CAIA L2 - 9.1 - Volatility as a Factor Exposure

Question 1

Contrast

Implied
vs.
Realized Volatility

9.1 - Volatility as a Factor Exposure

Answer 1

Implied return volatility
represents expected volatility going forward for the life of an option inferred from its option price and computed using market prices through an option pricing model. One example is the Black-Scholes option pricing model, which assumes that the underlying asset follows a geometric Brownian motion (GBM), which has instantaneous asset returns that
* have an unchanging (constant) variance through time (homoscedastic),
* are normally distributed, and
* are uncorrelated through time.
neither an accurate nor an unbiased estimate of realized volatility
‘–

realized return volatility
is the actual variation that occurs over a specific time period (typically measured as the standard deviation of returns) . Implied volatilities differ from expected realized volatilities because risk premiums are associated and are reflected in implied volatility. In practice, option returns are discrete and not continuous. Because realized return volatilities represent actual variation, they violate all three assumptions that the underlying asset returns follow a GBM, including that returns that have constant variance, are normally distributed, and are uncorrelated through time.

9.1 - Volatility as a Factor Exposure

Question 2

List

3 limitations
of realized volatility
as a measure of dispersion

9.1 - Volatility as a Factor Exposure

Answer 2

realized return volatility = ESTIMATES

1. It does not define the return distribution’s shape (assets with similar realized vol may experience very different actual returns)
2. Even when assets have identical realized volatility, they could differ if their returns are trending, mean reverting, or exhibiting minimal autocorrelation.
3. It does not indicate whether the dispersion of returns occurred near an underlying asset price or during a specific time period in the sample.

Examples of assets with same realized vol (but different behaviours)
Case 1: Asset A returns had alternating signs and tended to mean-revert, while asset B returns moved mostly in one direction.

Case 2: Asset C experienced mostly average size returns, and asset D experienced many very small returns and one large outlier return.

Case 3: Asset E had low volatility for most of the sample period with a few very large shocks at the end of the time, and asset F had a few very large shocks at the beginning followed by low volatility.

9.1 - Volatility as a Factor Exposure

Question 3

List

6 properties
of realized volatility

9.1 - Volatility as a Factor Exposure

Answer 3

1. Realized volatility is nonconstant and exhibits behavior of slow mean reversion and clustering. Common trading strategies use generalized autoregressive conditional heteroscedasticity (GARCH) or regime switching models.
2. Realized volatility is normally low until a market shock and then volatility rises for an extended time period.
3. Realized volatility can spike in the short term; however, in the long term, volatility levels are near long-term averages.
4. Higher realized volatility is negatively correlated with risky asset returns because investor risk aversion increases at higher volatility levels.
5. Equity market realized volatility is typically higher in bear markets and lower in bull markets. Higher volatility also results from decreases in the stock price (or index value), which increases leverage and risk. Therefore, there is negative correlation between the equity value and volatility.
6. Realized equity volatility has a greater rate of increase in bear markets than the rate of decrease in bull markets.

9.1 - Volatility as a Factor Exposure

Question 4

Formula

Vega

9.1 - Volatility as a Factor Exposure

Answer 4

Sensitivity of an option’s value
to a change in the underlying asset’s volatility,
holding all other variables constant

v = ∂p/∂σ = SN’(d)√T
vega per basis point = v/100

p = put or call option value
σ = underlying asset volatility
S = underlying asset price
N’(d) = probability density function for the normal distribution at d
T = time to expiration of the option (annual basis => 3 months = T= 0.25)

Obs:
* v >0 long calls and puts / v < 0 short calls and puts
* vcall = vput if same strike, maturity and implied vol
* As the time to expiration approaches zero, the option vega approaches zero
* v high value = at the money

Tip:
* Vega x stock graph => shape of a normal (higher vega = ATM)
* * The higher the implied vol σ, the higher the option price p

