# Delta Hedging – a Useful Options Trading Strategy

Delta hedging is an options trading strategy used to hedge or reduce the directional risk related to the price fluctuation of the underlying asset. In the delta hedge strategy, the trader aims to reach a delta neutral position by buying and selling call or put options and then offsetting the directional exposure by buying and selling an equal number of ETFs or stocks.

Delta is a risk variable that measures one of the many dimensions of risk in an option position. It is denoted by D. It is a theoretical estimate of how much the price of an option may change given a \$1 move in the underlying asset. In short, the delta of an option measures its price sensitivity.

The formula of an option’s delta is:

Where:

∂ = the first derivative, v = the price of the call option, s = the underlying stock’s price

Call options have a delta ranges between 0 and +1.0 whereas put options have a delta range between -1.0 and zero. For example, this means that the price of a call option with a delta of +0.50 will go up by \$0.50 if the price of the underlying asset increases by \$1 and that the price of a put option with a delta of -0.35 is expected to go down by \$0.35 if the price of the underlying asset falls by \$1.

### A Short Detour to Option Greeks

The term ‘Greeks’ is very widely used in the options market. The most common Greek variables include delta, vega, rho, gamma, and theta among others. These risk variables are named ‘Greeks’ because they are represented by Greek letters.

Different Greek variables calculate different dimensions of risk associated with taking an options position and are essential tools in risk management for options traders. For instance, delta measures the rate of change or sensitivity between the price of an option and the underlying asset. Vega measures the sensitivity to the implied volatility of an option. Rho represents how sensitive the price of an option is relative to interest rates. Gamma represents the rate of change of delta relative to the change of the price of the underlying security. Theta measures the option’s sensitivity related to time.

In essence, the Greeks show you how much exposure you will have, in other words, how much money you stand to lose or make if the market was to move in a particular direction and magnitude.

### Why Is Delta Hedging Important?

The delta of an option indicates how many options contracts are required to hedge a long or short position in the underlying asset. This strategy of buying options to create a risk-free portfolio is called delta hedging.

Every financial institution selling options to its clients needs to manage the associated risks. If the option in question is a standardized option (one that is traded on the exchange), then the institution can neutralize its position by buying the same option that it has sold on the exchange. However, if the option is a non-standard option, the financial institution will not be able to offset the risk easily. This could result in the financial institution having to face immense directional exposure risk.

Suppose the financial institution sells a European call option contract on 100,000 shares of non-dividend paying stock for \$350,000. The stock price is \$59, the strike price is \$60, the risk-free rate is 5% per annum, and the volatility of the stock price is 20% per annum. The time to maturity is 20 weeks. We can write it as follows:

Where:

• T = 0.3846 (Time to maturity in years)
• S = \$59 (Stock price)
• X = \$60 (Strike price)
• r = 0.05 (Risk-free rate)
• Ó = 0.2 (Volatility of the stock price)

According to the Black-Scholes model, the purchase price per share of the option is \$2.981 (i.e., the price of the option is \$298,100). This means that the financial institution has sold the call option for \$60,900 more than its value. However, it still needs to hedge its exposure.

What should the financial institution do then? One option is to do nothing – i.e., take a naked (uncovered) position. However, this position is only going to work for the financial institution if the stock price is less than the strike price at the time of maturity. This strategy will cost nothing to the financial institution.

However, this uncovered position will be very costly to the financial institution if the call option is exercised. At the maturity date, the institution will have to buy 100,000 shares at market price. If, for instance, the price of the stock at maturity is \$70, then the option will cost the institution \$1,000,000 (100,000 shares x (\$70 – \$60). This amount is significantly more than what the company had charged initially, i.e., \$350,000.

On the other hand, the financial institution can opt for a covered position by using delta hedging. In this case, the institution would buy 100,000 shares of the underlying asset at the same time it sold the option. However, this delta hedging strategy will only be beneficial to the institution if the option is exercised. If the option is not exercised, it will result in a significant loss. For instance, if the stock price goes down to \$50, the institution will lose \$90,000 (100,000 x (\$59-\$50) on its option.

Neither of the two positions offers an optimal hedge. Keeping the assumption of the Black Scholes model in mind, the cost should, on average, be around \$298,100. However, when looking at both positions, the cost of different hedge strategies is likely to range from \$0 to \$1,000,000. The cost of an optimal hedge should be around \$298,100. This is precisely what a sophisticated hedging method, such as delta hedging, aims to accomplish.

### Delta Hedging Example

Let’s try going over an example. Suppose the delta of a call option on a stock is 0.5. This means that if the stock price changes by \$1, the price of the option will change by 50% of that amount. Suppose the price of the stock is \$100, and the price of the option is \$10. You sell 30 call options (each option is for 100 shares), i.e., option on 3000 shares. How are you going to hedge your position? The answer is simple: by buying delta times the number of options – i.e., 0.5 x 3000 = 1500 shares.

How is this delta hedging strategy going to work? The gain or loss on the stock position would offset the loss or gain on the option position.

For instance, look at the condition here:

If the price of the stock goes up by a dollar

The increase in price will result in a profit of \$1500. The price of option will also go up by 0.5 x \$1 = \$0.5. But this will result in a loss of \$1500 on the sold options.

If the price of the stock goes down by a dollar

The decrease in stock price will result in a loss of \$1500. The price of the option will go down by \$0.5. This, however, will produce a profit of \$1500 on the options written.

Here your short position delta for 3000 options is 0.5 x (-3000) = -1500.

What this means is that you’ll lose \$1500 if the price of the stock goes up by \$1. Since the delta of one share equals1. This means that a long position in 1500 shares would have a delta of +1500. Therefore, the overall delta would equal 0. A position with a delta value equal to 0 is referred to as a delta neutral position.

### Rebalancing, Static Hedging, and Dynamic Hedging

One thing to keep in mind as an investor is that delta hedging only keeps your position covered (delta neutral) for the short term because the delta value of an option does not stay constant. However, to keep the value of your option as constant as possible, you will need to adjust your position (or hedge) periodically. This position adjustment is known as rebalancing.

For instance, suppose the stock price by the end of a day goes up to \$110. By now, you know that this increase in stock price will also increase the delta. If the value of delta increases from 0.5 to 0.55, this means that you’ll have to purchase an additional 0.05 x 3000 = 150 shares to maintain your position.

The delta hedging strategy of regularly readjusting the hedge is known as dynamic hedging. Contrary to it is static hedging, where you adjust your hedge one time and then never readjust.

### Pros and Cons of Delta Hedging

One of the drawbacks of delta hedging is the constant need to adjust your position. The trader must keep buying and selling stocks to avoid getting over-hedged. Constant hedging means that the trader would have to make several transactions to maintain the delta neutral position. Hence, delta hedging results in extra transaction costs.

On the contrary, delta hedging allows you to reduce the risk of significant price changes in your portfolio. It also protects your position from being affected by small price changes. Above all, it allows you to continue to produce gains despite the price movement of the stock.

Delta hedging is not a strategy that most investors should use. It is a complex strategy, that needs continuous adjusting, and therefore is mostly used by institutional traders and investment banks.

Vikram Raghavan is a value investor, technologist, and Finexy co-founder. In addition to stock market investing, Vik also invests and advises startups on growth marketing and product management. Vik's work is focused on themes of marketplaces, micro-entrepreneurship, marketing automation, and user growth. Previously, Vikram led product and growth teams at Overstock.com, focusing on efforts across acquisition, new user experience, churn, and notifications/email. He holds an MBA in Finance from Temple University and a B.S. in Computer Information Systems and Finance from Bemidji State University.