1) Analytical solution method:
Option pricing is actually a process of solving the function expression of this stochastic process according to the known stochastic differential equation (SDE) model. This is why stochastic calculus and Ito Lemma are one of the core knowledge of financial engineering, because Ito directly tells us the new SDE satisfied by the function of a stochastic process:
\rm{d}f(t，X_{t})=\frac{\partial f}
Then, if we can find the analytical solution of this SDE, then the price of a European pathless option is the expected value discounted at the final value. This is the analytical solution of option pricing. Of course, PDE can also be used to solve it. Because of Feynman Kac theorem, PDE and conditional expectation will have the same answer.
The advantages of this method are obvious. Once there is an analytical solution, the calculation speed of the option price formula will be very fast, and the efficiency will be qualitatively improved regardless of fitting or optimization. The disadvantage of this method is also obvious, that is, for most models and most exotic options, the analytical solution may not exist.
2) Tree method
Screaming Trees's method is not called binary tree, because we will also discuss the ternary tree model, but the essential idea is exactly the same.
If you are told about the volatility of the underlying assets, you can construct the volatility of n-segment binary trees by the following formula:
u = \rm{e}^{\sigma\sqrt{T/N}}，d = \rm{e}^{-\sigma\sqrt{T/N}}
Then the option price at the initial moment is obtained by reverse calculation.
What about the trident tree? First of all, we should understand a truth, except the trident tree model that meets the following conditions (U is the upper fork, D is the lower fork, and L is the middle fork)
The rest of the trident trees are incomplete markets. Under the rest of the tree model, we can only super-copy, but can't complete the perfect hedging. This unique trident tree model has also become one of the most commonly used tree models. Some people may wonder why there is a binary tree, and some people use a more troublesome ternary tree. This is because the convergence speed of ternary tree is higher than that of binary tree.
So what are the advantages and disadvantages of the tree model? The tree model has an advantage that no continuous-time model can replace, that is, every pricing, whether American, European, path-dependent or singular, is always accompanied by an explicit hedging strategy through the principle of backward induction. In the continuous-time model, the problem of obtaining the continuous-time hedging strategy is a backward stochastic differential equation (BSDE) problem, which is often not so easy to solve, especially when the option has singular or American attributes.
On the other hand, the shortcomings of the tree model are also obvious, and the tree model can't solve the high-dimensional problem. For the problem of multiple underlying assets, especially the problem of correlation coefficient, we can only resort to other methods. In terms of speed, the convergence speed of tree model is lower than that of PDE method.
3) Partial differential equation method
Many strangers in quantitative finance have also heard of Black Scholes PDE. In fact, different stochastic models will correspond to different PDE. BS PDE is just the PDE expression of the geometric Brownian motion stochastic model of a single asset. Because we often know the return on the final expiration date of options, we use the return function as the final value condition of this PDE.
If PDE has an analytical solution, the best way is naturally to find an analytical solution. However, if the analytical solution does not exist, we must resort to numerical methods. The most commonly used numerical solution method is finite difference, that is, all variables are constructed into a grid, and then the partial derivative is estimated by difference method on the grid, and then the PDE problem is transformed into an algebraic problem. For the partial differential equation of option pricing, we will get the final value condition and boundary value condition of partial differential equation according to the nature of option. However, sometimes according to different models, we may not get a simple PDE, but PIDE (Partial Integral Differential Equation), that is, there are more integral terms in PDE. At this time, numerical integration is needed to complete numerical calculation.
There are naturally many choices for numerical problems of PDE, including finite element method and spectral method. However, the option pricing PDE itself is not as nonlinear as many physical PDEs, and the boundary is not so strange, so basically the finite difference can solve most problems.
There are three finite difference methods: explicit difference, implicit difference and staggered difference. We don't study the algorithm in depth, but there are several points: in stability, the explicit difference is conditionally stable, and the other two are unconditionally stable; In terms of computational complexity, display is the simplest, implicit is the second, and interleaving is the most complicated; As far as accuracy is concerned, explicit and implicit are the same order of magnitude. In the special case of staggered difference, when the explicit difference and implicit difference are half, that is, krank-Liang Junnuo difference, the accuracy will also be improved by an order of magnitude in time.
