VaR and CVaR implied in option prices: the oil

marketGiovanni Barone-Adesi

The Swiss Finance InstituteUniversità 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.