An important purpose of derivatives modelling is to provide practitioners with actionable measures of risk. The Black and Scholes volatility remains a favourite on trading floors in spite of well-known biases. One popular extension is to make volatility a function of time and the underlying asset price, as in local volatility models. This paper presents an alternative extension, which produces volatility-like quantities to address the skews and smiles found in most derivatives markets.higher-order volatility; higher-order moments; volatility smile; S&P 500

Carey, Alexander

English

Research Papers in Economics

Page 1 of 14 
 
HIGHER-ORDER VOLATILITY 
 
 
 
 
 
Alexander Carey*
 
 
4 avenue de la Guillemotte 
78112 Fourqueux, France 
alex.carey@wanadoo.fr 
 
 
December 1, 2005 
 
 
 
 
* Alexander Carey is a graduate of Cass Business School, London. He has worked for Salomon Smith Barney 
New Zealand. 
 
 
 
 
An often overlooked, but nonetheless important purpose of derivatives modelling 
is to provide practitioners with actionable measures of risk, the “thinkable quantities” 
that Emanuel Derman has referred to.
1 Dollar prices generally convey little information in 
the world of derivatives, and option pricing models are used — and abused — to convert 
them to and from a view on the market. 
The historical risk measure, the Black and Scholes (1973) volatility, remains a 
favourite on trading floors in spite of well-known model-inconsistent biases, embodied in 
an implied volatility skew or smile. One popular approach to addressing these biases has 
been to make volatility a function of time and the underlying asset price, as in the local 
volatility models of Dupire (1994), Derman and Kani (1994) and Rubinstein (1994). This 
offers a model-consistent fit to market prices, without introducing fundamentally new or 
overly esoteric quantities into the risk interface, as can happen with other models. 
In this paper, we present an alternative extension of volatility. Working first with a 
general stochastic process, we define a sequence of statistical parameters with an 
                                          
1 Emanuel Derman, “A guide for the perplexed quant”, Quantitative Finance 1(5) (September 2001), page 477. Page 2 of 14 
intuitive gambling interpretation. We then derive moment formulae for the case when 
they are deterministic. Applied to a generic market quantity, the resulting risk interface 
features the familiar Black-Scholes handle on the variance of the underlying, along with 
“higher-order” analogues which capture departures from lognormality while retaining the 
look and feel of the original quantity. We provide snapshot implied values for the S&P 
500 index options market. 
 
 
1. j -TH ORDER VOLATILITY 
 
We begin by considering a general adapted process () t X  on a filtered probability 
space  () () Ω FF ,, , t Q . Let δ > 0 t  denote a finite period of time, and define 
δ δ + =− tt t t XX X , so that the relative change of the process over the interval t  to 
δ + tt  reads δ tt XX . We let  ⋅ t E  denote expectation conditional on  t F , and j  is a 
generic positive integer. 
Consider an agreement by which two parties undertake to exchange the amount 
() δ
j
tt XX , as yet unknown but to be revealed imminently, and a predetermined amount 
which we tentatively write in the form () δ Σ× ,
j
jt t . We suppose that both parties are in 
possession of the information    Ft , and that given this information, the agreement is a fair 
gamble under the probability measure Q , by which we mean: 
 
() , 0
j j
tt t jt EX X t δδ ⎡⎤ −Σ = ⎢⎥ ⎣⎦
. 
 
This leads us to formally define the quantity Σ , jt via the identity: 
 
() δδ Σ≡ ,
j j
tt t jt EX X t , 
 
with the convention that Σ , jt equals the nonnegative root when j  is even. Page 3 of 14 
For an intuitive interpretation of this quantity, consider the following 
extrapolation. Assume for convenience that  δ ×= 1 mt  for some integer m , and 
consider a second agreement by which the original arrangement is extended to m  
successive periods, but with the predetermined side remaining a fixed  δ Σ ,
j
jt t  per period. 
In other words, the parties agree to exchange an unknown ( ) δ
j
uu XX  and a 
predetermined  δ Σ ,
j
jt t  for each  δδ =+ + − , ,..., 1 ut t tt t , with the unknown leg to be 
revealed imminently.
2 Observing that the predetermined leg sums to  δ ×Σ =Σ ,,
jj
jt jt mt, 
suppose next that the process variable has changed exactly once between times t  and 
+ 1 t . Then if the relative change equals Σ , jt, the unknown leg equals the 
predetermined leg and the parties are even. Thus, for such an agreement the quantity 
Σ , jt can be viewed as a break-even relative change. We shall refer to Σ , jt as  j -th order 
finite-period volatility. 
Of particular interest will be the limit of vanishingly small δt , for which we define 
the quantity σ , jt via: 
 
