Today we cannot understand the world of investments without financial innovation. Organized markets and financial intermediaries constantly propose new alternatives to meet investment needs, risk management and financial resources management of governments, companies and individuals. In fact, intermediaries and financial markets exist, among other reasons, so that participants can redistribute their risk among those who are willing to take it.

Like technology, the evolution of the organized financial markets was relatively slow during the first 70 years of the 20th century. Until that time, it was only only possible to trade debt instruments (bonds), equity instruments (shares) and futures contracts subscribed mainly for agricultural raw materials. In 1972, the Chicago Board of Trade (or CBOT) launched the first financial futures contracts, which had underlying currencies setting the bid and ask exchange rate with the US dollar. A few years later, it began to trade interest rate futures. Also in the early 1970s, the Chicago Board Options Exchange (CBOE) began offering options on stock of 16 companies.

In Mexico, the organized derivatives market, as it is known today, was conceived in 1994 and materialized in 1998. Created by the Mexican Stock Exchange (BMV) and the S.D. Indeval, the Derivatives Exchange (MexDer) began operations in December 1998 with futures on US dollar.

Derivatives are financial instruments whose value and risk depend on another asset (underlying asset). For example, a call option is a contract that gives the investor the right to buy a certain asset at a certain price at a future date. By acquiring that right, the contract owner is willing to pay a premium in advance (call price). The price of the option depends on the time of expiration of the contract and the uncertainty about the evolution of the price of the underlying asset and interest rates. If the contract gives the investor the right to sell the underlying asset, then it is called a put option. The popularization of the use of derivative instruments and the search for options for hedging and investment mechanisms have resulted in the impressive development of markets and non-conventional financial instruments.

From a Price Model to a Volatility Indicator

In 1997, the Nobel Prize in Economics was awarded to Robert Merton and Myron Scholes for their method of determining the valuation of a stock option (which they developed with Fisher Black, who died in 1995). The method they proposed gave way to the creation of new instruments, as well as new mechanisms to determine the economic value of many other assets such as insurance contracts, guarantees and flexibility in an investment project. Black and Scholes presented their options pricing formula (later called “Black-Scholes”) in an article in 1973.

Before the Black-Scholes formula, the main problem in determining the theoretical value of an option was to know the attitude of the buyer of the option toward the risks he/she would run. A “very” risk-averse investor would be willing to pay more to eliminate it than a “less” risk-averse investor. The most important contribution of Black and Scholes is that in their method it is not necessary to know the attitude of the investors because the “risk premium” is included in the price of the underlying asset. That is, the risk is reflected in the evolution of the price of the underlying asset (in this case, the stock) and the price of the option depends on the expected price of the asset at the maturity of that option. The method starts from the possibility that a risk-less portfolio can be formed, because from the first moment its final value is known. The investor who has an option knows at what price he could buy or sell the underlying asset at the expiration of the option.

The variables of the Black-Scholes formula are the time to maturity of the option, the current price of the stock option, the risk-free interest rate, the exercise price^{1} and the volatility of returns of the underlying. The last variable represents a challenge to value the option. Volatility refers to how much the stock’s price changes from one moment to another: the “more volatile” the return of an asset is, the less is known what the price and the returns will be in the future. The main problem is that if the option gives the right to buy and sell the stock at a future date, the volatility to consider is the one between the time of purchase and the time to expiration. Obviously, we cannot know the future and, therefore, we cannot know what the volatility of the stock’s retunr will be at the expiration of the option. It is the only unknown variable in the Black-Scholes formula.

There are many stock options underwritten on other underlying assets that are traded in organized markets and of which the market price is known. Although we cannot know the future volatility of a stock’s performance, we can know the market price of the stock options. If we know the market price of the call and put options, solving the Black-Scholes formula for volatility could infer the volatility expected by the investors during the lifetime of the option. The volatility of the yield of an asset that is congruent with the quoted, or market, price of an option over the asset is known as “implied volatility.” If we know the option price, time to maturity, exercise price and interest rate, we can obtain information about the investors’ opinion about the future volatility of the yields of the underlying stock of the option. It is considered an ex ante measure of the perceived risk of an asset.

