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5. IMPLIED VOLATILITY

				XEO Calls
	E		Model	Actual	Lower/ Raise	Ꝺ
ITM CALL XEO		1145	42.95752053	37.25	Lower	16.03%
ATM CALL XEO		1170	24.11158	19.8	Lower	14.09%
OTM CALL XEO		1245	5.808925074	0.55	Lower	11.58%
				XEO PUT
	E		Model	Actual	Lower/Raise	Ꝺ
ITM PUT XEO		1245	79.1626283	75.20	Lower	14.89%
ATM PUT XEO		1170	22.4982	19.55	Lower	15.10%
OTM PUT XEO		1145	16.35510301	11.90	Lower	17.0812%

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	OEX Calls
	E	Model	Actual		Lower/ Raise	Ꝺ (%)
OEX ITM CALL	1145	42.47860321	38.00		Lower	17.10%
OEX ATM CALL	1170	28.508307	20.5		Lower	14.90%
OEX OTM CALL	1245	5.741794078	0.60		Lower	11.86%
	OEX PUT
	E	Model	Actual		Lower/Raise	Ꝺ
ITM OEX PUT	1245	79.85495443	78.20		Lower	18.62%
ATM OEX PUT	1170	27.654376	21.05		Lower	15.94%
OTM OEX PUT	1145	16.6356427	13.05		Lower	17.8355%

Black and Scholes Model, defines volatility as the "standard deviation of the continuously compounded return on an underlying asset". Volatility may be estimated using two approaches: historical volatility and implied volatility. The former assumes that volatility prevailing over recent past holds true in the future. In accordance to implied volatility, the price of a option is a reflection of the current volatility of the underlying asset. The implied volatility is "the "volatility that makes the theoretical value of the option equal to the actual value of the option"" (Chance and Brooks, 2013).

The implied volatility is calculated using a trial and error approach, however Excel’s Goal Seek method, allows easy computation of accurate volatility.

As shown above, the implied volatility amongst all the options are lower than the VXO value. It is inferred that options with high implied volatility are costlier as against options with lower implied volatility. Differences, in these volatilities may be viewed as differences in the relative cost of the options (Chance and Brooks, 2013).

Volatility is a representation of volatilities of underlying assets across option expirations. Thus, it is plausible, that across varying time periods, volatility could be different. Usually," longer the time to maturity of an option, the higher the volatility, however this may not always hold true" (Chance and Brooks, 2013).

5.1.Implied Volatility amongst options and VXO

In both American (OEX) options and European (XEO) options, the implied volatility is seen to be lower than the VXO, which indicates that these options are undervalued. The implied volatility, as discussed in the previous section, offers investors insight into an options cost. The general rule being, options with higher implied volatility are more expensive. Thus, there proves to be an opportunity for arbitrage; by buying at a lower price and selling them at higher prices.

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In the American (OEX) put option we can see that ITM and OTM options have the highest implied volatility in comparison to the lower implied volatility of the ATM option. This is because these options (ITM and OTM) represent higher risk of large movements on the underlying asset and are thus are more expensive.

5.2.Smiles

"For a specified exercise price, the relationship between implied volatility and option expiration is recognised as structure of volatility (Don and Robert)". The implied volatility smile is the correlation between implied volatility and the exercise price for a given option. Due to the 1989 market crash the implied volatility surface has become more skewed. Risk variables like Delta and Gamma can be approximated using the non-flat implied volatility surface (Don and Robert).

XEO CALL

	17.00%
Volatility(%)	16.00%
	15.00%
	14.00%
Implied	13.00%
	12.00%
	11.00%
	10.00%
	1100	1150	1200	1250

Excercise Price

Implied Volatility

XEO PUT

17.50%

17.00%

16.50%

16.00%

15.50%

15.00%

14.50%

1100 1150 1200 1250

Excercise Price

OEX CALL

Implied Volatility

18.00%

17.00%

16.00%

15.00%

14.00%

13.00%

12.00%

11.00%

10.00%

1100 1150 1200 1250

Excercise Price

The above graphs represent a reverse skew or otherwise known as a volatility smirk; the lower the strike price higher the implied volatility. Investors in such a scenario heavily trade on ITM call options and OTM put options. It occurs when investors are concerned about the market’ i.e.’ they are expecting a crash in the market. When such a scenario is anticipated, investors are often pressured to buy OTM puts to hedge their investment portfolios (Bamford, 2014).

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OEX PUT

Implied Volatility

19.00%

18.50%

18.00%

17.50%

17.00%

16.50%

16.00%

15.50%

1140 1160 1180 1200 1220 1240 1260

Excercise Price

In the above graph, implied volatility is clearly higher in ITM and OTM options but is lower at the ATM option. This implies there is a higher demand for ITM and OTM options. Investors tend to buy ITM options as these have a higher intrinsic value, whereas OTM options similarly have higher extrinsic values than ATM options. Speculative markets often see the presence of volatility smiles. Upon occurrence of large volatility shifts, investors buy ITM options to steady gains whereas they buy OTM options for speculative reasons (Bamford, 2014).

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