Chapter 7 - Pricing of Options (Done)

Question 1

What is delta?

Answer 1

Indicates how much the price of an
option is expected to change, given a
price change in the underlying interest.

Question 2

What is delta hedging?

Answer 2

Adjusting the number of contracts
used in an option hedge to reflect the
option’s delta.

Question 3

What is historical volatility?

Answer 3

Past or historical volatility of an
underlying asset, measured by taking
the standard deviation of price changes
over a set period of time.

Historical volatility measures how much the price of a stock has fluctuated over a specific period in the past.

In short, historical volatility tells you how erratic the stock’s price has been over the chosen past period, regardless of the current price or time of year.

Question 4

What is implied volatility?

Answer 4

The volatility implicit in an option’s
premium.

Implied volatility is a measure of the market’s expectation of how much the price of a stock will fluctuate in the future, as inferred from the option’s price.

So, implied volatility tells you how much the market thinks the stock’s price will move going forward.

Implied volatility is reflected in the price of the option.

Question 5

What is intrinsic value?

Answer 5

For a call option, intrinsic value is
calculated by subtracting the exercise
price from the market price. For a put
option, intrinsic value is calculated by
subtracting the market price from the
exercise price. For both calls and puts,
intrinsic value cannot be less than zero.

Question 6

What is time value?

Answer 6

The premium of the option less its
intrinsic value.

Question 7

What is volatility?

Answer 7

A statistical measure of how much
a given security or market index
fluctuated during a given time period.

Question 8

Describe the difference between intrinsic value and time value.

Answer 8

• Intrinsic value represents the real and tangible value of an option and is based solely on the relationship
  between the option’s strike price and the price of the underlying asset.
• Time value is the amount that an option is trading in excess of its intrinsic value. The time value of an option
  depends on several factors, as noted below. As the level of uncertainty about where the underlying price will
  be at expiration increases, the amount of time value built into an option’s premium increases.

Question 9

List the factors that affect intrinsic value and time value

Answer 9

• Intrinsic Value
  If an option is in-the-money, intrinsic value is the difference between the strike price and the underlying price.
  Prior to expiration, all American-style options and virtually all European-style options trade at a price that is
  at least equal to their intrinsic value.
  Occasionally (although very rarely), a European-style option will trade at a “discount” to its intrinsic value.
  The discount can be thought of as negative time value. The fact that a discount can arise is a direct result of
  the two different exercise features.
• Time Value
  The time remaining to expiration has an important effect on the time value of an option. The longer an option
  has until expiry, the more the option will be worth. However, the relationship between time and an option’s
  value is not linear. An option’s time value decays faster the closer the option is to expiration.
  In terms of dollars, the time value of an option is at its maximum when the underlying price is equal to the
  strike price (i.e., the option is at-the-money). As the option moves in-the-money or out-of-the-money, the
  dollar amount of time value decreases (although the percentage of an option’s premium represented by time
  value may go up or down).
  Perhaps the most important variable affecting time value is the expected volatility of the underlying asset. All
  else being equal, the more volatile the market expects the underlying asset to be over the life of the option,
  the more time value will be built into the option premium and the more the option will be worth. This is true
  for both calls and puts. Historical volatility is the actual past volatility exhibited by the price of the underlying
  asset. Traders will sometimes use historical volatility as an estimate of future volatility when using option-
  pricing models. To understand what level of volatility the market is building into the price of an option, the
  price of the option is fed into an option-pricing model, and the model is solved for volatility. The volatility
  figure arrived at through this process is known as the option’s implied volatility.

The net cost of carrying the underlying asset is also a determinant of the amount of time value built into an
option’s premium. The cost of carrying most underlying assets consists of the risk-free rate of interest and any
yield on the underlying instrument.

Question 10

Explain how changes in these factors affect an option’s value.

Answer 10

Holding all other variables the same, changes in the factors that affect an option’s value impact call and put
premiums as follows:
- An increase in the volatility of the underlying asset increases both call and put premiums. A decrease in the
volatility of the underlying asset decreases both call and put premiums.
- An increase in the risk-free interest rate increases call premiums and decreases put premiums. A decrease in
the risk-free interest rate decreases call premiums and increases put premiums.
- An increase in the yield of the underlying asset decreases call premiums and increases put premiums.
A decrease in the yield of the underlying asset increases call premiums and decreases put premiums.

