Volatility Smile is the implied volatility pattern that appears in the pricing of financial options. It is appropriate to find one single parameter (implied volatility) needed to be modified for the Black-Scholes formula to match the market price. Especially for the given expiration, the option whose strikes price is very different from the base asset price underlying the higher (and thus implied volatility) price of what the standard options pricing model suggests. These options are said to be in-money or out-of-the-money.
Graphing of the implied volatility of the strike price for the given expired produces a flatter "smile" rather than the expected flat surface. These patterns are different across markets. The equity options traded on the American market did not show a volatile smile before Crash of 1987 but started to show one after. It is believed that investors' reassessment of the tail-fat probability has led to higher prices for options running out of money. This anomaly implies a deficiency in the standard Black-Scholes option pricing model which assumes constant volatility and the log-normal distribution of underlying asset returns. However, the distribution of asset earnings that is empirical, tend to show the tail-fat (kurtosis) and setbacks. Modeling a volatility smile is an active area of ​​research in quantitative finance, and a better pricing model such as the stochastic volatility model partially overcomes this problem.
The related concept is that of the structure of the term volatility , which illustrates how (implied) volatility differs for the options associated with different maturities. An implied volatility surface is a 3-D plot depicting the volatility of a smile and the volatility term structure in a consolidated three-dimensional surface for all options on a given asset base.
Video Volatility smile
The implied volatility
In the Black-Scholes model, the theoretical value of the vanilla option is an increase in the monotonous function of the underlying asset volatility. This means it is usually possible to calculate the uniquely implied volatility of a given market price for an option. This implied volatility is best regarded as a rescaling of the option price which makes the comparison between different strikes, expirations, and different underlyings easier and more intuitive.
When implied volatility is plotted against the strike price, the resulting graph is usually sloping downwards for equity markets, or valleys shaped for the currency market. For markets with downward sloping graphs, such as for the equity option, the term " oblique volatility " is often used. For other markets, such as the FX option or equity index option, where a typical graph appears on both ends, the more familiar term " volatility smile " is used. For example, the implied volatility for an inverted equity option (ie high strike) is usually lower than the in-cash equity option. However, the volatility of options implied by the foreign exchange contracts tends to increase both on the downside and upside direction. In the equity market, small oblique smiles are often observed near money as the boldness in the implicitly downward sloping volatility graphs in general. Sometimes the term "smirk" is used to describe a crooked smile.
Praktisi pasar menggunakan istilah tersirat-volatilitas untuk menunjukkan parameter volatilitas untuk opsi ATM (at-the-money). Penyesuaian nilai ini dilakukan dengan memasukkan nilai-nilai Pembalikan Risiko dan Flys (Skews) untuk menentukan ukuran volatilitas aktual yang dapat digunakan untuk opsi dengan delta yang tidak 50.
Formula
dimana:
- adalah volatilitas tersirat di mana panggilan X% -delta melakukan perdagangan di pasar
- adalah volatilitas tersirat dari X% -dtata yang dimasukkan
- adalah volume At-The-Money Forward di Panggilan ATM (Dan Ditempatkan!) Address diperdagangkan di pasar
The risk reversal is generally cited as a reversal of x% delta risk and is essentially a Long X% delta call, and short short delta X%.
Butterfly, on the other hand, is a strategy consisting of: - Y% delta fly which means Long Y% delta call, Long Y% delta put, short one ATM call and short one ATM put. (small cap shape)
Maps Volatility smile
Implied volatility and historical volatility
It is important to note that the implied volatility is related to historical volatility, but both are different. Historical volatility is a direct measure of the underlying price movement (volatility realization) during recent history (eg, the 21-day trailing period). Implied volatility, in contrast, is determined by the market price of the derivative contract itself, and not the underlying one. Therefore, different derivative contracts on the same underlying have different implied volatility as a function of their own supply and demand dynamics. For example, the IBM call option, attacking $ 100 and expiring in 6 months, may have an implied volatility of 18%, while put option strikes at $ 105 and ending in 1 month may have an implied volatility of 21%. At the same time, historical volatility for IBM for the preceding 21-day period may be 17% (all volatility expressed as a percentage of annual percentage).
