Article Contents
Article Contents

# Total positivity and relative convexity of option prices

• *Corresponding author: Paul Glasserman
• This paper studies total positivity and relative convexity properties in option pricing models. We introduce these properties in the Black-Scholes setting by showing the following: out-of-the-money calls are totally positive in strike and volatility; out-of-the-money puts have a reverse sign rule property; calls and puts are convex with respect to at-the-money prices; and relative convexity of option prices implies a convexity-in-time property of the underlying. We then extend these properties to other models, including scalar diffusions, mixture models, and certain Lévy processes. We show that relative convexity typically holds in time-homogeneous local volatility models through the Dupire equation. We develop implications of these ideas for empirical option prices, including constraints on the at-the-money skew. We illustrate connections with models studied by Peter Carr, including the variance-gamma, CGMY, Dagum, and logistic density models.

Mathematics Subject Classification: Primary: 60E15, 60E07; Secondary: 60G42.

 Citation:

• Figure 1.  Implications among total positivity and relative convexity properties for out-of-the-money calls. The properties in the figure are stated for $x\ge 1$ and $K\ge 1$. An alternative to the implication (d) is discussed in Remark 2.2

Figure 2.  Implications among RR and relative convexity properties for out-of-the-money puts. The properties are stated for $x,K\in (0,1]$

Figure 3.  The ratio $f(x,\sigma_1)/f(x,\sigma_2)$, $\sigma_1 = 0.5<\sigma_2 = 1$, is increasing for $x\in(0,1]$ and decreasing for $x\ge 1$

Figure 4.  Points in the plane $(K_i, K_j)$ with $\delta_{ij}< 0$. They correspond to exceptions from TP$_2$ property for OTM call options (left) and from RR$_2$ property for OTM put options (right). Upper: $T_1 = 10/15/21, T_2 = 11/19/21$, middle: $T_1 = 10/15/21, T_2 = 12/17/21$ and lower: $T_1 = 11/19/21, T_2 = 12/17/21$. The vertical/horizontal red lines are at $K = S_0$. Pricing date 10/4/21

Table 1.  The first few expiries of the SPX index options, the forward price $F(T)$ and the number of strikes for which option pricing data is available. The pricing date is 4-Oct-2021, with closing price $S_0 = 4,300.46$

 Expiry $F(T)$ $n_K$ 15-Oct-2021 4,300.86 407 19-Nov-2021 4,295.84 367 17-Dec-2021 4,290.59 399
•  [1] O. E. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance and Stochastics, 2 (1998), 41-68.  doi: 10.1007/s007800050032. [2] F. Belzunce and M. Shaked, Stochastic comparisons of mixtures of convexly ordered, distributions with applications in reliability theory, Statistics and Probability Letters, 53 (2001), 363-372.  doi: 10.1016/S0167-7152(01)00030-X. [3] G. T. Cargo, Comparable means and generalized convexity, Journal of Mathematical Analysis and Application, 12 (1965), 387-392.  doi: 10.1016/0022-247X(65)90005-3. [4] P. Carr, H. Geman, D. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332.  doi: 10.1086/338705. [5] P. Carr and R. Lee, Put-call symmetry: Extensions and applications, Mathematical Finance, 19 (2009), 523-560.  doi: 10.1111/j.1467-9965.2009.00379.x. [6] P. Carr and D. Madan, Towards a theory of volatility trading, Volatility: New Estimation Techniques for Pricing Derivatives, Risk Books, (1998), 417-427. [7] P. Carr and L. Torricelli, Additive logistic processes in option pricing, Finance and Stochastics, 25 (2021), 689-724.  doi: 10.1007/s00780-021-00461-8. [8] C. Donati-Martin, R. Ghomrasni and M. Yor, On certain Markov processes attached to exponential functionals of Brownian motion; Application to Asian options, Revista Matematica Iberoamericana, 17 (2001), 179-193.  doi: 10.4171/RMI/292. [9] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [10] N. Fournier and J. Printemps, Absolute continuity for some one-dimensional processes, Bernoulli, 16 (2010), 343-360.  doi: 10.3150/09-BEJ215. [11] J. Gatheral, The volatility skew: Arbitrage constraints and asymptotic behaviour, preprint, 1999, Merrill Lynch. [12] P. Glasserman and D. Pirjol, W-shaped implied volatility curves and the Gaussian mixture model, to appear, Quantitative Finance. [13] G. H. Hardy,  J. E. Littlewood and  G. Pólya,  Inequalities, 2$^{nd}$ edition, Cambridge University Press, 1952. [14] F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and Associated Martingales, with Explicit Constructions, Springer-Verlag, 2011. doi: 10.1007/978-88-470-1908-9. [15] H. M. Hodges, Arbitrage bounds on the implied volatility strike and term structures of European-style options, Journal of Derivatives, 3 (1996), 23-35.  doi: 10.3905/jod.1996.407950. [16] S. Karlin, Total positivity, absorbtion probabilities and applications, Transactions of the American Mathematical Society, 111 (1964), 33-107.  doi: 10.1090/S0002-9947-1964-0168010-2. [17] S. Karlin,  Total Positivity, Stanford University Press, tanford, CA, 1968. [18] S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience Publishers, New York, 1966. [19] M. Keller-Ressel, Total positivity and the classification of term structure shapes in the two-factor Vasicek model, International Journal of Theoretical and Applied Finance, 24 (2021), 2150027, 27 pp. doi: 10.2139/ssrn.3441116. [20] M. Kijima, Monotonicity and convexity of option prices revisited, Mathematical Finance, 12 (2002), 411-425.  doi: 10.1111/j.1467-9965.2002.tb00131.x. [21] T. Klassen, Necessary and sufficient no-arbitrage conditions for the SSVI/S3 volatility curve, preprint, 2016, Available at SSRN. [22] A. E. Kyprianou, Introductory Lectures on Lévy Process Fluctuations and Applications, Springer-Verlag, Berlin, 2006. [23] J. Lynch, G. Mimmack and F. Proschan, Uniform stochastic orderings and total positivity, Canadian Journal of Statistics, 15 (1987), 63-69. doi: 10.2307/3314862. [24] D. B. Madan, P. P. Carr and E. C. Chang, The variance gamma process and option pricing, European Finance Review, 2 (1998), 79-105.  doi: 10.1023/A:1009703431535. [25] D. B. Madan and M. Yor, Representing the CGMY and Meixner Lévy procesess as time changed Brownian motions, Journal of Computational Finance, 12 (2008), 27-47. doi: 10.21314/JCF.2008.181. [26] D. B. Madan and E. Seneta, The variance gamma model for share market returns, Journal of Business, 63 (1990), 511-524.  doi: 10.1086/296519. [27] A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and its Applications, 2$^{nd}$ edition, Springer, New York, 2011. doi: 10.1007/978-0-387-68276-1. [28] W. Molenaar and W. R. van Zwet, On mixtures of distributions, Annals of Mathematical Statistics, 37 (1966), 281-283.  doi: 10.1214/aoms/1177699620. [29] J. A. Palmer, Relative convexity, preprint, 2003, Electrical and Computer Engineering Department, University of California at San Diego. [30] M. Shaked and J. G. Shanthikumar, On the first-passage times of pure jump processes, Journal of Applied Probability, 25 (1988), 501-509. doi: 10.2307/3213979. [31] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer Series in Statistics, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5. [32] P. H. Zipkin, The relationship between risk and maturity in a stochastic setting, Mathematical Finance, 2 (1992), 33-46.  doi: 10.1111/j.1467-9965.1992.tb00024.x.
Open Access Under a Creative Commons license

Figures(4)

Tables(1)