PRICING THE AMERICAN PUT

By: Piet Nova

The Binomial Tree Model

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-dividend-paying 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 a = e(r–q )∆t

1 – 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