Vega of equity index = 0

9.1 - Volatility as a Factor Exposure

Question 5

Formula

Gamma

9.1 - Volatility as a Factor Exposure

Answer 5

Degree of curvature of Delta
* Delta OTM = ~0 => Delta ATM = ~0.5 => Delta ITM = ~1 _/͡͞
* Gamma resembles normal (high point = ATM)

γ = N’(d) / (Sσ√T)
= v / (σ S^2 T)

σ = underlying asset volatility
S = underlying asset price
N’(d) = probability density function for the normal distribution at d
T = time to expiration of the option (annual basis => 3 months = T= 0.25)

Obs:
* γ >0 long calls and puts / γ < 0 short calls and puts
* γcall = γput if same strike, maturity and implied vol
* As the time to expiration approaches zero, the option gamma approaches zero
* γ high value = at the money
* low vol and low time to maturity => higher Gamma

9.1 - Volatility as a Factor Exposure

Question 6

Explain

The main reason
that options sell for a premium
that exceeds their intrinsic value

9.1 - Volatility as a Factor Exposure

Answer 6

Gamma (is the reason)

Because it creates a assymetric return characteristics
For a call option:
* increasing rates of returns for call options as the underlying asset increases in value
* decreasing rates of losses for call options as the underlying stock price decreases

Portfolios with positive gamma:
- Desirable
- tend to resemble a long call or put option
- have negative theta

9.1 - Volatility as a Factor Exposure

Question 7

Contrast

Long volatility
vs
Short volatility

9.1 - Volatility as a Factor Exposure

Answer 7

Long volatility (long vol)
when the investment returns are positively correlated with the volatility level of the broad index.

short volatility (short vol)
when the investment returns are negatively correlated with the volatility level of a broad index (e.g., equity options relative to a broad equity market index).

Long and short volatility is defined based on historical correlations with equity market indices.

All long (short) options have positive (negative) vega. Call options on equity indices that are deep in the money have positive vega but have short volatility because they are negatively correlated with equity market volatility

9.1 - Volatility as a Factor Exposure

Question 8

Explain

How a
long call option
can be a
short volatility position

9.1 - Volatility as a Factor Exposure

Answer 8

A long position in a deep in the money call option has more price influence of the delta than the influence of positive Vega. In this way, considering that the index is negative correlated with volatility, this effect dominates the positive effect of the positive Vega.

This is because a call option has a delta of almost 1, meaning that the call price return will be highly correlated to the index.

A long (short) at-the-money call option is long (short) volatility with respect to the underlying asset’s volatility. However, a deep-in-the money call option near maturity on an equity index has similar return behavior as the underlying equity index. The fifth property of realized volatility previously discussed states that an equity index value is negatively correlated with equity market volatility. Thus, a deep-in-the-money long call on an equity index is short volatility. This occurs because the option’s delta has more of an effect of decreasing the value of the option when the underlying equity index declines than the higher index value from the increased volatility related to the positive vega.

9.1 - Volatility as a Factor Exposure

Question 9

Explain

How
volatility
can be used to
Hedge risk

9.1 - Volatility as a Factor Exposure

Answer 9

Long call options on equity markets,
Because they are long vol
(they increase in value during crisis), and
long equities = short vol.

9.1 - Volatility as a Factor Exposure

Question 10

Explain

Why volatility
is considered a
unique risk factor

9.1 - Volatility as a Factor Exposure

Answer 10

Volatility is not highly correlated with other risk factors.
=> Less than 30%

Volatility products are a unique asset class as investor seek the risk premium linked to the volatility factor.

9.1 - Volatility as a Factor Exposure

Question 11

Explain

How
long vol products have
negative beta and
negative risk premium

9.1 - Volatility as a Factor Exposure

Answer 11

Long vol products are expected to be negative correlated to market index (vol go up in crisis). = negative beta and negative risk premium => return < Rf => VIX = decline in value over time.

Short vol = positive risk premium
Options writers => earn positive risk premium, reflecting the overpriced implied vol (vs realized vol)
Implied vol > Realized in ~90% of calendar quarters examined and volatility premiums are priced at approximately 10% of the level of implied volatility.

