Pricing the American Put
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Transcript Pricing the American Put
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