BINOMIAL TREES
The most basic method of presenting stock price movement is a binomial tree. It is very similar to the Brownian motion process we discussed in Chapter 3 except the step size can be of an arbitrary magnitude and the probability of an up or down movement does not necessarily have to be equal.
Figure 4.1 represents a basic recombinant binomial tree, a tree where two nodes can transition to the same node, with an initial starting stock price of S0 that transitions after step 1 to either price S1 with probability p1 or S2 with probability 1 – p1. Furthermore, you can see that from step 1, the price can take a total of 3 possible values at step 2. For the sake of simplicity let us start by examining the starting branch represented by S0, S1, and S2. Given this diagram, what would be the expected or average price of the stock? From statistics, we know the answer to be the value if the event occurs times the probability of occurrence plus the value if the event does not occur times the probability of the event not occurring (equation 4.1):
FIGURE 4.1 Basic recombinant binomial tree process where at each node the price can transition to two possible states. This is a recombinant tree because two nodes can transition to the same node in the next step.
If we were to sell an option, the most natural price would be E[S] – k, where k is the strike price, since any price set lower would mean we would lose money, and any price that is set higher we would not be able to find a buyer. This guarantees that on the average everyone breaks even.
This may seem extremely simple but is this the whole story? The answer is no because we do not know what S1, S2, and p1 are as they can be defined quite arbitrarily. However, there must be certain relationships among the prices of the different derivatives: The particular identity governing this process is known as “put-call parity.” Imagine for the moment that a seller of a call option makes a “guess” about the future prices, such that the option is priced in a manner where the return on the option plus the return on a risk-free bond is greater than the return of a stock plus a put option (i.e., priced too low). In this scenario, an investor can then sell the stock and put option to raise funds for the purchase of the calls and the bonds to earn a return without taking any risk, a process known as arbitrage. As we mentioned before, in order to facilitate an arbitrage-free market, there must be a mechanism by which we can anchor the option price that is common to all parties involved. It should be no surprise at this point that the anchoring mechanism is the bond.
Pricing Options
Exactly how are bonds used to fix the option price? Instead of pricing an option by using a portfolio consisting of just stocks, we can construct a portfolio consisting of both stocks and risk-free bonds with the same maturity as the option. The value of the call option is then the expected value of the portfolio minus the strike price. Therefore the value of the derivative should be exactly fixed to such a trading strategy. If a seller of an option decides to sell an option at a price that is less than the expected value of the portfolio (different from the expected value of just the stock as mentioned in the last section), then anyone can buy the option and trade based on the portfolio of stocks and bonds and keep the profit. In essence any market player who does not follow this rule will lose!
A question that arises from this phenomenon is, how does the incorporation of the bond affect the way in which we construct our binomial tree? Without going into too much mathematical detail or going through a step-by-step algebraic proof, we can conceptually say that by knowing that our portfolio of stocks and bonds is capable of completely defining our pricing point, we can see from equation 4.1 this also equates to being able to determine a fixed value for p1, the probability of the asset price reaching S1. Without the bond, as we mentioned before, p1 can be anything. Therefore by introducing the bond into our construction strategy we define the probability of transitions throughout our tree, which in essence will allow us to price any option given what we know of the bond market.
As it turns out, after all the algebra is done, we get a portfolio value of (equation 4.2):
Notice the similarity between this equation and equation 4.1. The probability here is represented by q, which is used to denote the change in p caused by introducing the bond into our portfolio. Also note that f is the value of the portfolio at each of the branching nodes rather than the asset price.
You may notice that there is nothing in the equations above that are unique to equities: Binomial trees can be used to price derivates and forecasting prices of all sorts of assets and rates. With that idea in mind, it should not surprise you that when discussing the Hull-White model, the tree is used to represent the rate movement rather than the stock movement and the valuation is discussed in terms of the bond price rather than the stock price. Of course this is not to say interest rate models are the only means by which options can be priced. Black-Scholes, for example, deals directly with the stochastic movement of the stock price. The construction strategy in the case of Black-Scholes is directly encapsulated in the methodology by which the stochastic differential equation (SDE) is solved.
Example of Pricing Options Using a Binomial Tree
As a quick example of the process of determining the price of an option, let us consider the branch shown in Figure 4.2.
Please note that for the sake of simplicity the probabilities are made up and we will pretend these are the true movements so that we needn't worry about using bonds to protect ourselves against arbitrage. What is important here is the methodology of backward inducing the price from the end nodes. According to Figure 4.3, the stock starts at a price of $2 and can transition to $4 with a probability of 1/3 or down to $1 with probability of 2/3. What would the price of option be if the strike price is $3?
If the stock ends up at $4, we win $1 in profit. If on the other hand, the stock goes down to $1, our option has lost all its value and it worth $0. The expected price is then V = 0*(2/3) + 1*(1/3) = $0.33. This result may seem trite, but the backward induction method shown here will be used heavily when discussing the Hull-White Trinomial tree.
FIGURE 4.2 Example branch of a stock after one tick.
FIGURE 4.3 Value of our portfolio with strike price of $3.
