Implied volatility Archives

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains how implied volatility is computed from option market prices and a option pricing model.

Introduction

Volatility is a measure of fluctuations observed in an asset’s returns over a period of time. The standard deviation of historical asset returns is one of the measures of volatility. In option pricing models like the Black-Scholes-Merton model, volatility corresponds to the volatility of the underlying asset’s return. It is a key component of the model because it is not directly observed in the market and cannot be directly computed. Moreover, volatility has a strong impact on the option value.

Mathematically, in a reverse way, implied volatility is the volatility of the underlying asset which gives the theoretical value of an option (as computed by Black-Scholes-Merton model) equal to the market price of that option.

Implied volatility is a forward-looking measure because it is a representation of expected price movements in an underlying asset in the future.

Computation methods for implied volatility

The Black-Scholes-Merton (BSM) model provides an analytical formula for the price of both a call option and a put option.

The value for a call option at time t is given by:

Call option value

The value for a put option at time t is given by:

Put option value

where the parameters d₁ and d₂ are given by:,

call option d1 d2

with the following notations:

S_t : Price of the underlying asset at time t
t: Current date
T: Expiry date of the option
K: Strike price of the option
r: Risk-free interest rate
σ: Volatility of the underlying asset
N(.): Cumulative distribution function for a normal (Gaussian) distribution. It is the probability that a random variable is less or equal to its input (i.e. d₁ and d₂) for a normal distribution. Thus, 0 ≤ N(.) ≤ 1

From the BSM model, both for a call option and a put option, the option price is an increasing function of the volatility of the underlying asset: an increase in volatility will cause an increase in the option price.

Figures 1 and 2 below illustrate the relationship between the value of a call option and a put option and the level of volatility of the underlying asset according to the BSM model.

Figure 1. Call option value as a function of volatility.

Source: computation by the author (BSM model)

Figure 2. Put option value as a function of volatility.

Source: computation by the author (BSM model)

You can download below the Excel file for the computation of the value of a call option and a put option for different levels of volatility of the underlying asset according to the BSM model.

We can observe that the call and put option values are a monotonically increasing function of the volatility of the underlying asset. Then, for a given level of volatility, there is a unique value for the call option and a unique value for the put option. This implies that this function can be reversed; for a given value for the call option, there is a unique level of volatility, and similarly, for a given value for the put option, there is a unique level of volatility.

The BSM formula can be reverse-engineered to compute the implied volatility i.e., if we have the market price of the option, the market price of the underlying asset, the market risk-free rate, and the characteristics of the option (the expiration date and strike price), we can obtain the implied volatility of the underlying asset by inverting the BSM formula.

Example

Consider a call option with a strike price of 50 € and a time to maturity of 0.25 years. The market risk-free interest rate is 2% and the current price of the underlying asset is 50 €. Thus, the call option is ‘at-the-money’. If the market price of the call option is equal to 2 €, then the associated level of volatility (implied volatility) is equal to 18.83%.

You can download below the Excel file below to compute the implied volatility given the market price of a call option. The computation uses the Excel solver.

Volatility smile

Volatility smile is the name given to the plot of implied volatility against different strikes for options with the same time to maturity. According to the BSM model, it is a horizontal straight line as the model assumes that the volatility is constant (it does not depend on the option strike). However, in practice, we do not observe a horizontal straight line. The curve may be in the shape of the alphabet ‘U’ or a ‘smile’ which is the usual term used to refer to the observed function of implied volatility.

Figure 3 below depicts the volatility smile for call options on the Apple stock on May 13, 2022.

Figure 3. Volatility smile for call options on Apple stock.
Apple volatility smile
Source: Computation by author.

We can also observe that the for a specific time to maturity, the implied volatility is minimum when the option is at-the-money.

Volatility surface

An essential assumption of the BSM model is that the returns of the underlying asset follow geometric Brownian motion (corresponding to log-normal distribution for the price at a given point in time) and the volatility of the underlying asset price remains constant over time until the expiration date. Thus theoretically, for a constant time to maturity, the plot of implied volatility and strike price would be a horizontal straight line corresponding to a constant value for volatility.

Volatility surface is obtained when values for implied volatilities are calculated for options with different strike prices and times to maturity.

CBOE Volatility Index

The Chicago Board Options Exchange publishes the renowned Volatility Index (also known as VIX) which is an index based on the implied volatility of 30-day option contracts on the S&P 500 index. It is also called the ‘fear gauge’ and it is a representation of the market outlook for volatility for the next 30 days.

Useful resources

Academic articles

Black F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities” The Journal of Political Economy, 81, 637-654.

Dupire B. (1994). “Pricing with a Smile” Risk Magazine 7, 18-20.

Merton R.C. (1973) “Theory of Rational Option Pricing” Bell Journal of Economics, 4, 141–183.

Business

CBOE Volatility Index (VIX)

CBOE VIX tradable products

About the author

The article was written in May 2022 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

VIX index

In this article, Youssef LOURAOUI (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021) presents the VIX index, which is a financial index that measures the uncertainty in the US equity market.

