The risk reversal premium manifests as an overpricing of out-of-the-money put options relative to out-of-the-money call options with equal expiration dates. The premium apparently arises from equity investors’ demand for downside protection, while most market participants are prohibited from selling put options. A typical risk reversal strategy is a delta-hedged long position in out-of-the-money calls and an equivalent short position in out-of-the-money puts. Historically, the returns on such a strategy have been positive and displayed little correlation with the returns of the underlying stocks. The strategy does incur gap risk with a large downside, however. The long-term profit of risk-reversal strategies reflects implicit market subsidies related to “loss aversion”.

Hull, Blair and Euan Sinclair (2021) “The Risk-Reversal Premium”.

The below quotes are mainly from the paper. Headings, cursive text, and text in brackets has been added. Some formulas have been replaced by explanatory text.

This post ties in with this site’s summary on implicit subsidies in financial markets, particularly the section “sources of implicit subsidies”.

### What is a risk reversal strategy?

“The risk reversal options trading strategy consists of __buying an out of the money call option and selling an out of the money put option in the same expiration month__…The investor who enters a risk reversal wants to benefit from being long the call options but pay for the call by selling the put…This eliminates the risk of the stock trading sideways but does come with substantial risk if the stock trades down…The maximum profit is unlimited as being long an upside call allows the investor to continue to make money as the stock trades higher. The maximum loss is also unlimited, at least down to zero, as the stock falls in price losses continue to build upon the short put.” [OptionStrategies Insider]

### What is the risk-reversal premium?

“The **risk-reversal premium** [means that] __out-of-the-money puts are overpriced relative to out-of-the-money calls__… [There have been significant] profits from trading risk-reversals, selling an out-of-the-money put and buying an out-of-the-money call with the same time to expiry.”

“__We view the risk-reversal premium as a sub-factor of the well-documented variance risk premium__: the tendency of implied variance to be higher than the subsequently realized variance. Just as an equity portfolio can benefit from exposure to smart equity beta factors, a volatility portfolio will benefit from including each of these variance premia.”

“Researchers have reached a consensus that __implied volatility is generally overpriced and that short positions in variance, either through swaps or delta-hedged options, produce positive returns__. However, far less is understood about what specific characteristics of implied volatility leads to these returns and how these characteristics relate to each other.”

### How does the risk reversal premium come about?

“The risk-reversal premium…is __driven by investors’ utility preferences__ which lead them to over-pay for the risk reduction benefits of long puts instead of valuing options on the basis of expected returns.”

“The effect can be re-expressed as an **implied skewness premium** or an **implied correlation premium**…Broadly, both of these explanations say that investor demand for options exceeds supply. Most __investors are long equities and would use options for downside protection__…This directly increases out-of-the-money put prices, which in turn increases the implied volatility of other options as market-makers spread their volatility risk. Further, many __retail investors are prohibited by their brokers from selling options at all, removing another possible source of supply__. Formally, the utility function of investors encourages them to buy options for protection rather than evaluating trades on the basis of their expected profitability.”

“Volatility as a function of strike forms the implied volatility curve, smile or smirk…Many studies have shown that __the implied risk-neutral density is more negatively skewed than the lognormal distribution__. Further, the implied risk-neutral skewness is __generally greater in magnitude than the subsequently realized skewness__…The implied skew premium has very little to do with the skewness of the underlying. Further, just as with the at-the-money volatility premium, the implied skew premium is time-varying and will sometimes trade at a discount.”

“The expected profit of the risk-reversal is given by

**vanna x beta x variance**

where

**vanna** is the first derivative of vega with respect to the underlying price,

**vega** is the change in option value with respect to change in volatility,

**beta** here measures the co-movement of price volatility and the underlying price, and

and **variance** is squared price volatility.”

“There have been many studies that show out-of-the-money put options are overpriced…However, risk-reversals have not been studied to the same extent. In the same way that investors’ utility preferences lead them to buy puts, __there are also many investors who sell calls as part of a covered call strategy__. Finally, __calls are not typically bought as hedges__, further lowering demand. Together, these effects create a risk-reversal premium which is distinct from the variance premium.”

### What is the empirical evidence for a risk reversal premium?

“__We construct a hedged risk reversal using SPY options__. We initiate the position by selling the 15-delta puts and buying the 15-delta calls with the expiration closest to 25 trading days until expiration. The initial hedge is to sell short 30 shares of SPY. The position is delta-hedged daily, and each week the strikes are adjusted to remain as close as possible to 15-delta. The position is closed when there are five trading days left, and a new 25-day position is entered. The account returns are calculated on the Regulation T margin required…and is normalized to an initial value of 100…Results without transaction costs are shown in [the table and graph below].”

“While this example shows the clear existence of a premium in the delta-hedged risk-reversal, this implementation is far from what traders would do in practice (Sinclair, 2013). __Common industry practice is to let the delta of a position fluctuate within a certain band and only re-hedge when those bands are crossed__. In our case, whenever the net delta of the options either drops below 20 or above 40, the portfolio is rebalanced by closing the position and re-establishing with the options that are now closest to 15-delta in the same expiration…Results without transaction costs are shown in [the table and graph below.”

### Do risk reversal positions benefit an equity portfolio?

“[Theoretically] the return of a hedged risk-reversal is not dependent on the returns of the underlying. And empirically the __returns of the hedged risk-reversal have low correlation to those of the underlying__…To an investor, the important aspect of instruments being ‘different’ is they can be combined in a diversified portfolio that is in some sense better than each individual asset. This is the case with the risk-reversals.”

“Our choice for portfolio construction is to find the optimal number of risk-reversals to hold with 100 shares of SPY. The hedged 15-delta risk reversal consists of one risk-reversal and short 30 shares. So, when combined with 100 shares of SPY a one risk-reversal portfolio would consist of 70 shares, one short 15-delta put and long one 15-delta call…The summary statistics of the ETF, risk-reversal, and the portfolio of the two are presented [below].”

“While the historically realized risk has been acceptable (similar to the underlying ETF), by adding the short puts, we have raised the probability and consequences of a catastrophic scenario. While some losses will be mitigated by the vega of the long calls responding to the corresponding increase in implied volatility, in the worst-case scenario when the ETF drops below the short put strike, a one-to-one portfolio will be long more delta than a pure equity position.”