Mar 23

Option Pricing Models (OPM). Part 2: Option Assumptions

by David Harper, CFA, FRM, CIPM

CFA | Quant |

This is part 2 of 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.

Don't skip the assumptions

Before we examine the Black-Scholes, it's good to understand the assumptions that underlie the model. When you know about the model's assumptions, you can avoid a common confusion. That is, you often hear somebody object to the Black-Scholes because "is just not accurate for such and such a purpose." This leads them to dismiss it as a mysterious black box that any magician can use to produce whatever desired value.

But the Black-Scholes is perfectly accurate, for the inputs provided, if the assumptions are met. If the assumptions are met, it deduces the best possible present value (PV) estimate of a stock option, given the inputs. But of course the assumptions are too rigorous. They can't really be met. This happens all the time: in removing the model from the clinical lab and applying it to real situations, we violate assumptions. But not all assumptions are created equal. We can violate certain assumptions and merely "bend the model" without harming its utility; other violations do in fact "break the model."

The good old-fashioned version is called the Black-Scholes-Merton, which hasn't changed for over thirty years. But lots of variations have since been developed. These variations mostly exist to handle the application of the model to certain circumstances; they cope with circumstantial violations of the assumptions.

So, if we apply the Black-Scholes to pricing an employee stock option (ESO) for example, we can do better than say "Black-Scholes doesn't work for ESOs." Better is to understand the two basic assumption violations. If we understand the assumption violations, we better understand the fix. And this let's us treat our estimate with proper care, rather than blindly dismiss the model.

Black-Scholes assumptions

When we teach the models, we refer to the Black-Scholes as deductive and the binomial as inductive. The Black-Scholes is a differential equation, it solves for (reduces to; inducts) the value of the option based on setting up a riskless portfolio that mimics the economics of the option. The binomial is more intuitive because it inducts; it maps out future scenarios.

Okay, so here are the onerous assumptions:

  • Stock price follows a particular random (stochastic) process
  • Short selling is allowed
  • No transaction costs and no taxes; securities are perfectly divisible
  • Dividends are not paid
  • There are no (risk-less) arbitrage opportunities
  • Security trading is continuous
  • The risk-free rate of interest is constant and the same for all maturities

The Black-Scholes works because a riskless portfolio can be created that matches the option's economics. But in order to maintain a riskless portfolio, the portfolio needs to be constantly rebalanced (i.e., that explains three of the assumption: short selling allowed, no transaction costs, security trading continuous).

Random process

The random process assumed by the model is called Geometric Brownian Motion (GBM). GBM assumes that the stock had "drift + shock." If we are thinking in terms of a discrete price path, that means, at each step the stock drifts up by some expected return plus some random shock (where greater volatility implies greater shock). As in the diagram below, imagine at each step, the stock will follow the linear drift but then it will end up "shocked" above or below the line.:

So, while there are fully seven assumptions, they mostly serve to:

  1. Allow for a continuous rebalancing of the riskless portfolio (that allows us to solve for the option value)
  2. Require an assumption about the random (stochastic) behavior of the stock

How an employee stock option (ESO) violates the assumptions

An employee stock option essentially violates the assumptions of the Black-Scholes for two broad reasons (neither of which are fatal). First, with contractual lives typically of ten years (or even five or seven years), they have long lives. Their long life puts undo stress on the assumption of constant volatility (and to a lessor extent, constant riskless rate). Second, ESOs are not liquid. They cannot be sold (in most cases, Googles' gooptions are an experimental quasi-exception) plus they at risk of forfeiture.

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