Calculation Error or High Vega? How to interpret?

Question

I am trying to calculate/interpret Vega. For the example below I get a Vega of ~36.36. I have checked my math multiple times, but would appreciate anyone pointing out any error that I have made. If the Vega is correct, I would appreciate any clarification on how to interpret such a high Vega.

Data: (Sorry in advance for the precision.)

Spot:       280
Strike:     275
Time 2 Mat: 0.1469746111428209 (53.6457/365)
Interest:   0.025
Volatility: 0.1024485798268360

My assumed formulae are as such:

$S \cdot N'(d_1)\sqrt{T-t}$

where

$d_1 = \frac{ln(\frac{S}{K})+(r+\frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$

Therefore my calculations look like so:

$d_1 = \frac{ln(\frac{280}{275})+(0.025+\frac{0.1024485798268360^2}{2})(0.1469746111428209)}{0.1024485798268360\sqrt{0.1469746111428209}} = 0.571956921435358$

$280 \cdot N'(0.571956921435358)\sqrt{0.1469746111428209} = 36.362426400092646$

Answer 1

As @AlexC mentions in comments above, Vega is truly a price sensitivity to a 100% move in volatility. It is commonplace for Vega to be scaled-down by dividing by 100 in order to find the price sensitivity to a 1% move in volatility.

In the above example the price sensitivity to a 1% move in volatility is therefore simply:

$\frac{36.36}{100} = 0.3636$

Here is a question/answer which (in the context of R) arrives to the same conclusion.

Answer 2

It is probably not a calculation error. The Black-Scholes model presumes that the parameters are known and not estimated. It is a model built around parameters, not parameter estimates. It also either assumes log-normality or that the traditional CAPM is true.

In 1958 a mathematician, John White, proved that models such as the CAPM have no solution in Frequentist or Likelihoodist interpretations of probability. More properly, there is a solution but it will never converge to a parameter. In 1934, Ronald Fisher criticized Pearson and Neyman's Frequentist methods using a similar model. His argument was that it was an example that would produce a result that was "accurate" but not "correct."

By accuracy, he pointed out that the parameter estimates would be symmetric around the true parameter, but would commonly be off by $\pm{100}\sigma.$ Economics never read the articles. In fact, this same topic was a source of a critique of a proof written by Laplace and reviewed Poisson. It keeps popping up in statistics and then everyone forgets about it. The great Augustin Cauchy used it to prove that there was an automatic source of failure for ordinary least squares at the time being pushed by Bienayme.

The issue has to do with parameters being "known" versus estimated. If the parameters are known with probability 1, then there is nothing at all wrong with Black-Scholes or the CAPM. If they are not known, then the Frequentist estimators have perfect asymptotic relative inefficiency compared to a valid estimation method. That is to say, the variance of the sampling distribution explodes to infinity as the sample size goes to infinity. This has been in the finance literature since 1963 when Benoit Mandelbrot wrote an article that basically said, "if this is your model, then this cannot possibly be your data, and this is your data." Population disconfirmation was performed by Fama and MacBeth in 1972.

There is a simple test of this you can perform yourself, two in fact. One you can perform in less than five minutes.

Make a histogram of your data. Get parameter estimates. Plug them in. Map the log-normal that is implied by them. Do not remove outliers. Then estimate another curve: $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\frac{\sigma}{\sigma^2+(x-\mu)^2}.$$

For the later, approximate $\mu$ as the empirical mode. For $\sigma$ approximate it with half the interquartile range. Both estimators will be very crude.

The proper estimate would be to use a Bayesian model test of the two. Give the log-normal 999:1 prior odds of being the correct model. Make it prejudiced in favor of Black-Scholes.

I have in fact done a population study on all end of day trades in the CRSP universe. Normality and log-normality are rejected in favor a truncated Cauchy distribution. It is in the bibliography below.

The presence of the truncated Cauchy is due to the fact that returns are a statistic and not data. Prices are data. Returns are a function of data. Under mild conditions, auction theory would require normally distributed prices for equity market securities. The ratio of two normals around the equilibrium would be a Cauchy distribution. Limitations of liability restrict that by truncation.

See the following articles:

Curtiss, J. H. (1941). On the distribution of the quotient of two chance variables. Annals of Mathematical Statistics, 12:409-421.

Fama, E. (1965). The behavior of stock market prices. Journal of Business, 38:34-105.

Fama, E. F. (1963). Mandelbrot and the stable paretian hypothesis. Journal of Business, 36:420 - 429.

Fama, E. F. and French, K. R. (2008). Dissecting anomalies. The Journal of Finance, LXIII(4):1653-1678.

Fama, E. F. and MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. The Journal of Political Economy, 81(3):607-636.

Fama, E. F. and Roll, R. (1968). Some properties of symmetric stable distributions. Journal of the American Statistical Association, 63(323):pp. 817-836.

Fama, E. F. and Roll, R. (1971). Parameter estimates for symmetric stable distributions. Journal of the American Statistical Association, 66(334):331 - 338.

Fisher, R. A. 1934. Two new properties of mathematical likelihood. Proc. Roy. Soc. Ser. A (144) 285-307.

Gull, S. F. (1988). Bayesian inductive inference and maximum entropy. In Erickson, G. J. and Smith, C. R., editors, Maximum-Entropy and Bayesian Methods in Science and Engineering: Foundations, volume 1 of Fundamental Theories of Physics, pages 53-74. Springer.

Gurland, J. (1948). Inversion formulae for the distribution of ratios. The Annals of Mathematical Statistics, 19(2):228-237.

Harris, D. E. (2017). The distribution of returns. The Journal of Mathematical Finance, 7(3):769-804.

Harris, David E., A Test of a Generalized Stochastic Calculus (November 27, 2018). Available at SSRN: https://ssrn.com/abstract=2653151 or http://dx.doi.org/10.2139/ssrn.2653151

Jaynes, E. T. (2003). Probability Theory: The Language of Science. Cambridge University Press, Cambridge.

Koopman, B. O. (1936). On distributions admitting a sufficient statistic. 39(3):399-409.

Lavine, M. and Schervish, M. J. (1999). Bayes factors: What they are and what they are not. The American Statistician, 53(2):119-122.

Mandelbrot, B. (1963). The variation of certain speculative prices. The Journal of Business, 36(4):394-419.

Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1):77-91. 30

Markowitz, H. and Usmen, N. (1996a). The likelihood of various stock market return distributions, part 1: principles of inference. Journal of Risk and Uncertainty, 13:207-219.

Markowitz, H. and Usmen, N. (1996b). The likelihood of various stock market return distributions, part 2: empirical results. Journal of Risk and Uncertainty, 13:221-247.

Marsaglia, G. (1965). Ratios of normal variables and ratios of sums of uniform variables. Journal of the American Statistical Association, 60(309):193- 204.

Roy, A. (1952). Safety first and the holding of assets. Econometrica, 20:431- 439.

Stigler, Stephen M. Studies in the History of Probability and Statistics. XXXIII: Cauchy and the witch of Agnesi: An historical note on the Cauchy distribution. Biometrika. 61(2). 1974. pp. 375-380

White, J. S. (1958). The limiting distribution of the serial correlation coefficient in the explosive case. The Annals of Mathematical Statistics, 29(4):1188-1197.

Yilmaz, B. Z. (2010). Completion, pricing and calibration in a levy market model. Master's thesis, The Institute of Applied Mathematics of Middle East Technical University.

If you need additional help, let me know in the comments section and I will find a way to get you my contact info.