ESTRUCTURALISMO E HISTORIA
V. La destrucción estructuralista de la historia
Model for European Options
This section describes some results presented by Madan, Carr and Chang28. The authors compare the variance gamma process with the symmetric vari- ance gamma and with the geometric Brownian motion using options on the Standard & Porr’s 500. Options from January 1992 to September 1994 are considered and several options prices on or around the close for each day are used to ensure liquidity and avoid issues of non synchronous trading. Up to fours strike for each of four maturities were used.
The test is realized both from a statistical point of view, using underlying prices to determine the values of the statistical parameters, and from a risk neutral point of view, by determining the risk neutral parameters implied by option prices. Moreover the pricing errors are studied to determine if the models are biased.
2.10.1
Skewness and Kurtosis Results
To compute statistical parameters for the variance gamma, maximum like- lihood estimation with the density function presented in equation (2.30) is used. The estimation of the symmetric variance gamma is realized in the same way, but constraining the ϑ= 0, while in the case of the Black Scholes model, the underlying process is the lognormal.
Statistical mean returns and volatilities estimated by the three models are very similar. As for the kurtosis, both symmetric and general variance gamma model, have an estimated statistical ν equal to 0.002. Remembering from equation (2.13) that, when ϑ = 0, ν provides a measure of percentage excess kurtosis over 3 for a unit time period, here one year, measuring the time in years. Therefore the corresponding daily kurtosis is given by 3∗(1 +
28Madan Dilib B., Carr Peter P. and Chang Eric C., ”The Variance Gamma Process
0.002∗365) = 5.19, where we converted years in days using calendar days instead of trading days. Finally we can say that statistical skewness for the variance gamma is insignificant.
Risk neutral parameters are obtained on a weekly basis, again using max- imum likelihood. The average estimates ofν for symmetric and general vari- ance gamma are much higher than the statistical counterparts, moreover a negative skew is showed in the case of (general) variance gamma.
2.10.2
Pricing Performance of the Variance Gamma
We saw that from a statistical point of view, kurtosis is an important factor in describing the dynamics of stock returns, while, from a risk neutral point of view, both skewness and kurtosis appear to be relevant. Madan, Carr and Chang try to give a more definitive opinion on the performance of vari- ance gamma, compared with its symmetrical counterpart and with geometric Brownian motion by studying the pricing error of the models. In particular a regression on the pricing errors from each model is realized to determined the presence of biases. Factors used are maturity, the moneyness, the square of the moneyness and the level of interest rate. Moneyness enters twice in the regression to allow volatility smiles where both out of the money puts and calls exhibit higher implied volatilities.
All four regressors together with the constant are significant in the case of the Black and Scholes model at a 5% level. Consistently with expecta- tion, the coefficient of the degree of moneyness is negative, while the one for the square of moneyness is positive. Positive are also the coefficient for the maturity and the interest rate. Adjusted R2 for this model is high at 16% and the F statistics takes to a conclusion that we must reject the hypothesis of orthogonality of errors to the regressors. The symmetric variance gamma performs only slightly better, with all the variables, but the interest rates significant at a 5% level. AdjustedR2 is high at 17% and F again requires to reject the hypothesis of orthogonality of the explanatory variables. The sym- metric variance gamma seems to have over adjusted the smile issue presented by Black and Scholes and now the smile is inverted with a positive coefficient for moneyness and a negative coefficient of the square of the moneyness.
The variance gamma model, on the contrary, produced a market improve- ment over the other two models, based on this data. Moneyness, square of moneyness and interest rate, together with the constant are now insignificant
even at a 1% level. Adjusted R2 is is 0.1% and the F statistic is not signifi- cant. The only problems which is not solved by the variance gamma model is the maturity bias: the model presents a negative coefficient for maturity which is significant at a 1% level.
Another empirical test on the pricing performances of the variance gamma model which is worth mentioning is the one realized by Lam, Chang and Lee29. In this paper the authors test both the symmetric and the general variance gamma on the pricing of the Hang Seng Index call options, which are European style options. The test of the model when the options fair values are computed using the closed form solution is better carried on Eu- ropean options like these ones rather than on American options, like the one used by Madan, Carr and Chang, because in this way no additional error related to the conversion of American to European prices is added. The test is realized on intraday prices over a three year time frame and the data are closely examined and matched so that they are as synchronous as possible. The presence of this large database allows for the use of robust statistical tests, with respect to outliers, without loosing too much test power. The conclusion found by Lam, Chang and Lee is that the variance gamma option pricing model performs someway better than the Black and Scholes model. Under the historical approach, the variance gamma can quite well correct some of the systematic biases of the Black and Scholes model. However, under the implied approach, the variance gamma continues to exhibit pre- dictable biases.
2.10.3
Conclusion
Based on the empirical tests realized on the model, especially on those ones presented by Madan, Carr and Chang, the variance gamma model presents a material improvement over the geometric Brownian motion and the sym- metric variance gamma model. The variance gamma is no more moneyness biased, however it still presents problems related with the estimation of op- tions with different maturities. As we will see in section (3.8), this can be corrected using a further extension of the model which presents stochastic volatility. The stochastic volatility model is however beyond the scope of this
29K. Lam, E. Chang and M. C. Lee, “An Empirical Test of the Variance Gamma Option
work and the attention, when presenting the numerical procedure for pricing European and American options in chapters 5, will be focus on the variance gamma model and not on the stochastic volatility extension.
Chapter 3
Extensions of the Variance
Gamma Model: CGMY Model
and Stochastic Volatility
Models
3.1
Introduction
As described in the previous chapter, stochastic processes describing the dy- namic of stock prices can exhibit infinite of finite variation and infinite of finite variation. The variance gamma, in particular, is a process of infinite activity and of finite variation. Carr, Geman, Madan and Yor1 presented an extension of the variance gamma which allows for both infinite and finite activity and for both infinite and finite variation. The name of the model, CGMY, is derived from the initials of the authors.
1The authors presented a series of articles on the subject, the most complete introduc-
tion to the model is probably in Peter Carr, H´elyette Geman, Dilip B. Madan and Marc Yor, “The Fine Structure of Asset Returns: An Empirical Investigation”,The Journal of Business, Vol. 75, No. 2, 2002, pages 305-332.