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**By Christina Lee and Ivana Lee**

THE VOLATILITY SKEW By Christina Lee and Ivana Lee

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**OVERVIEW Volatility Skew Black-Scholes Model Types of Volatility Skews**

Volatility Smile Volatility Smirks Calculations Analyzing Volatility Graphs Causes and Theories Crash-o-phobia

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THE VOLATILITY SKEW According 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 options In-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.

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BLACK-SCHOLES MODEL Calculates fair economic value of an option - the price at which both a buyer and seller would break even. On Maple: S = last trading price E = strike price r = riskless interest rate T = time until maturity σ = implied volatility

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**BLACK-SCHOLES MODEL Black-Scholes is technically inaccurate because:**

Implied volatility should be constant according to this model Implied volatility graph should be a horizontal straight line When implied volatility is graphed, it is presented in volatility skew Black-Scholes does not consider certain aspects that may alter the price, such as: Liquidity Supply and demand The 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 14 Black-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

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**VOLATILITY COMPARISONS**

Black-Scholes S&P Option Market

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THE VOLATILITY SMILE One 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.

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**THE 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.

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**THEORIES 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.

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**THE 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.

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**THEORIES 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.

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CALCULATIONS Find out the strike price and last traded price of AAPL on finance.yahoo.com (expiry 13 months)

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CALCULATIONS Use Black-Scholes formula to solve for the implied volatilities when market price is equal to stock price

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CALCULATIONS Create several amounts of data according to strike price and graph the implied volatilities based on the strike prices Notice an implied volatility skew

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EXXON MOBIL: 7 month

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EXXON MOBIL: 13 month

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**EXXON MOBIL: 7 month & 13 month**

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**MSFT Implied Volatility According to Strike Price**

(2 month expiry)

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**THE 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 phenomenon Rubinstein (1994) Hull and White (1987)

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**Binomial Tree Theory by Rubinstein (1994)**

Smile is caused by the presence of Jumps in the price of the underlying asset between successive opportunities to trade Determinants 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 restrictions Disturbances in the price process of the underlying assets that do not follow a constant volatility

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**STOCHASTIC 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 security The tendency of volatility to return to a long-run mean value The variance of the volatility process It is a more accurate way of modeling derivatives than the Black-Scholes model.

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AN ANALOGY Derman, 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.”

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“CRASH-O-PHOBIA” Before Black Monday (1987), the implied volatility graph was much flatter like the graph based on the BS model Following the crash, implied volatilities for out-of-the-money puts became much higher than for their at-the-money counterparts Out-of-the-money options are expensive to their at-the-money counterparts. A crash is more likely when the implied volatilites are more skewed Indicates 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.

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**Works Cited http://www.optiontradingpedia.com/volatility_smile.htm**

Market Crashes, Market Booms and the Cross-Section of Expected Returns, Jonathan F. Spitzer, September 19, 2006 Implied Binomial Trees, Mark Rubinstein, January 4, 1994 The Black-Scholes model as a determinant of the implied volatility smile: A similuation study, Gianluca Vagnani, June 9, 2009

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