Department of Mathematics 203 Avery Hall University of Nebraska Lincoln Lincoln, NE 68588-0323 402-472-3731 (voice) 402-472-8466 (fax)

Implied Volatility

Key Concepts

1. Historical volatility can be estimated by using the standard deviation estimator from statistics to the observations ln(S_i/S_i-1).
2. Implied volatility can be deduced by numerically solving the Black Scholes formula for .

Vocabulary

1. Historical volatility is defined as an estimate of the variance of the logarithm of the price of the underlying asset, obtained from past data.
2. Implied volatility is the numerical value of the volatility parameter that makes the market price of an option equal to the value from the Black-Scholes formula.

Mathematical Ideas

This section is adapted from: Quantitative modeling of Derivative Securities by Marco Avellaneda, and Peter Laurence, Chapman and Hall, Boca Raton, 2000, page 66,; and Options, Futures, and other Derivative Securities second edition, by John C. Hull, Prentice Hall, 1993, pages 229-230.

Historical volatility

Historical volatility estimates require the use of appropriate statistical estimators, usually one of the estimators of variance. One of the main problems in this regard is to select the sample size, or window of observations that will be used to estimate . Different time-windows tend to give different volatility estimates.

Another problem with using historical volatility is that it assumes that future market performance will be reflected in past market data. Although this is a natural scientific assumption, it does not take into account historical anomalies, or changes such as the October 1987 stock market jump.

To estimate the volatility of a stock price empirically, the stock price is observed at regular intervals, such as every day, every week, or every month. Define:

1. the number of observations n + 1
2. S_i, i = 0, 1, 2, 3,...n is the stock price at the end of the ith interval,
3. is the length of each of the time intervals (in years)

and let

for i = 1, 2, 3,... be the increment of the logarithms of the stock prices. We are assuming that the stock price acts as a Geometric Brownian Motion, so that ln(S_i) - ln(S_i-1) ~ N(r,²).

Since S_i = S_i-1e^u_i, u_i is the continuously compounded return, (not annualized) in the ith interval. Then the usual estimate s, of the standard deviation of the u_i’s is given by

where is the mean of the u_i’s. Sometimes it is more convenient to use the equivalent formula

As usual, we assume the stock price varies as a geometric Brownian motion, That means that the logarithm of the stock price is a Brownian motion with some drift and on the period of time , would have a variance ². Therefore, s is an estimate of . It follows that can be estimated as

Choosing an appropriate value for n is not easy. Because does change over time, and data that are too old may not be relevant for the present or the future. A compromise that seems to work reasonably well is to use closing prices from daily data over the most recent 90 to 180 days. Empirical research indicates that only trading days should be used, so days when the exchange is closed should be ignored for the purposes of the volatility calculation.

Implied Volatility

The implied volatility is the parameter in the Black-Scholes formula that would give the option price that is observed in the market, all other parameters being known.

Almost obviously, it is not possible to “invert” the Black-Scholes formula to explicitly express as a function of the other parameters. Therefore, one is reduced to numerical techniques to implicitly solve for . A simple idea is to use the method of bisection search to find . Here is an illustration: Suppose the value of a call on a non-dividend paying stock is 1.875 when S = 21, K = 20, r = 0.1, and T - t = 0.25. We could start by (more or less arbitrarily) guessing = 0.20, which gives a value of C = 1.76, which is too low. Since C is a increasing function of , this suggests we try a value of = 0.30. This gives C = 2.10, too high, so we try = 0.25, which gives a value of C = 1.92 , and then we try a value of = 0.225, which yields C = and then we try = 0.235 which is just about right. Therefore, we take = 0.235 or 23.5% per annum.

A faster procedure is to use the Newton method, which is iterative. Essentially we are trying to solve

, so from an initial guess ₀, we from the Newton iterates

This means one has to differentiate the Black-Scholes formula with respect to . This derivative is one of the “Greeks” known as vega which we will look at more extensively in the next section. A formula for vega for a European call option is

but still, one would probably program the iteration, with a computer rather than do it by hand.

Implied volatility is a “forward-looking” estimation technique, in contrast to the “backward-looking” historical volatility. That is, it incorporates the market’s expectations about the prices of securities and their derivatives, or more concisely, about risk. More sophisticated combinations and weighted averages combining estimates from several different derivative claims can be developed.

Problems to Work for Understanding

1. Suppose that the observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows: 30.25, 32, 31.125, 30.25, 30.375, 30.625, 33, 32.875, 33, 33.5, 33.5 33.75, 33.5, 33.25. Estimate the stock price volatility.

   Solution

2. A call option on a non-dividend paying stock has a market price of $2.50. The stock price is $15, the exercise price is $13, the ime to maturity is 3 months, and the risk-free interest rate is 5% per annum. What is the implied volatility?

   Solution

Reading Suggestion:

1. Quantitative modeling of Derivative Securities by Marco Avellaneda, and Peter Laurence, Chapman and Hall, Boca Raton, 2000, page 66,;
2. Options, Futures, and other Derivative Securities second edition, by John C. Hull, Prentice Hall, 1993, pages 229-230.

Outside Readings and Links:

1. Peter Hoadley, Options Strategy Analysis Tools has a Historic Volatility Calculator that calculates and graphs historic volatility using historical price data retrieved from Yahoo.com. Submitted by Bashar Al-Salim, Dec. 2, 2003.
2. Analysis of asset allocation A calculator to compute implied volatility using Black and Scholes. Submitted by Bashar Al-Salim, Dec. 2, 2003.
3. Jerry Marlow Free tutorials to compute implied volatility. submitted by Bashar Al-Salim, Dec. 2, 2003.
4. “Assessing the latest models developments for Stochastic (implied) Volatility”, Deutsche Bank A presentation by Deutsche Bank. (It is recommended to download the PDF file, since the link is very slow.) Submitted by Mansour Abdoli, December 3, 2003.
5. Federal Reserve Bank of New York Implied volatilityfor foreign currency options submitted by Mansour Abdoli, December 3, 2003.
6. MindXpansion,a Tool for Option TradersThis option calculator packages an enormous amount of functionality onto one screen, calculating implied volatility or historical volatility with Midas Touch. Submitted by Chun Fan, Dec. 3, 2003.

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