Notes on Option Valuation

1.

The Binomial Option Pricing Model

2.

How to Choose the Step Sizes

3.

The Black-Scholes Formula

4.

Monte Carlo Simulation

5.

Sensitivities for Option Value

6.

Common Types of Exotic Options

7.

Options on Futures

The Binomial Option Pricing Model

spot price of underlying asset

striking price of option

annualized continuously
compounded interest rate

annualized continuously
compounded yield from income
component

length of one period (in years)

With reinvestment of income component, one unit at

beginning of period grows to e y h units at end of
period.
equity

dividend yield

commodity

convenience yield

currency

foreign interest rate

Over each period,

uS
S
dS
For a call expiring in one period,

Cu

max [ u S - K, 0 ]

Cd

max [ d S - K, 0 ]

For a portfolio with units of the underlying asset

and B dollars in bonds,

e y h u S er h B
SB

e y h d S er h B

Choose and B so that

eyh u S er h B Cu
eyh d S er h B Cd
Solving these equations gives

Cu C d
yh
e (u d ) S
B

u Cd d Cu
e r h (u d )

With this choice of and B , the value of a European

call must be

C S B
Otherwise, there would be an arbitrage opportunity.

Writing and B in terms of Cu and Cd gives the

fundamental valuation equation

C [ p Cu ( 1 p ) Cd ] / e r h
where

(r y ) h

u d

The parameter p is the probability of an upward

move that would make the expected rate of return on
the underlying asset equal to the interest rate.

p e y h u (1 p ) e y h d e r h
It is the risk-neutral probability of an upward move,
not the actual probability.
For an American call, the current value must be

C max [ S K , S B ] .

The value of the option depended only on

Spot price of underlying asset
Striking price
Volatility (through u and d)
Interest rate
Income yield
Time to expiration

It did not depend on

Forecasting direction of future moves
Probabilities of upward and downward moves
Beta of the underlying asset
Investors attitudes toward risk

For a call with two periods until expiration,

u2 S
uS
S

udS
dS
d2 S
Cuu = max [u2 S K , 0]
Cu

C
Cd

Cud = Cdu= max [u d S K , 0]

Cdd = max [d2 S K , 0]

To replicate a multiperiod call, we need a dynamic

portfolio. The portfolio's composition will be different
at each node on the tree.
A call with any number of periods until expiration can
be valued by working backward from the end one step
at a time. Each step provides the information needed
for the next step.

For a European call,

Cu [ p Cu u ( 1 p ) Cu d ] / e r h
u

Cu u Cu d
e yh (u d ) u S

Cd [ p Cd u ( 1 p ) Cd d ] / e r h
d

Cd u Cd d
e yh (u d ) d S

C [ p Cu ( 1 p ) C d ] / e r h

Cu C d
e yh (u d ) S

By substitution we could write C as

C p 2 e 2 r h Cu u p (1 p ) e 2 r h Cu d
(1 p ) p e 2 r h Cd u (1 p )2 e 2 r h Cd d
This says that we could value the European call by
calculating the discounted payoff along each path,
weighting each result by the risk-neutral probability of
that path occurring, and then summing across all of
the paths.

For an American call,

Cu max u S K , [ p Cu u ( 1 p ) Cu d ] / e r h
Cd max d S K , [ p Cd u ( 1 p ) Cd d ] / e r h

C max S K , [ p Cu ( 1 p ) Cd ] / e r h

The value of h is determined by dividing the time to

expiration T by the number of periods chosen for the
calculation.
The particular payoff of a call option played no special
role. The same arguments apply to other derivatives
with payoffs that depend only on the value of the
underlying asset (but not on its previous path).
If V is the value of a general derivative receiving
periodic payouts of Xu or Xd , then the fundamental
equation becomes

V [ p ( Vu X u ) ( 1 p ) ( Vd X d ) ] / e r h

Creating a Synthetic Call with a Dynamic Portfolio of

Stock and Bonds

A Three-Period Example

S = 80

u = 1.5

d = .5

K = 80

h = 1

n = 3

er h = 1.1

ey h = 1

e(r - y) h = 1.1

e( r y )h d
u d

1.1 .5
1.5 .5

.6

Paths for Stock and Option Values

270
180
120
80

90
60

40

30
20
10

190
107.27
(1.00)
60.46
(.848)
34.07
(.719)

10
5.45
(.167)

2.97
(.136)

0
0
0.00
0

Step I - Stock Currently at 80

1)

Take 34.065 and invest it in a portfolio containing

= .719 shares of stock

2)

Finance the remainder by borrowing

.719 (80) 34.065 = 23.455

Step II - Stock Goes Up to 120

1)

Buy .848 - .719 = .129 more shares for a total cost

of
.129 (120) = 15.480

2)

Borrow to pay the bill. Total borrowing is now

23.455 (1.1) + 15.480 = 41.281

Step III - Stock Goes Down to 60

1)

Sell .848 - .167 = .681 shares, taking in

.681 (60) = 40.860

2)

Use this to repay part of the borrowing. This

reduces the total borrowing to
41.281 (1.1) - 40.860 = 4.549

Step IV u - Stock Goes Up to 90

1)

The shares owned are now worth

.167 (90) = 15

2)

Total borrowing is
4.549 (1.1) = 5

3)

Total value of the portfolio is 10, which is exactly

the value of the call.

