Computational Finance: Linking Monte Carlo Simulation, Binomial Trees and Black Scholes Equation
Linking Monte Carlo Simulation with Binomial Trees and the Black Scholes model
A binomial tree uses the same process to generate a path that the Monte Carlo simulation model uses which is also the same model that the Black Scholes solution integrates over an infinitely small interval. From node zero to the terminal node in a binomial tree is a single Monte Carlo simulation path. The more steps in a tree, the larger the base of paths that can be used to price an instrument. The average value across all paths adjusted for probabilities and interest rate is the value of the instrument in Binomial trees as well as Monte Carlo simulation. The Black Scholes equation provides an elegant closed from solution to the entire process that also serves as an exact answer for certain but not all types of derivative instruments.
Build a large enough tree, run 50,000 simulations or solve the Black Scholes equation for the price of a European call and chances are that the three results will converge to the same value.
The differences in the three approaches (Trees, Black Scholes and Monte Carlo Simulation) arise on account of the resources required to run the model and the time required to build the model. The Black Scholes model is a simple closed form equation. If you have all the relevant inputs you have an answer. The only issue is that closed form solutions only exist for a limited universe of instruments. Trees require limited resources but get complex very quickly for more complex structures. A Monte Carlo as we have seen only takes a trivial effort to setup but running 100,000 simulation on a large portfolio of exotic contracts may take more time and more hardware than you may have access to.
How are the following three equations related?
1. From the Black Scholes model for the price of a European call option
2. From the Binomial trees model determining the size of the up-jump and the down-jump in a binomial tree
3. And from the Monte Carlo Simulation model
The common theme in all three is the underlying distribution and the assumption with respect to the mean return and the standard deviation.
The return and standard deviation in all three equations is given by
Standard Deviation =