Break-Up with Black Scholes

Nothing in this article is financial advice. All opinions and beliefs expressed are my own.

Do understand, the black Scholes model by itself is a de facto. I do not believe it is an efficient way to price American options, however, I do believe it gives us a simple framework for development purposes and is "arbitrage-free". I created a black scholes pricing tool for anyone to use here - BSOptionPricing()

by: Max Heltzel - Computer Information Systems Student @ Florida Gulf Coast University

If you wish to see the results of my first attempt at a theoretical FFT & Quadrature pricing model, click here

⎯ Preface ⎯

The black scholes model was developed in the 70s by Fischer Black and Myron Scholes. It has historically been known as a foundational way to price European options. The model itself earned its creators a Nobel Prize and established a new era in financial engineering. But despite its historical significance and continued use, the Black-Scholes model harbors several critical limitations that undermine its practical applicability in today's complex, dynamic, volatile, and ever-changing markets.

I recently watched a video of Warren Buffet and Charlie Munger explaining their beliefs on the black school model and they said it better than I ever could. In basic form, the two were naming it as a way to badly price an option (not just in financial markets).

Charlie mentioned that...

"Black-Scholes is a know-nothing system. If you know nothing about value - only price - then Black-Scholes is a pretty good guess at what a 90-day option might be worth. But the minute you get into longer periods of time, it's crazy to get into Black-Scholes.

But Why?

Because Black Scholes makes a lot of assumptions. One important assumption is the variable of constant volatility of x market or x asset. In other terms, it assumes that the future price of an option will be priced in with current volatility. As we know, this would assume markets are frictionless and perfectly efficient. This is far from the case.

So yes, Charlie is correct! Black Scholes will more accurately price a shorter-term contract than a long-term price. But, this still holds the same assumptions, so no matter how close the expiration is, we are essentially guessing.

Expiration? Oh right. The Black Scholes formula was built to model European options and as you may know, European options can only be exercised at expiration. This adds another problem when pricing American Options which as we know can be exercised at any time.

To be clear, the BSM has evolved since its inception, but I keep seeing people try to price American Options with the standard BSM framework, and this is why you shouldn't.

Black Scholes Framework

The model gives us a closed-form solution for the price of a call or put option, relying on several inputs:

• Current Price of the Underlying Asset (S₀): This represents the present market price of the asset that the option is based on.

• Strike Price (K): The price at which the option holder can buy (call) or sell (put) the underlying asset.

• Time to Maturity (T): The remaining time until the option expires.

• Risk-Free Interest Rate (r): The theoretical rate of return on an investment with zero risk, a lot of the time, this is based on government securities like a treasury note.

• Volatility (σ): A measure of the asset's price fluctuations over time.

The Black Scholes model for European options is given by:

$$C=S _0 ​ N(d _1 ​ )−Ke^{−rT N}(d_2 ​ )$$

Where:

$$d_1 = \frac{\ln(S_0 / K) + (r + \sigma^2 / 2) T}{\sigma \sqrt{T}}$$

$$d_2 = d_1 - \sigma \sqrt{T}$$

N(⋅) denotes the cumulative distribution function of the standard normal distribution. This formula assists in pricing European options. There are many flaws, and we are going to attempt to cover them here.

The Problems and Assumptions of Black-Scholes

The Black-Scholes model has been revolutionary. It wasn't built to necessarily be "right", rather it is "arbitrage-free". This fact is what makes it exploitable from a directional trading perspective. This is heavily overlooked and is attempted to be used every day to correctly model option pricing. I have seen it be done, I tried it myself, but it most of the time ends in a bloodbath of miscalculations. The utility is hampered by several critical assumptions that often do not hold in real-world markets. These "lethal assumptions" can lead to significant discrepancies between theoretical prices and actual market prices.

Assumptions

You've probably heard the following points 100 times before, but it is important to re-iterate the incorrect assumptions. ChatGPT simplified everything to keep the points short and sweet. Thanks, ChatGPT!

1. Constant Volatility

Assumption: The model assumes that the volatility of the underlying asset is constant over the option's life.

Reality: Volatility is dynamic and can change due to market conditions, economic indicators, and company-specific events. Empirical observations, such as volatility clustering and mean reversion, contradict the assumption of constant volatility.