Step IV d - Stock Goes Down to 30

1)

The shares owned are now worth

.167 (30) = 5

2)

Total borrowing is
4.549 (1.1) = 5

3)

Total value of the portfolio is 0, which is exactly the

value of the call.

Long stock
Short stock
(less than one share) (less than one share)
+
+
+
+
Long
Short
Long
Short
bonds
bonds
bonds
bonds
(lending) (borrowing) (lending) (borrowing)
Long
Short
Buy
stock
stock
Sell
stock
(one
Long
Long
(one
stock
and
share)
one
one
share)
and
sell
+
call
put
+
buy
bonds
Long
Long
bonds
one put
one call
Long
Sell
stock
stock
(one
and
share)
buy
+
bonds Short
one call

Short
one
put

Short
one
call

Short
stock
(one
share)
+
Short
one put

Buy
stock
and
sell
bonds

As stock price falls

As stock price rises

STOCK-BOND PORTFOLIOS
EQUIVALENT TO OPTIONS

Futures contracts can be used as a substitute for holding

the underlying asset. Under the assumptions, the
proper futures price for a contract with time t until
delivery is
F

S e( r y ) t

Over each period, holding one unit of the underlying

asset with reinvestment of income is equivalent to
holding er(t - h) eyt futures contracts and placing the
amount S in a fixed-income account. At the end of the
period,
e r ( t h ) e y t (u S e( r y )( t h ) S e( r y ) t ) S e r h

u S ey h

e r ( t h ) e y t (d S e ( r y ) ( t h ) S e ( r y ) t ) S e r h

d S ey h

Instead of holding units of the underlying asset and

the amount B in bonds, one would hold
number of futures

= eyt r(t - h)

amount in bonds = S + B

How to Choose the Step Sizes

To get good results, h should be small and u and d

should reflect the volatility of the underlying asset, .
The parameter is the annualized standard deviation
of the continuously compounded rate of return of the
underlying asset.

Three different methods for specifying u and d may

be used. When h is very small, they all produce
nearly identical results, but each has some advantages
and disadvantages.
A. The method used in the software package provided
with the text is

u e

d e

e h
h
e h

(r y) h

p e
e

This method is easiest for sensitivity calculations, but

h must be small enough to have

e( r y ) h e

In the software, the term e ( r y ) h is referred to as

the growth factor per step.

B.

Another method is

u e (r y ) h
d e (r y ) h

h
1 e
p
h
e
e

Here sensitivity calculations are more awkward, but

there is no restriction on inputs.

C. A third possibility is

u e

(r y) h

d e

(r y) h

2
e

2
e

h e

h e

1
2

This method is similar to B, but with the additional

advantage that p is 1/2 and the relation between
the tree and standard Monte Carlo simulation
becomes clearer.

A Useful Generalization

With all three methods, r and y can be functions of

time, taking on different values in each period.
This will allow for a term structure of interest rates
and a term structure of income yields.
The proper values to use for r and y for any
particular period are the annualized forward rates for
that period expressed in decimal form.
Methods B and C can also accommodate discrete
income payments, where y is extremely large in a few
short periods and zero otherwise.
The software package provided with the text requires
that r and y be constant.
A term structure of volatility can be included by letting
h correspond to a different length of calendar time in
each period. The values of u and d remain constant
so that the branches of the tree will continue to
recombine.

The Black-Scholes Formula

For a European call, as h becomes small the results
produced by the binomial tree with all three methods
for choosing u and d converge to the Black-Scholes
formula,

C Se

yT

N ( d1 ) K e

rT

N ( d2 ) ,

where N is the standard normal distribution function

and

d1

log ( S e yT / K e rT )

d 2 d1

1
2

2
T

The fundamental valuation equation linking beginning

and end-of-period values becomes the Black-Scholes
partial differential equation
1
2

2 S 2 CSS (r y ) S CS CT r C

0 ,

where the subscripts indicate derivatives of C with

respect to S and T .
By using the put-call parity relation,

C P S e yT K er T ,
and basic properties of the standard normal
distribution function, we can find the Black-Scholes
formula for a European put,

P K e r T N( d2 ) S e y T N( d1)

Analytic formulas are also available for some exotic

options.

