What is the Black Scholes formula and why do option traders use it? Part 2 - Assumptions

Level 1 - Zero to Options Hero - A deeper dive into the assumptions behind Black-Scholes formula, and how those assumptions are not 100% true.

In part 1 of this series, I went over a little bit about the history of options pricing and the rapid adoption of the Black-Scholes model as the gold standard for options traders in the 1980s. In this post, I will go over the assumptions of the model, why all pricing models are flawed, and why it is critical for traders to understand the assumptions behind the model and real-world scenarios where those assumptions would not be true. Let’s dive in!

After spending enough time with the PhD-credentialed quants on Wall Street, a self-aware trader learns that all financial pricing models, no matter how fancy and packed with the latest innovations, have assumptions that deviate from reality. The job of the trader is to understand those assumptions and to treat all models with a healthy dose of skepticism. Heed the classic saying of engineers: Garbage In, Garbage Out!

Major Assumptions (possibility of major deviations in reality)

1. Black-Scholes assumes that an option seller can replicate/hedge the payoff of the option by delta hedging the option continuously, with zero transaction costs, regardless of the number of shares needed to hedge, through the lifetime of the option. The key point behind options pricing theory is that the cost of an option should be the cost of replicating the option payoff by delta hedging the underlying asset. Think of this as similar to the price of a product being the cost of manufacturing the good from parts plus a markup that businesses compete to reduce.
   
   This assumption is wrong on three counts:
   
   1) most asset markets are not open 24/7/365 to allow for continuous hedging.
   
   - Developed currency (also known as G10) FX markets are open 24 hours for 5.5 days a week and closed on a few major global holidays such as New Year’s Day and Christmas Day.
   - Equity markets are open for only about 7-8 hours a day (and in some markets such as in Hong Kong there is a 1-hour halt of trading at noon).
   - In fact, cryptocurrency markets which have a rapidly growing options market are the only markets that are truly 24/7/365, and it will be interesting to see how options are priced given the availability of delta hedging at all times.
   
   2) there are instances where delta hedging will not be possible in exceptional market circumstances.
   
   - During the 2008 Global Financial Crisis, many nations instituted a short-selling ban on equities. In the United States and the United Kingdom, a ban on short selling of equities on financial companies was instituted in 2008, so options traders would not be able to delta hedge short put or long call positions. Also, many equity markets have something known as a circuit-breaker, which means that if a stock moves an exceptional amount in one day, trading is halted for a period of time. For example, on the New York Stock Exchange (NYSE), there are 3 levels of circuit breaker for moves of 7%, 13%, and 20% within 5 minutes, which result in halts of trading of those stocks from 15 minutes up to the rest of the trading day.
   
   3) trading the underlying in order to delta hedge options will always have transaction costs
   
   - These transaction costs will include commissions on equities/ETFs, exchange fees on trading cryptocurrencies, and brokerage fees on trading futures, currencies, and CFDs (contracts for difference).
   - The market impact of delta hedging is also an inherent transaction cost, which is the spread from the mid-price to the execution price, as well as the impact of high-frequency traders (HFTs) and MEV extractors in crypto decentralized exchanges to front-run/adversely impact the mid-price ahead of the delta hedger.
   
   The good news is that as offerings for retail traders become more price competitive and as liquidity in financial markets becomes deeper with increased competition and innovations such as cryptocurrency markets, the costs of delta hedging have continued to decrease.
   
   

2. Black-Scholes assumes that the underlying asset’s price follows a Geometric Brownian Motion with constant drift and constant volatility for the lifetime of the option.
   
   - Don’t be intimidated by the term Geometric Brownian Motion, or if you Google it and see scary words like stochastic differential equations! The key concepts to understand here are that a Geometric Brownian Motion for the underlying price means that the percentage change of the asset in the next period of time has no connection to the percentage change of the asset in the previous period of time (returns are independent of each other) and that each return is a draw from the same normal distribution (returns are identically distributed).
   
   - As a toy example, we can think of a stock price in the Black-Scholes world to follow a Geometric Brownian Motion with zero drift and a constant volatility of 16% annualized. This means that the percentage change of the asset over the next period of time has an expected value of zero (which can be thought of as the best guess of the price of the asset in the next period of time to be the same as right now) and that the average daily percentage change of the asset to be 1% (see the post on the Rule of 16 for why this is the case!)

   
   The main flaw with this assumption is that volatility is not constant over time and is path-dependent. Typically, asset volatility ‘clusters over time’, i.e. it stays in a low volatility regime for a while before switching into a high volatility regime. Also, volatility in stocks and equity indices tends to be higher when the markets are going down for a variety of reasons that I will explain in a future post.
   

3. A point related to flaws 1 and 2 above is that the underlying prices that Black-Scholes assumes to be continuous are often not the case. Asset prices will ‘gap’ due to scheduled data releases that impact the valuation of the asset (central bank announcements, earnings releases) as well as unforeseen events (natural disasters, wars). Another way to think about this is that if an options trader leaves a stop loss (S/L) order on the underlying for delta-hedging purposes, if the market gaps through the S/L order, the fill can incur significant slippage.
   
   Minor Assumptions (more minor deviations in reality)
   

4. A more minor flaw in Black-Scholes is that it is assumed that the proceeds from shorting the underlying asset can be reinvested at the same interest rate as that used to borrow to long the underlying asset, i.e. zero bid-offer in interest rates. This can be more significant for assets denominated in emerging market currencies, or stablecoins in the case of cryptocurrencies.

5. Another minor flaw is that in the case of equities, dividends are modeled as a continuous yield, but in reality are paid discretely (on a fixed schedule).

6. The Black-Scholes model assumes an absence of taxes, when in reality different traders in the market may be facing different tax liabilities.

Again, this may seem like a lot of drawbacks to using Black-Scholes, but experienced traders have developed the intuition to think around the assumptions and prepare/hedge for scenarios where there are major deviations from these assumptions. As mentioned in part 1 of this series, the major advantages of Black-Scholes are that the prices and risks of a large options portfolio are easy to compute and also that the risks/Greeks in the model are fairly easy to understand and build intuition on.

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