Latané and Rendleman (1976) had the idea of calculating the volatility with the

Black-Scholes formula. Since then, numerous studies have been written in which the implied volatility has been used, especially as a parameter that provides market information. From being an equation to estimate the price of an option, the Black-Scholes model became a formula for calculating market expectations about future stock volatility.

The Volatility Indexes

In 1993, the CBOE introduced a volatility index, the VIX. Market indexes, such as the IPC in Mexico or the S&P500, provide information on the performance of shares in the market. They are representative indexes that indicate if at a particular moment the prices of shares rose or fell. The difference is that the VIX measures the expectation of volatility (uncertainty) of stock returns. The VIX is calculated from the implied volatility of the options on the S&P500 index. While the S&P500 is an index that measures the stock returns of the 500 largest and most tradable companies in the United States, the VIX represents the future volatility of those stocks that investors expect. Since the S&P500 is an index that represents market performance very well, the VIX accurately measures the expected volatility of that market ^{2}. Graph 1 shows the evolution of the VIX since 1986. Although introduced in 1993, the index was estimated since 1986 to cover the market crash of October 1987. The intention was to document the degree of market anxiety during the worst drop in the market since the Great Depression. This would give a framework of reference for measuring the magnitude of other cases of turbulence. Had the VIX existed when the stock market collapsed in October 1987, it would have reached its all-time high of 150 points. Since the appearance of the VIX in 1993, the highest level it has reached was in October and November of 2008, during the worst of the subprime mortgage crisis in the United States (80 points). At other times of turbulence, but less deep, the VIX tends to reach 30 or 40 points, while in normal times it is between 10 and 20 points.

The Purchase and Sale of Volatility

Being indexes, the VIX and VIMEX cannot be bought or sold in the financial markets; however, as with return indexes, there are derivatives and structured products that to trade in accordance with the expectations of the investors. In fact, one of the objectives in creating the VIX was to have a volatility indicator on which options and futures could be subscribed.

The CBOE offers options and futures on the VIX. How do futures and options on volatility work? For example, a long position (buyer) in a futures contract has gains when volatility increases. Suppose an investor suffers losses when market volatility increases; therefore, he wants to hedge against increases in volatility. The investor buys 100 futures contracts maturing in June 2017. The exercise price at the time of purchase is 13.0. The contract has a multiplier of 1,000 dollars, so the investor makes a hedge on 1,300,000 dollars (100 x 1,000 x 13). If the volatility index rises to 14.5, the investor will have a gain of 150,000 dollars (= 100 x 1,000 x (14.5 – 13.0). Conversely, if the index decreases to 12, the investor will lose 100,000 dollars (= 100 x 1,000 x (12.0 – 13.0). Graph 2 shows the gains or losses that the investor would have as the value of the VIX changes over the life of the contract. Futures contracts and other structured products subscribed on volatility indexes, allow investors to hedge their portfolios in times of uncertainty. With this type of strategies, the direction of market movements are not relevant because investors can hedge regardless if the market goes up or down.

Graph 2. Example of a transaction with a VIX futures contract.

Conclusions

Thanks to financial innovations, individuals, companies and governments increasingly meet their investment, financial and hedge needs. Prior to the introduction of volatility measurements, an investor could only hedge or make investment transactions based on expectations of rising or declining asset values. A producer of raw materials wins when the price of the product rises, but the consumer (a manufacturer who uses that raw material) loses. Likewise, most investors win when stock prices rise. However, prior to the arrival of structured financial instruments on volatility measures it was not possible to hedge or invest based on volatility expectations. Futures and options on volatility indexes allow transactions in which investors can make a profit in high or low volatility environments, depending on their risks and expectations.

References

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

Hull, J.C., 2015, Options, Futures and Other Derivates, 9th edition.

Latané, H.A. and Rendleman Jr., R. J., 1976, “Standard Deviations of Stock Price Ratios Implied in Option Prices,” The Journal of Finance, Vol. 31, No. 2.

Whaley, Robert E., 1993, “Derivatives on Market Volatility: Hedging Tools Long Overdue,” Journal of Derivatives 1, 71-84.

Whaley, Robert E., 2008, “Understanding VIX,” <http://ssrn.com/abstract=1296743>.