δ σ ≡Σ 20 ,, lim
jj
t jt jt, 
 
again with σ , jt nonnegative when j  is even. Equivalently,  δ σ =Σ 2 ,0 , lim jt t jt. We shall 
refer to σ , jt as j -th order instantaneous volatility. 
 
 
2. MOMENTS 
 
Now in the general case the volatilities we have defined are stochastic. In the 
appendix we show that when the finite-period volatilities of orders one to n  exist and are  
                                          
2 This variation on the first agreement need not itself be a fair gamble. Page 4 of 14 
deterministic, the n -th conditional moment of the process can be obtained as: 
 
  τ δ +
=
⎡ ⎤ ⎛⎞ ⎟ ⎜ ⎢ ⎥ ⎟ =+ Σ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎜ ⎝⎠ ⎣ ⎦
∑ ∏ ,
1
1
n
j nn
tt t ju
u j
n
EX X t j
, (1) 
 
for any positive multiple τ  of δt , where the product is over  δτ δ =+ + − , ,..., ut t tt t . 
If we next suppose that there exists a Δ>0 such that this assumption holds for every 
δ ≤Δ t , then clearly the instantaneous volatilities, when they exist, are also 
deterministic. Further, fixing τ > 0 and taking the limit δ 2 0 t  in (1) yields: 
 
  τ τσ +
=
⎡ ⎤ ⎛⎞ ⎟ ⎜ ⎢ ⎥ ⎟ = ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎜ ⎝⎠ ⎣ ⎦
∑
1
exp
n
j nn
tt t j
j
n
EX X j
, (2) 
 
where the σj , which are assumed to exist, are defined via: 
 
,
1 t jj
jj u t
du
τ
σσ
τ
+
≡ ∫ , 
 
again with the convention that σj  is nonnegative when j  is even. We shall refer to σj  as 
j -th order average volatility.
3 We emphasise here that we have not established that 
deterministic instantaneous volatilities imply (2) in and of themselves. However, since 
the upper bound Δ in the above argument can be chosen arbitrarily small, this would 
seem to be more of a technical than a conceptual issue. 
It is clear from (2) that the n -th moment of the process at time t  is fully 
determined by the average volatilities of orders one to n . Thus, first-order volatility σ1 
determines the first moment, and we may take the view that given σ1, second-order 
volatility  σ2 governs the second moment, and so on. It is also apparent that if σ = 0 j  
for every  > 2 j , then  () τ + ln tt XX  is normally distributed (with mean ( ) σσ τ − 2 1
12 2  
and variance στ 2
2 ).
4 Thus, nonzero values for higher-order average volatilities indicate 
                                          
3 Admittedly a misnomer since it is the  j -th power that is averaged. 
4 We caution that an ad-hoc, higher-order truncation of the sequence of volatilities may not be compatible with 
any probability measure. Page 5 of 14 
and quantify deviations from the lognormal distribution. In particular, σ3  and σ4  govern 
auxiliary skewness and kurtosis. 
 
