THE VOLATILITY SKEW By Christina Lee and Ivana Lee.

THE VOLATILITY SKEWBy Christina Lee and Ivana Lee

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

▫ Volatility Smile▫ Volatility Smirks

• Calculations• Analyzing Volatility

Graphs• Causes and Theories• Crash-o-phobia

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.

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

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.

VOLATILITY COMPARISONSBlack-Scholes S&P Option Market

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.

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.

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.

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.

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.

CALCULATIONS• Find out the strike price and last traded price

of AAPL on finance.yahoo.com (expiry 13 months)

CALCULATIONS• Use Black-Scholes formula to solve for the

implied volatilities when market price is equal to stock price

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

50 100 150 200 250 300 350 400 450 500 5500

0.10.20.30.40.50.6

Apple (AAPL) Implied Volatility According to

Strike Price13 month expire (Jan 2012)

Strike

Impli

ed V

ola

tili

ty (

%)

EXXON MOBIL: 7 month

50 55 60 65 70 75 80 85 90 95 1000

0.05

0.1

0.15

0.2

0.25

Exxon Mobil Implied Volatility7 month expiry (Jul 2011)

Strike Price

Impli

ed V

ola

tili

ty

EXXON MOBIL: 13 month

60 65 70 75 80 85 90 95 100 1050.165

0.17

0.175

0.18

0.185

0.19

0.195

Exxon Mobil Implied Volatility13 month expiry (Jan 2012)

Strike Price

Impli

ed V

ola

tilt

y

EXXON MOBIL: 7 month & 13 month

50 60 70 80 90 100 1100

0.05

0.1

0.15

0.2

0.25

Exxon Mobil Implied Volatility

Series1Series3

Strike Price

Imp

lied

Vola

tilt

y

7 month

13 month

24.00 25.00 26.00 27.00 28.00 29.00 30.00 31.0025.00%

25.50%

26.00%

26.50%

27.00%

27.50%

28.00%

28.50%

29.00%

29.50%

30.00%

Strike Price

Impli

ed V

ola

tili

tyMSFT Implied Volatility According to Strike

Price(2 month expiry)

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)

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

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.

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.”

“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.

Works Cited• http://www.optiontradingpedia.com/volatility_smile.htm• http://en.wikipedia.org/wiki/Volatility_smile• http://www.theoptionsguide.com/volatility-smile.aspx• http://efinance.org.cn/cn/FEshuo/18.pdf• 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

• http://www.aae.wisc.edu/ctaylor/SRC/SRC%20Bozic.pdfhttp://en.wikipedia.org/wiki/Stochastic_volatility