The term structure of volatility
For options of different maturities, we also see differences in characteristics in implied volatility. However, in this case, the dominant effect is related to the implied market impact of future events. For example, it was observed that stock price volatility increased significantly on a day when a company reported its earnings. Correspondingly, we see that the implied volatility for options will rise during the period before the earnings announcement, and then fall again as soon as stock prices absorb new information. The previously mature options show larger swings in implied volatility (sometimes called "vol vol)" rather than options with longer maturity.
Other option markets show other behaviors. For example, options on commodity futures usually indicate an increase in implied volatility just before the announcement of a crop estimate. Options on US Treasury Bill futures show an implied volatility increase just before the Federal Reserve Board meeting (when short-term interest rate changes are announced).
The market combines many other types of events into the volatility term structure. For example, the impact of future drug trials may lead to changes in implied volatility for the supply of medicines. The anticipated patent litigation settlement date may impact technology shares, etc.
The term volatility structure lists the relationship between implied volatility and time to expiration. The term structure provides another method for traders to measure cheap or expensive options.
Surface volatility implied
It is often useful to describe implied volatility as a function of strike price and time to maturity. The result is a two-dimensional curve surface plotted in three dimensions where the current market implies volatility (Z-axis) for all options on a basis plotted against price or delta (Y-axis) and time to maturity (X-axis "DTM" ). It defines the absolute implied volatility surface ; change the coordinates so that the price is replaced by delta yields the relative implied volatility of the surface .
Surface volatility implied simultaneously shows both a volatile smile and a volatility term structure. Options traders use an implied volatility plot to quickly determine the shape of the implied volatility surface, and to identify any areas where the plot's slope (and therefore relative implied volatility) appears to be out of line.
The graph shows the implied volatility surface for all put options on a particular base stock price. The Z axis represents the volatility in percent, and the X and Y axes represent the preferred delta, and days to maturity. Note that to maintain put-call parity, 20 delta put must have the same implied volatility as call 80 delta. For this surface, we can see that the underlying symbols have both tilted instability (slope along the delta axis), as well as the volatility long-term structure that signifies an anticipated event in the near future.
Evolution: Sticky
The implied volatility surface is static : this illustrates the volatility implied at a given time. How the surface changes due to a change of place is called the evolution of the surface of implied volatility .
Common heuristics include:
- "stick strings" (or "sticky-by-strike", or "stick-to-strike"): if a place change, the implied volatility of an option with an absolute strike given
- "sticky money" (aka, "sticky delta"; see money for why this is an equivalent term): if a change of venue, the implied volatility of an option with a given money ( delta) not unchanged.
So, if the spot moves from $ 100 to $ 120, a firm strike will predict that the implied volatility of the $ 120 attack option will be anything before moving (though it has moved from OTM to ATM), while the sticky delta will predict that the implied volatility of the $ 120 strike option would be anything with the $ 100 implied volatility predicted before the move (since both are ATMs at the time).
Volatility Modeling
Methods of smiling volatility include stochastic volatility models and local volatility models. For a discussion of the various alternative approaches developed here, see Financial Economics # Challenges and criticisms and Black-Scholes models # Smile volatility.
See also
- Volatility (finance)
- Stochastic instability
- SABR volatility model
- Vanna Volga Method
- Heston Model
- Implied binomial tree
- An implied trinomial tree
- Edgeworth Binomial tree
- Financial Economics # Challenges and criticism
References
External links
- Emanuel Derman, The Smile of Volatility and the Implied Trees (RISK, Feb. 7-22, pp. 139-145, pp. 32-39) (PDF)
- Mark Rubinstein, Implied Binomial Tree (PDF)
- Damiano Brigo, Fabio Mercurio, Francesco Rapisarda and Giulio Sartorelli, Volatility Smile Model with Mixed Differential Differential Equations (PDF)
- Visualization of volatility smile
- C. Grunspan, "Asymptotic Expansion for Inferred Lognormal Volatility: A Model-Free Approach"
- Y. Li, "Clear financial pricing and pricing options"
- examples of commodity/slope volatility smiles
Source of the article : Wikipedia
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