9.1 - Volatility as a Factor Exposure

Question 12

Define

Volatility clustering

9.1 - Volatility as a Factor Exposure

Answer 12

large changes (in a price time series)
is followed by large changes,

and small changes
is followed by small changes

Higher probability of staying in the current regime
than switching to a new regime.

9.1 - Volatility as a Factor Exposure

Question 13

List

2 reasons
why volatility strategies
tend to recover losses in less than a year
in most cases

9.1 - Volatility as a Factor Exposure

Answer 13

1. Realized volatility has a high degree of mean reversion
   The high volatility periods are typically short allowing suppliers of long volatility to start recovering losses more quickly. After a major event causes a spike in volatility, the level of volatility quickly dissipates and returning to normal levels if no additional unexpected events occur.
2. Demand for long volatility increases after a crisis event
   This increased demand with a limited supply of traders willing to take short volatility positions results in increased risk premiums for volatility exposure.

9.1 - Volatility as a Factor Exposure

Question 14

Identify

Reason
why volatility mean reversion
cannot be arbitraged

9.1 - Volatility as a Factor Exposure

Answer 14

realized volatility is not directly traded

Interest and inflation rates
also exhibit patterns of mean reversion
that traders are not able to arbitrage

9.1 - Volatility as a Factor Exposure

Question 15

Define

Volatility Skew

and

Volatility Smile (Smirk)

9.1 - Volatility as a Factor Exposure

Answer 15

Volatility skew
shows that options with different levels of moneyness
indicate different levels of implied volatility
‘–
volatility smile (Smirk) :
graphs the volatility skew for a single expiration and is present in a volatility skew when
out-of-the-money options have higher levels of implied volatility than other options.

“put vol > call vol” - Explanation:
Out-of-the-money call options typically have lower implied volatilities than out-of-the-money put options that are trading at a similar distance from the strike price.

The volatility smiles or smirks most likely occur due to
(1) negatively skewed return distributions,
(2) institutional investor demand for downside risk protection associated with long put options in the equity market, or
(3) negative correlation between returns and volatility changes.

9.1 - Volatility as a Factor Exposure

Question 16

List

4 reasons
that implied volatilities are different
for options with the same underlying asset
but different strike prices and maturity dates:

9.1 - Volatility as a Factor Exposure

Answer 16

1. Non-normal return distribution -
   Implied volatilities will vary even if expected realized volatility is constant over different time horizons. When return distributions are not normally distributed and are skewed or have non-zero excess kurtosis, variance measured by realized volatility does not sufficiently describe the return distribution and implied volatilities could differ from expected realized volatilities.
2. Seasonal effects or other expectations result in time-varying volatility in the underlying asset’s returns. Earnings announcements, holiday periods, or year-end tax effects are examples of seasonal effects impacting implied volatility.
3. The frequency of time used to measure volatility (days or weeks) or the length of time (different option maturities) changes the volatility due to the autocorrelation of returns of the underlying asset for different time series. The equity market results in positive (momentum or trending) or negative (mean reverting) autocorrelations based on the time interval.
4. Time-varying risk premiums - Changes in risk aversion results in differences between implied and expected realized volatility. Implied volatilities are typically lower than the expected realized volatility because a long option position results in a positive beta. Time-varying risk premiums result in differences in implied volatility structures and surfaces.

9.1 - Volatility as a Factor Exposure

Question 17

List

2 reasons
for high implied volatility
in out-of-the-money puts

9.1 - Volatility as a Factor Exposure

Answer 17

1. the negative return skewness in the underlying equity index—this negative skewness could be the result of high probabilities of an unfavorable market crash occurring; or
2. the significant risk premium for writing DOOM puts, given the demand from option writers who are willing to take the short position exposure.

9.1 - Volatility as a Factor Exposure

Question 18

Volatility diffusion risk and Volatility jump risk

Answer 18

Volatility diffusion risk represents continuous accrual of small changes in the volatility over time.

Volatility jump risk represents the sudden large periodic changes in volatility level

LO 9.1.4