Forward-looking estimators of risk measures

A first idea to obtain forward-looking estimators of our risk measures is the use of option-implied moments. The most prominent example of such a moment is the Black–Scholes implied volatility, which goes back to Latan´e and Rendleman (1976). It is obtained by inverting the Black–Scholes option pricing formula to back out the volatility parameter using observed option prices. Implied volatility is forward looking because it captures the expectations of market participants about future volatility. Moreover, it uses only current price information. The major disadvantage, however, is its dependence on a specific pricing model, the Black–Scholes model, which might not be adequate for a specific market and a specific time. To overcome this problem, the concept of model-free implied volatility has been developed (Britten-Jones and Neuberger (2000); Jiang and Tian (2005)), following the idea that complete markets allow for the recovery of the whole (risk-neutral) price distribution from observed option prices. Based on the same idea, the concept of model-free implied volatility has been extended to higher-order moments of the price distribution, such as implied skewness and implied kurtosis (Bakshi, Kapadia, and Madan (2003);

Neuberger (2012)).

If certain moments of the price distribution were unambiguously related to certain risk measures from the previous section, we could immediately apply model-free implied moment estimators. Intuitively, one could expect that higher variance has an increasing effect on the overall risk measure, greater (positive) skewness increases the probability and magnitude of a large positive price move, and higher kurtosis leads to a higher probability and magnitude of large price moves in general. However, as shown in the Appendix, these relations are not straightforward, because the resulting effects depend on the threshold levelA. Therefore, we follow a different route and develop direct model-free option-implied estimators of our risk measures.

The derivation of implied estimators for our risk measures follows the same idea that underlies model-free implied moments. The starting point is the assumption of complete markets. If markets are complete, we can apply the principle of risk-neutral valuation, which states that the price of any derivative equals its discounted expected payoff, using

risk-free interest rates and risk-neutral probabilities.¹¹ It is then our goal to express all risk measures in terms of the expected payoffs of portfolios of derivatives written on the commodity price. The (compounded) prices of these portfolios ultimately deliver the desired model-free implied estimates of the risk measures.

Consider the overall risk measureOM first. It can be written as the sum of two expecta-tions:

OM = E_t

|S¯|

= E_t[max[S_t+τ −K, 0 ] ] +E_t[max[K−S_t+τ, 0 ] ], (4.4) where K equalsE_t [S_t+τ], the expected price at t+τ. Equation (4.4) shows that OM is just the sum of the expected payoffs of a call option and a put option with the same strike priceK. It follows that

OM^imp = e^rτ[C(τ, K) +P(τ, K) ] (4.5) is the corresponding implied estimator, wherer denotes the risk-free interest rate for the period from t tot+τ, C(τ, K) is the price of a call option with time to maturity τ and strike priceK at time t, and P (τ, K) denotes the corresponding price of a put option.

Next, we express the probability of a large price move in terms of expected payoffs:

p_l = E_t

1_{{St+τ> K+A}}

+E_t

1_{{St+τ< K−A}}

, (4.6)

where 1{St+τ> K+A} (1{St+τ< K−A}) is an indicator function that takes a value of one if S_t+τ > K+A (S_t+τ < K−A) and zero otherwise. The indicator function 1_{{St+τ> K+A}} describes the payoff of a digital option that pays one currency unit if the priceS_t+τ exceeds the value of K +A and pays nothing otherwise. We use the expression “digital call”

for such a digital option because a payment occurs if prices are above a specific level.

Accordingly, we use the expression “digital put” for a digital option that makes a positive

11We return to the issue of potential risk adjustments that transform our risk measures into the corresponding real world risk measures in a later section.

payment if prices are below a certain level. The second term on the right-hand side of equation (4.6) is the expected payoff of such a digital put with strike price K−A.

Equation (4.6) therefore suggests the following implied estimator for the probability of a large price move:

p^imp_l = e^rτ

D^C(τ, K +A) +D^P (τ, K−A)

, (4.7)

where D^C(τ, K +A) andD^P(τ, K −A) are the prices of the corresponding digital call and put options, respectively. Digital options on commodity prices are usually not traded in liquid markets, which means that market prices are not directly available. Such options can be well approximated, however, by long and short positions of plain vanilla calls and puts. Consider a portfolio that consists of 1/k plain vanilla call options with strike price K+A and −1/k call options with strike price K +A+k. If k goes to zero, the payoff function of this portfolio converges to the payoff function of a digital call with strike price K+A. A digital put can be similarly approximated by a portfolio of plain vanilla put options.¹²

In a next step, we derive an implied estimator of the magnitude of large price moves. The risk measure LM_A can be written as:

LM_A=E_t

Equation (4.9) shows that LM_A equals the expected payoff of an options portfolio with four components. The first one is an out-of-the money call with strike priceK+A. The second one consists of a number ofAdigital call options with strike priceK+A. These two components capture the magnitude of large positive price moves. The last two components refer to large negative price moves. They consist of a plain vanilla put option with strike

12We use such an approximation of digital options withk= 0.001 in the empirical part of this article.

priceK −A and a number ofA digital put options with the same strike price. Finally, one has to divide by the probability of a large price move occurring because the measure is a conditional expectation. The resulting implied estimator reads

LM_A^imp = e^rτ C(τ, K +A) +A·D^C(τ, K+A) (4.9) +P (τ, K−A) +A·D^P(τ, K −A)

/p^imp_l .

An implied estimator of the magnitude of normal price moves (N MA) is obtained from equation (4.2) by applying the three estimatorsOM^imp, p^imp_l , and LM_A^imp. The resulting estimator becomes

N M_A^imp = OM^imp−p^imp_l LM_A^imp

/(1−p^imp_l ). (4.10)

Finally, we provide estimators of the magnitudes of positive and negative large price moves, respectively, and the probability that a large price move is positive. We write the risk measureLM_A+ as

LM_A+=E_tS¯|S > A¯

= (E_t[max[S_t+τ −(K +A), 0 ] ] (4.11) +E_t

A·1{St+τ> K+A}

/(p+|l·p_l)

and the probability that a large price move is positive as p+|l = E_t

1_{{St+τ> K+A}} E_t

1{S_t+τ> K+A}

+E_t

1{S_t+τ< K−A}

. (4.12)

Then the corresponding estimators are

LM_A+^imp = e^rτ C(τ, K+A) +A·D^C(τ, K +A)

p_+|^imp_l ·p^imp_l (4.13)

and

p_+|^imp_l = D^C(τ, K +A)

D^C(τ, K +A) +D^P (τ, K−A). (4.14) Using the relation in equation (4.3), we obtain an implied estimator of the magnitude of large negative price moves (LMA−) from the previous estimators as

LM_A−^imp =

LM_A^imp−p_+|l^impLM_A+^imp

/(1−p_+|^imp_l). (4.15)

In summary, we have shown that all risk measures can be expressed as the expected payoffs of portfolios of plain vanilla and digital options. The corresponding compounded prices of these options portfolios therefore provide implied estimates of the risk measures. These estimates are model free, in the sense that they do not rely on the validity of a specific option pricing model.