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The Risks and Rewards of Selling Volatility

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The Risks and Rewards of Selling Volatility SAIKAT NANDI AND DANIEL WAGGONER Nandi is a former senior economist at the Atlanta Fed and is currently a financial engineer at Fannie Mae. Waggoner is an economist in the Atlanta Fed’s research department. They thank Jerry Dwyer and Larry Wall for helpful comments. UYING AND SELLING CERTAIN KINDS OF VOLATILITY-SENSITIVE OPTIONS PORTFOLIOS IS A POP- ULAR PRACTICE EVEN THOUGH THIS ACTIVITY IS ASSOCIATED WITH SUBSTANTIAL RISK. WHEN SUCH PORTFOLIOS ARE FORMED, TWO FACTORS SIGNIFICANTLY INFLUENCE THE VALUE OF THE B OPTIONS: THE PRICE OF THE UNDERLYING ASSET AND THE FUTURE VOLATILITY EXPECTED TO prevail until the options expire. It is possible to form results in much higher implied volatility, the value of a portfolio of call and put options so that the portfo- the implicit short position in volatility could dra- lio’s payoff is very sensitive to the volatility of the matically decrease. For example, the former underlying asset but only minimally sensitive to Barings PLC sustained huge losses from short posi- changes in the level of the underlying asset. Traders tions in volatility (initiated by a trader, Nick and investors who frequently buy or sell such port- Leeson) on Nikkei futures as the Nikkei plunged in folios do so with a view of the volatility of the under- early 1995 (Jorion 2000). lying asset that does not correspond to the expected The objective of this article is to delineate the future volatility embedded in the option price, or risks and rewards associated with the popular prac- the implied volatility.1 For example, the former tice of selling volatility through selling a particular hedge fund Long Term Capital Management had portfolio of options called straddles. Toward this created portfolios of options on certain stock indexes end, the article first examines the statistical proper- based on its view that expected future volatility was ties of the returns generated by selling straddles on different from the prevailing implied volatilities the Standard and Poor’s (S&P) 500 index. Although (Dunbar 1999). it is theoretically possible to construct other options There is often substantial risk, however, in portfolios to make volatility bets (Carr and Madan options portfolios set up to exploit a perceived mis- 1998), straddles are by far the most popular type of pricing in the expected volatility of the underlying portfolio. The article also demonstrates that the asset without initial sensitivity to the level of the usual practice of selling volatility by comparing the asset. This risk stems from possible subsequent observed implied volatility (from option prices) abrupt changes in the level of the asset price. with the volatility expected to prevail (given the his- Changes in volatilities are highly negatively correlated tory of asset prices) could be flawed. This flaw with changes in levels in many asset markets. If a could arise if the underlying asset has a positive risk short position in implied volatility on a market is premium—that is, if its expected return over a created and a subsequent sharp market decline given horizon exceeds the risk-free rate over the Federal Reserve Bank of Atlanta ECONOMIC REVIEW First Quarter 2001 31 same horizon—and the returns of the underlying the chart clearly show, however, that the higher the asset are negatively correlated with changes in volatility of the underlying asset between the pres- volatility. Thus, basing the decision to sell a straddle ent time and option expiration, the higher the on a comparison of seemingly irrational high implied potential payoff from a long position in the straddle. volatilities with much lower expected volatility A straddle might thus seem to be an ideal vehicle could itself be an irrational choice. through which to express a view on the future volatility of the underlying asset without necessarily Straddles having an initial view on the future direction of the traddles are very popular but quite volatility- asset price—that is, a trader might buy a straddle if sensitive options portfolios. A straddle con- volatility is expected to be high or sell one if volatil- Ssists of a call and put option of the same strike ity is expected to be low. price and maturity so that the strike price is equal to To see how a straddle is a bet on volatility, (or very close to) the current price of the underly- assume that the implied volatility (using the BSM ing asset. The higher the volatility until the option model) from a straddle on an equity option that expires, the higher expires in one month is 40 percent (annualized). On the expected profits the other hand, assume that the maximum historical from buying the strad- volatility of the underlying asset that has been A straddle might seem to dle, and