 
3. AN IMPLEMENTATION 
 
We now take Q  to be an equivalent martingale measure, ()   Ft  to be the market 
information structure, and () t X  to be an adapted market process. For expediency we 
limit ourselves to the basic equity setting, with  t X  as the price at time t  of a stock or 
index, with the money-market account as numeraire, and with the instantaneous 
dividend yield and riskless rate of interest both deterministic. In this context, it is easily 
verified that first-order volatility σ1,t  equals the instantaneous cost of carry, which is 
itself deterministic. 
In what follows, we shall treat the volatilities exclusively as attributes of the 
market measure, to be implied from a set of option prices, rather than as inputs to a 
pricing scheme. We begin with the assumption that the volatilities of orders two through 
n  are deterministic as per the previous section, and thus that the first n  moments have 
the form (2) (no assumption is necessary regarding higher orders). Note here that the 
first two model moments are now identical to those of the Black-Scholes model, with 
second-order volatility in the role of Black-Scholes volatility. We then propose to 
substitute a known distribution in place of the unknown one, fit it to the option prices, 
and compute the volatilities from the fitted statistics. For this it will prove convenient to 
use the canonical Merton (1976) jump-diffusion, which offers both a ready-to-use pricing 
capability and a reasonably good fit to market prices. 
Under the cost-of-carry parameterisation of Bates (1991), the asset price 
experiences the usual standard diffusive innovations with coefficient σ t X , along with Page 6 of 14 
Poisson-driven jumps of size κ t X  at the rate λ. Here σ  and λ are nonnegative 
constants, and  ( ) κ + ln 1  is normally distributed with mean γδ − 2 1
2  and variance δ2 . 
Letting  n m  and  ′ n m  denote the n -th moments (about zero) of κ and  κ + 1  
respectively, we have  γ =− 1 1 me  and  () [] γδ ′ =+ − 2 1
2 exp 1 n mn n n , and the n -th 
conditional moment of the process reads: 
 
  () () () [] τ λτ σ τ λ τ + ′ =− + − + − 2 1
1 2 exp 1 1 nn
tt t n EX X n b m n n m , (3) 
 
where b  is the instantaneous cost of carry. For convenience we recall the Merton (1973) 
version of the Black-Scholes formula for the price at time t  of a plain vanilla option: 
 
() () 12
rb
t BSM e e X N d KN d ττ εε ε − ⎡ ⎤ =− ⎣ ⎦ , 
 
with: 
 
() () ντ
ντ
++
=
2 1
2
1
ln t XK b
d ,   ντ =− 21 dd , 
 
where K  is the exercise price, r  is the rate of interest, τ  is the time to expiration, ν  is 
the volatility, N ⋅ is the standard normal distribution function, and ε equals one for a 
call option, negative one for a put. The price of the same option under the jump-diffusion 
can then be written: 
 
()
0 !
n
n
n
e
BSM
n
λτ λτ
∞ −
= ∑ , 
 
where  n BSM  is the Black-Scholes-Merton formula, but with cost of carry 
1 n bbm n λγ τ =− +  and volatility  22
n n νσ δ τ =+ .
5 As for the relationship between 
                                          
5 This formula, found in Bates (1991), differs superficially from the one in Merton (1976), which involves the 
ordinary Black-Scholes formula, and applies only when the cost of carry equals the rate of interest. In the latter 
case, the two formulae are of course strictly equivalent. Page 7 of 14 
the jump-diffusion parameters and the volatilities, it is easily verified that (2) agrees with 
(3) when: 
 
σσ λ == ≥ =+ + 2
12 2
j
jjj j j bm 11 1 , 
 
where  ⋅ 1  is the indicator function, and  j m  can be computed from the  ′ k m  via: 
 
() −
=
⎛⎞ ⎟ ⎜ ⎟ ′ =− ⎜ ⎟ ⎜ ⎟ ⎜ ⎝⎠ ∑
0
1
j
jk
jk
k
j
mm
k
. 
 
