1, network analysis method
The main idea of network analysis is to deal with the stochastic process of satisfying the target assets in a decentralized way under the premise of risk neutrality, and get the price of derivatives of the target assets through dynamic programming.
At present, network analysis methods are divided into binary tree method, ternary tree method and multi-branch model. First of all, Cox, Ross and Rubinstein put forward CRR model and applied fork method to option pricing.
In the process of using CRR model, it is found that this method has the characteristics of vibration convergence, especially the estimation convergence speed of American option price is quite slow. Brin puts forward the method of accelerating binary tree, which improves the convergence speed. Broadie and Detemple put forward BSS method and BBSS method, and BSS method mainly faces binary tree method. BBSS method applies extrapolation method to BSS method. Parkinson first put forward the trident tree method, and Kamrad further deduced this method. Hull applied the trident tree method to Wa Siczek, and achieved good bidding results.
Grid analysis can be used to analyze the pricing of American options, but its vibration convergence is difficult to apply to high dimensions. Once the number of time nodes increases, the number of branches will explode exponentially. Although there are many improved models, this fundamental shortcoming is still difficult to change.
2. Finite gap method
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The main idea of finite gap method is to transform the micro gap satisfied by derivative into gap and pursue gap method repeatedly.
Brennan and Schwartz used finite difference method for option pricing for the first time. Marchuk and Shaidurov first applied Richardson's correlation extrapolation technique to the finite difference method. The finite difference method can be well applied to the prices of European options and American options, but the utility of this extrapolation method depends entirely on the expansion of a single discrete parameter. When the dimension increases, the amount of calculation is extremely large, and this problem is also difficult to overcome.
3. Monte Carlo simulation method
The main idea of Monte Carlo method is to sample in randomly distributed sample space, find the average value of samples, and replace the whole expectation with the expectation of random sample space.
Boyle first proposed to use Monte Carlo simulation method to price options. He further proposed the method of reducing variance to improve the simulation efficiency. According to the empirical analysis, Monte Carlo simulation method is very effective in demanding the value of European options. However, for American options, Monte Carlo simulation method can not directly solve the pricing problem because it needs repeated search. As for American options, Barraquand and Martineau separate the various states of the target asset price, get the probability of each path moving in each region, and use a method similar to grid analysis to solve it in reverse. Broadie, Glasserman and Jain put forward two estimation methods, and obtained two estimated values to estimate the trust region of options.