Implicit Volatilities

Contents

List of Abbreviations

Used Notations

List of Figures

List of Tables

1 Introduction

2 The Model of Black & Scholes
2.1 Implications & Outlook
2.2 Variations on Black & Scholes

3 Concept of Volatility

4 Implied Volatility Surface
4.1 Characteristics
4.2 Departure from Ideal Conditions

5 Impact of Changing Strike Prices on Volatility
5.1 The Shape of Implied Volatility Functions (IVF)
5.2 Influencing Parameters of the IVF
5.2.1 Jumps and Leverage
5.2.2 Fear of Negative Market Movements
5.2.3 Market Imperfection and Demand Pressure

6 Impact of Changing Maturity on Volatility
6.1 The Shape of the Term Structure
6.2 Influencing Effects on the Term Structure
6.2.1 Change in Basic Conditions
6.2.2 Mean Reversion

7 Option IV Model

8 Comparison
8.1 Concepts of Forecasting Volatility with Time Series
8.2 Model Approaches

9 Model-Free Implied Volatility
9.1 Updating Volatility Indexes and its Advantages
9.2 Calculation of Model-Free Volatility Indexes
9.3 The VIX® Term Structure
9.4 Relevance of the Model-Free

10 Conclusion

11 List of Tables

Appendix

Bibliography

List of Abbreviations

illustration not visible in this excerpt

List of Figures

Figure 1: Thesis overview

Figure 2: The relation between option value

Figure 3: Newton-Raphson Method

Figure 4: Call price vs. volatility

Figure 5: Average profile of implied volatility surface

Figure 6: Impact of kurtosis

Figure 7: Departures from lognormality

Figure 8: Different features of vega

Figure 9: Volga

Figure 10: Ideal type of smile and skew

Figure 11: Volatility smile and corresponding implied volatility vs. lognormal volatility

Figure 12: Volatility skew and its corresponding implied distribution

Figure 13: Volatility frown

Figure 14: Option supply

Figure 15: IV vs. maturity

Figure 16: Overview of volatility smile models

Figure 17: VIX Tem structure

Figure 18: Correlation of Daily Returns – S&P 500 and VIX

Figure 19: Term structures subprime crisis

List of Tables

Table 1: European call

Table 2: Comparison of volatility forecast categories

Table 3: Different options strategies with respect to the

Table 4: Option Values for Different Stock Prices and Maturities, by Standard deviation

Table 5: Ideal conditions (Black & Scholes)

Table 6: Option Values for Different Stock Prices and Standard Deviations,

Table 7: Explanations for different IVs

Table 8: Volatility Forecast Models

Table 9: Advantages of using IV as state variable

Table 10: Annual comparison of the range of closing prices

1 Introduction

Volatility is a crucial factor widely followed in the financial world. It is not only the single unknown determinant in the Black & Scholes model to derive a theoretical option price, but also the fact that portfolios can be diversified and hedged with volatility makes it a topic, which is crucial to understand for market participants comprising a wide group of private investors and professional traders as well as issuers of derivative products upon volatility.

The year 1973 was in several respects a crucial year for implicit volatility. The breakdown of the Bretton-Wood-System paved the way for derivative instruments, because of the beginning era of floating currencies.^[1] Furthermore Fischer Black and Myron Samuel Scholes published in 1973 the ground breaking Black & Scholes (BS) model in the Journal of Political Economy. This model was adopted in 1975 at the Chicago Board Options Exchange (CBOE), which also was founded in the year 1973, for pricing options.^[2] Especially since 1973 volatility has become a tremendously debated topic in financial literature with continually new insights in short-time periods.

Volatility is a central feature of option-pricing models and emerged per se as an independent asset class for investment purposes. The implicit volatility, the topic of the thesis, is a market indicator widely used by all option market practitioners.^[3]

In the thesis the focus lies on the implicit (implied) volatility (IV). It is the estimation of the volatility that perfectly explains the option price, given all other variables, including the price of the underlying asset in context of the BS model.^[4]

At the start the BS model, which is the theoretical basic of model-specific IV models, and its variations are discussed. In the concept of volatility IV is defined and the way it is computed is given as well as a look on historical volatility. Afterwards the implied volatility surface (IVS) is presented, which is a non-flat surface, a contradiction to the ideal BS assumptions. Furthermore, reasons of the change of the implied volatility function (IVF) and the term structure are discussed. The model specific IV model is then compared to other possible volatility forecast models. Then the model-free IV methodology is presented with a step-to-step example of the calculation of the widely followed CBOE Volatility Index VIX®. Finally the VIX® term structure and the relevance of the IV in practice are shown up. To ensure a good overview the structure of this thesis is presented in figure 1.

illustration not visible in this excerpt

Figure 1:^[5] Thesis overview

2 The Model of Black & Scholes

“Yet that weakness is also its greatest strength. People like the model because they can easily understand its assumptions. The model is often good as a first approximation, and if you can see the holes in the assumptions you can use the model in more sophisticated ways.”^[6] Black (1992)

This statement underlines the need to understand the implications of the BS model. Furthermore an outlook and a look at variations of the model are given in this chapter.