2. Lognormally Distributed Returns

Assumption: Asset returns follow a lognormal distribution, implying that prices can only move in continuous paths without sudden jumps.

Reality: Financial markets exhibit fat tails and skewness, indicating a higher probability of extreme events (both upward and downward) than predicted by a lognormal distribution. Events like market crashes or spikes violate this assumption.

3. No Transaction Costs or Taxes

Assumption: Trading the underlying asset or hedging the option incurs no transaction costs or taxes.

Reality: In practice, buying and selling assets involve costs such as bid-ask spreads, commissions, and taxes. These costs can erode profits and impact the hedging strategy's effectiveness.

4. Continuous Trading and Hedging

Assumption: The model requires continuous trading and the ability to adjust the hedge portfolio instantaneously and infinitely.

Reality: Continuous trading is impractical due to market hours, liquidity constraints, and the physical limitations of executing trades. Hedging strategies are implemented discretely, introducing additional risks.

5. No Dividends

Assumption: The underlying asset does not pay dividends during the option's life.

Reality: Many stocks pay dividends, which affect the option's price. While there are extensions to Black-Scholes that incorporate dividends, the original model does not account for them.

6. Efficient Markets with No Arbitrage Opportunities

Assumption: Markets are efficient, and there are no opportunities for riskless profit through arbitrage.

Reality: Markets are not perfectly efficient, and arbitrage opportunities can and do exist, especially in less liquid or emerging markets.

7. Geometric Brownian Motion

Assumption: The underlying asset's price follows a geometric Brownian motion with constant drift and volatility.

Reality: Asset prices can exhibit jumps, stochastic volatility, and other complex behaviors not captured by geometric Brownian motion.

These assumptions, while simplifying the mathematical framework, create a significant gap between the model's predictions and real market behaviors.

Incorrect Calculations

The mismatch between the Black-Scholes model's assumptions and real-world market conditions leads to various issues that cause the model to provide inaccurate option prices. Here, we explore the primary reasons why Black-Scholes often fail to deliver precise valuations.

1. Volatility Smile and Skew

Issue: The Black-Scholes model predicts that implied volatility is constant across different strike prices and maturities. However, in reality, implied volatility varies with strike price and expiration, forming patterns known as the "volatility smile" or "volatility skew."

Impact: Options with strikes significantly above or below the current price often have higher implied volatilities than predicted, leading to mispricing of out-of-the-money and in-the-money options.

2. Underestimation of Extreme Events (Fat Tails)

Issue: The model assumes lognormally distributed returns, which underestimates the probability of extreme price movements (fat tails).

Impact: During periods of high market stress or rapid movements, Black-Scholes significantly misprices options, as it fails to account for the increased likelihood of large price jumps.

3. Mispricing in Markets with Jumps or Stochastic Volatility

Issue: The model does not account for jumps in asset prices or changes in volatility over time (stochastic volatility).

Impact: Assets that exhibit sudden price movements or volatility shifts lead to option prices that deviate from Black-Scholes predictions, making the model unreliable for such securities.

4. Issues with Discrete Trading

Issue: Black-Scholes assumes continuous trading and hedging, but in reality, trading occurs at discrete intervals.

Impact: Discrete hedging introduces additional risk, known as hedging error, which the model does not account for, leading to inaccurate option valuations.

5. Impact of Liquidity and Transaction Costs

Issue: The original model ignores liquidity constraints and transaction costs.

Impact: In markets with low liquidity or high transaction costs, the model's assumption of frictionless trading is invalid, resulting in significant deviations from actual option prices.

6. Interest Rate Assumptions

Issue: The model assumes a constant risk-free interest rate.

Impact: Fluctuations in interest rates can affect option prices, especially for longer-dated options, leading to mispricing when rates are volatile.

7. Dividend Payments

Issue: The absence of dividends in the model.

Impact: For dividend-paying stocks, the omission leads to inaccuracies in option pricing, as dividends impact the expected future price of the underlying asset.

These factors collectively contribute to the Black-Scholes model's limitations, making it less reliable in accurately pricing options under real market conditions.

Can Black-Scholes Still Be Useful?

Well Yes! Nonetheless, it is still a somewhat effective anti-arbitrage model that allows market makers to price the option in a manner that makes it hard for others to make free money off of them.

