Mar 26

Option Pricing Models (OPM). Part 3: Binomial

by David Harper, CFA, FRM, CIPM

CFA |

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

 

The binomial is like a sideways tree, starting at time zero (T0). The binomial performs a giant, but intuitive calculation. Starting at today (time 0), the binomial:

  1. 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 sort of like saying to yourself, "here are all the places the stock could end up in ten steps, twenty steps, or one thousand steps.
  2. Using this array of potential future stock prices, it conducts a giant present value operation. That's called backward induction.

Step up or step down, to future stock prices

Start with today's stock price at $10. The key input into the binomial is a model of stock price behavior, including the probability that the stock will 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., this is annualized, so it will need to be converted based on the number of steps per year) and that a down-jump produces a 6% loss. In the standard binomial, the stock's volatility determines these probabilities for us.

If we zoom-in to a single step, we see that the $10 stock can either move up to $11.20 or down to $9.40 (note: we have really simplified by assuming that a single step is over an entire year; binomials are much more granular than this).

Expected future option value is weighted average

Now that 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 $10 strike price (i.e., the original stock price) produces an option value of $1.20 in the up path and zero in the down path. What is the expected future value of the option? It is weighted-average value; in this case, 50% multiplied by each of $1.20 and zero gets us $0.60.

Discount back to present value

So $0.60 is the weighted-average future value of the option. Now we simply discount that to the present value. Under a continuous compounding assumption, that is equal to $0.60 raised to the negative of the product 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.

For example, 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 $0.60e raised to the power -(5%)(1) = $0.57.

Backward induction is repeating this over many nodes

The above only showed one step. A binomial contains many steps and therefore many nodes. Backward induction is the process of starting from the array of final option values and working backwards one step at a time.

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