Following on from the previous article on Pricing a Call Option with Multi-Step Binomial Trees, we are now going to discuss what happens as we increase the number of steps, $N$. In particular, we will discuss what happens as $N\rightarrow\infty$. Some care must be taken at this stage, as we will be dealing with infinite series. We must progress in a manner that provides us with a sensible answer.
In order to obtain a meaningful price for our derivative we need to make an assumption about how the underlying asset ("the stock") moves as time increases. Fundamentally, this will involve precisely specifying the random process that governs the asset price.
These are the assumptions we will make for a first-go simple model. They are the same assumptions we have used in the previous articles, so you should be familiar with them:
- The interest rate will be zero
- The expectation of the asset price at the derivative expiry is equal to today's spot price
- For each step in the tree, the asset is able to move up or down with equal chance
We denote the spot price of the asset as $S_0$ and the expiry time to be $T$. The mean of the asset at expiry will also equal the spot price, $S_0$, and the variance at expiry will be $\sigma^2 T$. The next step is to split our interval from $0$ to $T$ into $k$ steps, of equal size. For each step we need to ensure that the mean change in the asset price is $0$ and that the variance is $\sigma^2 T/k$. This will guarantee that our overall variance is $\sigma^2 T$, since we are taking the mean and variance of a sequence of independent random variables, which simply sum up.
Thus our asset will move up or down with a probability of $p=0.5$ by a factor of:
\begin{eqnarray*} \sigma_k = \sigma \sqrt{\frac{T}{k}} \end{eqnarray*}Because of our assumptions on zero mean growth rate of the asset, as well as zero interest rates, we will utilise our argument of risk neutrality to state that the real world probabilities are equal to the risk neutral probabilities, in this case $p=0.5$. Recall that our investors do not need compensation for the extra risk they are taking on. Also recall that any other probability in this instance will lead to an arbitrage opportunity. If these two concepts are slightly unfamiliar, have a look at our risk neutral discussion for a quick refresher.
Ultimately, we wish to use our risk neutral pricing or hedging arguments to price our derivative. In order to do this we can use one of two methods:
- Backward propagation of the hedging or risk neutral argument through the tree
- Calculate the probabilities at each step of an asset price ending at a particular final value, in a forward manner, and then calculate the expectation of the derivative against this risk neutral probability density
In order to achieve the latter we need to know how the asset is distributed. Let us assume that we have reached step $k$. Consider a sequence of independent random variables that can take either $1$ or $-1$, each with probability half ("the coin toss"), which we will denote by $Z_i$. Then at step $k$ the asset price will be distributed by the following:
\begin{eqnarray*} S_0 + \sum^k_{l=1} \sigma_k Z_l \end{eqnarray*}Now that we have the asset distribution we can apply the risk neutral expectation argument. Given that the payoff function of the derivative is given by $f(S_T)$ at expiry T, we obtain the following expectation expression for the value of the derivative:
\begin{eqnarray*} \mathbb{E}\left(f\left(S_0 + \sum^k_{l=1} \sigma_k Z_l\right)\right) \end{eqnarray*}Our current goal is to see what happens as we let $k\rightarrow\infty$. However, we set up the model at the start of the article in such a way that as $k$ increases, the expression $\sigma_k \sum^{k}_{l=1} Z_l$ has mean $0$ and maintains a variance of $\sigma^2 T$. An application of the Central Limit Theorem shows that the following expression:
\begin{eqnarray*} \frac{1}{\sqrt{k}}\sum^{k}_{l=1} Z_l \end{eqnarray*}converges to $N(0,1)$, i.e. an element from a Guassian distribution with mean $0$ and variance $1$. Hence the asset price distribution will now converge to the following expression:
\begin{eqnarray*} S_0 + \sigma \sqrt{T} N(0,1) \end{eqnarray*}Leading to the derivative value converging to the following expression:
\begin{eqnarray*} \mathbb{E}(f(S_0 + \sigma \sqrt{T} N(0,1))) = \frac{1}{\sqrt{2\pi}}\int f(S_0 + \sigma \sqrt{T} x) e^{-\frac{x^2}{2}}dx \end{eqnarray*}So what are the shortcomings of this model in its current state?
- It allows a stock price to have a negative value, which is not in line with how a real-stock behaves.
- It considers absolute movements in stock prices, rather than relative movements. Share values move in relation to their current price, not in an absolute sense.
- We have yet to include interest rates
- We are neglecting the real world risk premium associated with taking on risk
We will improve all of these issues in the subsequent articles, eventually leading to a derivation of the famous Black-Scholes equation.