Despite its limitations, the Black-Scholes model retains significant value in financial markets. Its continued relevance can be attributed to several factors:

1. Analytical Tractability

Strength: The Black-Scholes model offers a closed-form analytical solution for option pricing, making it computationally efficient and easy to implement.

Benefit: Traders and analysts can quickly estimate option prices without resorting to complex numerical methods, facilitating timely decision-making. (Redundant if modeling/simulating exposure)

2. Foundational Framework

Strength: Black-Scholes serves as the foundation for more advanced option pricing models and financial theories.

Benefit: It provides a common language and starting point for exploring and developing sophisticated models that address its shortcomings.

3. Calibration and Benchmarking

Strength: The model is widely used as a benchmark to compare other pricing models and to calibrate market parameters.

Benefit: Even when more accurate models are employed, Black-Scholes provides a reference point to evaluate deviations and improvements.

4. Educational Value

Strength: Black-Scholes is integral to the education and training of financial professionals, offering fundamental insights into option pricing and risk management.

Benefit: Understanding the model's mechanics and limitations equips practitioners with the knowledge to apply it appropriately and recognize when adjustments are necessary.

5. Extensions and Adaptations

Strength: The Black-Scholes framework can be extended to incorporate additional factors such as dividends, stochastic volatility, and interest rates.

Benefit: These adaptations enhance the model's applicability, making it more aligned with real-world market conditions.

Extensions and Adaptations to Improve

Although our goal here isn't to find a way to make Black Scholes perfect, I want to reiterate the importance of mentioning our ability to leverage adaptations in Black Scholes.

We can HELP (not solve) fix a few real-world assumptions by adding adaptations.

Liquidity Constraints

I have formulated an ADJUSTED model to show how we can adapt to liquidity constraints. All we need to do is simply modify the traditional Black-Scholes equation by adjusting for bid-ask spread, market impact, and transaction costs. Here’s the resulting model:

$$C_{\text{liq}}(S, t) = \left[ S_0 \left( 1 + \lambda \cdot TS \right) N(d_{1,\text{adj}}) - K e^{-r(T-t)} N(d_{2,\text{adj}}) \right] - TC$$

Where:

$$C_{\text{liq}}(S, t) = \text{Liquidity-adjusted call option price}$$

$$TC = C_{\text{liq}} \cdot TS \cdot CR$$

In the example:

• ​S₀ is the current price of the underlying asset.

• λ is the market impact coefficient.

• 𝑇𝑆 is the trade size but as a fraction of market liquidity.

• 𝐶𝑅 is the cost rate but as a percentage of the trade value.

• 𝑁(⋅) is the cumulative distribution function of the standard normal distribution.

Adjusted Terms:

$$d_{1,\text{adj}} = \frac{\ln(S_0^{\text{adj}} / K) + \left( r + \frac{1}{2} \sigma_{\text{adj}}^2 \right)(T-t)}{\sigma_{\text{adj}} \sqrt{T-t}}$$

$$d_{2,\text{adj}} = d_{1,\text{adj}} - \sigma_{\text{adj}} \sqrt{T-t}$$

Definitions and Adjustments

Bid-Ask Spread: Adjusted Volatility

The bid-ask spread widens the effective volatility experienced by the trader, reflecting the difference between the buying and selling price.

$$\sigma_{\text{adj}} = \sigma (1 + BAS)$$

Where:

• 𝐵𝐴𝑆 is the bid-ask spread as a percentage.

Market Impact: Adjusted Spot Price

The spot price is impacted by large trades, making the effective price higher or lower than expected.

$$S_0^{\text{adj}} = S_0 \left( 1 + \lambda \cdot TS \right)$$

Where:

• λ is the market impact coefficient.

• 𝑇𝑆 is the trade size but as a fraction of market liquidity.

Transaction Costs

Transaction costs are proportional to the option price and the trade size.

$$TC = C_{\text{liq}} \cdot TS \cdot CR$$

Where:

• 𝐶𝑅 is the cost rate but as a percentage of the trade value.

Incorporating Adjusted Terms into Black-Scholes

The adjustments modify the traditional Black-Scholes 𝑑₁ and 𝑑₂​ terms to account for liquidity effects. The liquidity-adjusted call option price C_𝑙𝑖𝑞(𝑆, 𝑡) incorporates these factors, providing a more nuanced and accurate option valuation in markets where liquidity cannot be ignored.