Monte Carlo Simulation

The payoffs of some derivatives depend on the entire

path followed by the price of the underlying asset.
One example would be a security that pays on the
expiration date the maximum price achieved by the
underlying asset during the life of the contract. For
this derivative, a binomial tree cannot be used in the
usual way because the value at any node will depend
on the previous path followed in reaching that node.
Securities like this can still be valued by calculating the
discounted payoff along each path, weighting each
result by its risk-neutral probability of that path
occurring, and summing across all of the paths. The
recursive procedure we used earlier provided a
powerful way to effectively evaluate all of the paths,
but it cannot be used here because of the pathdependent nature of the payoffs. Without this help,
evaluating every path individually would become
prohibitively time-consuming.

The solution is to limit the scenarios considered to a

randomly chosen subset of the complete possibilities.
For example, using Method C of choosing u and d ,
we could construct a random path by letting the price
moves over each period be

S S e

(r y )h

2
e

e
h

~
hb

~
where b is a binomial random variable taking on the
value +1 if a coin flip is heads and - 1 if it is tails. If
the time to maturity T is divided into n periods, then
constructing each path would require n coin flips. By
building and using a sufficient number of paths, all of
which would have the same risk-neutral probability of
occurring, we can reach a reasonable estimate of the
derivative value.
Since there is now no advantage in having paths that
recombine, the binomial random variable is usually
replaced with a normal random variable and the
middle term becomes e

2 h / 2

. We then have

S S e ( r y ) h [e

h/2

] e

~
hn

~ is a normal random variable with a mean of

where n
zero and a standard deviation of one.
This description can be generalized in several ways.
The periods do not need to be of equal length, and r ,
y , and can be different in each period. In fact, y
and can also depend on current and past values of
the price of the underlying asset.
Monte Carlo simulation is well-suited for derivatives
with path-dependent payoffs, but it cannot handle
securities with American features. When making
recursive calculations on a binomial tree, we knew at
each node the value contributed by all of the future
possibilities that could occur, so it was easy to decide
about early exercise. With Monte Carlo simulation, we
would know at each point the value of only one of the
future possibilities.

A number of techniques are available to improve the

efficiency of a Monte Carlo simulation. One of the
simplest is the antithetic variable method. For each
random path constructed, a second path is built using
the same random drawings with their signs reversed.

Sensitivities for Option Value

It is often helpful to know how options will respond to

small changes in the inputs. These sensitivies are
normally defined as the derivatives with respect to the
inputs. The most widely used sensitivities are referred
to by Greek letters.

delta

value

asset price

gamma

delta

asset price

theta

value

time

kappa

value

volatility

rho

value

interest rate

Kappa is also known as vega.

With an analytic formula, the derivatives can be
calculated explicitly. For example, with the BlackScholes formula the deltas are

delta of call

= e y T N(d1 )

delta of put = e y T N( d1 )

With the binomial model, using method A for choosing

u and d , the sensitivities for calls are usually
calculated as
delta

gamma

theta

Cu Cd
e y h (u d ) S

u d
(u d ) S
Cu d C
2h

Rho and kappa (or vega) are calculated by running the

model a second time with a small change in the
corresponding input. Similar procedures are used for
other securities and when u and d are defined
differently.

A slightly different way of calculating delta and gamma

is used in the text and software,

Cu Cd
(u d ) S

delta

gamma

u d
=
(u 2 d 2 ) S / 2

With Monte Carlo simulation, all of the sensitivities are

calculated by running the process again with a small
change in the corresponding input. The same set of
random drawings is used in both runs.

Common Types of Exotic Options

1.

Barrier Options

2.

Asian Options

3.

Binary Options

4.

Lookback Options

5.

Exchange Options

6.

Compound Options

Barrier Options

Terms of option change when the underlying asset

price reaches a designated level called the barrier.
1)

What happens at the barrier?

knock-out option

option is cancelled

knock-in option

option is activated

mandatory exercise
option

option is exercised

2) Where is the barrier in relation to the current

asset price?

down-and-out option

option is cancelled at a
barrier below the
current asset price

up-and-out option

option is cancelled at a
barrier above the
current asset price

down-and-in option

option is activated at a
barrier below the
current asset price

up-and-in option

option is activated at a
barrier above the
current asset price

mandatory exercise call

option is exercised at a
barrier above the
current asset price

mandatory exercise put

option is exercised at a
barrier below the
current asset price

In some variations, the terms of the option change

only if the underlying asset price remains above (or
below) the barrier for a specified period of time.

Asian Options

Payoff depends on average price of underlying asset

over some period.

