In his recent report to the shareholders of Berkshire Hathaway (BRK.A, BRK.B), Warren Buffett explains why he’s buying and selling the very derivative securities he notoriously lambastes as weapons of financial mass destruction. It’s simple really. He thinks the Black Scholes model is wrong, and lots of these derivatives are mispriced as a result.
The guy’s got a point. Under the Black Scholes Model, the longer the time to exercise, the higher the price of the option. That makes some sense in the short term, but suppose you are trying to price a 100 year put option on the S&P 500? The Black Scholes Model would tell you that put option is vastly expensive, which is silly because it’s really, really unlikely that the S&P500 is going to be lower in 100 years.
As a refresher, here’s the basic Black Scholes Model for a European call option:
C = SN(d1) – Ke(-rt)N(d2)
C = the Call premium
S = current stock price
t = time to option expiration
K = strike price
r = risk-free rate of return
N = cumulative standard normal distribution
e = 2.7183
d1 = Log (S/K) + (r+S2/2)t
d2 = d1 - s√ t
So what’s the cause of the problem with the Black Scholes Model? Two things immediately leap out at you.
First off, the standard Model is measured in terms of a stock’s price, rather than a stock’s performance. That's a pretty static way of pricing anything and ignores the fact that over the long term, stock prices tend to perform at some average rate of return. Take the S&P 500, for instance. Over long stretches of time (as in, a century), the S&P 500 on average tends to go up about 7.5% each year, after inflation. The Black Scholes Model does not take that fact into account, because it simply looks at the standard deviation of a stock price (or equity index price) at a given moment of time, rather than over the time span from today until the option expiration date. That's the simple reason why the longer the time until expiration, the more wrong the Black Scholes Model is going to be. So, in that regard, how do you tweak the Model to get it right?
What you want to do is to look at the standard deviation of a stock's average returns, rather than its price. The reason why is that stock price is not static. The math is not really that tricky, either. For “S”, instead of using the stock’s price, just plug into the average rate of return of the equity (or equity index) for the time period during which the option may be exercised. For example, suppose you have a call option that provides the holder with the right, but not the obligation, to purchase one share of SPDR S&P500 Index ETF (NYSEARCA:SPY) in fifty years at a strike price equal to today’s closing price. To price this option under my revised Black Scholes Model, what I’m doing is that I am assuming a hypothetical call option where the strike price is a price that is equal to the future value of SPY in fifty years, calculated as a function of the price of SPY today, at an annualized rate of return of 7.5% - rather than the actual strike price of the call option which is simply today’s closing price of SPY. Then, I’m just letting the Black Scholes Model discount back to present value the difference between the hypothetical strike price and the actual strike price. By doing this, I’m accounting for the fact that over 50 years, SPY will, on average, return 7.5% a year, which is not something the original Black Scholes Model does (and which is why the original Model goes all pear shaped when you use it to price long term options).
There’s another problem with the original Black Scholes Model, which is that since it doesn’t account for average returns on equity prices, it can’t account for the law of averages. For example, we can expect SPY to return 7.5% a year on average, but in any given month or year, it’s not going to necessarily do that. In fact, the shorter the time frame we’re looking at SPY, the less likely it is to return 7.5%. But, the law of averages provides that longer the time frame we look at SPY, the likelier it gets that 7.5% is going to be the actual rate of return.
So, what is the second tweak for the formula? I’m thinking we should slap another variable (call it “β”) in front of N, which defines the law of averages in mathematical terms. There's a few potential formulas to do that, which I will explore in a follow-up article. Until then, I'm happy to let Warren Buffett arbitrage the Black Scholes Model on behalf of his appreciative shareholders.