3.2 Methodology
3.2.2 Functional data representation
Our analysis extends the prevailing literature on crude oil implied jump dynamics by com- bining FDA techniques with the Merton model to extract implied jump size probability and direction. FDA uncovers the smooth process underlying the data. This sets it apart
5The commentary presented here is for call options, the opposite eect is true for put options. 6The presence of a skew is also in line with studies by Mandelbrot (1963), Fama (1965), and Clark
Figure 3.1: Typical crude oil implied volatility curve (11th June 2010) 0.8 0.9 1.0 1.1 1.2 0.34 0.36 0.38 0.40 0.42 0.44 0.46
Typical Crude Oil Implied Volatility Curve (11th June 2010)
Moneyness
Implied V
olatility
from regular interpolation techniques which simply seek to nd the best t to the pos- sibly noisy data set. This true function is represented in an innite dimensional space over a continuum of values. A continuum is generally dened in terms of time or space; however, in this paper the functions are dened in the moneyness domain, as an option's implied volatility is related to how ITM [i.e., K/F (strike/forward rate)] it is. The FDA methodology has many advantages; it accurately captures volatility dynamics, there is no assumed parametric structure, it is computationally ecient, and the process can be evaluated on an arbitrarily ne grid. This FDA representation allows us to consistently identify the ATM (and ITM and OTM) implied volatility level for each day. It serves as an improvement over Yan (2011)'s near the money implied volatility calculation that only encompasses information obtained from -0.5 delta puts and 0.5 delta calls.
Our goal is to interpret the daily discrete option volatility data, x(mk), as a functional
data object or, more simply, as a function, denotedx˜(mk).7 When constructing a function,
a vector of n basis function, denotedφ1,....,n, must rst be specied. Functional structures
are approximated as a weighted linear combination of these bases: ˜
x(m) =c1φ1(m) +c2φ2(m) +...+cnφn(m),
wherec1, ..., cn represent the parameters of the expansion's coecients.
As in Ramsay and Silvermann (2005), the coecientscjare chosen in order to minimise: SSE(c1, ..., cn)≡ N X k=1 x(mk)− n X j=1 cjφj(mk) 2 (1).
The decision of which basis system to specify is driven by the underlying data's known characteristics. For instance, when modelling periodic data, a Fourier basis expansion, comprised of successive sine/cosine terms, is most commonly applied. However, an implied volatility process does not exhibit strong cyclical variation, so B-spline functions are chosen for the basis function system. B-splines are essentially a number of polynomials joined together smoothly at xed points called knots. The number and positioning of theN knots: mk : m1 ≤ ... ≤ mN, are derived from knowledge of the complexity of the underlying
process over particular ranges. The range of the various sub-intervals, [mk, mk+1], are dened through the placement of these knots. Within each sub-interval, the spline is simply a polynomial of order n. The order is calculated as:
order = 1 +degree of the polynomial.
B-spline representation is useful as it has a number of strengths. At any one point along the curve, it simplies to a polynomial that can be easily evaluated. Adjusting the order of the spline allows for the estimation of derivatives of any degree. In this paper, a fourth
7The standard notation utilised,x(t), signies a dependence on time; however, here the domain is that
order basis, or cubic polynomial is specied. Modelling the process as a cubic polynomial provides a good balance as it retains the function's continuous property up to the second derivative, while simultaneously smoothing the noise within the daily data.8 The knots
are placed at the discrete quoted option moneyness levels that are available from the data set, with polynomials describing the moneyness interval between the knots.
Given that a smooth function underlies the observed implied volatility curve a smooth- ing penalty is applied to remove noise/wiggles in the data. This can be caused by liquidity issues, misquotes or other data irregularities masking the true function. The eect of such non-trading and noise issues on the volatility function has been highlighted by Bannouh et al. (2013). Without the use of a smoothing penalty the uncovered function may simply converge to a tted spline interpolant of the data. In line with Ramsay and Silvermann (2005) a limitation is placed on the variation of the curvature. The total curvature of the process is found through integrating its squared second derivative:9
R(˜x)≡ ˆ d2 dm2x˜(m) 2dm.
This is also called the roughness of the function.
In an extension of (1), the coecients characterising the smoothed curve are found using the penalised sum of squared errors:
P EN SSE(c1, ..., cn)≡ N X k=1 x(mk)− n X j=1 cjφj(mk) 2 +λ∗ ˆ d2 dm2x˜(m) 2dm.
As λ∗ increases, more weight is placed on the roughness penalty and the uncovered
function converges towards a straight line, possibly missing some of the process' dynam- ics.10 Asλ∗ decreases, less weight is placed on the roughness penalty and only data tting matters in uncovering the function.
In order to balance the competing goals of retaining features and removing noise from the data, an optimal smoothing level,λ∗, must be selected. Inherent knowledge of variation
in the underlying process is useful here and can be used in conjunction various data-driven techniques such as information criteria and cross validation. Cross validation is based on the principal of removing one observation from the sample and using this sub-sample to see how well the removed observation can be predicted. We apply generalised cross validation (GCV), proposed by Craven and Wahba (1979). Without smoothing, noise in the relatively small number of discrete values available can distort the results for that range. This is particularly evident at extreme moneyness levels as prices may be contorted due to illiquidity.
8This is required as we examine the curvature of the implied volatility function. 9The second derivative is also denoted asD2
x(m)in the literature.
10The symbol λ∗ is used here to distinguish this smoothing weight for calculating coecients under penalised sum of squared errors, from the jump intensityλparameter specied in the Merton model.
FDA allows a systematic analysis of implied volatility curves over time. It provides a framework to obtain the slope and curvature at any point along the implied volatility curve. Therefore, the slope and curvature levels are evaluated consistently at three points along the daily implied volatility curve, namely, ATM, 10% ITM, and 10% OTM. The evolution of these levels over time is analysed in order to further understand how demand and supply side weaknesses alter its shape.