S&P 500: One Thousand Simulations
May 25, 2021
The value of the S&P 500 goes up an average of 8% per year. That statistic alone reveals little about the risk I take when jumping into the market. By simulating various paths the price could take, I can build some intuition around that uncertainty. The figure is intended to do just that.
Using the Simulation field I can select any of a thousand simulated price paths. For a more fulsome perspective, I tick the Show Percentiles box, allowing me to see what percentage of those 1,000 price paths fall below a given price on a given day in the future.
SPDR S&P 500 ETF Trust (SPY) is an exchange traded fund that attempts to mimic the price and yield performance of the S&P 500. It's a straight foward way for me to 'buy' the index. However, I prefer to trade related options.
SPY options are contracts for the right, but not the obligation, to buy or sell SPY shares (100 shares per contract) for a given price (strike) over a given period of time. That period of time is from now until the contract exipiration date. A SPY option is not the shares themselves but rather a derivative, whereby its value is dervied from SPY shares, the underlying security.
What's an option contract worth? For one, it's worth what people will pay. It could be worth nothing if it exipres out-of-the-money. For example, assume i bought a contract months ago that gives me the right to buy 100 SPY shares at $4 a pop. Further, imagine me still holding the contract on its expiration date but, unfortunatlely, SPY shares are trading at $3.90. That's 10 cents lower than my strike price. If I really wanted to own 100 SPY shares, it would be cheaper to buy them on the open market at $3.90 each rather than exercise my option to buy them at $4. In this scenario, my option contract is effectively worthless. In a happier scenario where SPY shares are trading at $4.10, I exercise my option and buy those 100 shares for $4 each, immediately turn around and sell them to the open market for $4.10 each and pocket the profits. These scenarios are just two in an infinite set of possibilities that collectively describe the risks and rewards of owning such an option.
The Black-Scholes formula (BSF) provides a theortical valuation for options contracts that attempts to account for the vast range of possibe price movements in their underlying security. If I plug-in the current price of SPY shares, a strike price, a time to expiration, and make a few assumptions, the BSF will give me a theoretical price for the option. Those assumptions I glossed over are 1) the returns I could get on a risk-free investment (arguably a US Treasury Bond) and 2) the future volatility of SPY between now and expiration of the option (which I can only guess at).
Personally, I'm unsatisfied with the BSF for a couple of reasons. BSF makes an assumption that option returns (security price at expiration divided by strike price) are lognormally distributed. Such a distribution is fat-tailed whereby the expectaion or average value is heavily influenced by a small number of high values. The implication for options trades is that, while the average return as estimated by the BSF might be attractive, it could be a frequent loser. The extent of that risk is not evident in the BFS output. While a problem in and of itself, this problem is exacerbated (at least for SPY options) by the choice of a lognormal distribution for returns. Looking at the history of the S&P 500, it's clear to me that these returns have a distribution that is even more fat-tailed than the lognormal distribution. This implies that the failure rate of these trades is even higher. Before taking the risk on a trade, I not only want to know the expected return but also have a sense of the number of losing investments I might have to make before landing on a winner, because it might take more than a single lifetime.
I use my S&P 500 simulations to generate a differentiated view of the value of SPY options. Here's how I do it. Every 15 minutes during market hours, I pull the option chain for SPY, giving me the asking price for all SPY options over various strike prices and expiration dates. Then, assuming SPY returns will match those of the S&P 500, I calculate the returns of each option for each simulation. For in-the-money options, the return is the simulated closing price of 100 SPY shares minus the strike price of the same all divided by the option ask price. In simulations where a given option expires out-of-the-money, the return is set to zero. For a given expiration date, statistics for all of these simulated returns over all of the available strike prices plot nicely on the interactive graphs above. With this perspective, I get a sense of expected returns and associated risk for the various options I could own, both calls and puts. Additionally, through my method of simulating the future by sampling the past, the return distribution I end up with is as fat-tailed as history implies.
- The future is not the past, yet my simulations in some respects look exactly like the past.
- While SPY is meant to imitate the S&P 500, they are not the same thing.
- Given these are American rather than European style options, I can exercise them at any time before the expiration date. However, I don't attempt to quantify any value that adds.
- My options return do not consider the time value of money (e.g. a dollar today is worth more than a dollar tomorrow).