Mar 28

Option Pricing Models (OPM). Part 5: Black-Scholes versus Binomial

by David Harper, CFA, FRM, CIPM

CFA |

This is part 5 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. 

 

In the prior posts, I sketched the binomial and the Black-Scholes. The binomial builds a tree or lattice. The binomial approach is said to be open or numerical: it "induces" the price by building a future scenario map of potential stock prices (then via backward induction, solves for the present value). The Black-Scholes is a solution to a partial differential equation (PDE). The Black-Scholes approach can be called closed or analytical: based on a riskless portfolio, it solves for the only possible option value under presumed no arbitrage conditions. In short, the binomial openly induces, the Black-Scholes restrictively deduces.

For both models, there is a generic version plus several variations including several advanced (even exotic) variations. For this reason, it is inexact to claim that one model has a consistent valuation bias (e.g., the oft-quoted cliche that binomials price employee stock options lower is imprecise).

In regard to the Black-Scholes,

  • The generic version is the Black-Scholes-Merton famously derived in 1973. It arguably includes the subsequent extension to account for dividends (where dividends are largely incorporated by a reduction in the stock price input equal to the present discounted value of the expected dividend stream). This generic Black-Scholes-Merton takes six inputs (stock, strike, option term, expected volatility, riskless rate, and expected dividend yield)
  • Many of the model's variations exist to overcome one or another of the several highly restrictive assumptions. For example, the Hull-White is one of a class that attempt to overcome the original's requirement that stock returns are lognormally distributed.

In regard to the binomial,

  • The generic binomial is called the Cox, Ross, Rubinstein (CRR) named after its inventors. Like the Black-Scholes-Merton, this binomial takes volatility as an input as uses it to compute the two key parameters: probability of an up-jump (i.e., odds of jumping up from one node to the next) and the magnitude of an up-jump (e.g., does $10 jump to $10.05 or $10.15).
  • The variations on the binomial are endless because the binomial is a numerical approach; you can pretty much "design in" your assumptions. Including, for example, the "tree nodes" can have three branches and it becomes a trinomial.

For ESOs, they use different methods to reduce value

As investors, a hot topic concerns the valuation of executive/employee stock options (ESOs). Against the goal of pricing ESOs, there can be no doubt the flexible binomial is better suited to the task (and buried in the transcripts of the erstwhile FASB expert deliberations on the topic, it's plain that most of the expert panel preferred the binomial, but they couldn't quite muster the will to insist on its use). Aside from long contractual lives, the major issue with ESOs is their lack of liquidity. Under FAS 123R,

  • The Black-Sholes treats the value reduction by replacing the contractual life with a shorter expected life. This is an understandable kluge, but it's all barb wire and bubble gum. The shorter life is a clunky way to approximate the necessary discount for a relative lack of liquidity.
  • The binomial treats the value reduction directly, with design modifications. This is more intuitive and realistic! The binomial can explicitly model the early-year vesting restrictions (e.g., four year ratable or cliff vesting). More importantly, an Exercise Trigger is introduced. For example, 1.5x. That means, we assume option holders will exercise if they are 1.5x in the money (if the stock reaches 150% of the strike price). That assumption about people's behavior can be directly built into the model. It is sort of like taking shear-clippers to the upper half of the model, it reduces the height to which the upper branches can grow!

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