Yield of the Underlying Asset
Impact:

An increase in the yield of the underlying asset decreases call premiums and increases put premiums.
A decrease in the yield of the underlying asset increases call premiums and decreases put premiums.

Why:

Call Options: If the underlying asset pays a higher yield (e.g., dividends for stocks), holding the asset becomes more attractive compared to holding the option. This is because owning the actual asset allows the holder to receive the yield, whereas holding a call option does not. As a result, the premium for call options decreases.
Put Options: Higher yields make holding the asset less attractive since the asset's price may drop after yielding a dividend or interest payment. This makes put options more valuable because there's a higher chance the asset's price will decrease, making it more likely the put will end up in-the-money.

Risk-Free Interest Rate
Impact:

An increase in the risk-free interest rate increases call premiums and decreases put premiums.
A decrease in the risk-free interest rate decreases call premiums and increases put premiums.Call OptionsCost of Carry:

Holding an asset requires funding. If you can invest money at a higher risk-free rate instead, the opportunity cost of holding the asset increases.
Why: Higher risk-free rates mean higher returns on alternative investments (like bonds), which makes holding the underlying asset (and thus holding the call option to buy it later) more attractive.Put OptionsCost of Holding the Underlying Asset:

If interest rates are higher, the opportunity cost of holding cash (rather than the underlying asset) increases.
Why: Investors might prefer to hold cash or risk-free assets that earn a higher interest rate rather than holding the underlying asset, making put options (the right to sell the asset) less valuable.

Question 11

Describe and explain the importance of an option’s delta

Answer 11

An option’s delta is a number that measures the approximate change in the value of an option premium given a
1-unit change in the price of the underlying asset.
Option deltas are important for a variety of reasons. First, option deltas allow option traders to estimate how
much an option value will change given a change in the price of the underlying asset. If an option trader has an
opinion on the price of the underlying asset, this can be used to estimate the return on an option position.
Second, an option’s delta acts like a hedge ratio, allowing hedgers to estimate the number of contracts required
to hedge a position in the underlying asset. To use the delta as a hedge ratio, divide the option-adjusted size
of the exposure being hedged by the option’s delta. The option-adjusted size of the exposure is the size of the
exposure divided by the option’s trading unit.
Because deltas change as the price of the underlying asset changes, a delta hedging program needs to be
constantly monitored for changes in option deltas.

Question 12

Calculate an option’s delta given all the inputs.

Answer 12

The inputs required to calculate an option’s delta are the change in the price of the underlying asset and the
corresponding change in the price of the option.

Delta = Change in Price of the Option / Change in Price of the Underlying Asset

Question 13

Calculate the expected change in an option’s price given the delta and the change in the underlying
asset’s price.

Answer 13

A simple rearrangement of the above formula provides the following formula for calculating the expected
change in an option’s price given the delta and the change in the underlying asset’s price:

Change in Price of an Option = Delta × Change in Price of the Underlying Asset

Question 14

DEF 50 call options are trading at $5. DEF stock is trading at $56. If DEF options have an American-style
exercise feature, which of the following steps must be taken to exploit the arbitrage opportunity?
a. Buy DEF 50 call options at $5 and sell DEF stock short at $56.
b. Sell DEF 50 call options at $5 and buy DEF stock at $56.
c. Buy DEF 50 call options at $5, sell DEF stock short at $56 and exercise the options.
d. Sell DEF 50 call options at $5, buy DEF stock at $56 and exercise the options.

Answer 14

c. Buy DEF call options at $5, sell DEF stock short at $56 and exercise the options.

Question 15

XYZ stock is trading at $40. Which of the following options (all expiring in the same year) will have the
greatest dollar amount of time value built into its price?
a. XYZ March 35 calls.
b. XYZ March 40 calls.
c. XYZ June 35 calls.
d. XYZ June 40 calls

Answer 15

d. XYZ June 40 calls.
All else being equal, at-the-money options have the greatest dollar amount of time value built into their
price. All else being equal, options with a longer time to expiration will have a higher premium than options
with a shorter time to expiration

Question 16

ABC Inc. has just announced an increase in its dividend payable. All else being equal, what effect will this have
on ABC option premiums?
a. ABC call option premiums will increase, and ABC put option premiums will decrease.
b. ABC call option premiums will decrease, and ABC put option premiums will decrease.
c. ABC call option premiums will increase, and ABC put option premiums will increase.
d. ABC call option premiums will decrease, and ABC put option premiums will increase.

