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Since the creation of the *Chicago Board Options Exchange* (CBOE) in
1973, options have played an important part in financial markets,
allowing investors to hedge their portfolios against drawdowns, obtain
leverage while managing risk, and express nuanced investment hypotheses.
Options' asymmetric nature means that buying options always comes with
limited risk, a desirable property when compared to shorting a stock or
selling futures, both of which have unlimited downside.

In this post we're going to talk about the mathematics of options, how they are priced, and how to potentially profit off of them.

## Black-Scholes Model

Before the advent of *Black-Scholes* (BS), options were not traded on an
exchange and instead they were bespoke contracts called warrants that
were traded over-the-counter (OTC). Only when a reasonable model (BS)
for pricing options was discovered did a market open for trading them,
the CBOE. Options can be broadly split into two categories:
American-style options, and European-style options. While European-style
options can be traded in for the underlying only at expiration,
American-style options can be exchanged for the underlying shares at any
time.

BS and its derivatives all concern themselves with European-style options as they are mathematically easier to work with, though as the name implies, are not very common in America. Even so, it's very rare for an investor to early exercise their American option, even when they have the right to do so. The reason for this is that an option's value can be decomposed into two parts: the intrinsic value of the option and the time value of the option. The more in the money an option is, the higher its intrinsic value, the longer the option has to maturity, the greater the option's time value. By choosing to exercise an option instead of merely selling it, an investor will generally make less money, as they are only capturing the intrinsic value of the option and missing out on the time value.

Because of this, American options are rarely exercised early, and thus, BS should give us a relatively reasonable approximation, even if it's not a completely mathematically correct thing to do.

The original Black-Scholes model has 6 principle inputs: the expected future volatility of the underlying asset, the current price of the underlying, the risk-free rate, the time to maturity, the strike price, and whether the option is a call or a put. Given these inputs, the model can generate a fair market price, or value, of the option. Unfortunately, unlike the other inputs, the expected future volatility of the underlying is unobservable and can only be estimated.

Using Black-Scholes and a root finding algorithm (such as Brent's
method), we can also go
the other way, finding the expected future volatility of the asset by
using the current market price. This expected volatility is called the
*implied volatility*, or simply IV. For an option to be fairly priced
from a mathematical perspective, the option's IV should be equal to the
realized volatility, which we can only observe after the option has
expired. Let's consider *SPY* (S&P 500 ETF) options that are
at-the-money (ATM) and also have the shortest time to maturity. Below is
a graph of the annualized IV of puts and calls vs the realized
volatility over a 63-day rolling window:

Note that because we're simply using a trailing window for the volatility calculation, this isn't an exact apples-to-apples comparison. Regardless, the results are interesting: the historical volatility is almost always lower than the IV for both puts and calls. From this chart, it's evident that options are rarely sold for a fair price, with a large premium on puts and a moderate premium on calls. And this is for close to or ATM options, which tend to have lower IVs than other options (see volatility smile for more information). This means that the further an option's strike price is from the current spot price, the more expensive it is compared to what Black-Scholes predicts. According to Black-Scholes, the IV of all options with the same time-to-maturity should have the same IV, regardless of strike price.

Starting in around 2018, the IV of calls and puts are actually lower than the realized volatility, implying that options (both put and calls) are cheaper than their mathematical fair value. Also note that puts almost always command a premium over calls. This should come as no surprise: there are many more natural buyers of puts (predominantly investors looking for portfolio insurance) than buyers of calls (primarily speculators).

## Implied Volatility/Realized Volatility Arbitrage

It's clear from the above graph of IVs that, most of the time, we are getting ripped off when buying options. So instead of buying options, we would want to sell them instead. Let's consider two simple strategies: one where we sell an ATM call everyday, and one where we sell an ATM put everyday. Below is a graph of the cumulative profit and loss of each strategy:

Surprisingly, even though puts have a higher IV on average, and thus command a higher price, we actually make more money selling calls than puts. We initially start making really good profits, but then things start to flatten out for both strategies, especially around 2018. Based on the IV graph we saw, this makes sense: we are selling options for below their fair mathematical value and losing money because of the positive difference between the realized volatility and IV.

Even ignoring the regime change after 2018, selling calls and puts like this without a hedge in incredibly dangerous. Remember that unlike with buying options, selling options puts us in a situation in which we are taking virtually unlimited risk. If the volatility of the market spikes unexpectedly, we could very easily get into hot water: the amount of long-tail risk taken is so staggering as to make all but the riskiest investors run for the hills. With one bad day, not only could we be wiped out, but we could be on the hook for money we do not have.

## Hedging Risk

In order to try and capture the option premiums while hedging away some our risk, we can hold the underlying ETF for the call strategy, and short the SPY ETF for the put strategy. While this hedges away the risk of large moves upward (downward) for the call (put) strategy, it still leaves us exposed to downward (upward) moves for the call (put) strategy. Unlike in the unhedged version where we sold a contract each day, we only sell one contract at a time to ensure that we are correctly hedged. When the option we sold matures, we buy the next available option that is closest to ATM, has the minimum maturity time, and has available bids. Note that we are not perfectly hedged because there will always be a small difference between the strike and spot price. Let's take a look at the return of both strategies:

The put strategy makes a marginal amount of return until 2018 when the IV of puts starts to trend lower than the actual realized volatility. Though we make money from the option premiums, it is barely enough to cover the persistent negative return from shorting the S&P 500 in good times and not enough in bad times, when the realized volatility outpaces the IV. The call strategy on the other hand does very well until 2018, improving upon the return of the S&P 500. Like with puts, we start to lose money after the start of 2018 and finish slightly below the returns of the S&P 500 on 2020-07-16. We do observe less of a drawdown when the market crashes though, as the premiums from the calls we write cushion the blows to a certain extent.

In general, it's clear that even though puts are less fairly priced than calls, it's generally a better idea to write calls instead of puts. Hedging by longing the underlying doesn't have the large negative carry associated with hedging puts in bull markets and if the IV becomes unfavorable, one can simply stop selling calls (while still holding the underlying) and hold a pure beta portfolio until the historical realized volatility drops below the IV.

However, if we tried to pause writing puts, we would still hold SPY short, costing us a lot of money in most market environments. We would be forced to unwind the hedge until the realized volatility drops. And even in favorable volatility environments, the premium from the puts was slight compared the cost of the short hedge, with only around a 15% return being realized from 2012 to 2018.

## Conclusion

Perhaps the only take away we can honestly put forth is that options are difficult to profit off of, for both the buyers, and the sellers. In most market environments, the premium you are paying for portfolio insurance by buying puts is simply too large to be worth it. Even calls are overpriced, though not to such a large extent. But while buying options isn't the best idea, selling them comes with their own set of challenges, especially if you want to sell puts. If you forecast a low volatility environment of steady but marginal upward growth, selling covered calls, even ATM, can enhance your returns substantially; they also help mitigate losses during bear markets as well. The only situation in which selling ATM covered calls would not be appropriate would be if a strong bull market is forecasted: the premiums from the options are unlikely to be large enough to offset the money left on the table when capping the upside.

Thanks for reading, hope you liked my first options post! I won't be sharing the code for this one as the option data isn't publicly available and without the data, the code doesn't have much value. If, however, you are interested in accessing the data, check out Algoseek for all of your option data needs (and more!).