If you’ve been reading our last few articles on option greeks (here and here) and volatility (here), then you’ll likely have a sense of how an underlying asset’s volatility is an extremely important factor in an option’s price. Today we’re going to look at an example of just how important this relationship is – an example that will hopefully also emphasize theory vs practice in option pricing.
The groundwork
We must first understand how an option’s moneyness affects the implied volatility (how volatile the market thinks the underlying stock is). Put simply, implied volatility tends to increase as the option becomes more in the money (profitable) or out of the money (literally worthless). Why? Deep in and out of the money options have not only higher degrees of uncertainty associated with them (investors don’t know if the underlying stock price is going to move favorably), but also, perhaps more importantly, larger potential for larger gains.
A deep in the money option can obviously be very profitable, but technically, so can deep out of the money options. Say you buy a call option with a strike price of $50 but the underlying stock is worth only $25 per share – this call currently possesses no intrinsic value and is hence significantly cheaper than a call option with a $20 strike price. But if the stock price jumps from $25 to $60, then the $50-strike-price call option can in fact generate a healthy profit should you choose to exercise it. This is why even out of the money options still sell for something, as it’s still possible for them to become in the money.
Conversely, implied volatility tends to be lowest with at the money (when the strike price is identical to underlying asset’s price) options; because they’re priced closest to the current stock price, this means they reflect the market’s assessment of the stock’s future volatility the most accurately (which is why the strike price and actual price match up).
The second principle is the Black-Scholes model. Without getting too lost in the weeds (for further details, here’s our article on the matter), it’s a formula used to calculate option prices and one of its assumptions is that stock prices follow a log normal distribution, as shown below:
However, it goes without saying that stock prices don’t actually follow a log normal distribution. Rather, they have a fat-tailed distribution wherein extreme moves are more likely than in a log normal distribution. In other words, large price movements occur more often than what we would expect based on the Black-Scholes model.
This also means that, using the Black-Scholes model, we should expect options of the same stock to have the same implied volatility attached to them regardless of their strike price. After all, the stock is going to have a fixed level of volatility, so why should we price options for that same underlying asset differently? Why do in and out of the money options have a higher implied volatility than at the money options? The answer leads to the crux of this article.
Grins and smirks
Turns out there lies a fatal flaw in the Black-Scholes model. Because it incorrectly assumes a log normal distribution of stock prices, the Black-Scholes cannot account for the “fat-tail” probabilities that are inherent to deep in and out of the money options. Remember, these options have higher implied volatility because of a larger potential for larger gains – price movements so substantial that they don’t show up on a log normal distribution. So when plotting the implied volatility of options at varying strike prices, the Black-Scholes model would have us predict a flat line, meaning that the implied volatility for each of the options is the same. But this isn’t the case, as seen below:
In case it doesn’t already look familiar, all the volatility smile depicts is the phenomenon discussed earlier, that is, how an option’s implied volatility increases as it becomes more in or out of the money. As the graph shows, volatility increases as calls become more in the money and puts become more out of the money (meaning the stock’s price rises above the strike price), and when calls become more out of the money and puts become more in the money (meaning the stock’s price drops below the strike price). Hence, when plotting implied volatility against various strike prices, we see a “smile” rather than a flat line.
But is the smile always symmetrical? Not at all. Indeed, financial analysts have observed that people are more willing to "overpay" for downside striked options (options that become more in the money as the stock price decreases). This makes sense because historically, stocks tend to fall with much greater speed and magnitude – higher volatility means higher option price. On the flip side, upside striked options (options that become more in the money as the stock price increases), the implied volatility is lower because stock prices tend to gradually increase. This is called a volatility “skew” or “smirk”:
Here we’re seeing that a put option, regardless of how in or out of the money it is, has a greater implied volatility than a call option. What’s important to note is the difference in volatility once the call option becomes out of the money and the put option becomes in the money (i.e. the market price drops below the strike price). One reason might be because investors seek protection against potential decreases in the stock price – with put options, you make more money the further the stock declines. If investors anticipate tumultuous market conditions, the demand for in the money puts will increase, thereby driving up their prices and with it, their implied volatility.
One thing to keep in mind, however, is that the smirk’s direction depends almost entirely on the kinds of options investors prefer at any given moment. If people had a very bullish outlook on asset prices, then it’s entirely possible for the graph to inverse, with the call option’s implied volatility suddenly increasing and exceeding the put option’s implied volatility as it becomes more in the money (i.e. the market price rises above the strike price).
Black Monday
It’s important to keep in mind that the volatility smile isn’t a theory as much as it is an empirical discovery. By reverse-engineering the Black-Scholes equation, investors calculated higher levels of implied volatility for in and out of the money options relative to at the money options. But these outcomes were only observed after that one dreadful day: October 19, 1987. Black Monday, people later called it, recorded a 22% plunge in the Dow Jones – the largest single-day drop in the index’s history. With worldwide losses totalling over $1.7 trillion, people feared another Great Depression ahead of them (which, thankfully, didn’t occur).
The exact causes of the crash are multifaceted, but perhaps the key triggers, as outlined by Nobel-prize winning economist Robert Shiller, were a general feeling of overvaluation in the stock market, trade and budget deficits, and rising interest rates. Shiller surveyed nearly 900 investors and found that almost all reported some gut feeling of a looming crash.
Now what does this have to do with the volatility smile? Black Monday showed investors just how much the market could suddenly move, prompting people to rethink their approach to risk management and consequently, option pricing.
Econ IRL
One popular but slightly controversial strategy in real estate is buy-to-let investing, which involves purchasing a property specifically to rent it out. Critics of this practice argue that it artificially inflates housing prices, as investors will compete among each other to buy properties with the hope that they can either be sold at a higher price or that the rental income will exceed the initial cost.
But how truly harmful is this investment technique to neighborhoods? This week’s paper features the Netherlands as a case study as they provide some insight into that question. In 2022, the Dutch government gave municipalities the authority to ban buy-to-let investing from their respective areas. Rotterdam was the first city to do so, and reduced the fraction of owner-occupied properties bought by investors by 23 percentage points.
Interestingly, the ban attracted wealthier inhabitants, with new residents having incomes 2-3% higher in the income distribution, and the entry of low-income residents being somewhat limited. Although the researchers observed no significant effect on prices, the change in residential composition suggests that any impact on housing prices may be less related to investors bidding up prices and more to how investors ultimately change who lives in these neighborhoods.
‘Till next time,
SoBasically