How to Price Options using a Binomial Tree (The Portfolio Approach)

How to Price Options using a Binomial Tree. The portfolio approach.

These classes are all based on the book Trading and Pricing Financial Derivatives, available on Amazon at this link. https://amzn.to/2WIoAL0
Check out our website http://www.onfinance.org/

Follow Patrick on twitter here: https://twitter.com/PatrickEBoyle

The Binomial Tree approach to options pricing involves constructing a diagram of the possible paths of the stock price over the life of the option and then calculating the present value of the final cash flows to determine the current option price.

We will start with a simplified view of the world and explain the approach, then we will slowly adjust the model to make it more and more realistic.

The Portfolio Approach
For our first example we will start with an underlying that has a price of $50, and we know at the end of three months that the underlying will be at one of two prices, either $70 or $30 (this "foreknowledge" is in fact a very big assumption, but stay with us for a while and we will improve this admittedly hugely unrealistic assumption). We will price a European Call with a strike of $50, and one month to expiration. The only additional piece of information that we need in order to solve this problem is the interest rate, which we will set at 5%.

The first step is to draw our tree putting in the spot price S0, the two ending prices of the underlying at T ST and the value of the call at expiration given the two ending prices c.

If we know with certainty (a big assumption) that there are only two outcomes for the stock price at T and we assume that no arbitrages are freely available in marketplaces (a much better assumption) we can set up a portfolio of S and the derivative c on that same underlying where there is no uncertainty about valuations at T maturity.

The portfolio will be some amount (delta) of S and short one call option c.

If we set the two portfolios as equal and solve for delta 
The portfolio is riskless if there is a value for delta where the two portfolios have an identical value at maturity in all possible scenarios. In either case above, the portfolio at expiration is worth $15. Because this portfolio is riskless we can discount it at the risk free rate (5%) for one months (1/12 of a year) to find the present value of the portfolio. 

So far, we have found the interesting result that if we know the two next possible steps in an underlying assets price and we know the risk free interest rate we can price a derivative. The only problem we have is that our first assumption is quite unrealistic, but as you will see, we can keep working with this approach and make more reasonable assumptions as the chapter progresses.
Notation

As we move forward with binomial valuations, we will always be assuming a portfolio at each node knowing that some value for delta makes the portfolios equivalent at time T. It is important to note that we are not valuing the option in absolute terms. We are calculating its value as implied by the price and volatility of the underlying and the risk free rate. The probabilities of up and down movements are already incorporated in these prices and we don’t need to take them into account again when pricing the option which is based on the stock. All of our methods of valuing derivatives share this approach. People's expected returns for underlyings are irrelevant in this calculation, as all we are saying is that assuming the price for the underlying is X, then Y is the only fair price for the option, any other price would allow for arbitrage opportunities between the price of the underlying and the derivative.

Watch tomorrows video to learn the risk neutral approach to pricing binomial trees.

Видео How to Price Options using a Binomial Tree (The Portfolio Approach) канала Patrick Boyle