Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2010

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Implied Volatility

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Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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QuestionofDay

Question of the Day

What are some methods you could use to find the solution of $f (x) = c$ for $x$ where $f$ is a function that is so complicated that you cannot use elementary functions and operations to isolate $x$ ?

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Key Concepts

We estimate historical volatility by applying the standard deviation estimator from statistics to the observations $ln (S_{i} ∕ S_{i - 1})$ .
We deduce implied volatility by numerically solving the Black-Scholes formula for $σ$ .

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Vocabulary

Historical volatility of a security is the variance of the changes in the logarithm of the price of the underlying asset, obtained from past data.
Implied volatility of a security 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.

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Mathematical Ideas

Historical volatility

Estimates of historical volatility of security prices use statistical estimators, usually one of the estimators of variance. A main problem for historical volatility is to select the sample size, or window of observations, used to estimate $σ^{2}$ . Different time-windows usually give different volatility estimates. Furthermore, for a lot of customized “over the counter” derivatives, the necessary price data may not exist.

Another problem with historical volatility is that it assumes future market performance is the same as past market data. Although this is a natural scientific assumption, it does not take into account historical anomalies such as the October 1987 stock market drop, which may be unusual. That is, computing historical volatility has the usual statistical difficulty of how to handle outliers. The assumption that future market performance is the same as past performance also does not take into account underlying changes in the market such as economic conditions.

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

the number of observations $n + 1$
$S_{i}$ , $i = 0, 1, 2, 3, \dots, n$ is the security price at the end of the $i$ th interval,
$τ$ is the length of each of the time intervals (say in years),

and let

u_{i} = ln (S_{i}) - ln (S_{i - 1}) = ln (\frac{S_{i}}{S_{i - 1}})

for $i = 1, 2, 3, \dots$ be the increment of the logarithms of the security prices. We are modeling the security price as a Geometric Brownian Motion, so that $ln (S_{i}) - ln (S_{i - 1}) \sim N (r τ, σ^{2} τ)$ .

Since $S_{i} = S_{i - 1} e^{u_{i}}$ , $u_{i}$ is the continuously compounded return, (not annualized) in the $i$ th interval. Then the usual estimate $s$ , of the standard deviation of the $u_{i}$ ’s is

s = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(u_{i} - ū)}^{2}}

where $ū$ is the mean of the $u_{i}$ ’s. Sometimes it is more convenient to use the equivalent formula

s = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} u_{i}^{2} - \frac{1}{n (n - 1)} {(\sum_{i = 1}^{n} u_{i})}^{2}} .

We assume the security price varies as a Geometric Brownian Motion. That means that the logarithm of the security price is a Wiener process with some drift and on the period of time $τ$ , would have a variance $σ^{2} τ$ . Therefore, $s$ is an estimate of $σ \sqrt{t}$ . It follows that $σ$ can be estimated as

σ \approx \frac{s}{\sqrt{τ}} .

Choosing an appropriate value for $n$ is not obvious. Remember the variance expression for Geometric Brownian Motion is an increasing function of time. If we model security prices with Geometric Brownian Motion, then $σ$ 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. [2, page 215]

Economists and financial analysts often estimate historical volatility with more sophisticated statistical time series methods.

Implied Volatility

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

The Black-Scholes formula is complicated to “invert” to explicitly express $σ$ as a function of the other parameters. Therefore, we use numerical techniques to implicitly solve for $σ$ . A simple idea is to use the method of bisection search to find $σ$ .

Example.

Suppose the value of a call on a non-dividend paying security is $1.85$ when $S = 21$ , $K = 20$ , $r = 0.10$ , and $T - t = 0.25$ and $σ$ is unknown. We start by arbitrarily guessing $σ = 0.20$ . The Black-Scholes formula gives $C = 1.7647$ , which is too low. Since $C$ is a increasing function of $σ$ , this suggests we try a value of $σ = 0.30$ . This gives $C = 2.1010$ , too high, so we bisect the interval $[0.20, 0.30]$ and try $σ = 0.25$ . This value of $σ$ gives a value of $C = 1.9268$ , still too high. Bisect the interval $[0.20, 0.25]$ and try a value of $σ = 0.225$ , which yields $C = 1.8438$ , slightly too low. Try $σ = 0.2375$ , giving $C = 1.8849$ . Finally try $σ = 0.23125$ giving $C = 1.8642$ . To $2$ significant digits, the significance of the data, $σ = 0.23$ , with a predicted value of $C = 1.86$ .

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

f (σ, S, K, r, T - t) - C = 0,

so from an initial guess $σ_{0}$ , we form the Newton iterates

σ_{i + 1} = σ_{i} - f (σ_{i}) ∕ (d f (σ_{i}) ∕ d σ) .

This means one has to differentiate the Black-Scholes formula with respect to $σ$ . 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

\frac{d f}{d σ} = S \sqrt{T - t} Φ^{'} (d_{1}) exp (- r (T - t)) .

A natural way to do the iteration is with a computer program rather than 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, market expectations about risk. More sophisticated combinations and weighted averages combining estimates from several different derivative claims can be developed.

Sources

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.

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Problems to Work

Problems to Work for Understanding

Suppose that the observations on a security 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 security price volatility.
A call option on a non-dividend paying security has a market price of $2.50. The security price is $15, the exercise price is $13, the time to maturity is 3 months, and the risk-free interest rate is 5% per year. What is the implied volatility?

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Books

Reading Suggestion:

References

[1] Marco Allavenada and Peter Laurence. Quantitative Modeling of Derivative Securities. Chapman and Hall, 2000. HG 6024 A3A93 2000.

[2] John C. Hull. Options, Futures, and other Derivative Securities. Prentice-Hall, second edition edition, 1993. economics, finance, HG 6024 A3H85.

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Links

Outside Readings and Links:

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.
Analysis of asset allocation. A calculator to compute implied volatility using Black and Scholes. Submitted by Bashar Al-Salim, Dec. 2, 2003.
MindXpansion,a Tool for Option Traders. This 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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Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1

Email to Steve Dunbar, sdunbar1 at unl dot edu

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