Binary Trees Pricing of Options - Precificação de Opções por Árvores Binárias

Let's consider the pricing of an european option on a non-dividend paying stock. The binary tree pricing model is a very common method of doing this. Two assumptions are made in the following discussion:

• There is a well defined risk-free rate of interest and one can always risklessly borrow or invest money at this rate,
• There are no arbitrage oportunities, which implies that every risk-free investment can only earn the risk-free return rate.

Let S(t) be the price of a share over time and C(t) be the price of an option on that share with strike price K and maturity T. For simplicity, let t=0 be the moment of the writing of the option. The first step it to discretize the time t in N equal steps:

$t_n = \frac{nT}{N}, \qquad n = 0,1,2..,N$

For a litle while lets consider N=1 so that there are only two moments being considered: 0 and T.

Let's consider the following portfolio: a short position in an option and a long position in A shares of the underlying stock. At t=T, the value of the option is well defined, it's the payoff S(T)-K if the option in in-the-money or 0 if it's out-of-the-money:

$C(T) = \max(S(T)-K,0)$

so that, at this time, the final value of the portfolio is:

$P(T) = AS(T) - \max(S(T)-K,0)$

Now we ask the question: is there a given number of shares A which makes this portfolio riskless? In other words: is there a given that makes the final value of the portfolio independent of the final price of the share? Of course it's impossible to do this with only two securities in the portfolio as we know that there are infinite possible values for P(T) given P(0) and only one constant A to be adjusted. But let us consider that there are only two possible outcomes given the price P(0) of the stock in t=0:

1. The stock price goes up by a factor u>1, that is, S(T) = S(0)u
2. The stock price goes down by a factor d, that is, S(T) = S(0)d

In this case we have two possible final values for the portfolio. If the price goes up:

$P_u(T) = AuS(0)-\max(uS(0)-K,0)$

and if the price goes down:

$P_d(T) = AdS(0)-\max(dS(0)-K,0)$

If the portfolio is to be riskless we have:

$P_u(T) = P_d(T)$

or:

$AuS(0)-\max(uS(0)-K,0) = AdS(0)-\max(dS(0)-K,0)$

giving:

$A = \frac{\max(u-K/S(0),0) -\max(d-K/S(0),0)}{u-d}$

or:

$A = \left\lbrace \begin{array}{ll} 1&\mbox{, if }uS(0)>K \mbox{ and }dS(0)>K \\ \frac{u-K/S(0)}{u-d}&\mbox{, if }uS(0)>K \mbox{ and }dS(0)\\ 0&\mbox{, if }uS(0) \end{array}\right.$

... em construção