2.1 Implications & Outlook

illustration not visible in this excerpt

Figure 2:^[7] The relation between option value

and stock price

In the article “The Pricing of Options and Corporate Liabilities” Black & Scholes presented a graphic seen in figure 2, which exemplified the relation between option value and stock price. Line A represents the maximum value and line B the minimum value of a call option. The value of the option can not be worth more than the stock and can not be negative. Moreover it can not be less than the stock price minus the exercise price.

Due to the concave upward curves (T1, T2, T3), representing the value of an option successively for shorter maturities, it can be seen that a given percentage change in the stock price results in a larger percentage change in the option value, holding maturity constant.^[8] Directly from this figure it can be concluded that the option value (c) depends on the stock price (S) and the time to maturity (τ) defined as

illustration not visible in this excerpt

These dependencies can be seen in table 4. If the tables are read horizontally, it is shown that the option will be worth more when maturity increases, indicating that τ has a higher value. Reading the tables vertically, the effect is shown that c increases when the stock price increases. Furthermore the dependence of another determinant, namely the interest rate (r), on the option price is presented. If r increases the observed call option will be worth more. The table also shows that the call option price increases, when volatility ( 𝜎) increases.

Taking into account that the option price also depends on the strike price (K) and of a possible dividend yield (δ) of the underlying asset, there are six fundamental direct determinants of the option value.

The option value can be written as: c (S, K, τ, r, δ, 𝜎) (2-2)^[10]

According to the ideal conditions in table 5, which form the basis of the Black-Scholes (BS) formula the stock pays no dividends and thus, only five variables are necessary to specify the model. The impact of (S, τ, r, 𝜎) on the option price was already shown in table 4. The only determinant, which is held constant in this table, is the strike price K. The impact of the strike price will be discussed within the surrounding of the implied volatility function (IVF).

The valuation formula of Fischer Black and Myron Scholes for the price of a European call (c) is as follows:

illustration not visible in this excerpt (2-3)^[11]

σ[2] à Variance rate of the return on the stock (a measure of the stock’s volatility)^[12]

illustration not visible in this excerpt (2-4)^[13]

Respectively the formula for the price of a European put is:

illustration not visible in this excerpt (2-5)^[14]

The equations (2-3) and (2-5) do not include the expected return of the underlying asset. Thus, risk preferences of investors have no impact on the value of the option when it is expressed as a function of the price of the asset. The derivative price is computed as the discounted expectation of the payoff under a risk neutral measure.^[15] This general result is known as risk-neutral valuation. Besides the risk neutral measure another methodology, namely the replication strategy, can be used for deriving prices of contingent claims.^[16]

The only unknown determinate in the formula is the volatility of the underlying asset, because exercise price and time to maturity are known, stock price and the interest rate can be seen from the market.^[17]

The first problem, which can be read out directly from the formula is that you can not analytically invert the formula to calculate the implied variance explicitly.^[18] There are some numerical methods for solving this mathematical problem, such as the Newton-Raphson method presented in figure 3.

illustration not visible in this excerpt

Figure 3:^[19] Newton-Raphson Method

In this figure the BS option value versus a graph of volatility can be seen. The y-axis represents the value of a European call option of a fixed strike price and expiration. The change of the option value with respect to volatility is examined by the graph. The relative steepness of the tangent line equals the instantaneous rate of change of the observed price with respect to volatility.^[20] The method provides an algorithm for locating the IV usually in no more than three iterations.^[21]

The derived IV of the BS formula can be regarded as a direct measurement of the market’s expectation of future volatility.^[22] Thus, if an inaccurate estimate of the variance is given the market’s opinion will be estimated incorrectly. But according to Black if an accurate estimate of the variance is given, the model works very well.^[23]

If the Gaussian distribution is used, risk measures will be distorted systematically. The fat tails observed in the true implied distribution of possible stock prices, which describe a higher probability for extreme movements, the higher peak at the mean and the fact that intermediate movements are less likely to occur are all neglected.

Using the BS model the volatility estimates from reported option prices vary systematically with the value of K and τ. Another approach takes this systematic behaviour into account, arguing that it is driven by changes in the volatility rate of asset returns, and hypothesizing that volatility is a deterministic function of asset price and time. This led to the increased use of binomial or trinomial option valuation procedures.^[24] Since the crash in 1987, the distributions exhibit negative skewness and leptokurtic behaviour. Due to the demand to capture these features, multiple jumps and stochastic volatility (SV) are incorporated in the data generating process.^[25]

The assumptions about the dynamics of the underlying asset are generalised with many variations on the BS formula to capture anomalies in option pricing. However non-trivial pricing puzzles still remain and even if the dynamics of underlying assets are quite similar, prices can display very different properties.^[26] Therefore whether a satisfactory description of option prices can be derived from the BS-model remains to be discussed.

