# Three pricing principles of American options

Because there is no suitable stochastic process to describe the process of underlying price volatility, there is no good pricing model for options. Until 1973, two scholars, black and Scholes, assumed that the price change process of the underlying price coincided with geometric Brownian motion, and applied it to European option pricing. From then on, the option market developed rapidly. However, unlike European options, American options have no explicit expression, and can only be solved numerically. From the perspective of numerical solution, it is mainly divided into three categories: grid analysis method, finite difference method and Monte Carlo simulation method.

(1) Network analysis
The main idea of network analysis method is: under the premise of risk neutrality, the stochastic process of the target asset matching is discretized, and then the dynamic programming is used to solve the problem to obtain the price of the underlying asset derivatives.

At present, network analysis methods can be divided into binary tree method, trigeminal tree method and more branch models. Firstly, Cox, Ross and Rubinstein proposed the CRR model, and applied the binomial tree method to option pricing.

In the process of using the CRR model, it is found that this method has the characteristics of oscillatory convergence, especially in the estimation of American option price. The accelerated binary tree method is proposed by Breen, which has improved the convergence rate. Broadie and detemple put forward BSS method and BBSs method. BSS method mainly aims at binary tree method, while BBSs method applies extrapolation method to BSS method. Parkinson first proposed the trigeminal tree method, which was further deduced by Kamradt. Hull applied the trigeminal tree method to Vasicek and obtained good estimation results.

The grid analysis method can stop Pricing American options, but its oscillatory convergence makes it difficult to apply to high-dimensional situations. Once the number of time nodes increases, the number of branches of the tree will show an exponential explosion state. Although there are many improved models, it is still difficult to change this basic defect.

(2) Finite difference method
The main idea of the finite difference method is to transform the differential equation satisfied by derivatives into a difference equation, and then stop evaluating the difference method by iteration.

Brennan and Schwartz first applied the finite difference method to option pricing. Marchuk and shaidurov first applied Richardson's extrapolation technique to the finite difference method. The finite difference method can be well applied to the pricing of European options and American options, but the effectiveness of the extrapolation method completely depends on the expansion of a single discrete parameter. When the dimension increases, the amount of calculation is very large, and the problem is difficult to restrain.

(3) Monte Carlo simulation
The main idea of Monte Carlo method is: Stop Sampling in a random sample space, then calculate the uniform value of the sample, and replace the overall expectation with the random space sample expectation.

Boyle first proposed to stop pricing options by Monte Carlo simulation. He further proposed the use of variance reduction to improve the efficiency of imitation. According to the empirical analysis, Monte Carlo simulation method is very effective for European option pricing. But as for American option, it needs backward iterative search, which makes Monte Carlo simulation method can not directly deal with the pricing problem. On American options,

Barraquand and Martineau separated each state of the underlying asset price, obtained the probability of each path moving in each region, and then used the method similar to the grid analysis method to stop the reverse solution. Broadie, glasserman, Jain put forward two methods to estimate the trust region of options.

The above two methods deal with the numerical solution of American option to a certain extent, but in practice, the effect is not very ideal. Longstaff and Schwartz proposed the least square Monte Carlo simulation method for American option solution. Because of the effectiveness of this method, American option has become the most popular pricing method.