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* 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

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,

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.

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.

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.

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 likelyinvolves:

- Selling 10,000 call options.
- Buying 20,000 call options.
- Selling 20,000 call options.
## Solution

The correct answer is C.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.*