Study Option Price Based On The Entropy Tree Method

Option has become the center of financial derivatives due to its various functions and therefore studied extensively. Option pricing involves the internal stochastic factors and external innovations, which make it become a hot research topic in recent years.This thesis combines tree option price model with the entropy and relevant optimization principles and whereby analyzes several important problems of option, such as constructing the entropy tree models for option price, analyzing the Greeks of option, pricing Asian option as well as option price under the incomplete market. The main investigations and achievements are listed as follows:1. Constructing the entropy tree option price modelConsidering the financial market as the information system, the entropy tree option price model is obtained based on the maximum entropy principle and the aggregate function smoothing the payoff function of the option under the underlying assets information constrains. The essence for measuring uncertainty of the entropy is employed to derive the probability as such the entropy is introduced to the binomial tree option price model. The numerical examples are calculated to verify the effectiveness of the entropy tree model. The conclusions are in the following:(1) Entropy and moments are closely related, so it can compute the binomial tree unbiased, impartial parameters under the incomplete information.(2) To use the aggregate function to smooth the payoff function, the rate of the maximum entropy tree option price model is O (1/n).2. The option price mode considering the underlying assets history informationGiven the underlying assets history information, the smooth minimum cross-entropy tree option price model is obtained based on the minimum cross-entropy principle and controlling the position of the strike in the last layer nodes. The conclusions are as follows:(1) The smooth minimum cross-entropy tree directly embodies the function of the underlying assets history information and avoids the convergent oscillations of CRR model.(2) It will benefit the accuracy interpolation in the smooth minimum cross-entropy tree.3. Analyzing the Greeks of the entropy tree modelGreeks of the entropy tree model reflect the factors which influence the option price risk, so that it can promptly adjust the investment strategy and observe the sensitivity of the entropy tree. The conclusions are in the following:(1) It computes the Delta of the entropy tree, and proves that the Delta of the entropy tree converges to the Delta of the B-S model.(2) Sensitivity for avoiding risk of the entropy tree is not lower than that of the B-S model.4. The separation entropy tree for the exotic option priceThis thesis transforms the arithmetic average Asian options to two assets option, and applies the entropy tree to compute the two assets option. The numerical examples show the new model is accurate and easy to compute. The conclusions are in the following:(1) The parameters of the entropy tree are unbiased, impartial for the two assets option.(2) It can avoid the choice of path function parameters.(3) It builds the relation between the arithmetic average Asian options and the two assets option.5. m-nomial tree option price model based on utility function in the incomplete marketThe hedging with m-nomial tree and maximum utility function derives the option price in the incomplete market. The numerical examples show the new method is feasible and effective. The conclusions are in the following:(1) During analyzing the maximum growth rate of wealth, the utility function is obtained by the generalized entropy.(2) Combining m-nomial tree with maximum utility function, the multi-object option price model is derived in the incomplete market.