CAIA L2 - 9.1 - Volatility as a Factor Exposure Flashcards
Contrast
Implied
vs.
Realized Volatility
9.1 - Volatility as a Factor Exposure
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. 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
List
3 limitations
of realized volatility
as a measure of dispersion
9.1 - Volatility as a Factor Exposure
realized return volatility = ESTIMATES
- It does not define the return distribution’s shape (assets with similar realized vol may experience very different actual returns)
- Even when assets have identical realized volatility, they could differ if their returns are trending, mean reverting, or exhibiting minimal autocorrelation.
- 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
List
6 properties
of realized volatility
9.1 - Volatility as a Factor Exposure
- 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.
- Realized volatility is normally low until a market shock and then volatility rises for an extended time period.
- Realized volatility can exhibit large variability in the short term; however, in the long term, volatility levels are near long-term averages.
- Higher realized volatility is negatively correlated with risky asset returns because investor risk aversion increases at higher volatility levels.
- 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.
- Realized equity volatility rises faster in bear markets and falls slower in bull markets.
9.1 - Volatility as a Factor Exposure
Formula
Vega
9.1 - Volatility as a Factor Exposure
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
Dica:
* Gráfico de vega x stock => formato de uma normal (maior vega = ATM)
* Quanto maior a vol implícita σ, maior o preço p da opção
Vega of equity index = 0
9.1 - Volatility as a Factor Exposure
Formula
Gamma
9.1 - Volatility as a Factor Exposure
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
Explain
The main reason
that options sell for a premium
that exceeds their intrinsic value
9.1 - Volatility as a Factor Exposure
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
Contrast
Long volatility
vs
Short volatility
9.1 - Volatility as a Factor Exposure
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.
9.1 - Volatility as a Factor Exposure
Explain
How a
long call option
can be a
short volatility position
9.1 - Volatility as a Factor Exposure
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
Explain
How
volatility
can be used to
Hedge risk
9.1 - Volatility as a Factor Exposure
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
Explain
Why volatility
is considered a
unique risk factor
9.1 - Volatility as a Factor Exposure
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
Explain
How
long vol products have
negative beta and
negative risk premium
9.1 - Volatility as a Factor Exposure
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.
9.1 - Volatility as a Factor Exposure
Define
Volatility clustering
9.1 - Volatility as a Factor Exposure
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
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
-
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. -
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
Identify
Reason
why volatility mean reversion
cannot be arbitraged
9.1 - Volatility as a Factor Exposure
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
Define
Volatility Skew
and
Volatility Smile (Smirk)
9.1 - Volatility as a Factor Exposure
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
