# Delta Hedging

Delta hedging involves adding up the deltas of the individual assets and options making up a portfolio. A delta hedged portfolio is one for which the weighted sums of deltas of individual assets is zero. A position with a zero delta is referred to as a delta-neutral position.

Denote the delta of a hedging instrument by $$Delta_{H}$$

The optimal number of hedging units, $$N_{H}=-\frac{\text{Portfolio Delta}}{\text{Delta}_{H}}$$

To achieve a delta hedged portfolio, short the hedging instrument if $$N_{H}$$ is negative and long the hedging instrument if  $$N_{H}$$ is positive.

A delta-neutral portfolio is one that does not change in value for small changes in the underlying price. Delta neutral implies that:

$$\text{Portfolio delta} +N_{H}Delta_{H}=0$$

The portfolio should be rebalanced regularly to ensure that the sum of deltas remains close to zero. Static delta hedging involves constructing an initial portfolio with a sum of deltas of zero, at time 0, and never adjusting it. On the other hand, dynamic delta hedging involves continuously rebalancing the portfolio to maintain a constant total portfolio delta of zero.

### Example 1

Consider a portfolio composed of 1,500 shares. Call options with a delta of +0.50 are used to hedge this portfolio. A delta hedge could be implemented by selling enough calls to make the portfolio delta neutral.

The optimal number of hedging units is determined as follows:

$$N_{H}=-\frac{\text{Portfolio delta}}{\text{Delta}_{H}}$$

Portfolio delta= 1,500

$$\text{Delta}_{\text{H}} = +0.50$$

Thus, $$\text{N}_{H}=-\frac{1,500}{0.50}=-3,000$$

This means that we must sell 3,000 calls to achieve delta neutrality.

### Example 2

Given the following information:

$$S-{0}=60$$

$$K=50$$

$$r=2\%$$

$$T=1$$

$$\sigma=20\%$$

$$Delta_{c}=0.537$$

$$Delta_{p}=-0.463$$

Assume that the underlying asset does not pay a dividend.

Consider a short position of 5,000 shares of stock.

###### Hedging using call options

The optimal number of hedging units,

$$N_{H}=-\frac{\text{Portfolio delta}}{\text{Delta}_{H}}$$

Where: $$\text{Portfolio Delta}=-5,000$$

$$\text{Delta}_{H}=0.537$$

$$N_{H}=-\frac{-(-5,000)}{0.537}=9,311$$

This means that we must buy 9,311 calls to make the portfolio delta neutral.

###### Hedging using put options

We have portfolio delta $$=-5,000$$

$$\text{Delta}_{H}=-0.463$$

$$N_{H}=-\frac{-5,000}{-0.463}=-10,799$$

This means that we must sell 10,799 put options.

## Question

An investor owns a portfolio with 10,000 shares of Contagia Inc. common stock currently trading at $30 per share. The investor wants to delta hedge the portfolio using call options. A call option on the Contagia shares with a strike price of$30 has a delta of 0.5.

The strategy to create a delta-neutral hedge most likely involves:

1. Selling 10,000 call options
3. Selling 20,000 call options

### Solution

Portfolio delta $$= 10,000$$

$$\text{Delta}_{H}=0.5$$

The optimal number of call options required to hedge against movements in the stock price is determined as:

$$N_{H}=-\frac{\text{Portfolio delta}}{\text{Delta}_{H}}$$

$$N_{H}=\frac{-10,000}{0.5}=-20,000$$

This means that the investor must sell 20,000 calls to achieve delta neutrality.

Reading 38: Valuation of Contingent Claims

LOS 38 (i) describe how a delta hedge is executed

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