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# Limitations of the BSM model

In this article, we go over some of the limitations of the BSM model and explain when it fails in practice. But first weβll go over some notes on volatility.

Volatility

We briefly discussed volatility in a previous post. Here we describe its estimation in the context of the BSM model. Recall that under the geometric Brownian motion assumption, we have:

where denotes the underlying asset price at time , is the expected return on the underlying (per year, say) is the variance of the return on the underlying per year, and is a normally distributed random variable with mean and variance .

 Day, TSLA stock price, Price Relative, Log of Price relative, 0 1114.00 1 1208.59 1.0849 0.03544 2 1172.00 0.9697 -0.01335 3 1213.86 1.0357 0.01524 4 1229.91 1.0132 0.00570 5 1222.09 0.9936 -0.00277

Suppose we have a time series of stock prices for some stock, say TSLA. Using the observations in the stock price data, and assuming the data are generated by a GBM, we can estimate the mean and variance of returns in reduced form. Hence we estimate by computing the sample mean of and by computing the sample variance or realized variance of . It's important to note that is notoriously difficult to estimate and the sample mean is typically not a good estimator for it. Luckily, in the BSM model, all that is needed from the price process parameters to price options is .

Implied volatility, on the other hand, is estimated by using the BSM formula. Recall that the BSM call price is a function of the strike price, current underlying price, risk-free interest rate, and time to expiration. For a given market price for the call option today, we can substitute the strike, underlying price, risk-free interest rate, and time to expiration, and with some calculations, recover the implied volatility for the BSM model. Formally, the implied volatility is the volatility that solves the following equation

where BSM stands for the BSM price given the fixed parameters , which is the current price of the underlying asset, and where is the market price of the option. In practice, we get different values of for different values of .

Problems with BSM

There are two main sources of problems with the Black-Scholes pricing model. The first has to do with assumptions about the probability distribution of the underlying (log-normal prices, constant vol, no jumps etc), and the second has to do with market frictions (no trading costs, perfect competition, equivalent borrowing and lending costs, etc).Β  Below we go over some of these in more detail.

# Probability Distribution of the Underlying

1. Constant volatility
2. Firstly, we assumed constant volatility in the BSM model. If volatility is time-varying,Β this can result in mispricing. Even in the case where volatility changes in a known fashion βan unrealistic assumption β the BSM model would have to be adjusted by summing (or integrating) over the different volatilities up until expiration of that particular contract. This would lead to changing implied volatility along the time dimension of the options contracts. See more about IV and model misspecification below.

4. In the BSM model, we assumed that the market was competitive and so the buying and selling of options by traders have no impact on the market price. Equivalently, this assumes that traders are always price takers, i.e. they must always accept the market price. This is not true in reality, where large market makers or other whales can have significantly higher market power than other participants.

5. No Bubbles
6. The assumption that the underlying (typically a stock) price follows a lognormal distribution precludes the existence of bubbles. However, we often see price bubbles in different markets. A price bubble is an event in which the market price of an asset deviates from its intrinsic value, the intrinsic value being the price paid for the asset if one were to not ever sell it after purchase.

7. No Jumps in prices
8. Jumps in prices can occur, for example, due to market manipulation, large trades, and news events.

Suppose the stock price has evolved over time and we are now day from expiry, so , and that at this time, we have an out-of-the-money call option because . We can use the BSM formula to show that the call option price and the option delta are both roughly . This makes intuitive sense because we are almost at expiration and the option is out-of-the-money. Suppose a market manipulator buys a large number of shares to cause the stock price to suddenly jump to . Then and is roughly , which will be different from what is predicted by the BSM model. For more information see, for instance, AΓ―t-Sahalia and Jacod.

# Market Frictions

2. The BSM model is a continuous time model.Β  In reality, prices are updated with trades that occur at discrete intervals.Β  In practice, uncertainty about price movements over discrete intervals results in traders and market makers picking optimal hedges that are different from the ones implied by Black-Scholes, and this can feedback into realized option prices.

3. Non-constant interest rates
4. The BSM model assumed constant interest rates. Long-lived options, whose underlying asset may be sensitive to interest rates movements, require further refinement of the BSM model for this to be accounted for.

5. Different rates for borrowing and lending
6. The BSM model assumes the same rates for both borrowing and lending. This spread between borrowing and lending rates is often referred to as the funding cost. This can be accounted for by extensions to the BSM model, and it can be shown that funding cost leads to an increase in the price of options.

7. Collateral requirements for borrowing and lending
8. The BSM model assumes that there is no credit or default risk. This is also known as default or counterparty risk - the risk that the entity exposed to financial risk may be unable to fulfill its obligation. In practice, financial regulations require the option seller to deposit cash or securities in their account, the margin requirement. Since there is a cost associated with locking up this capital, this can affect the profitability of different strategies and can feedback into realized option prices.

# Implied Volatility as an Indicator of Model Misspecification

Oversimplifying a bit, model misspecification is when your model does not accurately capture the data that you observe in options markets. In the BSM model, this can be seen by assuming that the model is true and seeing if the data observed in markets does not violate any of its assumptions. The easiest way to see this in the BSM model is to calculate implied volatility for all traded options contracts for a particular underlier. We do this by observing prices (or quotes) for each contract and calculating the associated volatility which makes the BSM model price match the observed price. Carrying out this computation almost always leads to an implied volatility surface that varies along strike and time to expiration is observed. As an example, check out the IV surface for GME back in March of 2021

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hat tip to u/indonesian_activist on wsb for the awsome graphic!! https://www.reddit.com/r/wallstreetbets/comments/m3zh49/gme_dd_friday_opex_312_post_mortem_warning_gamma/

If the BSM model was a complete description of the world, then DK would have 0 fun rolling around because the surface would be flat! So, what leads to this surface?

In reality, any of the above limitations could be to blame. Accounting for time varying volatility and jumps in prices helps capture some features of the IV surface, but no model has, as of yet, captured all of its features. This has led practitioners to instead treat IV as a statistical object to be modeled out while keeping the overall framework of the BSM model. This unfortunately doesn't always lead to good hedging strategies as it is unclear where the mis-pricings arise from and how to hedge those risks. (See section 20.5 in Jarrow and Chatterjea, 2nd ed)

For a good reference to learn more about the IV surface and how it is dealt with in practice see Jim Gatheral's Implied Volatility Surface . We will explore IV and its information content further in future posts!