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### Subject 7. The delta of an option and dynamic hedging

Options prices are extremely dependent on underlying prices, and the price of the underlying is the key determinant of an option price. Therefore, delta is the most important of all of the sensitivity measures that we consider in this chapter.

Delta is the first derivative of an option's price with respect to a change in the price of the underlying. As such, Delta measures the sensitivity of the option's price to changing underlying prices. This definition is exact. In fact, the delta can be obtained approximately from the Black-Scholes-Merton formula as the value of N(d1) for calls and N(d1) - 1 for puts.

Continue with the example we worked in f, where S0 = \$100, X = \$100, r(c) = 0.06, T = 1, and σ = 0.1. The call price is \$7.46 and the put price is \$1.64. The call delta N(d1) = 0.7422, and the put delta is 0.7422 - 1 = -0.2578. For a \$1 change in the price of the stock, we should expect that

• Change in call option price = \$0.7422, and the new call option price = 7.46 + 0.7422 = \$8.20.
• Change in put option price = -\$0.2578, and the new put option price = 1.64 - 0.2578 = \$1.38.

If we use the Black-Scholes-Merton model to re-calculate the option price, the new call option price is \$8.21, and the new put option price is \$1.39.

Therefore, the delta approximation is good but not perfect.

However, for a \$10 change in the price of the stock, the call price should change by 0.7422 x 10 = \$7.422, and the put option price should change by -0.2578 x 10 = -\$2.578. The new call option price would be 7.46 + 7.422 = \$14.882, and the new put option price would be 1.64 - 2.578 = -\$0.938 (this should not be true as the option price cannot be negative).

If we use the Black-Scholes-Merton model to re-calculate the option price, the new call option price is \$16.09, and the new put option price is \$0.26. The approximations based on delta are not very accurate. In general, the larger the move in the underlying price, the worse the approximation.

The delta of a call option is always positive as the value of a call increases with an underlying price increase.

• When the call option is deep-in-the-money, its delta approaches 1.
• When the call option is deep-out-of-the-money, its delta approaches 0.

The delta of a put option is always negative as the value of a put decreases with an underlying price increase.

• When the put option is deep-in-the-money, its delta approaches -1.
• When the put option is deep-out-of-the-money, its delta approaches 0.

When the underlying is near the exercise price (at-the-money), delta is most sensitive to a change in the underlying price.

Delta Hedging

Continue with the example above, where S0 = \$100, X = \$100, r(c) = 0.06, T = 1, and σ = 0.1. The call price is \$7.46 and the call delta N(d1) = 0.7422. Consider a portfolio, P, of a short position of one call on the stock combined with a long position of 0.7422 units of the stock. The portfolio would have the value: P = -c + N(d1) S0 = -7.46 + 0.7422 x 100 = \$66.76.

If the stock price were to suddenly change to \$101, the portfolio's value would be -8.21 + 0.7422 x 101 = \$66.7522.

Thus, the value of the portfolio would change by only \$0.0078 for a \$1 change in the stock price. If the change in the stock price were infinitesimal, the price of the portfolio would not change at all. If the change in the stock price were larger, the change of the portfolio would be larger, but it would still be quite small relative to the change in the stock price. For example, if the stock price rose from \$100 to \$110, the portfolio's value would be -16.09 + 0.7422 x 110 = \$65.552. In this case, a change of \$10 in the stock price caused a change of \$1.208 in the value of the portfolio.

A portfolio like the one we are considering is known as a delta-neutral portfolio. It is delta-neutral because an infinitesimal change in the price of the stock does not affect the price of the portfolio. We could also say that the delta of this portfolio is zero: the value of the portfolio is insensitive to the value of the stock.

Investors often use delta to construct hedges to offset the risk they have assumed by buying and selling options. For example, if a dealer sells 1,000 call options discussed above, he would buy 742 shares of the stock to construct a delta-neutral portfolio.

An option is delta-hedged when a position has been taken in the underlying which matches its delta. Such a hedge is only effective instantaneously, because the option's delta is itself altered by changes in the price of the underlying, interest rates, the option's volatility and time to expiry. A delta-hedge must, therefore, be rebalanced continuously to be effective. In fact, delta hedging is often referred to as dynamic hedging.