Recent empirical studies find that the BS-IV is an (upward) biased forecast, that usually has to be scaled to remove this bias,^[27] and what does not subsume all information contained in historical volatility (HISVOL).^[28]

The BS model was used as option pricing model in the old VIX® methodology till the year 2003. Thereafter, it was removed in order to derive volatility with a model-free IV approach.^[29] This development shows that the BS model is still crucial in the financial world, however new valuable insights continually arise as evidenced by an abundance of discussion, which result in alternatives to the BS model.

For European call and put options, which only can be exercised at the end of their lives, the model and its extensions can be used for valuation with the underlying assets of stocks, stock indices, currencies, and future contracts. Furthermore it can be used for American options, which can be exercised during their lives at any time, if the stock pays no dividend.^[30] Unfortunately most traded options are American style and pay dividends. Therefore other procedures must be used for these purposes.

2.2 Variations on Black & Scholes

illustration not visible in this excerpt

As already mentioned many variations on the BS model can be found in literature. The model can be diversified in a way that all six fundamental direct determinants of the option value, dividends included, can be taken into account. This also allows r and 𝜎 to be defined as functions of time.^[31] If quoted prices of options are compared to these variations of the BS model, a computing of the IVs will become possible. Due to the fact that most of the traded stocks pay dividends it is crucial to have a model, which takes dividends into account.

An example of a variation on the BS model with a dividend paying underlying for a European option (table 1) will now be presented.

illustration not visible in this excerpt

Table 1:^[32] European call

Assuming the stock price consists of a riskless and a risky component, the BS formula is correct if the riskless component is the present value of all dividends during the life of the option discounted at the risk free rate from the ex-dividend dates to the present.^[33]

Furthermore, S has to be substituted in the formula with the risky component.^[34]

An example of calculating the present value of the dividends (D), also denoted as (riskless component), is given:^[35]

illustration not visible in this excerpt

Thus the risky component illustration not visible in this excerpt of the dividends is 43.78€, denoted with illustration not visible in this excerpt. As mentioned before the BS formula is correct if the risky component is substituted for S.

In the first calculation in the appendix it can be seen how all six fundamental determinants (2-2) are taken into account and how the option price is calculated. The price of the European call option (table 1) is 8.41€.

In this variation on Black & Scholes it should be considered if 𝜎 should be the risky component volatility or the stock price volatility itself.

An adjustment, which takes the fragmentation in two components into account, is: 𝜎 risky component = illustration not visible in this excerpt(2-7), where D represents the present value of the dividends.^[36]

An American call option can be approximately calculated with the BS model if two prices of appropriate European call options are calculated and the American option price (C) is regarded to be the higher one of the prices of the calculated European options.^[37]

Taking the previous example the value of one of these European option is already calculated in the appendix (c = 8.41€), assuming that it can only be exercised at the end of its life.

The other option implies that it expires just before the final ex-dividend date to avoid unfavourable effects of the dividend.^[38]

Srisky component and τ is calculated as follows:

- Present value of the dividend (riskless component):

illustration not visible in this excerpt (2-8)

- illustration not visible in this excerpt (2-9)

- Final ex-dividend date in 6 months: τ = illustration not visible in this excerpt = 0.5 (2-10)

The value of the second European option is given by the second calculation in appendix. It is 8.30€. The price of the American option (illustration not visible in this excerpt) is the higher one of these two values, namely illustration not visible in this excerpt = 8.41€.^[39] Upon these variations on the BS-model it is possible to calculate the IV with the Newton-Raphson Method (figure 3), both for an option with a dividend paying underlying and for American-style options.

3 Concept of Volatility

All variables, which are necessary to specify the model, are directly observable except for the standard deviation of returns from the underlying stock.^[40] Owing to different kinds of volatility it is crucial to indicate what kind of volatility is meant exactly and in what context it is used.

