Improving the Black Scholes Model

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)

where

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

s√ t

and

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 (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.

Disclosures: Author owns BRK.B and SPY

Comments

[F. Bradeen:]

You are right for long term options of a year or longer to consider the long term trend, but there is a third very serious problem with the Black Scholes Model and that is the assumption of a Normal distribution. (Lets forget that stock returns are more log normal than normal for the moment.) The Normal or log normal distribution does not take into consideration the fat tails on both sides of the distribution. And let me tell you they are enormously fat. This will lead you to seriously underestimate the value of any option, whether long term or short term. In fact, I doubt any serious investor uses the Black scholes as it currently stands. If so, they WILL be wiped out on days like Oct. 19, 1987 where the market went down 25% in one day.

Mar 18 11:50 AM

[Alex Trias:]

That's a very good point - and your point about ultra high volatility days does should go against my point on mis-pricing.

Thanks for your comment!

On Mar 18 11:50 AM F. Bradeen wrote:

> You are right for long term options of a year or longer to consider
> the long term trend, but there is a third very serious problem with
> the Black Scholes Model and that is the assumption of a Normal distribution.
> (Lets forget that stock returns are more log normal than normal for
> the moment.) The Normal or log normal distribution does not take
> into consideration the fat tails on both sides of the distribution.
> And let me tell you they are enormously fat. This will lead you to
> seriously underestimate the value of any option, whether long term
> or short term. In fact, I doubt any serious investor uses the Black
> scholes as it currently stands. If so, they WILL be wiped out on
> days like Oct. 19, 1987 where the market went down 25% in one day.

Mar 18 12:32 PM

[IAmEric:]

If you are going to write something about the Black-Scholes option pricing formula, it might be a good idea to consult someone who actually understands it next time. There are certainly many valid reasons to criticize it, but what you present here is just silly. The most obviously valid criticism is that returns are not normally distributed. BUT if you assume they are, the average return drops out of the pricing formula. It is not that it doesn't take mean returns into consideration. It does, but due to the nature of normally distributed returns, the mean drops out of the formula. They teach that in the first week of any financial engineering course.

Mar 18 03:42 PM

[IAmEric:]

By the way, pricing formulas for options that drop the assumption of normality have been known for years too. Unlike what Taleb might have you believe, we've always known securities are not normally distributed and practioners deal with it in various legitimate ways.

If you picked up on the fact that my patience is thin regarding "quant bashing", you're right:

Disingenuous Quant Bashing
phorgyphynance.wordpre.../

As argued there, do you hear people complaining about the use of the "yield" of a bond? The concept of the yield of a bond is probably more bogus than the implied volatility of an option. Both concepts serve the same purpose. They each, based on some bogus model, convert the price of a security to something else that helps with relative value analysis. To compute the yield of a bond, you have to assume the issue is going to pay every coupon, i.e. no chance of default, and determine the discount rate that matches the market price to the discounted cashflows ASSUMING you can reinvest the coupons at the same rate. THAT is just as bogus as Black-Scholes. Maybe more, but people aren't on a rampage about the use of bond yields even though the very concept is pretty bogus if you think about it.

Nobody is blameless throughout this crisis, but grossly underestimating the value of quants and overestimating the influence they have had is misguided energy. Long-dated options are likely not priced according to a reasonable assumption of discounted cashflows, but there is a reason. Liquidity is high on the list. If you attempt to price straight bonds with discounted cash flows, you will often see significant differences compared to market prices as well for very similar reasons. Should you go out and buy all the long-duration corporate bonds you can find?

You can say what you like about quants, but suggesting that they've (we've) made this universal blunder by not taking into consideration mean returns is frustrating to see to say the least.

Mar 18 04:24 PM

[frank_j;sam...:]

Hey I normally really like your blog. But this is college level calculus--and this piece is way way out of its depth. It's not that I'm offended or piqued--rather I'm cringing in embarrassment for you at the innumeracy of this writing. I hope some undergraduate level first-year finance teacher will take time to walk through this nonsense--which I say with no spite or animosity at all. Good luck. And seriously consider getting some help with this post. (PS I doubt Buffet knows any more college level calculus himself either.)

PS And yes the B-S world of constant volatility, normal distributed returns, continuous prices and trading, etc, etc, etc contains all sorts of distortions and downright untruths. But even every investment bank drone who can't even differentiate knows that and makes various adjustments.

Apr 05 07:17 AM