Valuing stock options

(This may be an advanced topic.)

Most of the time, the Black-Scholes model will come very close to the “correct” price of an option (whatever it is).  Fundamentally, how the Black-Scholes model works is this:

1. It assumes that future stock prices are random.
2. It assumes that the future stock prices will follow a bell curve distribution.  Many random phenomena in nature follow a bell curve distribution.  An infinite series of 50/50 coin flips for example would result in a bell curve distribution.
3. It introduces the concept of dynamic hedging / delta hedging.  If you continually buy/sell the underlying security in a certain way, the direction of that security does not matter.  You can, in theory, perfectly hedge your options position.
4. Is assumes that future volatility will be the same as past volatility.

In reality, there are many assumptions underlying the Black-Scholes formula and all of them can be wrong.  However, the Black-Scholes model is not bad and it’s actually pretty close most of the time.  Many options traders use it.  To  handle faulty assumptions, options traders simply fudge the the volatility parameter of the Black-Scholes formula.

The more a stock fluctuates, the more an option is worth.  In the Black-Scholes model, the amount of money made on delta hedging is related to how volatile a stock is / how much the stock price moves up and down.  Plugging into a slightly higher or lower value of a stock’s expected volatility will affect the “correct” value of an option.  Fudging volatility is a way of adjusting what an option’s price should be.

The bell curve distribution / volatility smile

An infinite series of 50/50 coin flips would generate a bell curve / Gaussian distribution.  Do stock prices follow a random bell curve distribution?  Probably not!

Freakish outlier events in stock markets occur more often than what a bell curve distribution would predict.  This is what people refer to when they talk about “fat tails”.

To account for this, options traders will price far-out-of-the-money options more expensively than other options.  If you plotted strike price versus implied volatility on a graph, there would be a curve that looks like a smile.

Volatility smile

Because outlier events rarely happen, we usually don’t really know how often they happen.  So there is a huge degree of uncertainty as to the correct price of far-out-of-the-money options.

Skew

In some cases, large sudden increases in a stock’s price (e.g. short squeezes, takeover bids, asset bubbles/manias) are less likely than large sudden decreases in a stock’s price (fraud, natural disasters, bad news, etc.).  The stock option markets sometimes take this into account.  For example, many large cap stocks exhibit skew.

In the commodity futures market, it can be argued that the skew should go the other way.  Large increases in commodity futures are much more likely than large decreases in commodity futures.  Historically this has generally been the case.

Future volatility versus past volatility

Usually the future resembles the past, but not always.  Correctly valuing an option depends on predicting the future volatility… this can be hard.

There are some events that are likely to cause huge short-term volatility in a stock.  An earnings release or other important news will cause a huge short-term spike in volatility.  Usually markets anticipate earnings release as implied volatility on options will rise before earnings and drop afterwards.

Another phenomenon is seen when companies are being taken over.  If another company announces a cash takeover at \$X, a company’s share price will usually trade in a narrow price range close to \$X as the merger arbitrage folks do their thing.  Volatility is usually very low until the deal closes (after which there is zero volatility in the stock price), there is a competing takeover bid, the deal falls apart, or there is uncertainty over the deal.  Again, the options markets are usually pretty efficient here as implied volatility will drop dramatically for stocks with announced takeovers.  Buying/selling options on these stocks are a method of speculating on the outcome of the takeover.

Dividends

Changes in dividends will affect the value of an option and I don’t have any special insight in predicting these changes.

One thing to watch out for is that it is worth exercising some in-the-money call options early if the stock is paying a dividend.  Those options holders who fail to do so will miss out on the dividend and will make less money (on average) than if they exercised their option.  Some traders will sell many call options before a dividend payout as many retail investors will erroneously neglect to exercise their options.

Put/call parity and heavily shorted stocks

By buying/shorting the underlying stock, you can turn calls into puts and puts into calls.  In odd cases, this relationship breaks down.

Stocks that have very high short interest (e.g. over 10%) tend to have an elevated borrow cost.  For example, the small-cap stock ATPG had a borrow cost that exceeded 90% interest per year at one point in time.  Short sellers could avoid the borrow cost by buying a put and selling a call with the same strike price (and duration).  This would create a position very similar to shorting the stock.  However, they do not need to pay interest because they aren’t borrowing the shares at all.

In practice, everybody knows about this.  So what happens is that the puts get more expensive and the calls get cheaper.  One way to profit from this (in theory anyways) is to correctly predict the future borrow costs.

Practical considerations

Special dividends and special situations

For special dividends (and weird distributions like spinoffs), options may or may not have their strike price adjusted to reflect the special dividend or distribution.  This is a big deal for anybody who has bought or sold that option as one side will benefit at the expense of the other.

These rulings have a slight tendency to benefit the marker makers and/or others in the financial community (if they predominantly have one position).  Sometimes markets are unfair.  I would argue that this is a hidden transaction cost (albeit a minor one).

Yes there are folks who trade on inside information.  They tend to buy options since the upside is far greater than selling options.  Often these guys aren’t caught.  In rare cases, you are exposed to adverse selection if you sell options.

HFT arbitrage

Sometimes stocks will make a split-second move up/down several ticks (especially during the opening and closing hours).  High frequency traders with lightning-fast computers will pick off options orders that are updated by computers slower than theirs.  The HFT guys have a pretty big speed advantage when they pay exchanges money to co-locate their computers beside the exchange’s (as well as making other investments in speed).

This is detrimental to legitimate options traders.

Front running, rebates, price improvement

I believe rebates and price improvement are mechanisms whereby market makers can front run other orders on an options exchange.

In general

Your overall transaction costs may be slightly higher than you think.

Protracted trading halts (this happens commonly with Chinese reverse mergers

Your options may not be automatically exercised… remember to do it.

Exercising the options could also potentially force a margin call.  You need to carefully avoid that.

Counterparty risk

This is theoretically a risk but not something I worry about with exchange-traded options.

Closing remarks

Basically, valuing options is difficult because it involves predicting the future.  There is a high degree of uncertainty involved.

If you want to know more about this topic, a really good book is Nassim Taleb’s Dynamic Hedging: Managing Vanilla and Exotic Options.  It is a broad book that covers options in general and is written by somebody who knows what he is talking about and has a track record of success.  And unlike Taleb’s newer books, he is less opinionated and philosophical.

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