In addition, there are two kinds of partial differential equations in option pricing, forward and backward. The traditional BS PDE is a typical example of reverse, and its final condition is the payment function of options. The backward PDE corresponds to the forward PDE, which is no longer the PDE satisfied by the option price, but the PDE satisfied by the "price density" of this target. This "price density" is called national price, or Harold Blue price, or Green's function. And this has been introduced in my previous article.
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The shortcomings of PDE method are mainly two points: path dependence and high dimension. The PDE form of many path dependence problems is very troublesome and even impossible to express, such as Aristotle's options, such as the look back option. For high-dimensional problems, if the numerical method of PDE is upgraded from plane grid to space grid, it will not only be complicated, but also the boundary conditions will be more difficult to control. The advantage of PDE is that it is fast, and according to the numerical method of difference, there is no need to bump when calculating Greeks. For example, if the dimensions are not reduced, the finite difference of options with two assets is a cubic grid like this:
4) Monte Carlo method
Monte Carlo method is currently the most widely used method. Because the option price without the early exercise attribute is actually an expectation, we can estimate the real expectation by simulating many paths. However, the American or Bermuda option price with early exercise property is actually a stochastic optimization problem. We can use Monte Carlo based on regression, that is, least squares Monte Carlo, to estimate the conditional NPV through regression, and then use Monte Carlo to solve the present value.
Therefore, Monte Carlo method is the most common method. But the disadvantages of Monte Carlo are also obvious: because we have to simulate millions of paths, for exotic options, we have to calculate paths, and for the United States, we have to do regression, so Monte Carlo method has become synonymous with long calculation time. Fortunately, we have three acceleration methods: 1, which can reduce the simulation path by using variance reduction while keeping the variance unchanged; 2. Using multi-layer Monte Carlo method to reduce complexity; ; 3. Use GPU or supercomputer for parallel computing.
For the ordinary Monte Carlo method, the above three methods are feasible, and the speed of GPU is very significant. For variance reduction, it must be emphasized that, generally speaking, the simplest method is dual variables, followed by control variables, and then the use of conditional expectations. The most difficult thing is importance sampling, but in terms of effect and scope of application, their rankings are often just the opposite. For example, the least square Monte Carlo of American options, the most effective method to reduce variance is important sampling, and other methods have little effect.
Here, another emphasis is put on least square Monte Carlo. The flow of least square Monte Carlo is roughly as follows: firstly, the path of the target is simulated forward; Secondly, at each time node, all path values are regressed and conditional expectations are estimated until the initial time point; Finally, generally. So it is worth noting that here, if we simply use GPU cluster to accelerate, the effect is not very ideal, because path simulation is not the most time-consuming step, but regression of all paths. Even so, GPU clustering can be used to improve the regression accuracy. For example, you can classify the paths and then convert the global regression quantity into multiple local regression quantities.
Generally speaking, Monte Carlo method is the most widely used numerical method in option pricing, but it is also the slowest method. However, we can use variance reduction, complexity reduction and GPU calculation to optimize our Monte Carlo algorithm to speed up and increase accuracy.
5) Fourier method
Fourier method, also called characteristic function method, makes use of the fact that for many models, the characteristic function is often expressed explicitly, for example, the model determined by infinitely separable processes with independent increments, because in this case, we have Levy-Khintchine representation, and many processes with good fitting properties, such as variance Gamma and normal inverse Gaussian, belong to this category. The characteristic function can actually be regarded as the Fourier transform of a random variable, which is the origin of this name.
If we have an explicitly expressed characteristic function, we can get the density of the original random variable by inverse Fourier transform, and then achieve the purpose of solving the option price. Generally speaking, this method is faster than PDE method, because the speed of numerical integration is faster than that of numerical solution of differential equation. However, the shortcomings of this method are also obvious, such as path dependence and dimension problems, and the explicit expression of characteristic function is necessary.
Summary:
Only superficial things are mentioned here. I will talk about the specific and in-depth thing in the official account of WeChat: derivative finance.