This article is structured as follows: we begin by defining the grounding notions of the VIX index. We then explain the behavior of this index and its statistical characteristics. We finish by presenting its practical usage in financial markets.

Definition

The CBOE Volatility Index, abbreviated “VIX”, is a measure of the expected S&P 500 index movement calculated by the Chicago Board Options Exchange (CBOE) from the current trading prices of options written on the S&P 500 index.

Known as Wall Street’s “fear index”, the VIX is closely monitored by a broad range of market players, and its level and pattern have become ingrained in market discussion.

Figure 1 illustrates the evolution of the VIX index for the period from 2003 to 2021.
Figure 1 Historical levels of the VIX index from 2003-2021.

Source: computation by the author (Data source: Thomson Reuters).

VIX values greater than 20 are regarded to be high by market participants. If the VIX is between 12 and 20, it is considered normal; if it is less than 12, it is considered low. As it is the case with other indices, the VIX is computed using the price of a basket of tradable components (in this case, options expiring within the next month or so). The profit or loss that option buyers and sellers realize during the option’s life will depend, among other things, on how significantly the S&P 500’s actual volatility will differ from the implied volatility given by the VIX at the start of the period (S&P Global Research, 2017).

Behavior of the VIX index

Statistical distribution of the S&P500 index returns and VIX level

Figure 2 displays the statistical distribution of the price variations in the S&P500 index for different levels of the VIX index The higher the VIX index (by convention, greater than 20), the more severe the distribution tends to be, with negative skewness and high kurtosis indicating heightened volatility in the US market, therefore exacerbating both positive and negative swings. An opposite finding may be made for the VIX level at lower levels (often less than 12), when market swings are less evident due to less skewness and lower kurtosis (S&P Global Research, 2017).

Figure 2. The distribution of 30-day return in the S&P500 index for different VIX index levels.

Source: S&P Global Research (2017).

If the VIX is low, market players may benefit by purchasing options; conversely, if the VIX is high, market participants may profit from selling options. The specific utility of anticipated VIX is that it gives us with a more accurate assessment of whether VIX is high, low, or normal at any point in time (S&P Global Research, 2017). Thus, VIX may be regarded of as a crowd-sourced estimate of the S&P 500’s expected volatility. As with interest rates and dividends, one cannot invest directly in them, even though one can guess on their future worth, one cannot invest directly in VIX, and the significance of a specific VIX level is commonly misinterpreted (S&P Global Research, 2017).

Recent volatility in the S&P500 index and VIX level

Figure 3 demonstrates that the VIX index is strongly correlated with recent market volatility. However, there is considerable variance; for example, a recent volatility level of about 20% has been associated with a VIX level of 34 (point B, when VIX was very “high”) and with a VIX level of 12 (point C, when VIX was relatively “low”). Volatility (realized or implied) has a strong propensity to return to its mean. This insight is not especially original, despite its illustrious past. There is an enormous body of data demonstrating that volatility tends to mean revert across markets, and the pioneers of this field were given the Nobel Prize in part for incorporating their results into volatility forecasts and simulations (S&P Global Research, 2017).

Figure 3. Relation between VIX and recent volatility.

Source: S&P Global Research (2017).

Realized volatility in the S&P500 index and VIX level

Figure 4 represents the relationship between Realized volatility in the S&P500 index over a period and the VIX level at the begining of the period.

Figure 4. VIX versus next realized volatility.

Source: S&P Global Research (2017).

Mean reversion

Figure 5 shows how VIX index converge to a certain llong-term level as time passes. This finding is not due to 15% being exceptional in any manner; this figure for M was calculated using historical volatility levels for the S&P 500 and their evolution. It is not implausible that M (else referred to as long-term average volatility in the US equities market) may change over time; changes in the S&P 500’s sector weightings, trade All of these factors have the ability to influence both the pace and the volume and the point at which mean reversion occurs.

Figure 5. Mean-reversion dynamic in recent volatility.

Source: S&P Global Research (2017).

Use of the VIX index in financial markets

There are two methods for determining an asset’s volatility. Either through a statistical calculation of an asset’s realized volatility, also known as historical volatility, which serves as a pointer to the asset’s volatility behavior. This is a limited method that is based on the premise that past volatility tends to replicate itself in the future, without including a forward-looking study of volatility. The second technique is to extract an asset’s volatility from option prices referred to as “implied volatility”.

Why should I be interested in this post?

When investors make investment decisions, they utilize the VIX to gauge the degree of risk, worry, or stress in the market. Additionally, traders can trade the VIX using a range of options and exchange-traded products, or price derivatives using VIX values.

Useful resources

Business analysis

CBOE , 2021. VIX

Nasdaq, 2021. Realized Volatility

Nasdaq, 2021. Vix Index Volatility

S&P Global Research, 2017. Reading VIX: Does VIX Predict Future Volatility?

S&P Global Research, 2017. A Practitioner’s Guide to Reading VIX

About the author

The article was written in September 2021 by Youssef LOURAOUI (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021).