Option valuation

Black–Scholes model
In the previous section, we got an idea of the constraints on option prices imposed
by ruling out the possibility of arbitrage. For more specific results on option prices,
one needs either a market or a model. Options that trade actively are valued in the
market; less actively traded options can be valued using a model. The most common
option valuation model is the Black–Scholes model.
The language and concepts with which option traders do business are borrowed
from the Black–Scholes model. Understanding how option markets work and how market participants’ probability beliefs are expressed through option prices therefore
requires some acquaintance with the model, even though neither traders nor academics
believe in its literal truth. It is easiest to understand how asset prices actually
behave, or how the markets believe they behave, through a comparison with this
benchmark.
Like any model, the Black–Scholes model rests on assumptions. The most important
is about how asset prices move over time: the model assumes that the asset
price is a geometric Brownian motion or diffusion process, meaning that it behaves
over time like a random walk with very tiny increments.
The Black–Scholes assumptions imply that a European call option can be replicated
with a continuously adjusted trading strategy involving positions in the underlying
asset and the risk-free bond. This, in turn, implies that the option can be valued
using risk-neutral valuation, that is, by taking the mathematical expectation of the
option payoff using the risk-neutral probability distribution.
The Black–Scholes model also assumes there are no taxes or transactions costs,
and that markets are continuously in session. Together with the assumptions about
the underlying asset price’s behavior over time, this implies that a portfolio, called
the delta hedge, containing the underlying asset and the risk-free bond can be
constructed and continuously adjusted over time so as to exactly mimic the changes
in value of a call option. Because the option can be perfectly and costlessly hedged, it
can be priced by risk-neutral pricing, that is, as though the unobservable equilibrium
expected return on the asset were equal to the observable forward premium.
These assumptions are collectively called the Black–Scholes model. The model
results in formulas for pricing plain-vanilla European options, which we will discuss
presently, and in a prescription for risk management, which we will address in more
detail below. The formulas for both calls and puts have the same six inputs or
arguments:
Ω The value of a call rises as the spot price of the underlying asset price rises (see
Figure 1.9). The opposite holds for puts.

The value of a call falls as the exercise price rises (see Figure 1.10). The opposite
holds for puts. For calls and puts, the effect of a rise in the exercise price is almost
identical to that of a fall in the underlying price.
1.55 1.60 1.65 1.70
0 Strike
0.02
0.04
0.06
0.08
0.1
Call value
Figure 1.10 Call value as a function of exercise price.
Ω The value of a call rises as the call’s time to maturity or tenor rises (see Figure
1.11).
30 90 180 360
0 Days
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Call value
Figure 1.11 Call value as a function of time to maturity.
Ω Call and put values rise with volatility, the degree to which the asset price is
expected to wander up or down from where it is now (see Figure 1.12).
Ω The call value rises with the domestic interest rate: since the call is a way to be
long the asset, its value must be higher when the money market rate – the
opportunity cost of being long the cash asset – rises. The opposite is true for put
options, since they are an alternative method of being short an asset (see Figure
1.13).

Ω The call value falls with the dividend yield of the asset, e.g. the coupon rate on a
bond, the dividend rate of an equity, or the foreign interest rate in the case of a
currency (see Figure 1.13). The reason is that the call owner foregoes this cash
income by being long the asset in the form of an option rather than the cash
asset. This penalty rises with the dividend yield. The opposite is true for put
options.
This summary describes the effect of variations in the inputs taken one at a time,
that is, holding the other inputs constant. As the graphs indicate, it is important to
keep in mind that there are important ‘cross-variation’ effects, that is, the effect of,
say, a change in volatility when an option is in-the-money may be different from
when it is out-of-the-money. Similarly, the effect of a declining time to maturity may
be different when the interest rates are high from the effect when rates are low.

As a rough approximation, we can take the Black–Scholes formula as a reasonable
approximation to the market prices of plain vanilla options. In other words, while we
have undertaken to describe how the formula values change in response to the
formula inputs, we have also sketched how market prices of options vary with
changes in maturity and market conditions.
Implied volatility
Volatility is one of the six variables in the Black–Scholes option pricing formulas, but
it is the only one which is not part of the contract or an observable market price.
Implied volatility is the number obtained by solving one of the Black–Scholes
formulas for the volatility, given the numerical values of the other variables. Let us
look at implied volatility purely as a number for a moment, without worrying about
its meaning.
Denote the Black–Scholes formula for the value of a call by v(St ,t,X,T,p,r,r*).
Implied volatility is found from the equation C(X,t,T )óv(St ,t,X,T,p,r,r*), which sets
the observed market price of an option on the left-hand side equal to the Black–
Scholes value on the right-hand side. To calculate, one must find the ‘root’ p of this
equation. This is relatively straightforward in a spreadsheet program and there is a
great deal of commercial software that performs this as well as other option-related
calculations. Figure 1.14 shows that except for deep in- or out-of-the-money options
with very low volatility, the Black–Scholes value of an option is strictly increasing in
implied volatility.
There are several other types of volatility:
Ω Historical volatility is a measure of the standard deviation of changes in an
asset price over some period in the past. Typically, it is the standard deviation of
daily percent changes in the asset price over several months or years. Occasionally,
historical volatility is calculated over very short intervals in the very recent
past: the standard deviation of minute-to-minute or second-to-second changes
over the course of a trading day is called intraday volatility.
Ω Expected volatility: an estimate or guess at the standard deviation of daily
percent changes in the asset price for, say, the next year. Implied volatility is
often interpreted as the market’s expected volatility.
The interpretation of volatility is based on the Black–Scholes model assumption
that the asset price follows a random walk. If the model holds true precisely, then
implied volatility is the market’s expected volatility over the life of the option from
which it is calculated. If the model does not hold true precisely then implied volatility
is closely related to expected volatility, but may differ from it somewhat.
Volatility, whether implied or historical, has several time dimensions that can be
a source of confusion:
Ω Standard deviations of percent changes over what time intervals? Usually, closeover-
close daily percent changes are squared and averaged to calculate the
standard deviation, but minute-to-minute changes can also be used, for example,
in measuring intraday volatility.
Ω Percent changes averaged during what period? This varies: it can be the past day,
month, year or week.
Ω Volatility at what per-period rate? The units of both historical and implied volatility
are generally percent per year. In risk management volatility may be scaled to the one-day or ten-day horizon of a value-at-risk calculation. To convert an annual
volatility to a volatility per some shorter period – a month or a day – multiply by
the square root of the fraction of a year involved. This is called the square-rootof-
time rule.