How to Sell Volatility
I want to design systematic trading strategies that harvest the intermediary risk premium. In what follows, I describe four strategies for trading systematic risk. One can always rescale bets to achieve any desired level of returns by bearing greater risk. The risk of poor returns can be proxied by the volatility of the portfolio returns. Risk premium, ie compensation per unit of risk, is then given by the Sharpe ratio — the ratio of returns in excess of the risk-free rate to the volatility of the portfolio returns.
Three of our four strategies are passive holding strategies: (1) holding the SP500; (2) selling volatility — proxied here by the tradable version of the fear gauge, VIXY, so our portfolio weights are [-1,0] in [VIXY,VIXM]; (3) a passive hedging strategy that sells VIXY and holds VIXM — we use antisymmetric weights [-1,+1]. We believe that (3) allows us to harvest a considerable portion of the intermediary risk premium. And it is for a related reason that the volatility risk premium, the expected excess return on (2), exceeds the equity risk premium, the expected excess return on (1) — ie, why (2) makes more money than (1). Our fourth systematic strategy not only hedges the risk of selling volatility by going long on medium-term volatility, (4) tactically flips the logic if the predicted probability of a risk-off tomorrow exceeds a certain threshold. Ie, we go long on VIXY and short VIXM — we flip the weights to [+1,-1].
The first two strategies are chosen as benchmarks. The third, the passive hedging strategy, (3), is also a more sophisticated benchmark for strategies specifically designed to harvest the intermediary risk premium. The motivation for the specific form of the tactical reallocation strategy comes from the following considerations.
In order to construct our trading strategy, we need a bet-sizing function. Given our estimate of the probability of a risk-off tomorrow, we begin by examining the 3-parameter family of bet-sizing functions whose returns are given by,
where $latex \omega>0$ is a scale hyperparameter that allows us to lever up or down. We choose it achieve a desired level of portfolio volatility. Without loss of generality, we set it to unity. $latex \zeta>0$ is a shape parameter that controls how rapidly we move from holding [-1,+1] to [+1,-1]. The functional form above follows naturally from the nature of the trade. The hyperparameter $latex 0<\theta<1$ is a location parameter that controls the sign of $latex (p-\theta)$ and thus the direction of our bet.
We obtain daily closing prices of VIXY and VIXM from the Financial Times, returns on the SP500 from FRED, and the risk-free rate from Kenneth French's website. We obtain daily log returns from January 11, 2011 to November 27, 2019. In order to test the performance of the (4), we construct a predictive model based on intermediary asset pricing theory. We estimate the model using machine learning. We reserve the first 500 trading days for training our predictive model. Then we use expanding window — using data that is available on the previous day — 5-fold cross-validation to estimate the parameters of our model using Prado's log loss as the loss function. We then cross-validate the hyperparameters, again using 5-fold cross-validation. We find that the out-of-sample performance of our trading strategy is monotonically increasing in the shape parameter $latex \zeta$.
This considerably simplifies the search problem because of the following mathematical fact.
The one-parameter family characterized by a single location parameter thus turns out to be sufficient for our purposes. This is nothing but (4) above. Again, we cross-validate the location parameter out-of-sample.
The future will be unlike the past. The past decade of asset prices is simply one sample path. We can get a firmer handle on expected returns through random sampling of the dataset. We run 100,000 simulations of annual returns on the four systematic strategies by randomly choosing 252 trading days with replacement from our sample of 1,733 trading days after the initial training period. This gives us a better estimate of expected risk premia compared to realized risk premia. We report Sharpe ratios rescaled by the volatility of returns on the SP500.
We find a significant expected equity risk premium. The expected excess returns on holding the SP500 is 10.3 percent per annum. The volatility risk premium, the compensation for selling vol, ie excess returns on (2), is 14.0 percent, 3.7 percent higher than the market. The premium on the passive hedging portfolio is marginally higher still at 14.5 percent, or 4.2 percent above the market benchmark. The expected risk premium on the tactical hedging strategy is 16.3 percent, 6 percent higher than the market, 2.3 percent higher than the vol risk premium, and 1.8 percent higher than the passive hedging strategy. These are big differences given the power of compounding.
The realized risk premia on these four strategies over the past decade have been similar.
We also display the realized risk-adjusted returns on a rolling basis for the past few years since we need some initial data in order to stabilize the Sharpe ratios.
The results documented here should be of some interest to investment managers. Given the role that systematic daily volatility plays in dealer balance sheet management, our interpretation is that the vol risk premium is high because it contains a portion of the intermediary risk premium. The problem with selling volatility, however, is that you can lose your shirt when systematic volatility returns with a vengeance. The drawdowns can test the patience of all but the most patient investors. This is precisely what happened over the past few years.
The passive hedging strategy allows one to harvest a greater portion of the intermediary risk premium without risk of such horrible drawdowns. Meanwhile, the tactical hedging strategy strictly dominates the passive hedging strategy. It thereby allows us to harvest an even greater portion of the intermediary risk premium at daily frequency. Of course, we can do even better with a more sophisticated model of predicting risk-offs, or working at a higher frequency still; thereby allowing us to exploit intraday fluctuations.
Please let me know if you can get me access to expensive tick data. Or if you want me to work with your team on researching systematic trading strategies.
Postscript. Instead of examining expanding window Sharpe ratios, as in the penultimate graph, we are better off looking at fixed-window rolling Sharpe ratios. Here we look at annual rolling Sharpe ratios of our passive and tactical hedging strategies. The graphs below clarify why the tactical hedging strategy strictly outperforms the passive hedging strategy. Basically, we flip from holding the portfolio [-1,+1] to [+1,-1] in [VIXY,VIXM] when we are near-certain that tomorrow will be a risk-off. This ensures that, instead of losing money in a risk-off we can predict the day before, we make money. The next figure displays daily year-on-year risk-adjusted returns — the moving 252-day Sharpe ratio rescaled to market volatility.
The next figure displays the 252-day moving average of the same. From both graphs, we can see that the outperformance of the tactical strategy is due entirely to our ability to predict risk-offs on the next day. This is secret sauce of the extraordinary risk-adjusted returns documented here.
Since we use up another year to initialize the rolling Sharpe ratios, we again report the realized risk premia on the four systematic strategies. The time window here is Jan 13, 2014 to Nov 27, 2019, so we are talking about the performance of these strategies over the past six years. On a risk-adjusted basis, tactical hedging has outperformed passive hedging by 2.2 percent; selling vol by 2.3 percent; and the US stock market by 5.5 percent per annum. If you'd invested $1 on Jan 13, 2014 in these four strategies with weights chosen to target market levels of volatility, as of Nov 27, 2019, you'd have $1.65 if you held the market portfolio; $1.97 if you sold volatility; $1.98 if you held the passive hedging portfolio; and $2.22 if you obeyed the tactical hedging strategy.
