Hedging the sale of a Call Option with a Two-State Tree

In the Introduction to Option Pricing with Binomial Trees we showed that if a stock was worth 100 today, and could be worth either 110 or 90 (respectively the $U$ or $D$ state) tomorrow, a call option struck at $K = 100$ on the stock had to be worth between 0 or 10. In this article we will be attempting to put a *no-arbitrage price* on the option by means of a *hedging argument*.

The main problem is that we are unsure whether the stock will rise or fall. Thus we need to purchase some quantity, $\Delta$, of the stock today in order to eliminate the risk of selling the option. Let us first consider our portfolio. If we purchase £$\Delta$ of the stock and sell an option worth $C$, our portfolio can be written:

\begin{eqnarray*} \Delta S - C \end{eqnarray*}Now let us consider what happens to the value of this portfolio if we enter our world states $U$ and $D$. First, consider $U$. Our portfolio will be worth $\Delta S - C = 110\Delta - 10$. Now, consider $D$. Our portfolio will be worth $\Delta S - C = 90\Delta - 0 = 90\Delta$.

The essence of the hedging argument is that the portfolio should be equal in any state of the future, since this will remove all uncertainty. Since our portfolios must be equal, we can set the values of the portfolios from $U$ and $D$ to be equal to each other and solve the equation for our quantity of stock, $\Delta$:

\begin{eqnarray*} 110 \Delta - 10 = 90\Delta \end{eqnarray*}This leads to a value of $\Delta = 0.5$. We have already discussed that our mathematical treatment of options pricing would allow us to hold fractional quantities of shares. If this seems unrealistic, then the same argument can work if all quantities in this example are multiplied by one million, in which case we would be holding 500,000 shares, a non-fractional amount.

Given that $\Delta = 0.5$, we can show that our portfolio is worth $110\cdot 0.5 - 10 = 45$ in state U and $90 \cdot 0.5 = 45$ in state D. Since interest rates have been set to zero, our portfolio contains no risk. This leads us to the conclusion that today's portfolio must be worth 45 as well, because we could synthesise the same riskless portfolio by purchasing 45 worth of zero-coupon bonds expiring tomorrow. So what does this mean for our option price? Well, the value of today's portfolio is $\Delta\cdot S - C = 0.5 \cdot 100 - C = 45$. This implies that $C=5$.

Let us summarise the main thrust of the argument. The condition of no-arbitrage forces the portfolio today to be worth 45, as any other value would lead to an *arbitrage opportunity*, and thus the option must be valued at 5.

Notice here that the probabilities of the stock going up or down tomorrow have not even entered the picture! Our next method of pricing the option, that of risk neutral pricing will consider these probabilities, however, and see how they affect the price.

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