The expression IV can be used in different contexts:

- Statistical measure
- Parameter of an option pricing model
- Independent asset class

Furthermore, it will give a good overview on IV to see how it is embedded in volatility concepts. Besides IV the historical volatility and the instantaneous volatility are defined. Generally volatility can be seen as a measure of the uncertainty of the return realized on an asset.^[41]

Historical (realized) volatility (HISVOL) is an estimate of the standard deviation based upon the ex-post continuously compounded stock returns measured over a specific sample period in the past.^[42] Normally the standard deviation is annualized. HISVOL is also an important concept of volatility for option pricing models, because these models can only be estimated with a volatility input factor like HISVOL to compute the BS option price.^[43]

The instantaneous (actual) volatility refers precisely to the volatility that appears in the stochastic differential equation, which describes the evolution of the underlying asset.^[44] It measures the instantaneous standard deviation of the return process of the log-asset price.^[45] The instantaneous volatility depends on the share price, time and possibly on other state variables.^[46]

Latané and Rendleman already introduced the concept of IV in 1976.^[47] They stated that if the BS assumptions were valid and the option market was completely efficient, all options on a particular stock would be priced with the same standard deviation at any moment during the lifetime of the option. But even in a highly efficient market this is unlikely.^[48] The implied variance is the value of the instantaneous variance of the stock’s return which employed in the BS formula, results in a model price equal to the market price.^[49] In other words the implicit volatility (IV) is the estimation of the volatility that perfectly explains the observed option price, given all other variables, including the price of the underlying asset in context of the BS model.^[50] Normally the IV is expressed as implied standard deviation (ISD).^[51] It is the average expected volatility of the asset returns until the option expires.^[52] The IV depends on the strike price, the expiry date of the financial contract and on the time which has already passed in its lifetime.

The IV for a European call option, priced at the market price (illustration not visible in this excerpt), is the number σimplied, which matches the following equation:

illustration not visible in this excerpt (3-1)^[53]

Therefore, the option price implicitly contains the market perception of the expected volatility of the underlying asset over the option’s life.

The implied volatility depends on the strike price and maturity date.

Thus, it can be written as:

illustration not visible in this excerpt (3-2)^[54]

If observed market prices for put and call options conform to the put-call parity, the IVs of put and call options are equal.^[55] In the following put-call parity it is demonstrated step-by-step that a European call option and a European put option, both with same strike price and time to maturity, have the same IV value in the BS world.^[56] For BS-prices (cBS, pBS) it can be written:

illustration not visible in this excerpt (3-3)

The put-call parity also holds for market prices (illustration not visible in this excerpt:

illustration not visible in this excerpt (3-4)

With subtraction equation (3-4) from equation (3-3) it follows:

illustration not visible in this excerpt (3-5)

This equation shows that in the BS world the difference between observed market prices and the BS-prices are exactly the same. Hence, theoretically the IV of a European call is the same as for the European put if both have the same maturity date and strike price.^[57]

illustration not visible in this excerpt

Figure 4:^[58] Call price vs. volatility

In reality however, a call and a put with the same K and illustration not visible in this excerpt can show different ISDs.^[59] If IVs are unequal, the normal approach should be to average the IV of the call and the put. This approach to averaging across call and put IV reduces any measurement error from no synchronous asset and option prices, because of the negative correlation between the typical call and put error.^[60] Considering this formula it is obvious that there are many existing IV for one underlying asset, namely one for each traded option.^[61] When τ converges to zero (t ↑ t*) it can be derived from stochastic IV models that IV converges to the instantaneous volatility if the strike price (K) equals the forward price for the underlying asset at that time (at-the-money options).^[62] Moreover, under certain conditions (e.g. ATM strikes), IV can be interpreted as an average volatility over the remaining life time of the option.^[63] Between IV and option prices there is a one-to-one map as shown in figure 4 and equation 3-6. In the figure the behaviour of a call price with respect to volatility is presented. If the BS price is strictly increasing, it will define a one-to-one map between option prices and volatility.

illustration not visible in this excerpt (3-6)^[64]

This one-to-one-map can also be seen in a data collection of Black (1975) in table 6.

If volatility increases the prices of a call option and put option increase as well, because the owner of a call benefits from price increases and the owner of a put from price decreases, while having a limited downside risk.^[65]

A higher (lower) value of IV involves a higher (lower) option price. Conclusions from prices of options with different K and 𝜏 can not be drawn to derive statements about the expense of options. But in comparing the IVs of the options it can be said that an option is respectively expensive or cheap in relation to another option with different strike price and/or time to expiration. Moreover, traders and brokers frequently quote option prices using IVs, because they usually do not have to be updated as often as prices themselves.^[66]

4 Implied Volatility Surface

The behaviour of the IVS was till now a topic of much research. This research was focused on three different aspects:^[67]

- Profile across strikes (smile pattern)
- Profile across maturities
- Time series behaviour of IVs.

