Mathematical Modeling in Finance with Stochastic Processes

254 CHAPTER 7. THE BLACK-SCHOLES MODEL

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

s=

√√

√

√^1

n− 1

∑n

i=1

u^2 i−

1

n(n−1)

( n

∑

i=1

ui

) 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 timeτ, would have a varianceσ^2 τ. Therefore,
sis an estimate ofσ

√

t. It follows thatσcan be estimated as

σ≈

s

√

τ

.

Choosing an appropriate value fornis not obvious. Remember the vari-
ance expression for Geometric Brownian Motion is an increasing function of
time. If we model security prices with Geometric Brownian Motion, then
σdoes 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. [22, page 215]
Economists and financial analysts often estimate historical volatility with
more sophisticated statistical time series methods.

Implied Volatility

Theimplied volatility is the parameterσ 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 ex-
pressσas a function of the other parameters. Therefore, we use numerical
techniques to implicitly solve forσ. A simple idea is to use the method of
bisection search to findσ.

Example.Suppose the value of a call on a non-dividend paying security is
1 .85 whenS= 21,K= 20,r= 0.10, andT−t= 0.25 andσis unknown.
We start by arbitrarily guessingσ= 0.20. The Black-Scholes formula gives