Binomial Pricing

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Saurabh Jain

A useful and very popular technique for pricing a stock option involves constructing a binomial tree. This is a diagram that represents different possible paths that might be followed by the stock price over the life of the option.

Example

We start by supposing that we are interested in valuing a European call option to buy a stock for $21 in three months. A stock price is currently $20. We make a simplifying assumption that at the end of three months the stock price will be either $22 or $18. This means that the option will have one of two values at the end of the three months. If the stock price turns out to be $22, the value of the option will be $1; if the stock price turns out to be $18, the value of the option will be zero. The situation is illustrated in Figure.

It turns out that an elegant argument can be used to price the option in this situation. The only assumption needed is that no arbitrage opportunities exist. We set up a portfolio of the stock and the option in such a way that there is no uncertainty about the value of the portfolio at the end of the three months. We then argue that, because the portfolio has no risk, the return it earns must equal the risk-free interest rate. This enables us to work out the cost of setting up the portfolio and therefore the option's price. Because there are two securities (the stock and the stock option) and only two possible outcomes, it is always possible to set up the riskless portfolio. Consider a portfolio consisting of a long position in A shares of the stock and a short position in one call option. We calculate the value of A that makes the portfolio riskless. If the stock price moves up from $20 to $22, the value of the shares is 22 A and the value of the option is 1, so that the total value of the portfolio is 22A - 1. If the stock price moves down from $20 to $18, the value of the shares is 18 A and the value of the option is zero, so that the total value of the portfolio is 18 A. The portfolio is riskless if the value of A is chosen so that the final value of the portfolio is the same for both alternatives. This means

22A - 1 = 18A or A = 0.25

A riskless portfolio is therefore Long: 0.25 shares Short: 1 option

If the stock price moves up to $22, the value of the portfolio is 22 x 0.25 - 1 = 4.5 If the stock price moves down to $18, the value of the portfolio is 18x0.25 = 4.5

Regardless of whether the stock price moves up or down, the value of the portfolio is always 4.5 at the end of the life of the option. Riskless portfolios must, in the absence of arbitrage opportunities, earn the risk-free rate of interest. Suppose that in this case the risk-free rate is 12% per annum. It follows that the value of the portfolio today must be the present value of 4.5, or

4.5e-0.12*3/12=4.367

The value of the stock price today is known to be $20. Suppose the option price is denoted by /. The value of the portfolio today is 20 x 0.25 - / = 5 - / `It follows that 5 - / = 4.367 or / = 0.633 This shows that, in the absence of arbitrage opportunities, the current value of the option must be 0.633. If the value of the option were more than 0.633, the portfolio would cost less than 4.367 to set up and would earn more than the risk-free rate. If the value of the option were less than 0.633, shorting the portfolio would provide a way of borrowing money at less than the risk-free rate.

References

Option, Future and Derivatives 5th edition Writer John C. Hull