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Deeper Analysis of Adaptions

The Liquidity-Adjusted Black-Scholes model introduces several modifications to the original Black-Scholes framework, when we add these adaptations, we can clearly show its versatility and applicability in real-world trading environments. These modifications are designed to incorporate liquidity constraints that significantly impact option pricing. But, with these adaptations, must also come to the understanding that our goal still is not to effectively price an option, but rather improve the framework which provides simple implementation for other purposes.

1. Bid-Ask Spread

Impact on Volatility:

The bid-ask spread reflects the difference between the price at which traders can buy (ask) and sell (bid) an asset. A wider spread indicates lower liquidity and higher trading costs, effectively increasing the volatility experienced by traders.

$$\sigma_{\text{adj}} = \sigma \times (1 + BAS)$$

Explanation:

• σ is the original volatility.

• 𝐵𝐴𝑆 is the bid-ask spread expressed as a percentage.

• The adjusted volatility σ_adj increases with a wider spread, capturing the additional uncertainty and cost introduced by the spread.

2. Market Impact

Effect on Spot Price:

Large trades can significantly impact the underlying asset's price, especially in markets with lower liquidity. The liquidity-adjusted model accounts for this by modifying the spot price based on trade size.

$$S_0^{\text{adj}} = S_0 \times (1 + \lambda \cdot TS)$$

Explanation:

• S₀ is the original spot price.

• λ is the market impact coefficient, representing the sensitivity of the asset's price to trade size.

• 𝑇𝑆 is the trade size as a fraction of market liquidity.

• The adjusted spot price S₀^adj​ increases with larger trades, reflecting the price movement caused by significant buying or selling pressure.

3. Transaction Costs

Reduction in Option Price:

Transaction costs encompass fees, commissions, and other costs associated with trading. These costs are proportional to the option price and trade size, reducing the overall profitability.

$$TC = C_{\text{liq}} \times TS \times CR$$

Explanation:

• C_𝑙𝑖𝑞 is the liquidity-adjusted call option price.

• 𝑇𝑆 is the trade size as a fraction of market liquidity.

• 𝐶𝑅 is the cost rate, representing transaction costs as a percentage of the trade value.

• Transaction costs 𝑇𝐶 are subtracted from the option price, reflecting the real-world expenses involved in trading the option.

4. Adjusted d1d_1d1​ and d2d_2d2​

Incorporation of Adjusted Volatility and Spot Price:

The d₁​ and d₂​ terms in the Black-Scholes formula are recalibrated to incorporate the adjustments for volatility and spot price.

$$d_{1,\text{adj}} = \frac{\ln(S_0^{\text{adj}} / K) + \left( r + \frac{1}{2} \sigma_{\text{adj}}^2 \right)(T - t)}{\sigma_{\text{adj}} \sqrt{T - t}}$$

$$d_{2,\text{adj}} = d_{1,\text{adj}} - \sigma_{\text{adj}} \sqrt{T - t}$$

Explanation:

• d_1,adj and d_2,adj​ are recalculated using the adjusted spot price S₀^adj and adjusted volatility σ_adj.

• These adjustments will help us ensure that the option pricing formula reflects the impact of liquidity on both the asset's price dynamics and the trading costs.

Summary of Modifications

So finally, here is our summary...

Our Liquidity-Adjusted Black-Scholes model integrates the following key modifications:

1. Bid-Ask Spread: Increases the effective volatility to account for the cost of trading.

2. Market Impact: This will adjust the spot price based on the size of the trade relative to market liquidity.

3. Transaction Costs: Deduct costs proportional to the trade size and option price.

4. Adjusted d1d_1d1​ and d2d_2d2​: Recalculates these terms to incorporate the adjusted volatility and spot price.

While not perfect by any means, I am hoping you now understand how we can make slight modifications to enhance the original Black-Scholes model by incorporating factors that reflect REAL market conditions!

Hate the Black Scholes Model? Try this!

The following is only a concept, and I am hoping someone can collaborate on this. I am by no means a math genius and am a professional plug and chugger.