Volatility smiles and term structures can be combined to implied volatility surfaces (IVS)^[68] to plot volatilities adequate for pricing options with any strike price and any maturity.^[69]

Thus, IVS defines IV as a function of both of K and τ. In practice the ideal conditions of the BS world can not be reached. In the IVS all deviations from these assumptions can be summarized.^[70]

In figure 5 a typical profile of IV for SP500 options as a function ofillustration not visible in this excerpt and moneyness (m) in March 1999 can be seen. If time is measured in years, then illustration not visible in this excerpt will measure the fraction of a year remaining in the life of the option,^[71] for instance nine months are denoted as illustration not visible in this excerpt = ¾.

illustration not visible in this excerpt

Figure 5:^[72] Average profile of implied volatility surface

A surface can be presented in different ways. It is possible to use other notations such as moneyness (m), which reflects the degree to which an option is in-the-money (ITM) or out-of-the-money (OTM). Accordingly, not K, but its relation to the underlying price is crucial.^[73] Moneyness (m) is defined in the figure as the ratio of strike price and the actual price of the underlying asset, because in the relevant journal the expression in the BS formula of (illustration not visible in this excerpt) is defined as (–ln m) with illustration not visible in this excerpt (4-1).^[74]

Representing the IVS in relative coordinates, it is possible to define it as a function of m and illustration not visible in this excerpt:

illustration not visible in this excerpt (4-2)^[75]

In this case, call options are OTM (ITM) for m > 1 (m < 1). Moreover, this definition of moneyness measure is often used by traders to quote option prices.^[76]

An industry standard has become the standardized measure illustration not visible in this excerpt (4-3) to make the scaling of m comparable with illustration not visible in this excerpt across different maturities. Moreover, with the use of some volatility measure 𝜎 the measure can be made comparable across different securities of different volatility levels.^[77]

With any given strike price and time to maturity a unique and nearly always different IV can be seen at the surface. Preparatory work for creating such a surface is a matrix of option contract parameters e.g. (K, τ). Note, that the different matrix-numbers presenting IVs, might expected to be equal if traders are acting and behaving on the assumptions of the BS model, whereas in empirical matrices the numbers are different.^[78]

At any point on the surface today representing a specific BS IV, the value of vanilla options^[79] quoted on the market given the current term structure of interest rates and dividends can be derived, due to the one-to-one relationship between IV and option price. Thus the IV surface today can be seen as a snapshot of today’s market prices of vanilla options.^[80]

If market data do not exist for a given coordinate, which means that options are not traded with the corresponding tuple (K, τ), it can be found out while using linear interpolation.^[81] The effect that ctBS c.p. increases, when illustration not visible in this excerpt increases can be seen empirically in table 6.

4.1 Characteristics

The IV changes over time, which implies that day by day the surface is subject to an evolution, however some general features persist. Some sources in literature state that equity index option smiles are not constant over time and that there is a flattening-out effect^[82] if smiles are expressed in K, which says that the pronouncement of the smile pattern gets reduced if τ increases.^[83] A summary of explanations for different IVs can be seen in table 7.

It is not only the smile - term structure feature that raises the interest of researchers but also the fact that the IVS changes itself randomly with time, due to supply and demand on the options market, draws attention.^[84] This observation has led to the development of simple rules to estimate the evolution in order to have deterministic laws regarding motions of the IVS, namely sticky-strike, the sticky-moneyness (sticky-delta) and the sticky implied-tree rule, which are all associated with a different market regime. The first two mentioned rules can be defined as:^[85]

- Sticky strike rule: illustration not visible in this excerpt (4-4)
- Sticky moneyness rule: illustration not visible in this excerpt (4-5)

The first rule of thumb indicates that the IV, for an option with given strike price and maturity, will be unaffected by changes in the price of the underlying asset.^[86] The second rule assumes that the volatility for a particular maturity depends only on moneyness.^[87] Thus, not only the shape, but also the time evolution of the IVS has to be taken into account by risk models.

Realistic future scenarios of the IVS are crucial. Unfortunately when a model can perfectly fit the IVS it does not yet imply that it will also fit the future IVS perfectly.^[88] There are different stock option price affections, which are not yet fully understood.^[89] But there are well known anomalies in option pricing, which many studies have attempted to capture by generalizing the assumptions about the dynamics of the underlying asset without eliminating non-trivial pricing puzzles.^[90]

4.2 Departure from Ideal Conditions

In the ideal conditions of the BS world (table 5; assumption 2):

The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices at the end of any finite interval is lognormal.

The variance rate of the return on the stock market is constant.

The assumption of the idealised BS-world states that the asset price follows a GBM with constant volatility.^[91] In this chapter the departure from the lognormal distribution and from constant volatility is shown. The stock price should follow a random walk implying that proportional changes in the stock price in a short time period are normally distributed.^[92] This implies that the stock price should have a lognormal distribution at any time in the future, because a log-normally distributed variable can not take negative values.^[93]

Equivalently its continuously compounded rate of return should be normally distributed in any given time interval.^[94]

Due to the “ideal conditions” the distribution of possible stock prices at the end of any finite interval is lognormal. An asset price would follow a lognormal distribution if the volatility of the asset was constant and its price changed smoothly with no jumps.^[95] But the returns are not normally distributed, instead empirical distributions of returns show a generally leptokurtic behaviour.