^{1} The selling or buying price that is agreed upon at the expiration of the option.

^{2} The index was proposed by Whaley (1993). For details about the calculation and the index history, go to Whaley (2008; <http://ssrn.com/abstract=1296743>).

## Can You Buy and Sell Volatility?

Administration Department, ITAMBy: Renata Herrerías

Today we cannot understand the world of investments without financial innovation. Organized markets and financial intermediaries constantly propose new alternatives to meet investment needs, risk management and financial resources management of governments, companies and individuals. In fact, intermediaries and financial markets exist, among other reasons, so that participants can redistribute their risk among those who are willing to take it.

Like technology, the evolution of the organized financial markets was relatively slow during the first 70 years of the 20th century. Until that time, it was only only possible to trade debt instruments (bonds), equity instruments (shares) and futures contracts subscribed mainly for agricultural raw materials. In 1972, the Chicago Board of Trade (or CBOT) launched the first financial futures contracts, which had underlying currencies setting the bid and ask exchange rate with the US dollar. A few years later, it began to trade interest rate futures. Also in the early 1970s, the Chicago Board Options Exchange (CBOE) began offering options on stock of 16 companies.

In Mexico, the organized derivatives market, as it is known today, was conceived in 1994 and materialized in 1998. Created by the Mexican Stock Exchange (BMV) and the S.D. Indeval, the Derivatives Exchange (MexDer) began operations in December 1998 with futures on US dollar.

Derivatives are financial instruments whose value and risk depend on another asset (underlying asset). For example, a call option is a contract that gives the investor the right to buy a certain asset at a certain price at a future date. By acquiring that right, the contract owner is willing to pay a premium in advance (call price). The price of the option depends on the time of expiration of the contract and the uncertainty about the evolution of the price of the underlying asset and interest rates. If the contract gives the investor the right to sell the underlying asset, then it is called a put option. The popularization of the use of derivative instruments and the search for options for hedging and investment mechanisms have resulted in the impressive development of markets and non-conventional financial instruments.

From a Price Model to a Volatility IndicatorIn 1997, the Nobel Prize in Economics was awarded to Robert Merton and Myron Scholes for their method of determining the valuation of a stock option (which they developed with Fisher Black, who died in 1995). The method they proposed gave way to the creation of new instruments, as well as new mechanisms to determine the economic value of many other assets such as insurance contracts, guarantees and flexibility in an investment project. Black and Scholes presented their options pricing formula (later called “Black-Scholes”) in an article in 1973.

Before the Black-Scholes formula, the main problem in determining the theoretical value of an option was to know the attitude of the buyer of the option toward the risks he/she would run. A “very” risk-averse investor would be willing to pay more to eliminate it than a “less” risk-averse investor. The most important contribution of Black and Scholes is that in their method it is not necessary to know the attitude of the investors because the “risk premium” is included in the price of the underlying asset. That is, the risk is reflected in the evolution of the price of the underlying asset (in this case, the stock) and the price of the option depends on the expected price of the asset at the maturity of that option. The method starts from the possibility that a risk-less portfolio can be formed, because from the first moment its final value is known. The investor who has an option knows at what price he could buy or sell the underlying asset at the expiration of the option.

The variables of the Black-Scholes formula are the time to maturity of the option, the current price of the stock option, the risk-free interest rate, the exercise price

^{1}and the volatility of returns of the underlying. The last variable represents a challenge to value the option. Volatility refers to how much the stock’s price changes from one moment to another: the “more volatile” the return of an asset is, the less is known what the price and the returns will be in the future. The main problem is that if the option gives the right to buy and sell the stock at a future date, the volatility to consider is the one between the time of purchase and the time to expiration. Obviously, we cannot know the future and, therefore, we cannot know what the volatility of the stock’s retunr will be at the expiration of the option. It is the only unknown variable in the Black-Scholes formula.There are many stock options underwritten on other underlying assets that are traded in organized markets and of which the market price is known. Although we cannot know the future volatility of a stock’s performance, we can know the market price of the stock options. If we know the market price of the call and put options, solving the Black-Scholes formula for volatility could infer the volatility expected by the investors during the lifetime of the option. The volatility of the yield of an asset that is congruent with the quoted, or market, price of an option over the asset is known as “implied volatility.” If we know the option price, time to maturity, exercise price and interest rate, we can obtain information about the investors’ opinion about the future volatility of the yields of the underlying stock of the option. It is considered an ex ante measure of the perceived risk of an asset.