Combining FFT and Quadrature Methods

While the Liquidity-Adjusted Black-Scholes model offers an example of making improvements over the traditional framework, it still relies on analytical solutions that may not capture all complexities of real-world markets. To further enhance pricing accuracy, a theoretical approach involves combining Fast Fourier Transform (FFT) techniques with Quadrature Methods- Yep, like the Longstaff-Schwartz algorithm! This hybrid idea leverages the efficiency of FFT and the flexibility of Quadrature Methods to create a seamless (??maybe lol.) But overall robust option pricing model.

So without further ado, here is what we are working with:

Fast Fourier Transform (FFT)

FFT efficiently converts data between time and frequency domains. In option pricing, it's used to quickly compute prices by leveraging the asset price's characteristic function, especially in complex models.

Quadrature Methods (Longstaff-Schwartz)

Quadrature Methods approximate integrals, and the Longstaff-Schwartz algorithm is specifically for American options. It uses regression to estimate when it's optimal to exercise an option early, making it ideal for options with early exercise features.

Benefits of the Combined Approach

1. Computational Efficiency: FFT accelerates the calculation of integrals, significantly reducing the time required for option pricing, especially in high-dimensional settings.

2. Flexibility: Quadrature Methods can handle a wide range of option types, including American options and those with path-dependent features.

3. Accuracy: The combination ensures precise valuation by accurately capturing the nuances of asset price dynamics and option features.

4. Scalability: This approach can be extended to complex derivatives and multi-asset options, making it suitable for diverse financial instruments.

Here is how we could implement them:

Implementation Strategy

1. Characterize the Asset Price Dynamics: Define the stochastic processes governing the underlying asset's price, incorporating factors like liquidity, jumps, and stochastic volatility.

2. Develop the Characteristic Function: Utilize FFT to compute the characteristic function of the asset price distribution, facilitating the calculation of option prices through Fourier inversion.

3. Employ Quadrature for Integration: Use Quadrature Methods to numerically integrate the option pricing formula, ensuring accurate valuation even for complex option structures.

4. Optimize the Integration Process: Leverage FFT's speed to perform rapid integrations, enhancing the overall efficiency of the pricing model.

Theoretically, this would be running in the background as if we were pricing with a simulation trading system.

Example Implementation

Now for my favorite part. Figuring it out...

Setup

Here is what I did (This is a very simple explanation of what I actually tried to do, but for example purposes, this will suffice): We will use numpy for numerical computations, scipy for integration and Fourier transforms, and matplotlib for plotting.

import numpy as np
from scipy.fft import fft, ifft
from scipy.integrate import quad
from scipy.stats import norm
import matplotlib.pyplot as plt

Defining the Characteristic Function

The characteristic function is (what I figured out) an important component in FFT-based option pricing. For the Black-Scholes model, the characteristic function of the log asset price ln( 𝑆_𝑇 ) under the risk-neutral measure is given by:

$$\phi(u) = \exp\left(iu\left(\ln(S_0) + (r - \frac{1}{2} \sigma^2)T\right) - \frac{1}{2} \sigma^2 u^2 T\right)$$

def characteristic_function(u, S0, K, T, r, sigma):
    return np.exp(1j * u * (np.log(S0) + (r - 0.5 * sigma**2) * T) - 0.5 * sigma**2 * u**2 * T)

Implementing the FFT-Based Option Pricing

In basic terms, FFT can be used to compute the option price by evaluating the integral in the option pricing formula efficiently. So to implement it, here is a simplified version of what I did.

def option_price_fft(S0, K, T, r, sigma, N=4096, alpha=1.5):
    # Define integration range and step size
    eta = 0.25
    lam = 2 * np.pi / (N * eta)
    b = N * lam / 2
    k = -b + lam * np.arange(N)
    
    # Compute characteristic function
    u = np.arange(N) * eta
    cf = characteristic_function(u - (alpha + 1) * 1j, S0, K, T, r, sigma)
    
    # damping factor
    cf = cf * np.exp(-r * T)
    
    # Simpson's rule weights
    Simpson_weights = np.ones(N)
    Simpson_weights[1:N-1:2] = 4
    Simpson_weights[2:N-2:2] = 2
    
    # Compute FFT
    integrand = np.exp(-1j * u * b) * cf * Simpson_weights
    fft_values = fft(integrand).real
    
    # Compute option price
    C = np.exp(-alpha * k) / np.pi * fft_values
    return C

Implementing the Quadrature Method (Longstaff-Schwartz)

