Fundamentals of Volatility Smile
Introduction
The Options market is huge with trading happening on regulated exchanges as well as on the over-the-counter (OTC) market. The volume of contracts traded on the OTC is much higher vis-à-vis the volume on the exchange. Options are a major part of the trading portfolio of a wide range of market participants including — banks, investment banks, financial institutions, retail investors etc. Therefore, activities including — pricing, hedging and trading are done by market participants to manage their options portfolio.
Readers familiar with options pricing and models will recognize the following relation.
V = f (S, K, T, r, σ)
Where,
V: Option price
S: underlying asset price
K: strike price
T: time to maturity
r: risk free rate of interest
σ: volatility
For the benefit of our readers who may not have had the opportunity to work on options in the past, here is a brief explanation of the volatility term mentioned above. Volatility is one of the most important parameters that has an impact on option pricing. There are a variety of models used by industry participants to estimate the “correct” volatility number that goes into options pricing models.
This post focuses on a fundamental concept called volatility smile which is constructed using market data from the options market. A volatility smile can be visualized as a 2-D plot of volatility vis-à-vis Strike (or delta). The diagram below is a sample of vol smile that is often observed in the market.
As per market convention:
a. For Equity options: The market data is quoted as Volatility Vs strike
b. For FX options: The market data is quoted as Volatility Vs delta
In this article, we will follow the convention of volatility Vs strike. For readers who would like to apply similar ideas to the market data quoted in the FX options market, they will need to perform one extra step to convert from vol-delta quote to vol-strike quote using a standard transformation from delta to strike.
While smile analysis is a quantitatively heavy topic, I have attempted to keep the amount of maths limited in this article, so that it becomes a easy read for a wide audience.
Analyzing the smile
Using the two charts below, we will discuss the relation between implied volatility smile and the implied distribution; we will also compare it with respect to lognormal distribution. Note, that implied distribution can be extracted from traded options data quoted in the market, whereas, lognormal is a common assumption used in modelling.
In the subsequent section of this article, we will discuss a technique to build the implied distribution using traded options data.
First things first, lets interpret the shape of the vol smile and link it with implied distribution and also compare the same with lognormal distribution.
Below given is a “textbook” smile that we often observe in FX market. However, the shape of the vol smile keeps on evolving throughout the trading day. We may imagine this to be a snapshot taken at a certain instant of time. For Equity options, we typically observe a vol skew wherein the smile does not appear to be symmetric as the one given below, rather it shows a characteristic skew in one direction. Please note, in this article we will use a symmetric smile only for analysis.
The first chart is the vol smile we had seen earlier in this article. A options pricing model like the popular Black Scholes Merton (BSM) may be used to arrive at implied volatilities that will be mapped on the chart which will be useful to construct the smile. A few of the points on the vol smile can be extracted from traded option prices from which we can get implied vols through the BSM model. The other points in the smile can be plotted using different approaches ranging from interpolation-extrapolation right through till more sophisticated models. Please note, in this article, we will not be discussing the maths behind the smile construction. This will be discussed in a separate article.
The second chart gives two distributions namely: implied and lognormal. Let’s interpret the above two charts and understand the link between them.
K1 and K2 are two strike prices that we will use for analysis. Consider an out-of-the-money option with strike price K1. This option pays off only if the underlying asset price falls below K1. Focus on the tails of the two distributions to the left of strike price K1. We observe that the implied distribution shows a thicker tail vis-à-vis the lognormal distribution. The interpretation of the thick tails is the probability of tail events i.e., extreme events is higher under the implied distribution as compared to the lognormal distribution. Higher probability of extreme outcomes can result in market participants buying more options to hedge against extreme movements thereby pushing up the option vols and as a result pushing up the option prices too. This can be seen in the upward sloping curvature on the left-hand side of the vol smile chart. A similar analysis can be applied to the right-hand side of the smile as well. WE assume K1 as the strike price of an out-of-the-money Call option. We observe that the implied distribution has fat tails thus, we have an upward sloping curvature on the right as well.
A similar analysis can be applied to the Equity Skew chart and link it with implied distribution. We should note that for equity skew analysis, the lognormal and the implied distributions would not appear superimposed on top of each other.
We note that we are interested in the implied distribution as that is the one we extract from traded options in the market.
Extracting the implied distribution:
Lets introduce a few variables:
S: underlying asset price
K: strike price
T: Time to expiry
r: risk free rate
Let g represent the risk neutral density function of S.
The price of the call option can be given as below:
We note the limits of integral are from K to ∞ and thus we do not require a max(S-K,0) condition. Differentiating the above twice w.r.t. strike K gives:
We intend to use the traded options data from the market to construct the implied distribution. We attempt to approximate the differential on the RHS as below:
We choose three options with strike K, K+δ, K+δ which are very closely packed with each other. We assume c1, c2, c3 as the three option prices for the above-mentioned three set of strike prices. Then g(K) is approximated as below:
The above can be thought of as a butterfly strategy which is a very liquid strategy traded by market participants. We use a series of traded options contracts and use it to extract the implied distribution g(K) by solving the above equation. Similar analysis may be applied for equity and fx options.
Incorporating smile into pricing of non-vanilla options:
The smile impact needs to be considered when we move away from European options. For example, consider a first-degree exotic like barrier options. We may continue to use the BSM framework and adjust the smile cost into the pricing model. To this end, we may use the Traders Rule of Thumb in order to incorporate the smile cost. While there are other more sophisticated models out there, the traders rule of thumb enables us to stay in the familiar BSM framework and enhance it by incorporating the smile cost rather than using a constant vol.
Extending the idea to Volatility surface:
While volatility smile is a 2-D plot of volatility versus strike, the volatility surface extends to 3-D which incorporates volatility, strike and time to maturity. Vol surface construction increases the computational complexity further. However, vol surface get extensively used by market participants for their decision-making purposes.
Conclusion
This article discussed the fundamentals of volatility smiles and explained the linkage of the smile with the implied distribution that is extracted from traded option price data. These insights will help readers further build on the concept of vol smiles (and surfaces) and provide an inspiration to readers to further explore the quantitative modelling aspects that are used extensively in volatility smile modelling and analysis.
Looking to learn about volatility models?
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Thank you for reading!
