Remember: there are 3 basic ways to price options: closed formulas (like Black-Scholes), some kind of trees or Montecarlo simulations. During this class, we reviewed how to price a vanilla call with a binomial tree.
We assume a very simple world where the price of the underlying asset can only have two potential scenarios: up or down.
- First we expand the price of the underlying asset in the tree assuming a certain percentage for upward and downward movement.
- Then we price the option at the end of the tree. Given that we are at maturity, we can basically apply the formula for the payoff of the option (the outcome of exercising the option).
- We form a riskless portfolio by buying Delta shares and selling an option (a replicating portfolio). We calculate Delta making the portfolio riskless: it does not matter where the share goes, the value of my portfolio will always be the same.
- If the portfolio is riskless, we can bring it to present value using a risk-free rate. Now we know the value of the portfolio today and the value of the Delta shares today, so we can solve for the price of the option.
One important feature is that it does not matter what the real world probabilities are as we are working in a risk neutral world. In fact, a different way to price the option would be to solve for the expected value of the option using risk-neutral probabilities. The expected price for the share using this risk-neutral probabilities will be the forward.
We extended this methodology by increasing the number of branches in the tree.
Have a look at the presentation here and at the examples we saw in class here.
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