Latané and Rendleman (1976) had the idea of calculating the volatility with the

Black-Scholes formula. Since then, numerous studies have been written in which the implied volatility has been used, especially as a parameter that provides market information. From being an equation to estimate the price of an option, the Black-Scholes model became a formula for calculating market expectations about future stock volatility.

The Volatility IndexesIn 1993, the CBOE introduced a volatility index, the VIX. Market indexes, such as the IPC in Mexico or the S&P500, provide information on the performance of shares in the market. They are representative indexes that indicate if at a particular moment the prices of shares rose or fell. The difference is that the VIX measures the expectation of volatility (uncertainty) of stock returns. The VIX is calculated from the implied volatility of the options on the S&P500 index. While the S&P500 is an index that measures the stock returns of the 500 largest and most tradable companies in the United States, the VIX represents the future volatility of those stocks that investors expect. Since the S&P500 is an index that represents market performance very well, the VIX accurately measures the expected volatility of that market

^{2}. Graph 1 shows the evolution of the VIX since 1986. Although introduced in 1993, the index was estimated since 1986 to cover the market crash of October 1987. The intention was to document the degree of market anxiety during the worst drop in the market since the Great Depression. This would give a framework of reference for measuring the magnitude of other cases of turbulence. Had the VIX existed when the stock market collapsed in October 1987, it would have reached its all-time high of 150 points. Since the appearance of the VIX in 1993, the highest level it has reached was in October and November of 2008, during the worst of the subprime mortgage crisis in the United States (80 points). At other times of turbulence, but less deep, the VIX tends to reach 30 or 40 points, while in normal times it is between 10 and 20 points.The Purchase and Sale of VolatilityBeing indexes, the VIX and VIMEX cannot be bought or sold in the financial markets; however, as with return indexes, there are derivatives and structured products that to trade in accordance with the expectations of the investors. In fact, one of the objectives in creating the VIX was to have a volatility indicator on which options and futures could be subscribed.

The CBOE offers options and futures on the VIX. How do futures and options on volatility work? For example, a long position (buyer) in a futures contract has gains when volatility increases. Suppose an investor suffers losses when market volatility increases; therefore, he wants to hedge against increases in volatility. The investor buys 100 futures contracts maturing in June 2017. The exercise price at the time of purchase is 13.0. The contract has a multiplier of 1,000 dollars, so the investor makes a hedge on 1,300,000 dollars (100 x 1,000 x 13). If the volatility index rises to 14.5, the investor will have a gain of 150,000 dollars (= 100 x 1,000 x (14.5 – 13.0). Conversely, if the index decreases to 12, the investor will lose 100,000 dollars (= 100 x 1,000 x (12.0 – 13.0). Graph 2 shows the gains or losses that the investor would have as the value of the VIX changes over the life of the contract. Futures contracts and other structured products subscribed on volatility indexes, allow investors to hedge their portfolios in times of uncertainty. With this type of strategies, the direction of market movements are not relevant because investors can hedge regardless if the market goes up or down.

Graph 2. Example of a transaction with a VIX futures contract.Conclusions

Thanks to financial innovations, individuals, companies and governments increasingly meet their investment, financial and hedge needs. Prior to the introduction of volatility measurements, an investor could only hedge or make investment transactions based on expectations of rising or declining asset values. A producer of raw materials wins when the price of the product rises, but the consumer (a manufacturer who uses that raw material) loses. Likewise, most investors win when stock prices rise. However, prior to the arrival of structured financial instruments on volatility measures it was not possible to hedge or invest based on volatility expectations. Futures and options on volatility indexes allow transactions in which investors can make a profit in high or low volatility environments, depending on their risks and expectations.

References^{1}The selling or buying price that is agreed upon at the expiration of the option.^{2}The index was proposed by Whaley (1993). For details about the calculation and the index history, go to Whaley (2008; <http://ssrn.com/abstract=1296743>).