vice versa. observed over any one-month horizon is 30 percent For example, suppose (annualized). Based on this observation, would one be an ideal vehicle through an investor buys a sell the straddle—in other words, sell volatility— which to express a view on European straddle because it appears to be high? If volatility is con- the future volatility of the that matures in fifty stant or evolves deterministically, that is, if the days, the current assumptions of the BSM model hold true, it may be underlying asset without price of the underly- tempting to sell the straddle. It has been well docu- necessarily having an initial ing stock is $100, and mented, however, that in most asset markets volatil- view on the future direction the strike prices of ity evolves randomly through time (Bollerslev, both the European Chou, and Kroner 1992). Even if volatility were con- of the asset price. call and put options stant or predictable, the price of the underlying are $100. Under the asset could periodically undergo an abrupt level (BSM) Black-Scholes- shift—sometimes referred to as a jump in the price Merton model (out- process.2 Because of these factors, a risk premium lined in Black and Scholes 1973 and Merton 1973) related to the randomness of volatility or unpre- with an annualized implied volatility of 20 percent dictable jumps in asset prices could get incorporated and a risk-free rate of 5 percent, the price of the call into the price of an option (Bates 1996). The implied option is $3.29, and the put option price is $2.61. volatility determined from the price of an option The cost of the straddle is thus $5.90 ($3.29 + using the BSM model would thus be contaminated $2.61). If the stock price at the expiration of the with the relevant risk premium, making any com- option is at least $105.90 (payoff from the call parison with historical volatility problematic.3 option is $5.90, and payoff from the put option is Box 1 gives an example in which the volatility of zero) or $94.10 (payoff from the put option is $5.90, the underlying asset evolves randomly and the and payoff from the call option is zero), the buyer changes in volatility and returns are negatively cor- breaks even. The greater the stock price is than related. The box shows that as long as the underlying $105.90 or the lower it is then $94.10, the greater asset has a positive risk premium, the implied volatil- the buyer’s profits are. Conversely, if the stock price ity from the options could be higher than a historical is greater than $94.10 but less than $105.90, the measure of volatility. Because the BSM model may seller of the straddle turns in a profit because the be an inadequate description of the observed option revenue generated by the sales exceeds the losses prices, caution is warranted in using it to compare when the straddle expires. Of course, the narrower implied volatility to historical volatility. the dispersion of the possible stock prices at matu- rity, the higher the seller’s profits. The chart shows The Implied and Historical Volatility the payoff from buying the straddle when the of the S&P 500 options expire. n trading straddles on a market index such as the When the straddle is created, the change in its S&P 500, it is important to have a good under- value is not very sensitive to the change in the value Istanding of the dynamics of volatility through of the underlying asset. The previous example and time. This section offers a computation of the implied 32 Federal Reserve Bank of Atlanta ECONOMIC REVIEW First Quarter 2001 A Hypothetical Payoff Scenario 20 Call Profit 10 Put Profit Straddle Payoff Profits 0 80 90 100110 120 Stock Price Note: Chart shows profits at option expiration from buying the straddle as well as the profits at expiration from buying a call and put option with strike prices of 100. volatilities from near-the-money options (options in imum and maximum implied volatilities. Someone which the strike price is as close as possible to the comparing the implied volatility with the simple price of the underlying asset) on the S&P 500 and a measure of historical volatility might often be simple but frequently used measure of historical tempted to sell straddles. Box 1 shows, however, that volatility for each trading day over a six-year horizon, implied volatilities can be above historical volatilities 1990 through 1995.4 On each day, historical volatility without any trading opportunities. is computed as the standard deviation from a sample of the last thirty days of daily S&P 500 returns The Impact of Correlation and (close-to-close) and then multiplied by the square Implied Volatility Skew root of 252 (as there are about 252 trading days in a lthough it is possible to sell straddles based on year with daily returns assumed to be independently an incorrect comparison of volatilities, the and identically distributed).
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