{"id":7922,"date":"2022-05-15T20:52:08","date_gmt":"2022-05-15T19:52:08","guid":{"rendered":"https:\/\/www.simtrade.fr\/blog_simtrade\/?p=7922"},"modified":"2022-05-15T20:52:08","modified_gmt":"2022-05-15T19:52:08","slug":"implied-volatility","status":"publish","type":"post","link":"https:\/\/www.simtrade.fr\/blog_simtrade\/implied-volatility\/","title":{"rendered":"Implied Volatility"},"content":{"rendered":"

\"Jayati<\/a><\/p>\n

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

Introduction<\/h2>\n

Volatility is a measure of fluctuations observed in an asset\u2019s 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.<\/p>\n

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.<\/p>\n

Implied volatility is a forward-looking measure because it is a representation of expected price movements in an underlying asset in the future.<\/p>\n

Computation methods for implied volatility<\/h2>\n

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

The value for a call option at time t is given by:<\/p>\n

\"<\/p>\n

The value for a put option at time t is given by:<\/p>\n

\"Put<\/p>\n

where the parameters d1<\/sub> and d2<\/sub> are given by:,<\/p>\n

\"call<\/p>\n

with the following notations:<\/p>\n

St<\/sub> : Price of the underlying asset at time t
\nt: Current date
\nT: Expiry date of the option
\nK: Strike price of the option
\nr: Risk-free interest rate
\n\u03c3: Volatility of the underlying asset
\nN(.): 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. d\u2081 and d\u2082) for a normal distribution. Thus, 0 \u2264 N(.) \u2264 1<\/p>\n

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.<\/p>\n

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.<\/p>\n

Figure 1. Call option value as a function of volatility.
\n \"Call
\nSource: computation by the author (BSM model)<\/p>\n

Figure 2. Put option value as a function of volatility.
\n \"Put
\nSource: computation by the author (BSM model)<\/p>\n

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.<\/p>\n

\"Excel<\/a><\/p>\n

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.<\/p>\n

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.<\/p>\n

Example<\/h3>\n

Consider a call option with a strike price of 50 \u20ac 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 \u20ac. Thus, the call option is \u2018at-the-money\u2019. If the market price of the call option is equal to 2 \u20ac, then the associated level of volatility (implied volatility) is equal to 18.83%.<\/p>\n

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.<\/p>\n

\"Excel<\/a><\/p>\n

Volatility smile<\/h2>\n

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 \u2018U\u2019 or a \u2018smile\u2019 which is the usual term used to refer to the observed function of implied volatility.<\/p>\n

Figure 3 below depicts the volatility smile for call options on the Apple stock on May 13, 2022.<\/p>\n

Figure 3. Volatility smile for call options on Apple stock.
\n\"Apple
\nSource: Computation by author.<\/p>\n

\"Excel<\/a><\/p>\n

We can also observe that the for a specific time to maturity, the implied volatility is minimum when the option is at-the-money.<\/p>\n

Volatility surface<\/h2>\n

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.<\/p>\n

Volatility surface is obtained when values for implied volatilities are calculated for options with different strike prices and times to maturity.<\/p>\n

CBOE Volatility Index<\/h2>\n

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 \u2018fear gauge\u2019 and it is a representation of the market outlook for volatility for the next 30 days.<\/p>\n

Related posts on the SimTrade blog<\/h2>\n

   ▶ All posts about Options<\/a><\/p>\n

   ▶ Akshit GUPTA Options<\/a><\/p>\n

   ▶ Jayati WALIA Brownian Motion in Finance<\/a><\/p>\n

   ▶ Jayati WALIA Brownian Motion in Finance<\/a><\/p>\n

   ▶ Youssef LOURAOUI Minimum Volatility Factor<\/a><\/p>\n

   ▶ Youssef LOURAOUI VIX index<\/a><\/p>\n

Useful resources<\/h2>\n

Academic articles<\/h3>\n

Black F. and M. Scholes (1973) \u201cThe Pricing of Options and Corporate Liabilities\u201d The Journal of Political Economy, 81, 637-654.<\/p>\n

Dupire B. (1994). “Pricing with a Smile” Risk Magazine 7, 18-20.<\/p>\n

Merton R.C. (1973) \u201cTheory of Rational Option Pricing\u201d Bell Journal of Economics, 4, 141\u2013183.<\/p>\n

Business<\/h3>\n

CBOE Volatility Index (VIX)<\/a><\/p>\n

CBOE VIX tradable products<\/a><\/p>\n

About the author<\/h2>\n

The article was written in May 2022 by Jayati WALIA<\/a> (ESSEC Business School, Grande Ecole<\/i> Program – Master in Management, 2019-2022).<\/p>\n","protected":false},"excerpt":{"rendered":"

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\u2019s returns over a period of time. The standard deviation of historical asset returns … Read more<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[5,10],"tags":[102,105,114,305,437,475,505,621],"class_list":["post-7922","post","type-post","status-publish","format-standard","hentry","category-contributors","category-financial-techniques","tag-brownian-motion","tag-bsm","tag-call-option","tag-implied-volatility","tag-options","tag-put-option","tag-risk","tag-volatility"],"yoast_head":"\nImplied Volatility - SimTrade blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.simtrade.fr\/blog_simtrade\/implied-volatility\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Implied Volatility\" \/>\n<meta property=\"og:description\" content=\"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. 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