Catastrophe Risk and the Implied Volatility Smile

• Date: 01 June 2020
• DOI: http://doi.org/10.1111/jori.12268
• Published date: 01 June 2020
• Author: Semir Ben Ammar

©2018 The Journal of Risk and Insurance. Vol.XX, No. XX, 1–25 (2018).

DOI: 10.1111/jori.12268

Catastrophe Risk and the Implied Volatility Smile

Semir Ben Ammar

Abstract

Property–casualty insurers are exposed to rare but severe natural disasters.

This article analyzes the relation between catastrophe risk and the implied

volatility smile of insurance stock options. We find that the slope is signifi-

cantly steeper compared to the rest of the economy and exhibits a seasonal

pattern due to hurricanes. We are able to link the insurance-specific tail risk

component derived fromoptions with the risk spread from catastrophe bonds

and global economic losses caused by catastrophes. Our results provide an

accurate, high-frequency calculation for catastrophe risk linking the tradi-

tional derivatives market with insurance-linked securities.

The hurricane does not know the rate that was charged for the hurricane policy, so

it’s not going to respondto how much you charge. And if you charge an inadequate

premium, you will get creamed over time.

–Warren Buffett

June 9, 2014, Las Vegas

Introduction

Options allow us to evaluate the expectation of market participants regarding extreme

events (Backus, Chernov, and Martin, 2011). As property–casualty (P&C) insurance

companies are exposed to natural and man-made catastrophes, options written on

P&C insurance stocks should exhibit a catastrophe risk premium in the tail of their

density function. This risk premium should be in excess of the tail risk in stock prices

induced by market events, given that P&C insurance companies are also exposed

to the overall economic development and thus the same market events. This article

Semir Ben Ammar is at the School of Finance, Institute of Insurance Economics, University of

St. Gallen, Rosenbergstrasse 22, CH-9000 St. Gallen, Ben Ammar can be contacted via e-mail:

semir.benammar@unisg.ch or semirbenammar@gmail.com. I would like to thank two anony-

mous referees,Christian Biener, Patricia Born, Martin Boyer, Alexander Braun, GeorgesDionne,

Randy Dumm, David Eckles, Martin Eling, Martin Halek, Charles Nyce, Hato Schmeiser, Jan

Wirfs,and seminar participants at the University of St. Gallen for helpful comments and sugges-

tions. I am also grateful for comments from participants at the 2015 German Finance Association

meeting, the 2015 World Risk and Insurance Economics Congress, the 2016 Western Risk and

Insurance Association (WRIA) meeting, the 2014 Asian-Pacific Risk and Insurance Association

meeting, and the 2016 meeting of the German Insurance Economics Association. This article

received the 2016 Dorfman Awardby the WRIA as the best PhD paper presented at the WRIA

meeting, for which I am deeply grateful.

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analyzes the slope of the implied volatility, that is, the absolute difference between

out-of-the-money (OTM) and in-the-money (ITM) put options, as a measure of tail

risk to identify a catastrophe risk premium. The idea behind this approach is that

OTM options provide more effective protection against rare events than ITM options

(Kelly, P´

astor, and Veronesi, 2016).

There are at least three motivating aspects in analyzing tail risk specifically using

options on P&C insurance stocks to identify inherent catastrophe risk.1First of all,

catastrophes can cause great damage to specific regions. Risk-averse households are

interested in offloading such risks but face high insurance premiums for this type

of risk (see Froot, 2001; Zanjani, 2002). Any insight into catastrophe risk can thus

further enhance our understanding of risk-adequate compensation for catastrophe

risk. Second, some market participants specifically securitize part of their tail risk

(i.e., catastrophe risk) in financial markets by means of insurance-linked securities

(ILS). This allows us to verify our results for catastrophe risk in another market with

readily available prices and establish a link between the two. Third, in contrast to

previous studies that investigate the entire economy or the banking sector, it is the

unique business model of P&C insurers to assume risk from other companies and

individuals to reduce overall (tail) risk exposure. As such, P&C insurers are a natural

fit for analyzing the impact of tail risk assumption on the implied volatility smile.

No previous studies on options written on insurance stocks exist. However, the fi-

nance literature focuses on two aspects closely related to ours. First, the determinants

of the implied volatility smile are important to explain the anomaly of the implied

volatility smile itself (Dennis and Mayhew, 2002; Bollen and Whaley, 2004). Second,

the relation between the implied volatility smile and tail risk has recently gained much

attention with regard to financial guarantees (Kelly, Lustig, and Nieuwerburgh, 2016)

and political uncertainty (Kelly, P´

astor, and Veronesi, 2016). Our article adds an im-

portant perspective to the discussion between tail risk and the implied volatility smile

by linking catastrophe risk with the steepness of the implied volatility smile.

The contribution of this article is fourfold. First, we derive an option pricing model

unique to P&C insurers that accounts for catastrophe risk and uses the derivatives

market for accurate pricing of catastrophe risk. To further enhance our knowledge of

catastrophe risk and its pricing, new methods and perspectives can reduce market

imperfections. Second, fair pricing for catastrophe reinsurance can affect the capital

requirements for catastrophe risk and thus reduce the cost of capital (Zanjani, 2002).

1We define catastrophe risk as a specific and independent component of the overall tail risk

to which companies are exposed. Thus, catastrophe risk is one of many potential sources of

distress to a firm (here,the P&C insurer). We follow Froot’s (2001) definition of catastrophe risk,

which relates to all events linked to natural hazard (e.g., hurricanes, earthquakes, wind and ice

storms, floods, etc.) causing financial losses. In theory, our definition expands to man-made

disaster such as terrorist attacks. However, such events not only affect the P&C insurance

industry but also the rest of the economy (see, e.g., Brounen and Derwall, 2010; Thomann,

2013). As our research design relies on identifying differencesbetween insurers and the rest of

the economy,man-made disaster can only have a marginal impact on our analysis. Therefore,

the focus of this article is on natural disasters.

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