In figure 6, it can be seen how a distribution changes if its corresponding value of kur (ut) is higher than a normal distribution. A common kurtosis measure is:

illustration not visible in this excerpt, (4-6)^[96]

illustration not visible in this excerpt

Figure 6:^[97] Impact of kurtosis on the distribution

where ut is a random variable and illustration not visible in this excerpt is its expected value. These higher values can be seen from financial data, indicating that values close to the mean and extreme positive and negative outliers (heavy tails) appear more frequently.^[98] A common skewness measure is:

illustration not visible in this excerpt, (4-7)^[99]

where sym (ut) is the degree of symmetry. As a result of the symmetry of the Gaussian distribution the skewness measure is sym (ut) = 0. For a skewed distribution to the right (left) sym (ut) will have a positive (negative) value.

The features of different kurtosis and values of skewness can often be observed from financial data. The differences between the lognormal distribution, one assumption in the BS world, and possible true distribution of stock prices with conclusions of the mispricing of the BS model can be seen in figure 7.

illustration not visible in this excerpt

Figure 7:^[100] Departures from lognormality

Already Black (1975) stated that the actual prices differ from prices derived by the BS-model in certain systematic ways. ITM call options tend to be underpriced with respect to the figure 7 (B).^[101] Moreover, the BS model tends to overestimate the value of an option on a high variance security and tends to underestimate the value of one on a low variance security.^[102] For foreign currency options the appropriate distributions tend to have the shape of the true distribution in figure 7 (D), indicating that extreme market movements in both directions are more likely with regard to the lognormal distribution.

Another assumption of Black & Scholes “ideal conditions” shows that the actual standard deviation of returns from the stock is constant through time.

This assumption of constant volatility can not be met in practice, because volatility changes over time. Therefore, the value of an option is subject to the change of volatility, next to the change of asset price and the life time of an option. The assumption of constancy is thought to lead to the failed description of the structure of reported option prices with the BS model fails.^[103] Due to the fact that movements in volatility are crucial for option pricing, attempts are made in literature to explain these movements.

illustration not visible in this excerpt

Figure 8:^[104] Different features of vega

Values of ATM options are very sensitive to small changes in volatility due to the high vega in absolute terms.^[105] If the stock price approaches the strike price, the value of vega will increase. The Greek vega principal component analyses are shown in figure 8. The vega of an option is the rate of change of the value of an option with regard to the volatility of the underlying asset.^[106] The vega is defined for a call option as: illustration not visible in this excerpt. (4-8)^[107]

[...]

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^[2] CBOE – History of CBOE, CBOE Chicago Board Options Exchange.

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^[4] Brenner, M. & M. Subrahmanyam. “A Simple Formula to Compute the Implied Standard Deviation.” Financial Analysts Journal. 44 (1988): 80.

^[5] Own creation.

^[6] Fengler, M. R.. Semiparametric Modeling of Implied Volatility. (Berlin: Springer-Verlag, 2005), 1.

^[7] Black, F. & M. Scholes. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy. 81 (1973): 638.

^[8] Black, F. & M. Scholes. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy. 81 (1973): 639.

^[9] Fengler, M. R.. Semiparametric Modeling of Implied Volatility. (Berlin: Springer-Verlag, 2005), 20.

^[10] Black, F.. “Fact and Fantasy In the Use of Options.” Financial Analysts Journal. 31 (1975): 36.

^[11] Black, F. & M. Scholes. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy. 81 (1973): 644. Black, F. & M. Scholes. “THE VALUATION OF OPTION CONTRACTS AND A TEST OF MARKET EFFICIENCY.” Journal of Finance. 27 (1972): 401. With regard to the original notation the formula is adjusted to the notation used in the thesis.

^[12] Black, F.. “Fact and Fantasy In the Use of Options.” Financial Analysts Journal. 31 (1975): 65.

^[13] Black, F. & M. Scholes. “THE VALUATION OF OPTION CONTRACTS AND A TEST OF MARKET EFFICIENCY.” Journal of Finance. 27 (1972): 401.

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^[15] Fengler, M. R.. Semiparametric Modeling of Implied Volatility. (Berlin: Springer-Verlag, 2005), 11.

^[16] Fengler, M. R., Semiparametric Modeling of Implied Volatility, 10-11.

^[17] Black, F., Fact and Fantasy In the Use of Options, 36.

^[18] Cox, J.C. & M. Rubinstein. Options Markets. (NJ: Prentice-Hall Inc, 1985), 278.

^[19] Criss, N. A.. Black-Scholes and beyond: option pricing models. (Boston: McGraw-Hill, 1997), 337.

^[20] Criss, N. A., Black-Scholes and beyond : option pricing model, 337.

^[21] Taylor, S., Asset price dynamics, Volatility, and Prediction, 278.