 
4. AN EXAMPLE 
 
For an example we turn to the S&P 500 (European-exercise) index options from 
the Chicago Board Options Exchange. The option price data is sourced from the 
exchange’s web-based quote service on May 13, 2005, time-stamped 14:43 ET (fifteen 
minutes delayed), and consists of a snapshot of bid-ask quotes for all call and put 
contracts with under one year to expiration. The index level is 1,152.66. We discard all 
strikes for which the bid of either option is less than one-half a point, upon which for 
each contract the bid and ask are averaged to produce a price estimate. Rates of interest 
are linearly interpolated from the most recent Libor curve. For each expiration, the 
estimation of the jump-diffusion parameters (cost of carry included) is carried out by 
minimizing the sum of squared errors between market and model prices. Table 1 reports 
the estimated jump-diffusion parameters, and table 2 the corresponding average 
volatilities up to order four. Figure 1 shows the market price data in the form of implied 
Black-Scholes volatilities (computed with the cost of carry from table 1), and figure 2 
plots the implied average volatilities as per table 2. 
  Four aspects of these results stand out. The first is that the volatility estimates of 
orders one and two are broadly as expected, given that they correspond respectively to Page 8 of 14 
the cost of carry and to the Black-Scholes volatility.
6 A second feature is that the two 
higher-order volatilities show themselves to be of the same order of magnitude as 
second-order volatility, making good on their initial promise. In absolute value, all three 
are within a two-percent range at the 18-week mark. A third feature is that third-order 
volatility is markedly negative, which is to be expected since equity indices tend to drop 
more sharply than they rise. Fourth and finally, the third and fourth-order volatility 
estimates show a marked expiration dependence, with sharply lower absolute values at 
the short end compared with the long one. On this feature we limit ourselves to two 
comments. First, our assumption that the volatilities are deterministic is clearly 
counterfactual, and should be expected to result in parameter bias. This is the unhappy 
lot of most financial models. However, as our second comment we ask whether it is 
reasonable to expect perfect rationality from derivatives markets. For example, the 
writing of short-dated, out-of-the-money vanilla options is an ordinarily profitable 
operation, which could lead some participants to take chances, wittingly or not, for 
comparatively less remuneration tha n i s de m a nde d a t  l o nge r  e xpi r a tio ns. Th i s ki nd o f 
misjudgement is all the more plausible in the absence of suitable risk metrics beyond 
those measuring and pricing ordinary market variability. 
 
 
5. CONCLUSION 
 
  We have introduced a set of risk measures which translate and convey the 
information in option market prices in a new way. Whether skew and smile exposures 
can be managed effectively via these quantities remains to be seen. The primary function 
that is envisaged for them is as an alternative to implied volatility surfaces for the 
                                          
6 To be precise, under our estimation methodology the second-order volatility estimate is unchanged whether 
we assume zero average volatilities for  2 j >  (tantamount to the Black-Scholes specification) or not. Page 9 of 14 
monitoring of market conditions. This risk interface is suitable for other products besides 
equity derivatives. It applies in a straightforward way to foreign exchange options, as 
well as interest-rate caplets and swaptions, subject to the standard parameterisations 
and assumptions. 
  In the implementation of this interface, we have treated the volatilities as 
attributes of the market measure, and presented snapshot estimates for the first four. A 
potential next step is to investigate the volatilities as inputs into a pricing scheme. While 
those of order higher than four can be expected to have an impact on valuations, it could 
be that for some purposes, they need not be known with precision. A number of models 
offer enhanced versions of the Black-Scholes formula based on higher-order moments, 
for example the Edgeworth expansion of Jarrow and Rudd (1982). Page 10 of 14 
APPENDIX 
 
To establish (1), we note that: 
 
  () () () τ δδ +
=
⎡ ⎤ ⎛⎞ ⎟ ⎜ ⎢ ⎥ ⎟ =+ =+ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎜ ⎝⎠ ⎣ ⎦
∑ ∏∏
1
11
n
nn j
t t uu uu
uu j
n
X X XX XX j
, (A1) 
 
using the binomial theorem. Now  δτ δ ++ − ⋅= ⋅ ... tt t t t t EE E E , and assuming that the Σ ,
j
ju 
exist and are deterministic, taking expectations in (A1) and simplifying yields (1). To 
derive (2) we note that: 
 
() δδ δ
==
⎡ ⎤ ⎛⎞ ⎛⎞ ⎟⎟ ⎜⎜ ⎢ ⎥ ⎟⎟ +Σ = +Σ ⎜⎜ ⎟⎟ ⎢ ⎥ ⎜⎜ ⎟⎟ ⎜⎜ ⎝⎠ ⎝⎠ ⎣ ⎦
∑∑ ,,
11
1e x p
nn
jj
ju ju
jj
nn
to t t jj
, 
 
where  () δ ot  represents terms which vanish with δt  faster than δt  (that is, 
() δδ → 0 ot t  as δ 2 0 t ). Replacing in (1), we obtain: 
 