Answer 16

d. ABC call option premiums will decrease and ABC put option premiums will increase

Question 17

If the delta of a call option is 0.6 and the price of the underlying asset rose from $45 to $50, what would you
expect to happen to the price of the call option?
a. Increase by $0.60.
b. Increase by $1.50.
c. Increase by $3.00.
d. Increase by $8.33.

Answer 17

c. Increase by $3.00.
($50 − $45) × 0.6

Question 18

The price of JKL stock has increased from $20 to $22. At the same time, the price of JKL 20 put options
decreased by $0.40. What was the put option’s delta prior to the change in the stock price?
a. −1.0
b. −0.4
c. −0.2
d. 0.2

Answer 18

c. −0.2
−$0.40 ÷ ($22 − $20)

Question 19

PQR May 35 calls have a delta of 0.6. What is the corresponding delta of PQR May 35 puts?
a. −0.6
b. −0.4
c. 0.4
d. 0.6

Answer 19

b. −0.4
The sum of the absolute values of the deltas of a call and a put with the same expiry and the same strike
price is 1.

Question 20

When would a hedger use a delta hedging technique?

Answer 20

A hedger would utilize a delta hedging technique when dollar-for-dollar protection is required from an adverse
change in the price of the underlying asset.

Question 21

A portfolio manager owns 10,000 shares of MNO Inc. The stock is currently trading at $65. The manager wants
to buy MNO 65 put options to protect the portfolio from a sharp decline in the price of MNO. If the MNO
65 puts have a delta of −0.5, how many option contracts will give the portfolio manager dollar-for-dollar
protection from a decline in the price of MNO?
a. 50
b. 100
c. 200
d. 500

Answer 21

c. 200
(10,000 ÷ 100) ÷ −0.5

Question 22

Explain how arbitrage with respect to American and European options differ.

Answer 22

Arbitrage opportunities differ between American and European style options mainly because of the timing of when they can be exercised.

American Style Options:

Can be exercised at any time before expiration.
Example: If you can buy and exercise an option for $21.50 when the stock is trading at $22, you can immediately do so to make a profit of $0.50 per share.

European Style Options:

Can only be exercised at expiration.
Example: If the option to buy at $21.50 exists but can only be exercised at expiration, you can't take advantage of the current $22 stock price unless it's still above $21.50 at expiration.

In short, American options allow for immediate arbitrage, while European options require waiting until expiration, potentially missing out on immediate profit opportunities.

Question 23

Explain why European options can sometimes be under priced relative to the stock.

Answer 23

European options can sometimes be underpriced relative to the stock due to their restriction on exercising only at expiration. Here are the key reasons:

Limited Exercise Flexibility:
    European options cannot be exercised before expiration, reducing their flexibility and potential profitability compared to American options.

Time Value Impact:
    The inability to exercise early means that even if the stock price moves favorably before expiration, the option holder cannot capitalize on it until the set expiration date. This can make the option less valuable.

Dividends and Interest Rates:
    If a stock pays dividends, the holder of a European call option cannot exercise early to receive the dividend, unlike with American options. This can lower the option's price.
    Interest rates also affect option pricing differently. The longer the time to expiration, the more the present value of the strike price is discounted, potentially leading to a lower option price.

Example:
Suppose a stock is trading at $100. A European call option with a strike price of $95 expiring in one month may be priced lower than an American call option with the same terms because the European option holder cannot exercise early if the stock price rises significantly before expiration.

In summary, European options might be underpriced relative to the stock because they lack the flexibility of early exercise, which can limit their potential profit opportunities.

Question 24

Why is time value graetest at the money?

Answer 24

Time value is at its highest level when an option is at the money because the potential for intrinsic value to begin to rise is greatest at this point. Time-value decreases as an option gets deeper in the money; intrinsic value increases.

Question 25

Why does the difference between strike and underlying price affect both time and intrinsic value of an option?

Answer 25

Effect:

Affects both the intrinsic and time value of the option.

Why:

Intrinsic Value: This is directly the difference between the strike price and the underlying price.
Time Value: It influences the likelihood that the option will end up in the money by expiration.

Question 26

Explain delta hedging.

Answer 26

Definition:

A strategy to hedge the risk of price movements in the underlying asset by offsetting long and short positions.

How to Do It:

Calculate the option's delta (rate of change of option price relative to the underlying asset's price).
Take an opposite position in the underlying asset proportional to the delta. For instance, if you hold a call option with a delta of 0.5, you would short 0.5 units of the underlying asset to hedge your position.