower bounds are very close to the Monte Carlo values and that one of M. Vanmaele et al. and Tom Hoedemakers and al. techniques leads to very satisfying upper bounds, see Theorem 3.3.3. Several sets of numerical results are included.