^[22] Figlewski, S.. “Forecasting Volatility.” Financial Markets, Institutions & Instruments. 6 (1997): 83.

^[23] Black, F. & M. Scholes. “THE VALUATION OF OPTION CONTRACTS AND A TEST OF MARKET EFFICIENCY.” Journal of Finance. 27 (1972): 416.

^[24] Dumas, B., J. Fleming & R. E. Whaley. “Implied Volatility Functions: Empirical Tests.” Journal of Finance. 53 (1998): 2101-2103.

^[25] Doran, J. S. et al. “IS THERE INFORMATION IN THE VOLATILITY SKEW?” Journal of Futures Markets. 27 (2007): 921.

^[26] Gârleanu, N. et al. “Demand-Based Option Pricing.” Rodney L White Center for Financial Research -Working Papers-. (2006): 35.

^[27] Taylor, S.. Asset price dynamics, Volatility, and Prediction. (NJ: Princeton Univ. Press, 2007), 420.

^[28] Jiang, G. J. “The Model-Free Implied Volatility and Its Information Content.” Review of Financial Studies. 18 (2005): 1323.

^[29] VIX® Volatility Index, CBOE Chicago Board Options Exchange, 1.

^[30] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 308.

^[31] Ibid., 237.

^[32] Own calculation. Cp. Hull, J., Introduction to futures and options markets, 246.

^[33] Hull, J., Introduction to futures and options markets, 246.

^[34] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 246.

^[35] Ibid.

^[36] Ibid.

^[37] Ibid., 247.

^[38] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 247.

^[39] Own calculation. Cp. Hull, J., Introduction to futures and options markets, 247.

^[40] Latané, H. A. & R. J. Rendleman. “STANDARD DEVIATIONS OF STOCK PRICE RATIOS IMPLIED IN OPTION PRICES.” Journal of Finance. 31 (1976): 369.

^[41] Hull, J.. Options, futures and other derivatives. (NJ: Prentice Hall, 1997), 713.

^[42] Hafner, R.. Stochastic implied volatility. (Heidelberg: Springer, 2004), 33.

^[43] Poon, S-H. & C. Granger. “Practical Issues in Forecasting Volatility.” Financial Analysts Journal. 61 (2005): 46.

^[44] Ibid.

^[45] Fengler, M. R.. Semiparametric Modeling of Implied Volatility. (Berlin: Springer-Verlag, 2005), Notations, XI.

^[46] Ibid.

^[47] Fengler, M. R.. Semiparametric Modeling of Implied Volatility. (Berlin: Springer-Verlag, 2005), 19.

^[48] Latané, H. A. & R. J. Rendleman. “STANDARD DEVIATIONS OF STOCK PRICE RATIOS IMPLIED IN OPTION PRICES.” Journal of Finance. 31 (1976): 370-371.

^[49] Manaster, S. & G. Koehler. “The Calculation of Implied Variances from the Black-Scholes Model: A Note.” Journal of Finance. 37 (1982): 227.

^[50] Brenner, M. & M. Subrahmanyam. “A Simple Formula to Compute the Implied Standard Deviation.” Financial Analysts Journal. 44 (1988): 80.

^[51] Ederington, L. H. & W. Guan. “MEASURING IMPLIED VOLATILITY: IS AN AVERAGE BETTER? WHICH AVERAGE?” Journal of Futures Markets. 22 (2002): 812.

^[52] Nandi, S. & D. F. Waggoner. “Issues in Hedging Options Positions.” Economic Review. 85 (2000): 24.

^[53] Taylor, S.. Asset price dynamics, Volatility, and Prediction. (NJ: Princeton Univ. Press, 2007), 378.

^[54] Cont, R. & J. da Fonseca. “Dynamics of implied volatility surfaces.” Quantitative Finance. 2 (2002): 45.

^[55] Hull, J.. Options, futures and other derivatives. (NJ: Prentice Hall, 1997), 331.

^[56] Hull, J.. Fundamentals of Futures and Options Markets. (NJ: Pearson/Prentice-Hall, 2005), 371-372.

^[57] Ibid., 372.

^[58] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 191.

^[59] Ederington, L. H. & W. Guan. “MEASURING IMPLIED VOLATILITY: IS AN AVERAGE BETTER? WHICH AVERAGE?” Journal of Futures Markets. 22 (2002): 812.

^[60] Taylor, S.. Asset price dynamics, Volatility, and Prediction. (NJ: Princeton Univ. Press, 2007), 407.

^[61] Cont, R. & J. da Fonseca. “Dynamics of implied volatility surfaces.” Quantitative Finance. 2 (2002): 47.

^[62] Fengler, M. R.. Semiparametric Modeling of Implied Volatility. (Berlin: Springer-Verlag, 2005), 95.

^[63] Ibid., 39.

^[64] Cont, R. & J. da Fonseca. “Stochastic Models of Implied Volatility Surfaces.” Economic Notes. 31 (2002): 363.

