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# Delta Hedging

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

Denote the delta of a hedging instrument by $$\text{Delta}_H$$ and 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 as a result of small changes in the underlying price. Delta neutral implies that:

$$\text{Portfolio delta}+N_H {\text{Delta}_H}=0$$

A portfolio should be rebalanced regularly to ensure that the sum of deltas remains close to zero. Static delta hedging involves the construction of an initial portfolio with a sum of deltas of zero, at time 0. In fact, the sum of deltas is never adjusted. On the other hand, dynamic delta hedging involves continuously rebalancing the portfolio to maintain a constant total portfolio delta of zero.

#### Example: Delta Hedging #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

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

Therefore, $$N_H =-\frac{1,500}{0.50} =-3,000$$

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

#### Example: Delta Hedging #2

Given the following information:

$$S_0 = 60$$

$$K = 50$$

$$r = 2\%$$

$$T = 1$$

$$\sigma=20\%$$

$$\text{Delta}_c=0.537$$

$$\text{Delta}_p= – 0.463$$

Assume that the underlying asset does not pay a dividend consider a short position of 5,000 shares of stock.

Delta hedge this portfolio using call options and put options.

#### Hedging Using Call Options

The optimal number of hedging units,

$$N_H = -\frac{\text{Portfolio Delta}}{\text{Delta}_H}$$

Where:

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.

Note to candidates: The amount of call options to be bought is not the same amount as the number of put options to be sold.

## 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:

\begin{align*} N_H &=-\frac{\text{Portfolio Delta}}{\text{Delta}_H} \\ N_H &=\frac{-10,000}{0.5} =-20,000 \end{align*}

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

Reading 34: Valuation of Contingent Claims

LOS 34 (l) Describe how a delta hedge is executed.

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