Implicit Volatilities

Diplomarbeit von Robert Schott

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. 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. 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.

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.

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.

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Chapter 5.2, Influencing Parameters of the IVF: As discussed previously, the inappropriate assumptions of stochastic movements of asset prices and volatility lead to shapes in the IVF, which differ from being flat and constant through time with respect to the ideal conditions of the BS model. Besides the calculation method, IV is also affected by data problems e.g. bid-ask spreads in option and underlying assets, non-synchronous prices, transaction costs and from imbalanced option supply and demand, which is not arbitraged away.

Chapter 5.2.1, Jumps and Leverage: In the right pattern of figure 11, extreme movements of exchange rates are more likely, because of the fatter tails of the implied distribution. The reason for smiles for foreign currency options is neither that the volatility of the exchange rate is constant nor that the rate changes smoothly with no jumps. Jumps can occur in response of actions of central banks, which have an impact on the exchange rate. This implies that the price process is no longer continuous. Sources in literature state that if τ increases, jumps tend to get averaged out, so that the distribution with jumps becomes approximately a distribution with smooth changes. Thus the percentage impact of jumps on the smile gets bigger for short dated options. This is consistent with the argumentation that non-lognormal price jumps do not have much substantial influence on longer maturity contracts, but well for an option valuation for OTM options close to expiration.

In financial literature a large jump premium in the short term can be found as general conclusion for the negative skew for short maturity options.

Another possible argumentation can explain the skew, if we consider the volatility of the companies’ equity to be a decreasing function of the price. If equity declines in value the leverage of a company will increase, holding its total liabilities constant. Due to more uncertainty the volatility will increase. The other way around leverage will decrease c.p. if the value of the equity increases. This leads to a decreasing volatility, because the risk is taken out. In terms of this argument there is a negative correlation between volatility and stock price.

Chapter 5.2.2, Fear of Negative Market Movements: One possible reason for the volatility skew can be crashophobia. Traders are worried that a crash like on October the 19th, 1987 could occur again. It was identified that before the crash the implied distribution resembled the lognormal distribution, whereas after the crash the post-crash implied distribution was left-skewed and showed leptokurtosis. In using the implied distribution extreme price drops will be more likely as if the lognormal distribution is used. This results in the fact that traders make adjustments in pricing options. The fear of a higher possibility of a negative market movement results in the fact that investors pay a higher premium to buy the put option’s „insurance” attribute. As already mentioned the general conclusion of a large jump premium for short-dated options best explains the significant negative skew for short maturity option.

This is consistent with the shown example in figure 12 that the IVs of OTM put options with a relatively low K have higher values than of IVs of ATM or ITM put options. It is stated that all 12 examined major equity indexes around the world show downward-sloping smile shapes. Moreover, for any given maturity smiles close to one another can be observed, suggesting that in the different equity markets are similar asymmetries for the risk-neutral return distributions. Due to the fear of negative market movement the market will charge a higher premium on all options if the fear of crashes increases.

If the slope of the volatility skew was considered to have predictive power in forecasting market declines in the short run, then it would be possible of hedging depending on the level of risk-aversion.

Chapter 5.2.3, Market Imperfection and Demand Pressure: Financial literature points out the argument that IV is the expected market volatility. This statement is not completely true, because IV is also influenced by other factors. Parameters, which can influence the value of the IV, can be market imperfection and demand pressure effects. Market imperfection can take form in bid-ask spreads, discrete prices and the fact that price observations sometimes can not be completely synchronous. The simple fact alone, that measuring IV with bid prices or ask prices results in different IV values. These imperfections can be seen as departures of the ideal conditions in the BS world. The relaxations of these ideal assumptions take shape in the deviation of the IVS from a flat line. Supply shortages, demand shocks as well as widening of spreads, which can result in market illiquidity, are reasons for the skew discussed in literature.

The net buying pressure defined by Bollen & Whaley (2004) is the difference between the number of buyer-motivated contracts and the seller-motivated contracts traded each day, where buyer-motivated trades (seller-motivated trades) are regarded to be trades executed at a price above (below) the prevailing bid/ask midpoint. In the idealised Black-Scholes world, where all assumptions are held, market makers earn the risk-free rate.

Derivative prices are determined by the no-arbitrage theory, which is crucial because its effect is that prices can not differ from fundamental values in a considerable way and markets are kept to be efficient. Theoretically market makers can supply an unlimited number of contracts in a particular option series in creating a delta-hedge portfolio of the underlying asset and a risk free asset. The delta hedged portfolio has to be rebalanced continuously. But in the idealised world the supply curve for each option series is a horizontal line, because suppliers of option market liquidity can perfectly and costlessly hedge their inventories.

This horizontal supply curve can be seen in the left pattern of figure 14.

The red line in the left pattern illustrates the demand curve (Demand 1). Assuming that new information will be available, which indicates a negative expectation of the market, the demand curve of put options in order to hedge portfolios shifts to the right (Demand 2) and the price of the option remains unchanged due to the horizontal supply curve. No matter how large the demand it does not affect its price, because the BS-price of a call option is derived from a formula (2-3), in which demand issues and expectations are not included.

In reality, however, exposure of volatility-risk and hedging costs, namely transaction costs like margin requirements, broker commissions and bid-ask spreads can face a market maker. Rebalancing portfolios is only possible at discrete intervals of time. According to imbalances and large positions in market makers’ portfolios the desired compensation for risk and increasing hedging costs lead to higher prices of options, implying higher IV. In the right pattern it can be seen that the initial horizontal supply curve is upward sloping at a specific quantity. Given the preceding example of a negative expectation of the market the demand shifts again to the right (Demand 2).

However, due to the part of the upward sloping supply curve it results in a higher price of the option. Thus, it can not be said that IV represents the expected market volatility, because IV is influenced from other parameters as well. Note, that for the presented supply curve, presenting the feature of demand pressure, other shapes of supply curves are also possible. With the idea of an upward sloping supply curve, different shapes of the IVF can be explained.

In contrast to the model of arbitrage, based on the assumption that a lot of arbitrageurs take favourable infinitesimal positions in different markets, arbitrage is carried out by few professionals taking large positions with mostly other peoples’ money. This implies two features, namely capital constraints and agency problems. With capital-free arbitrage the fact is described that an arbitrageur has enough money to take favourable positions, however, in practice arbitrageurs can run out of money, which can result in the possibility of liquidating of loss-making positions. When losses occur an arbitrageur can be regarded as not being competent anymore with the effect that people are unwillingly to provide more money or even withdraw their capital. The limits of arbitrage can result in the fact that market prices can be different from model values.

With the net buying pressure Bollen and Whaley (2004) also found a possible explanation for different shapes of index IVF and stock IVF of S&P 500 options documenting that trading in stock options (index options) involves mostly calls (puts).

Thus, due to different demand an explanation is found for the feature that the index IVF is more negatively sloped than the individual equity option IVFs.