Pricing The American Put

1. Pricing The American Put The Binomial Tree Model By: Piet Nova

2. 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 The pricing of options

3. Binomial Trees Trinomial Trees Monte Carlo Simulation Other pricing methods

4. 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?

5. 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 What is an option?

6. 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? European options vs. American options

7. 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 What about the American call?

8. 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 The American put

9. 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. Binomial Trees

10. 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

11. 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) Risk-Neutral Valuation Principle

12. 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

13. 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. Generating the tree

14. 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 Pricing the American put

15. 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. Checking for early exercise

16. 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 Continuing to price the nodes

17. Binomial trees in Mathematica, Excel, and r

18. 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. Problems with the binomial method

19. Monte Carlo Implementation • Least-Squares Approach • Exercise Boundary Parameterization Approach • Measures of accuracy Next time…