For European options, the Longstaff-Schwartz method can be used to perform numerical integration. But, it is important to note that it is more commonly applied to American options. Here, we'll use it to complement the FFT approach for enhanced accuracy.

def option_price_quadrature(S0, K, T, r, sigma):
    # Define the integrand for the Black-Scholes formula
    def integrand(x):
        return np.exp(-0.5 * x**2) * (S0 * np.exp(x * sigma * np.sqrt(T)) - K) * norm.cdf(x)
    
    # Perform numerical integration
    price, error = quad(integrand, -np.inf, np.inf)
    return price * np.exp(-r * T)

Combining FFT and Quadrature Methods

We can now combine both methods to leverage FFT's speed and Quadrature's accuracy!

def combined_option_price(S0, K, T, r, sigma):
    fft_price = option_price_fft(S0, K, T, r, sigma)
    quad_price = option_price_quadrature(S0, K, T, r, sigma)
    # Combine the prices (this is a simplistic combination; when doing it for real, more in-depth weighting will need to be applied)
    combined_price = (fft_price + quad_price) / 2
    return combined_price

Running the Combined Model

Let's run the theory:

# Basic Sample Parameters
S0 = 100    # Current stock price
K = 100     # Strike price
T = 1       # Time to maturity in years
r = 0.05    # Risk-free interest rate
sigma = 0.2 # Volatility

# Now calculate!
fft_price = option_price_fft(S0, K, T, r, sigma)
quad_price = option_price_quadrature(S0, K, T, r, sigma)
combined_price = combined_option_price(S0, K, T, r, sigma)

print(f"FFT-Based Price: {fft_price}")
print(f"Quadrature-Based Price: {quad_price}")
print(f"Combined Price: {combined_price}")

Output:

FFT-Based Price: [-1.73191818e-01 -1.69138491e-01 -1.76492062e-01 ... -7.49317720e-18
 -8.03076077e-18 -7.51324568e-18]
Quadrature-Based Price: 16.086944138688843
Combined Price: [7.95687616 7.95890282 7.95522604 ... 8.04347207 8.04347207 8.04347207]

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How does it compare to Black Scholes?

I created a test to compare the two. You can view the test here. The tests I ran were based on the simplified versions that I showed in this article.

This is how I calculated the accuracy of both models:

• Computed the absolute percentage error of each method relative to the Black-Scholes price. This is done using the formula:

$$\text{Error (\%)} = \left| \frac{\text{Method Price} - \text{Black-Scholes Price}}{\text{Black-Scholes Price}} \right| \times 100$$

• Then I stored the errors in errors_fft, errors_quad, and errors_combined.

Results:

Results

Top Panel: Shows the option prices computed by each method across different strike prices, including the Black-Scholes benchmark.

Bottom Panel: Displays the absolute percentage errors of each method relative to Black-Scholes, allowing for easy visual comparison of accuracy.

As you can see, as of now, there needs to be a few tweaks (e.g. we should not get negative prices) but that is why it is only a concept.

Code Output

----- Option Pricing Comparison at K=100 -----
Black-Scholes Price: 10.4506
FFT-Based Price: -0.0000
Quadrature-Based Price: 16.0869
Combined FFT & Quadrature Price: 8.0435

----- Accuracy (Relative to Black-Scholes) -----
FFT Error: 100.00%
Quadrature Error: 53.93%
Combined Method Error: 23.03%

Interpretation of Results

Below is an in-depth version of the results and some adjustments that could be made.

Click here to see the overview.

or download it:

58KB

Interpretation of Results.pdf

pdf

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Conclusion

As much as I give the BS formula a hard time for being b.s., it is a cornerstone of financial theory. But, it is just that, a theory. Its practical application necessitates careful consideration of its limitations. By embracing advanced methodologies that build upon and enhance the foundational principles of Black-Scholes, financial practitioners can achieve more accurate and reliable option valuations. The future of option pricing lies in the continuous evolution of models that integrate computational efficiency, numerical precision, and real-world market dynamics, ensuring that financial markets remain resilient and adaptive in the face of ever-changing economic landscapes.

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Stay Curious.