Volatility smiles areimplied volatility patterns that arise in pricing financialoptions. It is aparameter (implied volatility) that needs to be modified for theBlack–Scholes formula to fit market prices. Generally, for a given expiration, options whosestrike price differs substantially from the underlying asset'sforward price tend to have prices that deviate from their expected prices using a constant-volatility model based on the at-the-money (strike price near the underlying'sforward price).
Graphing implied volatilities against strike prices for a given expiry produces a skewed "smile" instead of the expected flat surface. The pattern differs across various markets. Equity options traded in American markets did not show a significant volatility smile before theCrash of 1987 but began showing one afterwards.[1] It is believed that investor reassessments of the probabilities offat-tail have led to higher prices for out-of-the-money options. This anomaly implies deficiencies in the standardBlack–Scholes option pricing model which assumes constant volatility andlog-normal distributions of underlying asset returns. Empirical asset returns distributions, however, tend to exhibit fat-tails (kurtosis) and skew. Modelling the volatility smile is an active area of research inquantitative finance, and better pricing models such as thestochastic volatility model partially address this issue.
A related concept is that ofterm structure of volatility, which describes how (implied) volatility differs for related options with different maturities. Animplied volatility surface is a 3-D plot that plots volatility smile and term structure of volatility in a consolidated three-dimensional surface for all options on a given underlying asset.
In theBlack–Scholes model, the theoretical value of avanilla option is amonotonic increasing function of the volatility of the underlying asset. This means it is usually possible tocompute a unique implied volatility from a given market price for an option. This implied volatility is best regarded as a rescaling of option prices which makes comparisons between different strikes, expirations, and underlyings easier and more intuitive.
When implied volatility is plotted againststrike price, the resulting graph is typically downward sloping for equity markets[citation needed], or valley-shaped for currency markets. For markets where the graph is downward sloping, such as for equity options, the term "volatility skew" is often used. The shape of the skew can also be described as a "half frown" when "implied volatilities stop increasing and tend to flatten out".[2] For other markets, such as FX options or equity index options, where the typical graph turns up at either end, the more familiar term "volatility smile" is used. For example, the implied volatility for upside (i.e. high strike) equity options is typically lower than for at-the-money equity options. However, the implied volatilities of options on foreign exchange contracts tend to rise in both the downside and upside directions. In equity markets, a small tilted smile is often observed near the money as a kink in the general downward sloping implicit volatility graph. Sometimes the term "smirk" is used to describe a skewed smile.
Aninvestment skew arises from structural factors such as institutional hedging strategies, while ademand skew results from concentrated buying or selling interest in specific strikes or maturities, often driven by speculative positioning.[3] Understanding whether observed skew is investment- or demand-driven can be important for interpreting market sentiment and relative value opportunities.
Market practitioners use the term implied-volatility to indicate the volatility parameter for ATM (at-the-money) option. Adjustments to this value are undertaken by incorporating the values of Risk Reversal and Flys (Skews) to determine the actual volatility measure that may be used for options with a delta which is not 50.
where:
Risk reversals are generally quoted asx% delta risk reversal and essentially is Longx% delta call, and shortx% delta put.
Butterfly, on the other hand, is a strategy consisting of:−y% delta fly which mean Longy% delta call, Longy% delta put, short one ATM call and short one ATM put (small hat shape).
Whilehistorical volatility may influenceimplied volatility, the two are distinct. Historical volatility is a direct measure of the movement of the underlying’s price (realized volatility) over recent history (e.g. a trailing 21-day period). Implied volatility, in contrast, is determined by the market price of the derivative contract itself, and not the underlying. Therefore, different derivative contracts on the same underlying have different implied volatilities as a function of their ownsupply and demand dynamics. For instance, the IBM calloption, strike at $100 and expiring in 6 months, may have an implied volatility of 18%, while the put option strike at $105 and expiring in 1 month may have an implied volatility of 21%. At the same time, the historical volatility for IBM for the previous 21 day period might be 17% (all volatilities are expressed in annualized percentage moves).
For options of different maturities, we also see characteristic differences in implied volatility. However, in this case, the dominant effect is related to the market's implied impact of upcoming events. For instance, it is well-observed that realized volatility for stock prices rises significantly on the day that a company reports its earnings. Correspondingly, we see that implied volatility for options will rise during the period prior to the earnings announcement, and then fall again as soon as the stock price absorbs the new information. Options that mature earlier exhibit a larger swing in implied volatility (sometimes called "vol of vol") than options with longer maturities.
Other option markets show other behavior. For instance, options on commodity futures typically show increased implied volatility just prior to the announcement of harvest forecasts. Options on US Treasury Bill futures show increased implied volatility just prior to meetings of the Federal Reserve Board (when changes in short-term interest rates are announced).
The market incorporates many other types of events into the term structure of volatility. For instance, the impact of upcoming results of a drug trial can cause implied volatility swings for pharmaceutical stocks. The anticipated resolution date of patent litigation can impact technology stocks, etc.
Volatility term structures list the relationship between implied volatilities and time to expiration. The term structures provide another method for traders to gauge cheap or expensive options.
It is often useful to plot implied volatility as a function of both strike price and time to maturity.[4] The result is a two-dimensional curved surface plotted in three dimensions whereby the current market implied volatility (z-axis) for all options on the underlying is plotted against the strike price (y-axis) and time to maturity (x-axis "DTM"). This defines theabsolute implied volatility surface; changing coordinates so that the strike price is replaced bydelta yields therelative implied volatility surface.
The implied volatility surface simultaneously shows both volatility smile and term structure of volatility. Option traders use an implied volatility plot to quickly determine the shape of the implied volatility surface, and to identify any areas where the slope of the plot (and therefore relative implied volatilities) seems out of line.
The graph shows an implied volatility surface for all the put options on a particular underlying stock price. Thez-axis represents implied volatility in percent, andx andy axes represent the option delta, and the days to maturity. Note that to maintainput–call parity,[note 1] puts must have the same implied volatility as calls of the same strike and expiration date. For this surface, we can see that the underlying symbol has both volatility skew (a tilt along the delta axis), as well as a volatility term structure, indicating an anticipated event in the near future.
An implied volatility surface isstatic: it describes the implied volatilities at a given moment in time. How the surface changes as the spot changes is called theevolution of the implied volatility surface.
Common heuristics include:
So if spot moves from $100 to $120, sticky strike would predict that the implied volatility of a $120 strike option would be whatever it was before the move (though it has moved from being OTM to ATM), while sticky delta would predict that the implied volatility of the $120 strike option would be whatever the $100 strike option's implied volatility was before the move (as these are both ATM at the time).
Since theBlack-Scholes model assumes the log returns of the spot are arandom walk, Black-Scholes assumes that the forward distribution is a log-normal distribution. However, in practice, this assumption is inconsistent with the behavior of many securities.[5] If the forward risk-neutral probability distribution deviates from a log-normal distribution, the IVs will vary across the strikes of a given expiration, in other words, a non-constant volatility surface.
Given a volatility surface, it is possible to determine the implied forward, risk-neutral probability distribution.
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Methods of modeling the volatility smile includestochastic volatility models andlocal volatility models. For a discussion as to the various alternate approaches developed here, seeFinancial economics § Challenges and criticism andBlack–Scholes model § The volatility smile.