VaR and CVaR implied in
option prices: the oil
market
Giovanni Barone-Adesi
The Swiss Finance Institute
Università della Svizzera italiana
IRMC, Hebrew University, Jerusalem, June 14 2016
baroneg@usi.ch

The limits of statistical risk
measures
• Risk measures are traditionally based on the application of economic
and statistical theory to a sample of observations.
• Therefore, they are based on past observations. They do not
necessarily reflect current risk. To remedy that, they can be extrapo-
lated into the future through a time-series model, such as a GARCH.
• Most of the time this procedure leads to risk estimates supported by
new data. However breaking points in the time series, likely to occur
in market crashes, can be identified only ex-post.
• Risk managers may be left without a working model when they need
it most.

Advantages and problems of market-based risk
measures
• Market-based risk measures are based on current derivative prices.
• Measures as VIX and SKEW are increasingly popular.
• The fact that VaR and CVaR, the most common measures in risk
management, look at the left tail of possible return outcomes allows
them to be related to out-of-the money options.
• By doing so, investors’beliefs on future risk are directly elicited,
without statistical estimations.
• These relationships are not used much currently because they seem
to require a change in probability measure and the specification of an
option pricing model. In practice, that is often unnecessary.

Problems with VaR and CVaR
• VaR is defined as a quantile of the distribution of possible returns.
• CVaR as the conditional expectation computed on the left of VaR.
• VaR is relatively easy to estimate, but it provides no information on
the distribution of largest losses. It is not subadditive, creating some
problems in portfolios.
• CVaR is a coherent risk measure, but it is hard to make good
inferences about it. The inference problem is complicated by the
scarcity of observations on the tail of the distribution.
• Option-based measures bypass inferencing. To illustrate how, start by
defining VaR and CVaR under the pricing measure.

VaR as option strike price

CVaR as option price

Pricing and objective probability measures
• Risk managers are concerned with future losses, not with market
prices associated with them. The objective, or historical measure, is
appropriate to study their distribution.
• Option prices are current values of future outcomes. They are often
computed under a different measure, consistent with risk neutrality
(pricing measure). That is convenient and it does not affect the option
price.
• Option-based risk measures should be adapted to the objective
measure for consistency. The adaptation is mechanical if both
measures are known. That is often not the case, but luckily it does not
matter much in many applications of interest.

Changes of distribution under continuous
or discrete probability measures
• If both distributions are known and absolutely continous the Radon-
Nikodym theorem (or Girsanov’s theorem in case of diffusion
processes) allows to move from one to the other.
• If both distributions are discrete, the change of distribution is given
for each outcome by the state price compounded at the risk-free rate
divided by the objective probability.
• The above results are well-known. Unfortunately, in practice we see
only option prices in the market. We do not know objective
probabilities or Radon Nikodym derivatives, unless we choose a
model. Our model choice undermines the robustness of our results.

When is the change in measure
relevant

Numerical examples
• The example in the paper shows the computation of VaR and CVaR
under both measures in detail.
• The results for the oil market, which crashed over the last two years,
our second example, evidence also the changing skewness and
kurtosis of the return distribution under the pricing measure.
• Our results suggest that the hypothesis of a return distribution of
constant form under the physical measure and constant risk aversion
is untenable.

VaR/CVaR in the oil futures market, 2014-
2015
(3-month horizon)

The changing intensity of the jump
process
•

Conclusions
• Option prices have been shown to convey substantial information about VaR and
expected shortfall.
• They do not rely on past observations, which in any case do not lead to good
inferences about expected shortfall.
• The pricing measure differs from the objective measure because of risk
premiums. The difference vanishes at shorter horizons, commonly used in the
management of market risk.
• Inferences from options allow for the study of the changing shape of the
distribution of security returns. Investors appear to move the probability mass
toward the extremes only at a few dates, characterized by great uncertainty.
Otherwise, the ratio CVaR/VaR moves slowly through time. That is consistent
with a GARCH,to which a jump process is added only occasionally. Contrary to
current models, it appears to rule out the possibility of large jumps most of the
time.