^[65] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 189.

^[66] Daglish, T., J. Hull & W. Suo. “Volatility Surfaces: theory, rules of thumb, and empirical evidence.” Quantitative Finance. 7 (2007): 507.

^[67] Cont, R. & J. da Fonseca. “Stochastic Models of Implied Volatility Surfaces.” Economic Notes. 31 (2002): 365.

^[68] Fengler, M. R.. Semiparametric Modeling of Implied Volatility. (Berlin: Springer-Verlag, 2005), 19.

^[69] Hull, J.. Fundamentals of Futures and Options Markets. (NJ: Pearson/Prentice-Hall, 2005), 377-378.

^[70] Fengler, M. R.. Semiparametric Modeling of Implied Volatility. (Berlin: Springer-Verlag, 2005), 19.

^[71] Cox, J.C. & M. Rubinstein. Options Markets. (NJ: Prentice-Hall Inc, 1985), 34.

^[72] Cont, R. & J. da Fonseca. “Dynamics of implied volatility surfaces.” Quantitative Finance. 2 (2002): 48.

^[73] Hafner, R.. Stochastic implied volatility. (Heidelberg: Springer, 2004), 85.

^[74] Cont, R. & J. da Fonseca, “Dynamics of implied volatility surfaces,” 46-47.

^[75] Ibid., 47.

^[76] Hafner, R., Stochastic implied volatility, 42.

^[77] Foresi, S. & L. Wu. “Crash-O-Phobia: A Domestic Fear or a Worldwide Concern?.” Journal of Derivatives. 13 (2005): 13.

^[78] Taylor, S.. Asset price dynamics, Volatility, and Prediction. (NJ: Princeton Univ. Press, 2007), 379.

^[79] A vanilla option is an option with no special or unusual features.

^[80] Cont, R. & J. da Fonseca. “Stochastic Models of Implied Volatility Surfaces.” Economic Notes. 31 (2002): 363.

^[81] Hull, J.. Options, futures and other derivatives. (NJ: Prentice Hall, 1997), 336.

^[82] Hafner, R.. Stochastic implied volatility. (Heidelberg: Springer, 2004), 39.

^[83] Ibid., 39.

^[84] Cont, R. & J. da Fonseca. “Stochastic Models of Implied Volatility Surfaces.” Economic Notes. 31 (2002): 362.

^[85] Hafner, R., Stochastic implied volatility, 53-55.

^[86] Daglish, T., J. Hull & W. Suo. “Volatility Surfaces: theory, rules of thumb, and empirical evidence.” Quantitative Finance. 7 (2007): 508.

^[87] Ibid.

^[88] Cont, R. & J. da Fonseca. “Stochastic Models of Implied Volatility Surfaces.” Economic Notes. 31 (2002): 364.

^[89] Hull, J.. Options, futures and other derivatives. (NJ: Prentice Hall, 1997), 507.

^[90] Gârleanu, N. et al. “Demand-Based Option Pricing.” Rodney L White Center for Financial Research -Working Papers-. (2006): 35.

^[91] Dumas, B., J. Fleming & R. E. Whaley. “Implied Volatility Functions: Empirical Tests.” Journal of Finance. 53 (1998): 2060.

^[92] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 229-230.

^[93] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 230.

^[94] Ibid., 328.

^[95] Hull, J.. Options, futures and other derivatives. (NJ: Prentice Hall, 1997), 333.

^[96] Von Auer, L.. Ökonometrie, Eine Einführung. (Heidelberg: Springer-Verlag, 2005), 418.

^[97] Ibid., 419.

^[98] Franke, J., et al. Statistics of Financial Markets, An Introduction. (Heidelberg: Springer, 2004), 41-42.

^[99] Von Auer, L., Ökonometrie, Eine Einführung, 418.

^[100] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 329-330.

^[101] Black, F.. “Fact and Fantasy In the Use of Options.” Financial Analysts Journal. 31 (1975): 64.

^[102] Black, F. & M. Scholes. “THE VALUATION OF OPTION CONTRACTS AND A TEST OF MARKET EFFICIENCY.” Journal of Finance. 27 (1972): 416-417.

^[103] Dumas, B., J. Fleming & R. E. Whaley. “Implied Volatility Functions: Empirical Tests.” Journal of Finance. 53 (1998): 2060.

^[104] Franke, J., et al. Statistics of Financial Markets, An Introduction. (Heidelberg: Springer, 2004), 109.

& Hull, J.. Introduction to futures and options markets. 297-298.

^[105] Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 296.

^[106] Ibid.

^[107] Fengler, M. R., Semiparametric Modeling of Implied Volatility, 17. & Hull, J.. Introduction to futures and options markets. (NJ: Prentice-Hall Internat., 1991), 297-298.