By: Piet Nova The Binomial Tree Model

 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

 Binomial Trees  Trinomial Trees  Monte Carlo Simulation

 An option is a financial contract between a seller (writer) and a buyer (holder).  Basic Components:  Option Price  Value of Underlying Asset (Stock Price, S 0 )  Strike Price (K)  Time to Maturity (T)

 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 – S T

 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?

 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

 Optimal to exercise early if it is sufficiently deep in the money.  Extreme situation: K=$10, S 0 =$  When is it optimal to exercise?  In general, when S 0 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

 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: 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.

 This principle leads to the calculation of the following crucial aspects of the binomial tree:  u = e σ*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)

 At T=0, S T is known. This is the “root” of the tree.  At T=1∆t, the first step, there are two possible asset prices:  S 0 u and S 0 d  At T=2∆t, there are three possible asset prices:  S 0 u 2, S 0, and S 0 d 2  And so on. In general, at T=i∆t, there are i+1 asset prices.

 To generate each node on the tree:  S 0 u j d (i–j),j=0, 1, …, i  Where T=i∆t is time of maturity (final node)  Note u = 1/d  S 0 u 2 d = S 0 u  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.

 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 – S T, 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

 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.

 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

 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.

 Monte Carlo Implementation  Least-Squares Approach  Exercise Boundary Parameterization Approach  Measures of accuracy