In a new working paper, a scholar at the School of Finance, at the University of St Gallen, Switzerland, Semir Ben Ammar, has analyzed the relationship between catastrophe risk and the “implied volatility smile” in stock options. Along the way, he has proposed what may be a novel type of arbitrage, and he makes a point that may bear on debates over climate change.
This is a long-time quant concern. The “vol smile” is the u-shaped figure that comes of plotting strike price against the implied vol of a group of actions with the same expiration date. Vol increases as one moves in either direction away from the strike price.
This is a bit of a mystery. The Black-Scholes model (through which, after all, the “implied” volatility is found) predicts a flat line, not a smile. So: why the smile?
One common explanation (though not the only one) is that a big divergence between the spot price of the underlying commodity and the strike price of market-clearing options itself suggests an extreme market move, a “black swan” event.
Hurricanes Don’t Know
Black swan events, such as natural disasters, are exogenous shocks. Ammar’s paper begins with a quote from Warren Buffett, “The hurricane does not know the rate that was charged for the hurricane policy, so it’s not going to respond to how much you charge.”
Thus, it is natural to look at catastrophe risk as it appears specifically in the steepness of the smile, on options written for property and casualty insurance, as a special case of the vol smile. This was the tack Ammar pursued.
The paper begins by deriving an option pricing model specific to property and casualty insurers, using the derivatives market price in determining the valuation of their exposure to catastrophe risk. In this connection Ammar makes the point that fair pricing can affect the cost of capital for such businesses. Third, he addresses the theoretical question of why there is a smile at all.
Here is where he mentions a potential new arb play. He asks that his readers consider the link between the traditional derivatives market and insurance-linked securities suggesting that for investors “this link might be an indicator for potential arbitrage opportunities if expectations on catastrophe risk in the two markets significantly diverge.”
Ammar’s findings confirm the hypothesis that “the implied volatility slope of P&C insurers is related to risk premiums from the cat bond market with a correlation of 49.4%.” He also reaches an important negative conclusion, that is, a conclusion about something that does not affect the steepness of the smile: the reinsurance cycle and its pricing dynamics.
Changes in the Slope
Further, Ammar sees changes in the slope over time. Katrina, the 2005 hurricane that devastated the Gulf Coast in August 2015, provides a natural experiment here, and Ammar divides his data into pre and post Katrina halves. The slope in the pre-Katrina period (the nine years ending in 2004) is considerably less steep than the slope has become since. There are at least three possible explanations for this. The first is that Hurricane Katrina could have created enhanced risk awareness on the f the pre-existing smile. The second is that there has been an “overall increase of natural disasters in the post-2005 years.” A third explanation, though, would turn coincidence. Katrina happened to come three years before a global financial crisis, and that crisis is what has driven the steeper slope of the post-Katrina period.
To test for and eliminate that third possibility, Ammar introduces a dummy variable and gives it the value of 1 for a period beginning in February 2008 and ending in July 2009. Those are start and end dates for the crisis according to the National Bureau of Economic Research.
With the benefit of the dummy variable, Ammar did a new regression on the post-Katrina period. He found that the steeper slope of P&C insurers was not driven by the financial crisis. Thus, one is left with either of the two other choices: post-Katrina heightened awareness in the market, or climate change and the volatility it brings.