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1By Christina Lee and Ivana Lee THE VOLATILITY SKEWBy Christina Lee and Ivana Lee
2OVERVIEW Volatility Skew Black-Scholes Model Types of Volatility Skews Volatility SmileVolatility SmirksCalculationsAnalyzing Volatility GraphsCauses and TheoriesCrash-o-phobia
3THE VOLATILITY SKEWAccording to the Black- Scholes model, we should expect options that expire on the same date to have the same implied volatility regardless of the strikes.The implied volatility actually varies among the different strike prices. This discrepancy is known as the volatility skew.At-the-money options tend to have lower volatilities that in- or out-of-the-money optionsIn-the-money: for call options, the market price is below the option’s strike price.At-the-money: The market price is the same as the option’s strike price.Out-of-the-money: for call options, the market price is above the option’s strike price.
4BLACK-SCHOLES MODELCalculates fair economic value of an option - the price at which both a buyer and seller would break even.On Maple:S = last trading priceE = strike pricer = riskless interest rateT = time until maturityσ = implied volatility
5BLACK-SCHOLES MODEL Black-Scholes is technically inaccurate because: Implied volatility should be constant according to this modelImplied volatility graph should be a horizontal straight lineWhen implied volatility is graphed, it is presented in volatility skewBlack-Scholes does not consider certain aspects that may alter the price, such as:LiquiditySupply and demandThe Black-Scholes model performs a sort of regulation of the market itself, with traders adapting themselves to it which causes the volatility skew.The Truth on Implied Volatilities – Slide 14Black-Scholes is technically inaccurate because if this model was true, the implied volatility as a function of the exercise price, we should obtain a horizontal straight line. This implies that all options for buying or selling the same underlying asset with the same expiration date, but with different exercise prices, should have the same implied volatility. This, however, does not occur because the implied volatility presents a strong U-shaped pattern which is the .. VOLATILITY SMILE
7THE VOLATILITY SMILEOne type of volatility skew is the volatility smile. The volatility smile is the U-shaped curve that often occurs when strike prices for a group of options expiring on the same date are plotted against their implied volatilities.The volatility smile for equity options traded in American markets did not appear until after the Crash of 1987.Basically, what the volatility smile shows is that implied volatility is higher as the options go more and more In The Money (ITM) and Out Of The Money (OTM), forming a curve in the shape of a smile.
8THE VOLATILITY SMIRK – REVERSE SKEW Another type of volatility smirk is the reverse skew. The reverse skew is a more common skew pattern. It usually appears for longer term equity options and index options.The implied volatilities for options with lower strikes are higher than the those with higher strikes.
9THEORIES FOR THE REVERSE SKEW One explanation for the reverse volatility skew is that investors are usually worried about market crashes, so they buy puts for protection. This notion is supported by the fact that the reverse skew was not apparent until after the Crash of 1987 (Crash-o- phobia).Buying in-the-money calls have become popular alternatives to buying stocks since they offer leverage, which increases the rate of interest. This causes a greater demand for in-the-money calls, which therefore increases implied volatilities at the lower strikes.
10THE VOLATILITY SMIRK – FORWARD SKEW The other type of volatility smirk is the forward skew. Here, the implied volatility increases as the strike price increases.This suggests that out-of-the-money calls and in-the-money puts are in greater demand compared to in-the-money calls and out-of-the- money puts.
11THEORIES FOR THE FORWARD SKEW The forward skew pattern is more apparent for options in the commodities market. When supply is low, businesses would rather pay more to secure supply than to risk supply disruption.For example, if weather reports indicates a heightened possibility of an impending frost, fear of supply disruption will cause businesses to drive up demand for out-of-the-money calls for the affected crops.
12CALCULATIONSFind out the strike price and last traded price of AAPL on finance.yahoo.com (expiry 13 months)
13CALCULATIONSUse Black-Scholes formula to solve for the implied volatilities when market price is equal to stock price
14CALCULATIONSCreate several amounts of data according to strike price and graph the implied volatilities based on the strike pricesNotice an implied volatility skew
18MSFT Implied Volatility According to Strike Price (2 month expiry)
19THE CAUSES OF VOLATILITY SKEWS There are many ongoing studies to reason why the implied volatility is a U-shaped, but there are many theories to explain this phenomenonRubinstein (1994)Hull and White (1987)
20Binomial Tree Theory by Rubinstein (1994) Smile is caused by the presence ofJumps in the price of the underlying asset between successive opportunities to tradeDeterminants of volatility smile is affected by market participants’ assessment of crash risk (Hafner and Wallmeier 2001)Market imperfections and frictions, such as transaction costs, illiquidity, and other trading restrictionsDisturbances in the price process of the underlying assets that do not follow a constant volatility
21STOCHASTIC VOLATILITY MODELS Stochastic volatility models are used to evaluate derivative securities, such as options.It treats the volatility of the underlying security as a random process, depending on certain aspects, like:The price level of the underlying securityThe tendency of volatility to return to a long-run mean valueThe variance of the volatility processIt is a more accurate way of modeling derivatives than the Black-Scholes model.
22AN ANALOGYDerman, E. (2003): “From a behavioral point of view, it seems likely that implied volatilities are greatest where market movements are likely to cause the greatest shock and awe. In index markets, that’s the downside; In the gold market, since gold is more likely to be a haven, that jumps up when stocks move down, in recent years, a positive volatility skew has occurred in that market.”
23“CRASH-O-PHOBIA”Before Black Monday (1987), the implied volatility graph was much flatter like the graph based on the BS modelFollowing the crash, implied volatilities for out-of-the-money puts became much higher than for their at-the-money counterpartsOut-of-the-money options are expensive to their at-the-money counterparts.A crash is more likely when the implied volatilites are more skewedIndicates a strong negative skewness (very kertosis) in the physical stock returns distribution, which implies that a large decrease in stock prices is highly more probable than a large increase in stock prices.Put options are used to protect against large decreases in stock prices. This demand by investors has increased the price of put options, causing the left tail of the implied distribution to have more weight.
24Works Cited http://www.optiontradingpedia.com/volatility_smile.htm Market Crashes, Market Booms and the Cross-Section of Expected Returns, Jonathan F. Spitzer, September 19, 2006Implied Binomial Trees, Mark Rubinstein, January 4, 1994The Black-Scholes model as a determinant of the implied volatility smile: A similuation study, Gianluca Vagnani, June 9, 2009