Estimation of risk-neutral measures using quartic B-spline cumulative distribution functions with power tails

Abstract

In this paper, we propose the B-spline (BSP) method, which overcomes problems with the smoothed implied volatility smile (SML) method for estimating option implied risk-neutral measures (RNMs). We model the risk-neutral cumulative distribution function (CDF) using quartic B-splines with power tails so that the resulting risk-neutral probability density function (PDF) has continuity and arbitrage-free properties. Since the number of knots is selected optimally in constructing the quartic B-spline risk-neutral CDF, our method avoids both overfitting and oversmoothing. To improve computational efficiency and accuracy, we introduce a three-step RNM estimation procedure that transforms a nonlinear optimization problem into a convex quadratic program. Monte-Carlo experiments and applications to S&P 500 index options suggest that the BSP method performs considerably better than the SML method. The BSP method always produces arbitrage-free RNM estimators and almost perfectly recovers the actual risk-neutral PDFs for various hypothetical distributions.

Keywords:

• Risk neutral distribution
• Numerical method
• Option pricing
• Optimization

JEL Classification:

• C1
• C14
• G1
• G13

Acknowledgements

The author is grateful to J. Huston McCulloch for his invaluable comments and advice. The author also thanks P. Evans, P.S. Lam, and R. Kimmel for helpful comments.

Notes

¹The term non-parametric is not meant to imply that such models completely lack parameters, but that the number and nature of the parameters are flexible and not fixed in advance.

²The linearity of the SLM method outside the range of observed strike prices does not imply a flat extrapolation scheme. The SML method extrapolates outside the range of available data with second-order smoothness conditions as in Jiang and Tian (2007). Jiang and Tian (2007) improve the flat extrapolation scheme of Jiang and Tian (2005) and Carr and Wu (2009), which satisfies first-order smoothness, resulting in the implied volatility function being flat beyond listed strike prices.

³The B-spline CDF with power tails has infinite-dimensional control points since the number of knots is not fixed in advance. However, it also has finite-dimensional tail parameters and thus it is not a pure non-parametric method. A model is called semi-non-parametric if it has both finite-dimensional and infinite-dimensional unknown parameters of interest.

⁴The curve fitting method based on spline functions was first used by McCulloch (1971, 1975) in finance for modeling the term structure of interest rates.

⁵The use of the Black--Scholes formula to transfer between the call price and implied volatility domains does not require that the Black--Scholes model is true. The Black--Scholes formula is used as a translation device that allows us to interpolate implied volatilities rather than the observed option prices themselves for computational convenience.

⁶It should be recalled that , where d is the dividend rate of the underlying asset.

⁷Transforming each strike into a delta using the at-the-money implied volatility has the advantage that the ordering of deltas is always the same as that of the strikes. Panigirtzoglou and Skiadopoulos (2004) pointed out that using the implied volatilities that corresponds to each strike could change the ordering in the delta space, in cases where steep volatility skews are observed. This would result in generating volatility smiles with artificially created kinks.

⁸Bliss and Panigirtzoglou (2002) discussed different types of weighting schemes and how the weighting can account for different sources of pricing errors.

⁹The objective function suggests that the degree of freedom for the estimation is also related to the smoothing parameter. In particular, the maximum degree of freedom is achieved when , which amounts to fitting a straight line to the data, whereas when , the cubic spline provides an exact fit to the data. Fisher et al. (1995) give a rigorous definition of the effective number of parameters of the regression.

¹⁰The control points form a sequence that is known as the control polygon which is often visualized by joining them in sequence by straight lines. This set of straight lines is in fact the B-spline curve of order 1 defined by that set of control points. For more details, see de Boor (1978).

¹¹

¹²Monterio et al. (2008) estimate the cubic spline risk-neutral PDF only for the traded strike range by truncating the tails of the pdf without estimating them from observed option prices. Fengler (2009) proposed an algorithm for estimating the implied volatility smile under suitable linear inequality constraints, ensuring non-negativity of the risk-neutral PDF, but this algorithm does not guarantee the resulting risk-neutral PDF to be integrated to unity.

¹³We set the smoothness penalty parameter w equal to 10⁻³, which minimizes the RMISE of SML under the log-normal risk-neutral PDF.

¹⁴Monte-Carlo simulation shows that this iteration reduces only about 0.001% of RMISE, but significantly increases the computational burden. This implies that the simple three-step estimation procedure is considerably accurate and the improvement by iteration is negligible.

¹⁵The risk-neutral PDF at maturity can be inferred from the stochastic process. The risk-neutral PDF of the stochastic process is obtained by inverting its characteristic function. In order to find examples of the jump-diffusion process and stochastic volatility models, see Bakshi et al. (1997) and Carr and Wu (2003).

¹⁶In our experiment we generate the OTM option prices for 25 strikes in the interval [1100, 1800].

¹⁷Kullback--Leibler divergence is a measure of the difference between two probability distributions: from a ‘true’ probability distribution to an arbitrary probability distribution (Kullback and Leibler 1951). Although it is often interpreted as a distance metric, the KL divergence is not a true metric since it is not symmetric (hence ‘divergence’ rather than ‘distance’).

¹⁸The maximum bid--ask spread permitted by the exchange is linked to the option quotes. For instance, the CBOE rules state that the maximum bid--ask spread is 1/4 for options with bid quote below $2, 3/8 for bid quotes between $2 and $5, 1/2 for bid quotes between $5 and $10, and so on

The function is constructed to represent such rules. See Bondarenko (2003) for details on the construction of .

¹⁹As Buraschi and Jackwerth (2001) pointed out, most of the trading activity in S&P 500 options is concentrated in the nearest (0--30 days to maturity) and second nearest (30--60 days to maturity) contracts. Options that expire within 30 days are excluded from the sample because of abnormal trading volumes for options close to expiry reported by Stoll and Whaley (1987).

²⁰Since the BSP and SML methods are always capable of producing an exact fit to the data by decreasing the smoothing parameter (w) or increasing the number of knots (n), we do not compare the goodness-of-fit of the two methods.

²¹The convolution of f and g is written as . If X and Y are two independent random variables with probability distributions f and g, respectively, then the probability distribution of the sum is given by the convolution

²²The current level of is unobservable. Thus, the state variable is essentially treated as another free parameter in the RNM estimation procedure (Carr and Wu 2003).