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Summary Topic 9 CAIA Level 2 summaries
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Topic 9: Volatility and complex strategies9.1 Volatility as a factor exposure
9.1.1 Measures of volatility
Investors are familiar with common positions with returns that are tied to the directional re-
turns of underlying real or financial assets. This session discusses products and strategies with
returns that are tied to changes in levels of asset volatilities, correlations, or dispersions. Fur-
ther, volatility itself is increasingly being considered as an important factor in investment and
risk management.
Implied return volatility is the volatility over the remaining life of an option that is inferred
from an option price under assumptions including risk-neutrality, the validity of the specified
option pricing model, and the accuracy of the model’s inputs other than volatility. Further,
the implied volatility does not generally represent an accurate and unbiased consensus of the
realized volatility expected by market participants due to risk premiums. The realized return
volatility of an asset is the actual variation (typically measured as the standard deviation of
returns) that occurs over a specified time period using a specified return measurement inter-
val (e.g., daily or weekly return granularity). In practice, observed returns are discrete, not
continuous, and all three of the properties of a GBM process (homoscedastic, normally dis-
tributed, and uncorrelated returns) are violated in practice.
As standard deviations, realized return volatilities are simply single estimated measures de-
scribing the dispersion of a frequency distribution of a sample of outcomes. Assets with simi-
lar realized return volatilities may have experienced tremendously different returns. Note
these three limitations on realized volatility: (1) realized volatility does not describe the shape
of the return distributions; (2) assets with identical realized volatilities may differ with respect
to whether their underlying returns exhibited trending, mean-reversion, or minimal autocor-
relation; and (3) realized volatility does not describe whether the dispersion primarily oc-
curred near a particular price of the underlying asset or during a particular time period within
the sample. Most volatility strategies will perform very differently across these six assets even
though all of the assets experienced the same realized volatility.
The following summarizes the key observations.
1. Realized volatility is not constant. Realized volatility slowly mean-reverts and clusters.
As such, many traders model volatility using a variety of generalized autoregressive
conditional heteroscedasticity (GARCH) and regime switching approaches.
2. Realized volatility tends to stay low for some extended period of time until a market
shock occurs and volatility transitions to a higher level for some period of time.
3. The volatility of short-term changes in realized volatility can be high, but in the long
run, volatility tends to revert toward some long-term average level.
, 4. Empirical and laboratory evidence suggests that higher volatility increases investors’
risk aversion. This indicates that higher realized volatility tends to be negatively corre-
lated with returns on most risky assets.
5. Equity market realized volatility tends to increase in bear markets and decrease in bull
markets. In addition, a decline in the stock price of a firm increases its leverage and
riskiness, resulting in higher volatility.
6. The rate at which realized equity volatility rises in bear markets exceeds the rate at
which realized volatility falls in bull markets.
9.1.2 Volatility and the Vegas, Gammas and Thetas of options
The risk exposures of various option strategies and option portfolios are often expressed us-
ing the Greeks.
Vega
Vega indicates the sensitivity of an option value to a change in the volatility of the option’s
underlying asset. Strictly speaking, the vega of a portfolio describes the response of the port-
folio’s model value to a change in the volatility of the underlying asset assumed within the
model while holding all other variables constant. In practice, vega may be somewhat loosely
viewed as measuring the response of the current price of a portfolio to a change in the mar-
ket’s anticipated volatility of the returns of the portfolio’s underlying asset. Some traders may
seek to earn a volatility risk premium by implementing positions that short vega.
Where p is the option value (call or put), σ is the volatility of the underlying asset, N’(d) is the
(noncumulative) probability density function of the normal distribution at d, which means
that N’(d) is the density function (where d is the same as d as discussed in Level 1) and T is the
option’s time to expiration or tenor.
there is an important scaling issue involved in actual vega usage. While Equation 1 expresses
the “textbook” vega (i.e., the actual partial derivative of an option value with respect to its
volatility), practitioners adjust the formula in Equation 1 to scale vega to indicate the risk of a
one basis point change in volatility. In other words, practitioners and financial data providers
implicitly report vega as SN'(d)√T/100; a measure that might better be described as vega per
basis point.
