Trading Options

In this post I will analyze some statistical properties of the series of returns on my portfolio.

While the return earned by holding a share of stock is often assumed to have a normal (Gaussian) distribution, a stock option is a non-linear asset. Even under the simplest Black-Scholes framework, the return earned by holding a single option has a very complicated distribution. Then, of course, the distribution of returns of an actively traded portfolio is yet another completely different subject.

In what follows I present statistical properties of the series of daily returns for

• My entire portfolio
• My portfolio, without the contribution of trades in futures.
• The S&P 500 index, for which each raw daily return will be incremented by 1.79%/250, the assumed contribution from a 1.79% per annum dividend yield, divided over approximately 250 trading days per year.

A return over a time interval with the starting value S[i] and an ending value S[i+1] is defined as log(S[i+1]/S[i]), where “log” is the natural logarithm. So, these are series of “continuously compounded” returns. The convenience of this definition, as compared to using simple returns (S[i+1]-S[i]) / S[i] is that the former is additive when applied to consecutive time periods. For example, the (continuously compounded) annual return for a given year is simply 250 times the average daily return shown in the table for this year.

For simplicity, I focus on the average and the standard deviation of a daily return. The Sharpe ratio is defined as the ratio of average to standard deviation. Therefore, for a risk-averse speculator, a “better” portfolio will have a higher Sharpe ratio.

Start of trading to end of 2003:

	Full	Ex-Futures	S&P 500
Average	0.214%	0.065%	0.087%
St.Dev.	0.77%	1.22%	1.08%
Sharpe Ratio	0.278	0.054	0.080

2004:

	Full	Ex-Futures	S&P 500
Average	0.039%	-0.020%	0.041%
St.Dev.	1.55%	1.19%	0.70%
Sharpe Ratio	0.025	-0.017	0.059

2005:

	Full	Ex-Futures	S&P 500
Average	-0.057%	0.049%	0.019%
St.Dev.	1.66%	1.24%	0.65%
Sharpe Ratio	-0.034	0.039	0.029

Year 2006 to-date (from January 1, 2006 to November 24, 2006):

	Full	Ex-Futures	S&P 500
Average	0.107%	0.066%	0.058%
St.Dev.	1.33%	0.82%	0.64%
Sharpe Ratio	0.080	0.080	0.090

Total: considering the entire trading history from the day trading commenced to November 24, 2006:

	Full	Ex-Futures	S&P 500
Average	0.059%	0.040%	0.051%
St.Dev.	1.42%	1.14%	0.79%
Sharpe Ratio	0.041	0.035	0.065

Why are the averages in the above tables so much smaller than standard deviations? Consider the return for S&P 500 over a single trading day. The index will either go up, or it will go down. For the sake of example, it might move up or down by 1% - in which case the standard deviation will be exactly 1%. However, the average of signed returns would be much smaller. If equally-sized up- and down-moves were equally likely, the average would in fact be zero, despite the fact that the series will have a significant standard deviation.

The other way of looking at the relative magnitudes of average and standard deviation of a daily return is by realizing that, at least for S&P 500, one could annualize them, by multiplying the average by 250 (the number of trading days in a year), and the standard deviation by about 16 (approximately the square root of 250). Taking the numbers from the last table, covering the entire period from the day the trading began in 2002 to November 24, 2006, the annualized average return for S&P 500 is 0.051%*250=12.75%, whereas the annualized standard deviation is 0.79%*16=12.64%. So, the mean grew relative to the standard deviation. As the time horizon gets longer, the likely total returns on the index are dominated by the mean (which scales linearly with time), while the standard deviation, scaling as the square root of time, increases in absolute terms but declines relative to the mean total return. This behavior is nothing but the central limit theorem of statistics.

The scaling argument just applied to explain the order of magnitude of average and standard deviation of daily returns of S&P 500 cannot be applied to returns on my portfolio, because it is actively traded. However, one can still compare them to the respective numbers for the index. If I am able to attain a higher mean return, this is good. If the same could be attained while keeping the standard deviation lower than that of the index, this would be still better. This is a high hurdle to overcome. I believe that the change over time of numbers in the above tables shows an improvement. For example, in the latest period, year 2006 to-date (to November 24), the mean return on the portfolio was higher than that of the index and while the “noise” (the standard deviation) of the portfolio was also higher, its Sharpe ratio at 0.08 was only a little lower than the value of 0.09 for the index.