Upon exercise, an average price call pays

average price striking price
and an average price put pays
striking price average price

Terms will specify

1)

period over which average is calculated

2)

frequency of observations used in calculating

average

Binary Options

1)

2)

3)

Asset-or-nothing call
S* K
S* K

Payoff at expiration:

S*
0

Current value:

S e y T N(d1 )

if
if

Asset-or-nothing put
S* K
S* K

Payoff at expiration:

S*
0

Current value:

S e y T N( d1 )

if
if

Cash-or-nothing call
S* K
S* K

Payoff at expiration:

K
0

Current value:

K e r T N(d 2 )

if
if

4)

Cash-or-nothing put
S* K
S* K

Payoff at expiration:

K
0

Current value:

K e r T N( d 2 )

if
if

The analytic formulas apply at any time before the

expiration date.

Ordinary options are portfolios of binary options.

ordinary call

(asset-or-nothing call)
(cash-or-nothing call)

ordinary put

(cash-or-nothing put)
(asset-or-nothing put)

Lookback Options

Payoff depends on maximum or minimum price of

underlying asset over some period.

Upon exercise, a lookback call pays

final asset price minimum price
and a lookback put pays
maximum price final asset price

Terms will specify

1)

period over which maximum or minimum is

calculated

2)

frequency of the observations used in calculating

maximum or minimum

Exchange Options

An exchange option gives its holder the right to

exchange one asset for another. The current value of
a European option to exchange B for A is

C S A e y A T N ( x1 ) S B e y B T N ( x 2 )
where

x1

log (S A / SB ) ( y B y A

x 2 x1

2 T
2 T

2 A2 2 A B B2
and is the correlation between A and B .

1 2
)T
2

A European exchange option can be evaluated with

the Black-Scholes formula by replacing K with SB , r
with yB , and with .
An ordinary option is a special case of an exchange
option where B is a zero coupon bond with a principal
of K and a time to maturity of T .

Compound Options

A compound option is an option on an option.

Both the underlying option and the compound option
may be either a call or a put.
The terms of the contract specify the expiration dates
and strike prices of both options.
An analytic formula is available for European
compound options, but it is quite complicated.
The software package includes European compound
options.

Options on Futures

A futures option pays upon exercise the difference

between the futures price and the striking price.
A futures price is not an asset price and does not
behave in the same way as an asset price. Valuing
options on futures is fundamentally different from
valuing options on assets.
If F is the futures price of a contract with time t until
delivery, then with our assumptions
F S e(r y )t

Given the movement in the price of the underlying,

over each period
u S e ( r y )( t h ) e ( r y ) h u F

F
d S e ( r y )( t h ) e ( r y ) h d F

A portfolio with futures contracts and B dollars in

bonds requires a current investment of only B dollars.
This is because the futures position can be established
with no initial cost. The value of the portfolio at the
end of the period will be

(e ( r y ) h u F F ) e r h B
B

(e ( r y ) h d F F ) e r h B
The contribution of each futures contract is simply the
change in the futures price over the period. We want
the end-of-period values of the portfolio to match the
end-of-period values for the option, so

(e ( r y ) h u F F ) e r h B Cu
(e ( r y ) h d F F ) e r h B Cd
Solving these equation gives

Cu Cd
e ( r y ) h (u d ) F

(e ( r y ) h d ) Cu (u e ( r y ) h ) Cd
B

(
u

d
)

er h

With this choice of and B , the beginning-of-period

value of the call must be the same as the beginning-ofperiod value of the portfolio. The value of the
portfolio is B , so

C [ p Cu (1 p) Cd ] / e r h ,
where

e(r y )h d
.
p
ud
For an American call, we would have

C max F K , [ p Cu (1 p) Cd ] e r h

At this point, we could use any of the three methods

for choosing u and d . With our assumptions, the
volatility of a properly priced futures is the same as

that of the underlying asset, so the same applies to

both.
Although all three methods converge to the same
conclusions when h becomes small, there is a special
advantage in using Methods B or C for a futures
option. The benefit is that y does not have to be
specified separately as an input. All of the relevant
information is contained in the futures price. For
example, if Method B is used, with

u e( r y ) h

d e( r y ) h

then we have

e (r y )h u F e

e (r y )h d F e

and

1 e h
p h
e
e

The calculations made for a futures option using

Method B are identical to those for an option on an

asset when F replaces S , r replaces y , and Method

A is used to specify u and d . This is the procedure
used in the software.
For European options, as h becomes small the results
produced by the binomial tree converge to the Black
formulas for calls and puts on futures,

C F e r T N(a1 ) K e r T N(a2 )
P K e r T N( a2 ) F e r T N( a1 ) ,
where N is again the standard normal distribution
function and

a1

log (F / K )

1 2
T
2

a2 a1 T

Since pricing a futures option is formally identical to

pricing an option on an asset with an income yield
equal to the interest rate, both American calls and

American puts will be worth more than their European

versions.