#### Binomial tree

Binomialby David Harper, CFA, FRM

This is part 3 in a series of highlights from our 50-minute screencast called Option Pricing Models: Binomial and the Black-Scholes. This screencast earns 1.0 credit hours under the Professional Development (PD) program at CFA Institute.
Binomial: first walk forward, then backward induct
Thebinomial is like a sideways tree, starting at time zero (T0). Thebinomial performs a giant, but intuitive calculation. Starting at today(time 0), the binomial:
• Builds a tree of future price paths,one node a time, until it gets to the end. The end result is an array(from highest to lowest) of future, terminal stock prices. It is sortof like saying to yourself, "here are all the places the stock couldend up in ten steps, twenty steps, or one thousand steps.
• Usingthis array of potential future stock prices, it conducts a giantpresent value operation. That's called backward induction.
Step up or step down, to future stock pricesStartwith today's stock price at \$10. The key input into the binomial is amodel of stock price behavior, including the probability that the stockwill move up or down at each node. In this example, we assume a 50%likelihood of an "up-jump" and therefore a 50% of a "down-jump."Further, we will assume that an up jump produce a 12% gain (i.e., thisis annualized, so it will need to be converted based on the number ofsteps per year) and that a down-jump produces a 6% loss. In the standard binomial, the stock's volatility determines these probabilities for us.
Ifwe zoom-in to a single step, we see that the \$10 stock can either moveup to \$11.20 or down to \$9.40 (note: we have really simplified byassuming that a single step is over an entire year; binomials are muchmore granular than this).

Expected future option value is weighted averageNowthat we know future stock prices, we subtract the exercise (or strike)price to produce future option values. For the down path, typically,this produces a worthless option. In this example, subtracting the \$10strike price (i.e., the original stock price) produces an option valueof \$1.20 in the up path and zero in the down path. Whatis the expected future value of the option? It is weighted-averagevalue; in this case, 50% multiplied by each of \$1.20 and zero gets us\$0.60.

Discount back to present valueSo\$0.60 is the weighted-average future value of the option. Now we simplydiscount that to the present value. Under a continuous compoundingassumption, that is equal to \$0.60 raised to the negative of theproduct of the rate (r) and the time (T). Note, without the negative,we would be continuously compounding \$0.60 forward; with the negative,we are discounting \$0.60 back to the present.

Forexample, if our riskless rate is 5% and our time period (T) is one (1),then the present value of the option is about \$0.57. Thats \$0.60eraised to the power -(5%)(1) = \$0.57.
Backward induction is repeating this over many nodesTheabove only showed one step. A binomial contains many steps andtherefore many nodes. Backward induction is the process of startingfrom the array of final option values and working backwards one step ata time.