Stock market returns follow a normal distribution with a positive bias (skew) and an increased number of outliers than a textbook, symmetrical distribution. The outlier characteristic may be described as fat tails, with those to the downside typically more pronounced than those to the upside. Regardless, the distribution provides traders the ability to estimate fair option prices via a variety of models.
The Black-Scholes model from Fischer Black and Myron Scholes is one of the best known for option pricing. The collaborative effort between Black, Scholes and Robert Merton earned Scholes and Merton the 1997 Nobel Prize in Economics a couple of years after Black’s death. Merton’s 1973 paper, “Theory of Rational Option Pricing”, in the Bell Journal of Economics and Management, discussed a new method for determining derivative pricing.1
Reading that includes statistics and market returns is provided at the end of the article. It includes Jerry Marlow’s, “Option Pricing: Black-Scholes Made Easy”, which provides much more in-depth coverage for option pricing models.
Figure 1, which displays a histogram for ten-day holding period returns, was generated using StatSoft, Inc.’s Statistica© from data downloaded from Worden Brothers, Inc.’s TeleCharts® package. The data uses rolling ten-day periods from 1975 through 2004 for the Dow Jones industrial AverageSM and was used to approximate an apples-to-apples comparison last week, but does provide a basic view of our expectations for US stock market returns.
Figure 1 INDU 10-day Rolling Returns for All Days, All Months – Histogram (1975-2004)
The data is reasonably contained by the distribution curve, with more positive returns than negative ones and outliers on both sides, the left of which has a greater range.
Option Pricing Model Assumptions
There are different assumptions made in the modeling process, three of which focus on return and volatility characteristics. These assumptions are also made in the default settings of the probability tools found in Optionetics Platinum, which will be covered in greater detail next week. The model assumptions include:
- Prices follow a lognormal distribution
- Volatility remains constant through expiration
- The mean return for the underlying through expiration is zero
The lognormal distribution for prices results given the lower boundary for prices at 0; since prices cannot go below this level there is a different type of positive bias that requires adjusting the underlying data.
Traders can work with the lognormal price distribution (see Platinum probability formulas) or they can apply an appropriate calculation on each of the data points – such as a rate of change or a basic return as shown here – to transform the data. The result is a normally distributed data series which satisfies the first assumption.
Mean Return of Zero
Perhaps a more difficult assumption to be satisfied with is the mean return of zero for the life of the option. To provide a visual for this assumption, Figure 2 displays a weekly bar chart for SPY, the exchange traded funds that tracks the S&P 500® Index, with a linear regression channel.
Figure 2 Weekly Bar Chart for SPY with Linear Regression Channel ProfitSource
Regression analysis is a forecasting tool; however, it addresses expectations not guarantees. Included in those expectations is to see price move around the center regression line in Figure 2 rather than closing each day on the line.
The channel construction shown on the weekly chart has a strong bullish bent, but a different channel displayed in Figure 3 allows us to better visualize the mean return = 0 assumption. On this monthly bar chart for SPY, the predominant direction is sideways with a middle regression line that includes:
- A monthly high of 119.23 in July 1998 and
- A monthly high of 123.40 in Sep 2011.
While the 3.5% move in 23 years makes a bit of a case for the mean return = zero assumption, my intention is really geared towards visually price movement that can occur with a mean return that approximates zero. From this point adjustments in expectations can be made.
Figure 3 Monthly Bar Chart for SPY with Linear Regression Channel ProfitSource
Next week changes to the return assumption will be added.
Constant Volatility
The constant volatility assumption seems to be a necessity – it’s difficult to imagine a process that would systematically change volatility over time. Once again, simply understanding this assumption is in place and the potential for reality not tracking the model allows traders to assess option valuations. This is primarily accomplished by considering relative volatility levels, both statistical and implied.
Understanding the nature of returns and what goes into pricing models and probabilities is a useful step in assessing opportunities. As mentioned previously, market participants always need to recognize that probabilities never provide guarantees.
1 “Robert C. Merton - Autobiography". Nobelprize.org. 7 Feb 2012 http://www.nobelprize.org/nobel_prizes/economics/laureates/1997/merton.html
Brightman, H.J. (1986). Statistics in Plain English. Cincinnati, OH: South-Western Publishing Company
Kase, C. (1996). Trading with the Odds:Using the Power of Probability to Profit in the Futures Market. New York, NY: McGraw Hill
Marlow, J. (2001). Option Pricing: Black-Scholes Made Easy. New York, NY: John Wiley& Sons, Inc.
Taleb, N.N. (2005). Fooled By Randomness. New York, NY: Random House, Inc. (Original work published in 2004)
Vince, R. (2007). The Handbook of Portfolio Mathematics: Formulas for Optimal Allocation & Leverage. Hoboken, NJ: John Wiley& Sons, Inc.
Clare White, CMT
Contributing Writer and Options Strategist
Optionetics.com ~ Your Options Education Site
Questions for Clare? Please visit the discussion board on the homepage of Optionetics.com.
