PRICING THE AMERICAN PUT
The Binomial Tree Model
By: Piet Nova
THE PRICING OF OPTIONS
Important problem in financial markets today
 Computation of a particular integral
 Methods of valuation

 Analytical
 Numerical
Integration
 Partial Differential Equation (Black-Scholes)

However…
 Multiple
dimensions cause PDEs and numerical
integrals to become complicated and intractable
OTHER PRICING METHODS

Binomial Trees

Trinomial Trees

Monte Carlo Simulation
WHAT IS AN OPTION?

An option is a financial contract between a
seller (writer) and a buyer (holder).

Basic Components:
 Option
Price
 Value of Underlying Asset (Stock Price, S0)
 Strike Price (K)
 Time to Maturity (T)
WHAT IS AN OPTION?

Other components that determine price of
option:
 Volatility
of Asset (σ)
 Dividends Paid (q)
 Riskless Interest Rate (r)

Writer Profit vs. Holder Profit
 Option
Price
 Put Option: K – ST
EUROPEAN OPTIONS VS. AMERICAN OPTIONS

European Options
 May
only exercise at expiration date
 Black-Scholes

American Options
 May
be exercised at any time before maturity
 Majority of options traded on exchanges
 A choice exists: exercise now or wait?
WHAT ABOUT THE AMERICAN CALL?

In theory, an American call on a non-dividendpaying stock should never be exercised before
maturity.
 When
out of the money
 When in the money
 Extrinsic
or time value
Thus, pricing the American call is essentially
the same as pricing a European call.
 Exceptions

THE AMERICAN PUT

Optimal to exercise early if it is sufficiently deep
in the money.
 Extreme

situation: K=$10, S0=$0.0001
When is it optimal to exercise?
 In
general, when S0 decreases, r increases, and
volatility decreases, early exercise becomes more
attractive.
 When exercise is optimal, the value of the option
becomes the intrinsic or exercise value
BINOMIAL TREES

A diagram representing different possible paths
that might be followed by a stock price over the
life of an option.

Assumes stock price follows a random walk.
 In
each time step, stock price has a certain
probability of moving up by a certain percentage
and a certain probability of moving down by a
certain percentage.
RISK-NEUTRAL VALUATION PRINCIPLE

Risk-Neutral Valuation Principle: An option can
be valued on the assumption that the world is
risk neutral.
 Assume
that the expected return from all traded
assets is the risk-free interest rate r.
 Value payoffs from the option by calculating their
expected values and discounting at the risk-free
interest rate r.

This principle underlies the way binomial trees
are used.
RISK-NEUTRAL VALUATION PRINCIPLE

This principle leads to the calculation of the
following crucial aspects of the binomial tree:
u
= eσ*sqrt(∆t) (amplitude of up movement)
 d = e-σ*sqrt(∆t) (amplitude of down movement)
 p = (a – d) / (u – d) (probability of up movement)
 Where
1
a = e(r–q )∆t
– p = (probability of down movement)
GENERATING THE TREE
At T=0, ST is known. This is the “root” of the
tree.
 At T=1∆t, the first step, there are two possible
asset prices:

 S0u

and S0d
At T=2∆t, there are three possible asset prices:
 S0u2,

S0, and S0d2
And so on. In general, at T=i∆t, there are i+1
asset prices.
GENERATING THE TREE

To generate each node on the tree:
 S0uj d(i–j),
j=0, 1, …, i
 Where T=i∆t is time of maturity (final node)

Note u = 1/d
 S0u2d
= S0u
An up movement followed by a down movement
will result in no change in price.
 The same goes for a down followed by an up.

PRICING THE AMERICAN PUT
Once every node on the tree has an asset value,
the pricing of the option may begin.
 This is done by starting at the end of the tree and
working backwards towards T=0.
 First, the option prices at the final nodes are
calculated as max(K – ST, 0).
 Next, the option prices of the penultimate nodes
are calculated from the option prices of the final
nodes:

Suppose penultimate node is S
 (p * Su + (1 – p) * Sd)e-r∆t

CHECKING FOR EARLY EXERCISE
The reason why binomial pricing methods are
commonly used to price the American put.
 Once the option prices for these nodes are
calculated, we must then check if the exercise
price exceeds the calculated option price.
 If so, the option should be exercised and the
correct value for the option at this node is the
exercise price.
 This check must be carried out for every node
except the final nodes.

CONTINUING TO PRICE THE NODES
Option prices at earlier nodes are calculated in
a similar way.
 Working back through the tree, the value of the
option at the initial node will be obtained.
 This is our numerical estimate for the option’s
current value.
 In practice:

 Smaller
∆t value
 More nodes
BINOMIAL TREES IN MATHEMATICA, EXCEL, AND R
PROBLEMS WITH THE BINOMIAL METHOD
Only factor treated as unknown is the price of the
underlying asset.
 Other determining factors are treated as
constants.

Interest Rates
 Dividends
 Volatility


Stochastic factors cannot be computed because
the number of nodes required grows exponentially
with the number of factors.
NEXT TIME…

Monte Carlo Implementation
 Least-Squares
Approach
 Exercise Boundary Parameterization Approach

Measures of accuracy