Interactive pricing tool for American Barrier options:
Please be patient as it can be extremely slow in the browser.
The graphed result of trinomial price simulation is interactive.
American Barrier Options are a type of option contract that can be exercised at any point before expiry. These options include barriers that either activate them (knock-in) or render them worthless (knock-out) when the price of the underlying asset crosses a specified level.
These Options are difficult to price accurately because the optimal exercise strategy must be continuously evaluated. Since barrier options pricing also depends on the crossing of the barrier, path dependence must be monitored at all times, significantly increasing computational complexity.
I used a trinomial tree to model the possible movements of the underlying asset's price. This involved a discretisation of time (lest my computer explode). at each time step the price moves in 3 directions and creates the possible prices for the next time step. This is an improvement on binomial tree modeling because of better granularity on price behaviour. The price is modeled as moving either up, down, or nowhere. The size of the price movements in each direction, as well as their probabilities, are chosen to approximate the mean and variance of a distribution modeled by Geometric Brownian Motion (GBM), which is commonly used to represent asset price behavior. There are a few ways to choose these values, I found mine in a book called 'Option Pricing Formulas' by Espen Gaarder Haug. The probabilities are chosen to reflect the weighted likelihood of price movements in a risk-neutral world where all investors expect a return equal to the risk-free rate.
At each node, the price of the option depends on the discounted expected values from the next time step, accounting for the possibility of early exercise if it results in a higher value than continuing to hold the option.
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