#### Practice Question 1

Delta measures ______.

A. the sensitivity of the option's price to changing underlying prices.
B. the sensitivity of the underlying's price to changing option prices.
C. the sensitivity of the option's price to changing interest rates.

#### Practice Question 2

Consider a call option, with X = \$40; r = 0.06; T = 90 days; σ = 0.1; and S0 = \$70. The delta of this call option should be close to:

A. -1.
B. 1.
C. Cannot determine but it is very sensitive to a change in the underlying price.

As the option is deep-in-the-money, its delta should be close to 1.

#### Practice Question 3

From the Black-Scholes-Merton formula, we can obtain a put option's delta approximately as

A. N(d1) - 1.
B. N(d2) - 1.
C. 1- N(d1).

#### Practice Question 4

Consider a put option, with X = \$40; r = 0.06; T = 90 days; σ = 0.1; and S0 = \$40. The delta of this put option should be close to:

A. 0.
B. 1.
C. Cannot determine but it is very sensitive to a change in the underlying price.

The delta of a put option is always negative so B is not correct. When the put option is deep-in-the-money, its delta approaches -1. When the put option is deep-out-of-the-money, its delta approaches 0. As the put option is at-the-money, delta is most sensitive to a change in the underlying price.

#### Practice Question 5

For a call option with a delta of 0.5, a \$0.5 increase in the underlying price (current price: \$72) will cause the price of the call option to

A. increase by \$0.25.
B. decrease by \$0.25.
C. increase by \$1.

The value of a call increases with an underlying price increase: the increase = 0.5 x \$0.5 = \$0.25.

#### Practice Question 6

For a put option with a delta of 0.2, a \$0.5 increase in the underlying price (current price: \$72) will cause the price of the put option to (increase? decrease?) ______ by ______.

Correct Answer: This question itself is wrong. A put option cannot have a positive delta!

#### Practice Question 7

Consider a put option, with X = \$40; r = 0.06; T = 90 days; σ = 0.1; and S0 = \$70. The delta of this put option should be close to:

A. 0.
B. 1.
C. Cannot determine but it is very sensitive to a change in the underlying price.

When the put option is deep-out-of-the-money, its delta is close to 0.

#### Practice Question 8

For a call option with a delta of 0.2, a \$0.90 decrease in the underlying price (current price: \$2.37) will cause the price of the call option to

A. increase by \$0.18.
B. decrease by \$0.18.
C. it's hard to determine given the information.

In this case the approximation based on delta would be not accurate as the price change is pretty large (38% price change).

#### Practice Question 9 Examining the above diagram, which of the following statements would be false?

A. The time value of this option becomes the lowest when the option is at-the-money.
B. The delta of this options value can never be negative.
C. This is a long call position. This diagram is a snapshot of the option on a day other than its expiration date.

The fact is that the time value of an option, be it a call or a put, is at its highest when the option is at the money.

Furthermore, we know that this diagram is a snapshot of the option on a day other than its expiration date because it still has time value. On expiration day, the time value is zero.

#### Practice Question 10

A call option with an exercise price of \$25.00 is currently valued at \$3.22. The price of the underlying asset itself is \$24.75. If the price of the underlying asset appreciates to \$25.50, and the delta of the option is 0.35, what will be the return from holding the call option?

A. 3.45%
B. 5.43%
C. 8.15%

Delta = change in the value of the call / Change in the value of the asset

0.35 = Δc/(25.50 - 24.75) => Δc = 0.2625. Call value should increase by 0.2625 or 0.2625/3.22 = 8.15%.

#### Practice Question 11

Which of the following statements is (are) true with respect to option deltas?

I. The sum of the deltas for a long call and a long put that have the same parameters, should be equal to zero.
II. The delta for a call is at its highest when the call is at-the-money.
III. For a portfolio to be considered as delta neutral, the fraction of shares that must be bought and sold for each call that is shorted, must be equal to the call's delta.
IV. The delta for a short put will always be positive.

A. III and IV
B. II and III
C. I, II, and IV