Hedging Call Options in a Binomial Model: A Delta Hedging Approach

The Art of Hedging in a Binomial World

Imagine you're an investor about to buy a call option on a non-dividend-paying stock. You're aware that the stock price could either double or halve over the next year. How do you determine the current fair price of this call option, given these assumptions? This is where binhedging, or hedging in a binomial model, comes into play.

I-A A Single Period Binomial Setup

In a single period binomial model, we assume zero interest rates and no payouts from the underlying security. If S is the initial spot price of the asset, then at the end of the period, the spot price will either be Su or Sd, where u and d represent the upward and downward factors, respectively. For example, if S = $40, u = 2, and d = 1/2, then the possible ending prices are $80 (if the stock goes up) or $20 (if it goes down).

I-B Spanning the Payoffs: The Delta of a Call

By constructing a portfolio consisting of shares and bonds with specific face values, we can replicate the payoff of any European-style contingent claim. In our example, to replicate a European call struck at $50 maturing in one year, we would need to hold 1/2 of a share and short 1/5 of a bond. This relationship is known as delta hedging, where the slope (delta) of the graph depicting the value of the replicating portfolio against stock prices equals the required number of shares.

Portfolio Implications: C, TIP, GS, QUAL

For this specific call option, holding 1/2 of a share and shorting 1/5 of a bond is the optimal delta-hedged position. This strategy minimizes risk and replicates the payoff of the call option in a frictionless market with zero interest rates. Other assets like C (iShares Core U.S. Aggregate Bond ETF), TIP (iShares TIPS Bond ETF), GS (Goldman Sachs Group Inc.), or QUAL (First Trust NASDAQ-1