Following a visual comparison of my portfolio to the S&P 500 index, a summary is in order. What the graph shows is that, over the past 4 years, the entire portfolio rose in value by about 65%, nearly matching the total return on the index, including the dividends earned on stocks that comprise the index. However, the portfolio experienced significantly more noise along the way. As for the performance of stock and options trades only - that is, excluding the contribution from trades in futures, the chart shows about 50% total return.

It is perhaps worth noting that, in plotting the performance of my portfolio, transaction costs and commissions have been taken into account, just as one would expect. For the chart of S&P 500, raw index values (with returns incremented by the assumed dividend yield) have been used. In practice, an index-tracking mutual fund will probably charge a management fee of at least 0.2% per annum. This fee has not been included in the chart.

Before going into further detail, let’s compare the performance of my portfolio to S&P 500 index.

Imagine that one dollar was invested on the day the trading commenced. The value of the resulting portfolio is shown by the dark blue line. This line is built without any assumptions; it is merely a rescaled chart of the actual dollar value of the account. Positions were marked at mid-market to arrive at this value.

As the trading of futures is not done according to any model, it makes sense to isolate its effect and consider the portfolio value net of the contribution of trades in futures. The value of the one dollar under this assumption is shown by the purple line. The series, plotted by this line, is a series of mark-to-market values of a portfolio that includes trades in stocks, stock options, interest amounts, stock splits and dividend payments. The mark-to-market calculation is done using the daily closing prices I maintain in a database.

Lastly, the yellow line represents the value of one dollar invested in S&P 500 index. As this index is not adjusted for dividends, I make an assumption about the annual dividend yield. Prof.
Aswath Damodaran of NYU gives an estimate of 1.79% annual yield. Accordingly, each daily return of the series of index returns, used to build the yellow line, is incremented by 1.79%/250, assuming 250 business days per year.

At the start of the chart, and throughout much of 2003, the dark blue and purple lines follow each other closely. The reason is that at that time, futures were not traded. At that time, relatively simple pricing models were used, as well as very simple ways to select new trades. The next phase, in 2004, was characterized by a lot of noise, as large market moves exposed the simplicity of the models being used, necessitating further development. In 2005, the whole portfolio experienced a lot of P&L noise, first mainly positive, then mainly negative. This was driven by large directional moves of futures prices and somewhat large position sizes. As new rules for the selection of trading strategies have been instituted around March 2006, the subsequent evolution of the stocks-and-options portfolio (the purple line) has become more orderly. The program used to select new trading ideas has not been showing too many of them lately, causing the relatively flat part in the purple line at the end of the chart (that is, in recent months).

A future post will show various statistical measures of the series of returns on my portfolio, and will also compare them to the same measures for S&P 500.

I trade options on the most liquid US-listed stocks. Over the years, this turned out to be a somewhat rewarding venture, so I continue it as an ongoing business. The trading is done in the style of Black and Scholes. An option position is initiated at the start of a strategy, and is held until the strategy is unwound. This position remains unchanged unless changes in the margin requirements cause it to be reduced. The shares of the underlying stock are traded during the same time frame so that the position is approximately delta-hedged, given the delta computed under an option-pricing model. This trading style is highly unusual for an individual trader. Most individuals cannot pursue it due to lack of knowledge and IT infrastructure needed to compute risk, generate trade ideas and manage other aspects of the business. On the other hand, this trading style is bread-and-butter of derivative trading desks at investment banks, hedge funds and other institutions, and is practiced in trading of options on bonds, foreign exchange, interest rates and, of course, common stock.

My trading business is unique in that I alone manage every aspect. In contrast, in an institutional trading context many people and departments are involved in different aspects such as research, collection of historical data and its storage in a database, placing the trades and managing the aggregate risk of a portfolio, P&L accounting and others.

The business started almost 5 years ago. Several months were spent putting together the first modeling framework and writing programs that keep track of positions. Trading commenced about 4 years ago. I used various models to trade the underlying stock as a hedge for one or more options on that stock. This delta-hedging approach is still being used. It is just like the Black-Scholes framework, except a more sophisticated non-Black-Scholes model is used. So far, this project resulted in a positive total return with an improvement seen over time. The Sharpe ratio, defined as the ratio of the average daily return to the standard deviation of a daily return, has also shown improvement.

I trade options. In the past four years of active operation, this business has produced a return of approximately 50%. Over time, the performance, as measured by the average daily return, and by the standard deviation of a daily return, has shown improvement. The purpose of this site is to talk about various aspects of the trading activity, and describe its future progress. I welcome further inquiries from interested and qualified readers.