Joey

April 13, 2022

In this week’s option learning blog, we introduce another critical component of option trading as volatility. Volatility in general sense is defined as price fluctuation level in observed time scope. For example, stock volatility is usually quantified as standard deviation of equity price change in a given timeframe.

**Classification on terminology of volatility**

Volatility in option trading can be mainly categorized into historical volatility, implied volatility and future volatility. Historical volatility measures stock price fluctuation in selected historical interval, usually calculated as annualized standard deviation of price change. Implied volatility is a theoretical value as a calculated parameter assuming the option market price is approximately equal to the theoretical value of Black-Scholes pricing model. Future volatility in contrast to historical volatility is expected price change level in the future.

The definition tells the difference and relevance between implied volatility and historical volatility. Implied volatility is a subjective theoretical value aimed to make fair option pricing given by group expectation on a commonly recognized model. Historical volatility is more of an objective value on equity price fluctuation calculated by real historical data. On the one hand, the estimation of implied volatility can be highly correlated with historical volatility of underlying equity especially within a near shorter period. On the other hand, implied volatility compared with historical volatility is more of subjective constructed confidence widely accepted by option traders. The aim of implied volatility is to make estimation towards future price volatility level. For example, we may observe the deviation between historical volatility and implied volatility right before a fundamental event such as financial release. When the market expectation is largely divided, stock traders take cautious attitude towards stock buy-in. We may observe declining historical volatility as reflection of hesitated buy-in action. However, option traders may expect a rising stock price volatility right after financial release. Implied volatility as expectation of near-future rate of change can be higher than historical volatility as reflection of current action.

There are a range of different calculating methods on volatility. The mostly applied method is annualized standard deviation on close-to-close daily price return. Other methods include Parkinson estimator, Garman-Klass estimator, Roger-Satchell estimator and Yang-Zhang estimator. Different methods may choose different price components to construct price return. For example, Parkinson estimator applies high-to-low rate of change to calculate daily price return. For more introduction on volatility calculation methods and difference among them, please check book references and online resources such as *Volatility Trading* (Euan Sinclair) for more details.

**Volatility surface --- termstructure and smile/skew curve**

The volatility surface is a three-dimensional plot of implied volatility. It describes impacts of termstructure and strike price change on implied volatility. Volatility termstructure shows how the implied volatility of an option changes with different time to maturity. The other dimension of volatility surface as strike price shows how implied volatility distribution evolves with strike price change under equal time to maturity. The volatility surface also indicates a more practical version of Black-Scholes model. If the Black-Scholes model is ideally true, then the implied volatility surface would stay flat across strike price and term structure. In practice, this is not the case. Volatility pattern in real practice is often deviated from “perfect flat” into three shapes as “volatility smile”, “volatility forward skew” and “volatility reverse skew”.

**Figure 1：volatility surface**

（source：https://pic1.zhimg.com/v2-a884d4744c0093d5b5a9557cc5022815_720w.jpg?source=172ae18b）

Let’s first understand how “volatility smile” and “volatility skew” are formed. In Black-Scholes model, the stock price movement is basically assumed to be log-normal distribution. In practice however, the price return curve of underlying equity is generally fat-tailed. That is to say the average probability of extreme values is higher than that in the normal distribution. Therefore, the BS model underestimates the probability of option value being in deeply out-of-the-money (OTM) and deeply in-the-money (ITM) at expiration. In the same way, it underestimates the average price level of option in deeply OTM and ITM. Beyond this clarification, traders can also have asymmetric preference betting price being ITM or OTM in different markets. “Volatility forward skew” indicates that the higher the strike price in the contract with the same expiration date, the higher the implied volatility will be. Market traders of this type shows higher demand for the underlying out-of-the-money call options than out-of-the-money put options as well as higher demand for in-the-money put options than in-the-money call options. Therefore, the actual price of an out-of-the-money call option is often higher than the theoretical value of the BS model. “Volatility forward skew” tends to occur in commodity market, because underlying asset price in commodity market tends to have higher upside risk, and option is often used as means of hedging for risk management. Buyers of commodities are willing to pay an "additional premium" to protect against price upward risk. The second type of volatility pattern is “volatility smile”. It indicates that the implied volatility of deeply out-of-the-money option is higher than that of out-of-the-money option, which is usually found in the foreign exchange options market. As foreign exchange bets on two currency pairs and their eco-political development, there is often no unilateral bullish or bearish bias preference.

**Figure 2：Volatility Forward Skew**

The third type is “volatility reverse skew”, usually found in the stock option market. It indicates that traders demand more for the underlying out-of-the-money puts than in-the-money puts as well as more for the in-the-money calls than out-of-the-money calls. Stock traders tend to buy in stocks and hold them, thus willing to pay more premiums to safeguard against downward risk from price pull-back. In other words, people tend to have higher positive preference on stock market development and confidence towards economics than negative downside across average whole market periods. We then observe higher price of out-of-the-money puts than the theoretical value of its pricing model.

**Figure 3: Volatility Reverse Skew**

（Source：https://www.theoptionsguide.com/images/reverse-vertical-volatility-skew.gif）

Volatility termstructure is another parameter to describe implied volatility change. It reflects implied volatility change across different date to expiration and the volatility difference between near-month contracts and back-month contracts. In normal case, the termstructure of volatility tends to be upward. This means near-month implied volatility is lower than far-month, however the rate of change would attenuate with time. When market observes volatility “spike” or sudden price movement, it may finally lead to higher near-month implied volatility than far-month level forming a downward volatility termstructure curve. Based on the value difference across different term periods and the distribution of implied volatility across moneyness level, we can further construct trading strategies around volatility characteristics.

**The practice of volatility in option trading**

Volatility prediction is widely applied in delta neutralization strategies. The aim of volatility trading is to profit from expected price difference between future realized volatility and implied volatility. To take an example of at-the-money (ATM) straddle, the strategy constructs an option portfolio by selling or buying equal amount of ATM options with same strike price and date to expiration at the same time. The long ATM straddle expects gamma return from higher future realized volatility than implied volatility. The profit from gamma exposure is expected to overweigh loss from theta attenuation. On the contrary, the short ATM straddle would sell equal amount of options with same strike price and date to expiration at the same time. The trader expects future volatility of underlying asset fall in a narrow range or equity price movement evolving into a small trading range. The short strategy profit from higher theta return than gamma loss that traders expect a lower future realized volatility than implied volatility. ATM straddle strategy is only a small example of volatility trading, there are a wide range of different strategies around volatility prediction to explore.

Volatility trading is the “crown” and almost the most critical aspect of option trading. In this week we have a basic review over definition of volatility, different categories of volatility, termstructure of implied volatility, smile and skew curve of implied volatility and the real practice of volatility trading. Thank you for reading over this blog and welcome any remarks and comments below!