Vega, as shown in Equation 1, measures the instantaneous rate of response of an option value
to a change of one full unit in volatility, such as a change from 0.20 (i.e., 20%) to 1.20 (i.e.,
,120%). Vega per basis point divides the vega in Equation 1 by 100 so that it measures the
instantaneous rate of response of an option value to a change of one basis point in volatility,
such as a change from 0.20 (i.e., 20%) to 0.21 (i.e., 21%). If the vega per basis point of an
option is $0.20 it means that for an infinitesimal increase in its implied volatility, the option’s
value will rise at the rate of $0.20 for every basis point of implied volatility.
Assuming that N’(d) is 0.20 for both options, the “textbook” vega of both options (based on
Equation 1) would be $50*0.20*√0.25 or $5.00. The much more common measure of vega
would be $0.05, which is the vega per basis point found by dividing the “textbook” vega by
100. Each option would rise towards a value increase of $0.05 (i.e., $5.00 × 0.01) as the op-
tion’s implied volatility rose towards an increase of 0.01 from, say, 0.25 to 0.26.
The previous paragraph described that a value movement “would rise towards a value in-
crease of $0.05” rather than simply saying that the value would move by $0.05. The reason is
that the relation between value and volatility is nonlinear, so vega (as a partial derivative) is
only a precise measure when dealing with infinitesimal shifts. For a large (discrete) shift,
higher-order derivatives would be necessary in order to generate an accurate approximation.
For the purposes of this section, higher-order effects are ignored. Equation 2 depicts a linear
(first-order) approximation that illustrates the use of vega to estimate option value changes.
Viewing ν in Equation 2 as “vega per basis point”, for a vega of $0.30, a change in volatility of
0.02 (e.g., two basis points from 0.20 to 0.22) would cause a call or put to rise in value by
approximately $0.60 ($0.30 × 2).
1. Vega is always positive for a long position in a call or put option because all three terms
on the right side of Equation 1 are positive.
2. The vega of a call and a put with the same underlying asset, strike price, time to expi-
ration, and implied volatility must be equal because they share the same formula for
vega.
3. The vega of an option approaches zero as the time to expiration approaches zero, as
seen in Equation 1 with T approaching zero.
4. The vega of an option approaches zero as the value of the underlying asset approaches
zero or infinity. Therefore, vega approaches zero for deep in-the-money and out-of-
the-money options because the normal probability density function, N’(d) in Equation
1, approaches zero going out either tail of the normal distribution.
The differences in the time value of the three call options (i.e., the excess of the call option
values above the lower bound) are primarily driven by volatility and time to expiration
through the quantity σ√T. In other words, the three curves may be viewed as indicating dif-
fering tenor, differing implied volatility, or a combination.
, A shift upward or downward in volatility can be viewed as moving the option value to a higher
or lower curve in the previous exhibit. As time passes, everything else equal, option values
will move downward to lower curves.
Gamma
An option’s gamma is the second-order partial derivative of its value with respect to the value
of the underlying asset (i.e., it is also the first-order partial derivative of an option’s delta with
respect to the value of the underlying asset since delta is the first-order partial derivative of
the option’s value with respect to the value of its underlying asset). Gamma is the same for a
call or put option with the same strike price, tenor, and underlier.
Gamma and vega must share the same positive sign for simple calls or puts. The long volatility
exposure of a long option position is represented by its positive gamma; short option posi-
tions are short volatility and have negative gamma.
Graphically, the gamma of an option indicates the degree of curvature in the relation be-
tween option price and the price of the underlying asset. Note in the previous exhibit that the
lowest curve has sharp curvature near the money—indicating a high value of gamma. The
highest curve is less sharp and indicates low (but positive) gamma. All three curves have
greater gamma near the money and little or no gamma far into or out of the money.
Gamma is the degree of nonlinearity of long positions in options with respect to the price of
their underlying asset (i.e., is a measure of the degree of curvature) that provides call option
owners (who are long gamma) with the highly desirable combination of experiencing increas-
ing rates of gain as the underlying asset moves up and decreasing rates of loss as the
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