Question

What is the objective of a hedge portfolio in the Binomial Option Pricing Model?

Answer 1

Suppose that a market maker sells an option (on a stock). He is on the hook to sell (or buy) shares of the stock if the call (or put) buyer decides to exercise (i.e. when the share price of the underlying stock is above (or below) the strike price). He can hedge the risk of a short option position by creating a long synthetic option, i.e. creating a portfolio that replicates the same payoff of the option he sold. This replicating portfolio consists of shares of the stock and an appropriate amount of lending or borrowing. The is also called the hedge ratio and is the number of shares in the replicating portfolio to hedge away the risk from selling an option. Let’s discuss through two examples.

Example
Suppose that the future prices for a stock are modeled with a one-period binomial tree with 1.3 and 0.8 and having a period of 6 months. The current price of the stock is $50. The following is the binomial tree shows the future state of the stock prices.

The stock pays no dividends. The annual risk-free interest rate is 4%. Determine the price of a European 55-strike call option on this stock that will expire in 6 months. What is the replicating portfolio for this call option.

This is Example 1 in the post #1 on binomial model. At the end of 6 months, the stock price is either $65 or $40 and the value of the option is either $10 (if stock price goes up) or $0 (if price goes down). According to the calculation in the previous post, the replicating portfolio consists of holding 0.4 shares of the stock and $15.6832 in borrowing. The price of the call option is 50(0.4) – 15.6832 = $4.3168.

The market maker makes $4.3168 per call option sold. But the market maker is also at risk of losing $10 (selling a share at $55 for a share that is worth $65) when the call buyer decides to exercise. To hedge this risk, the market maker can buy a synthetic call option that replicates exactly the call option he sold.

In this example, the hedge ratio is 0.4, which is the ratio of the range of the values of the call to that of the stock across two possible outcomes. In this example, the calculation of is:

=0.4

For every call option written by the market maker, 0.4 shares of stock must be held to hedge away risk. The reason is that the strategy of holding 0.4 shares and the borrowing of $15.6832 has the same payoff as the call option as indicated by the following two equations. Note that $16.00 is the end of period value of $15.6832.

The above two equations show the payoff of the replicating portfolio of holding 0.4 shares and the borrowing of $15.6832, which is exactly the same as the payoff of the call option in the example. By selling a call option in this example, the market maker is at risk of losing $10 when the stock price goes up. He can offset the loss by creating a replicating portfolio that gains $10. So a market maker can hedge away the risk from selling a call by buying a synthetic call (the replicating portfolio).