Thursday, September 19, 2013

What is Black-Scholes, Anyway?

By Tim Dulaney, PhD, FRM and Tim Husson, PhD

In the past, we have reviewed the basics of options as well as included some discussion of more exotic options, such as binary options and barrier options, but we haven't talked in detail about option pricing.  There are a lot of great models for valuing options, but they can be a bit intimidating for the uninitiated, even though the underlying ideas are simple.

Any option's value is dependent upon the probability and timing of payouts.  For example, how much would you be willing to pay for an option that will pay $100 immediately with a 50% probability and pay$0 with a 50% probability?  You certainly shouldn't pay more than $50, and the seller would likely not take anything less than$50.  So the value of the option is $50 = 50%*$0 + 50%*\$100.  What if this option paid the owner a year from now?  Certainly you'd pay less for the option since there is some opportunity cost for the funds that will be invested in the option.  These ideas are the essence of option pricing, and really securities valuation in general.

European options are perhaps the most simple type of option (see our post on option basics for a review).  To value these options, we need to have a distribution of values for the underlying asset at the end of the option's term.  This distribution will give us the probability of a given payout and, just like the example used above, we use these probabilities to determine the present value of the option.

One of the most simple and straight-forward models for the distribution of asset prices is the Black-Scholes model, also known as the Black-Scholes-Merton model.  The model assumes that asset returns are independent and identically distributed (IID) and that asset prices follow a log-normal distribution.  The IID assumption simply means that, at any given time, the likelihood of the stock increasing or decreasing by some amount is the same as, and independent of, any other time.  The log-normal distribution for asset prices is motivated by both the empirical distribution of stock returns as well as the simple fact that asset prices cannot be negative.

The value of an option in the Black-Scholes model depends critically on one parameter that is not directly observable: implied volatility.  This parameter determines the width of the log-normal distribution of asset prices (put differently, it determines how uncertain future prices are).  Practitioners will typically solve for this parameter after observing the price of an option as well as the characteristics of the underlying asset and the option contract.

The Black-Scholes model can be used to derive equations for the value of European options.  They require that we specify properties of the underlying asset (current price and the dividend yield), properties of the option contract (option type, strike price, and time to maturity) as well as current interest rates (usually a risk-free rate).  The resulting equations are a little too complex for pen-and-paper calculations, but are easy to program and fast to compute -- so fast, in fact, we've embedded a Black-Scholes calculator into this very post.

Talking through the example in the tool, let's imagine we have a European call option with a strike price of , expiring in  months, on an asset with a current price of .  Assume the underlying asset has a dividend yield of  and the risk-free rate is currently .  Using the Black-Scholes model with an implied volatility of , the value of this  call option is .

SLCG Option Value Calculator (Black-Scholes)

 Underlying asset price: Strike price: Risk-free rate: Dividend yield: Implied volatility: Term (months):

 European call price: European put price: (at-the-money) (at-the-money)

Feel free to play around with the calculator to get a sense for how the price of calls and puts is dependent upon each of the different factors.  Black-Scholes isn't perfect -- the IID and log-normal assumptions are often criticized -- but it does provide good intuition for the sensitivity of option prices to asset and option specific features.