  () () τ δδ +
=
⎡ ⎤ ⎛⎞ ⎟ ⎜ ⎢ ⎥ ⎟ =+ Σ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎜ ⎝⎠ ⎣ ⎦
∑∑ ,
1
exp
n
n j
tt t ju
uj
n
EX X ot t j
. (A2) 
 
Now by definition  σ Σ→ ,,
jj
ju ju as δ 2 0 t , hence  δσ τ Σ→ ∑ ,
jj
ju j
u
t , and since 
() () δδ τ δ =→ ∑ 0
u
ot ot t , taking limits in (A2) yields (2). Page 11 of 14 
REFERENCES 
 
 
Bates, D. (1991) The crash of ’87: Was it expected? The evidence from options markets, 
Journal of Finance 46(3), 1009-1044. 
 
Black, F. and M. Scholes (1973) The pricing of options and corporate liabilities, Journal of 
Political Economy 81(3), 637-654. 
 
Derman, E. and I. Kani (1994) Riding on a smile, Risk 7(2), 32-39. 
 
Dupire, B. (1994) Pricing with a smile, Risk 7(1), 18-20. 
 
Jarrow, R. and A. Rudd (1982) Approximate option valuation for arbitrary stochastic 
processes, Journal of Financial Economics 10(3), 347-369. 
 
Merton, R. (1973) Theory of rational option pricing, Bell Journal of Economics and 
Management Science 4(1), 141-183. 
 
Merton, R. (1976) Option pricing when underlying stock returns are discontinuous, 
Journal of Financial Economics 3(1-2), 125-144. 
 
Rubinstein, M. (1994) Implied binomial trees, Journal of Finance 49(3), 771-818. Page 12 of 14 
Table 1 
Jump-diffusion parameters 
 
 
Weeks to 
expiration 
b (%)  σ (%)  λ  γ (%)  δ (%) 
1  0.73 11.38 9.09 -3.06 2.23 
5  1.25 10.13 2.47 -6.46 5.18 
9  1.46  10.16  1.12 -10.33 7.38 
18 1.53  9.79  0.68  -13.71  10.04 
31 1.65  9.64  0.46  -18.22  12.65 
44 1.71  9.64  0.33  -22.13  15.51 
 
 
 
Table 2 
Average volatilities (%) 
 
 
Weeks to 
expiration 
σ1  σ2  σ3   σ4  
1 0.73  15.95  -8.56  7.74 
5 1.25  16.05  -11.83  11.97 
9 1.46  16.15  -13.48  14.40 
18 1.53  16.09  -14.87  16.56 
31 1.65  16.45  -16.45  18.81 
44 1.71  16.58  -17.50  20.51 
 Page 13 of 14 
Figure 1 
S&P 500 implied Black-Scholes volatilities by expiration 
 
 
1 week
0%
5%
10%
15%
20%
25%
30%
35%
700 800 900 1000 1100 1200 1300 1400
Exercise price
call put
 
5 weeks
0%
5%
10%
15%
20%
25%
30%
35%
700 800 900 1000 1100 1200 1300 1400
Exercise price
call put
 
9 weeks
0%
5%
10%
15%
20%
25%
30%
35%
700 800 900 1000 1100 1200 1300 1400
Exercise price
call put
 
18 weeks
0%
5%
10%
15%
20%
25%
30%
35%
700 800 900 1000 1100 1200 1300 1400
Exercise price
call put
 
31 weeks
0%
5%
10%
15%
20%
25%
30%
35%
700 800 900 1000 1100 1200 1300 1400
Exercise price
call put
 
44 weeks
0%
5%
10%
15%
20%
25%
30%
35%
700 800 900 1000 1100 1200 1300 1400
Exercise price
call put
 Page 14 of 14 
Figure 2 
Implied average volatilities 
 
 
0%
5%
10%
15%
20%
25%
0 1 02 03 04 0
Weeks to expiration
s1 s2 -s3 s4
 
 
 

Higher-order volatility

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