Options and fractals
Option pricing is a fundamental concept in the world of finance, essential for both investors and traders. It involves determining the fair value of an option contract, which gives the holder the right to buy or sell an underlying asset at a predetermined price within a specified time frame. The price of an option, known as the premium, is influenced by various factors, including the current price of the underlying asset, the strike price, the time until expiration, volatility, and interest rates. Several mathematical models, such as the Black-Scholes model and the binomial options pricing model, have been developed to estimate option prices based on these factors. Understanding option pricing is crucial for making informed investment decisions and managing risk effectively in the derivatives market.
Table of contents
- What are the options?
What are the options?
Options are financial instruments that give the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price within a specified time frame. They are commonly used for speculation, hedging, and risk management in financial markets.
There are two main types of options: call options and put options.
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Call Options: A call option gives the holder the right to buy the underlying asset at a predetermined price, known as the strike price, before the option’s expiration date.
Let’s illustrate with an example:
Suppose you purchase a call option on Company XYZ stock with a strike price of $100 and an expiration date of one month from now. This means you have the right to buy Company XYZ stock for $100 per share until the option expires. If, at expiration, the price of Company XYZ stock is above $100, say $110, you can exercise your option and buy the stock at $100, even though it’s currently trading at $110. You could then sell it at the market price, earning a profit of $10 per share (minus the cost of the option).
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Put Options: A put option gives the holder the right to sell the underlying asset at a predetermined price before the option’s expiration date.
Let’s use an example to clarify:
Suppose you purchase a put option on Company ABC stock with a strike price of $50 and an expiration date of one month from now. This means you have the right to sell Company ABC stock for $50 per share until the option expires. If, at expiration, the price of Company ABC stock is below $50, say $40, you can exercise your option and sell the stock at $50, even though it’s currently trading at $40. You could then buy it at the market price, earning a profit of $10 per share (minus the cost of the option).
Options can also be categorized based on their expiration date:
- European Options: These options can only be exercised at expiration.
- American Options: These options can be exercised at any time before expiration.
Additionally, options can be classified based on their underlying asset, including stock options, index options, and commodity options, among others.
Some technical terms
Now let’s delve into some additional technical terms commonly associated with options contracts:
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Premium: The premium is the price paid by the option buyer to the option seller (also known as the writer) for the right to buy or sell the underlying asset at the agreed-upon price. The premium is determined by various factors, including the current price of the underlying asset, the strike price, the time until expiration, volatility, and interest rates.
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Strike Price (Exercise Price): The strike price is the price at which the underlying asset can be bought or sold when the option is exercised. It is predetermined and agreed upon by the buyer and seller at the time the option contract is initiated.
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Expiration Date: This is the date when the option contract expires and becomes void. After this date, the option holder no longer has the right to exercise the option. Options can have expiration dates ranging from days to years, depending on the contract specifications.
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In-the-Money (ITM), At-the-Money (ATM), Out-of-the-Money (OTM): These terms describe the relationship between the current price of the underlying asset and the strike price of the option.
- In-the-Money (ITM): For call options, if the current price of the underlying asset is higher than the strike price. For put options, if the current price is lower than the strike price.
- At-the-Money (ATM): When the current price of the underlying asset is equal to the strike price.
- Out-of-the-Money (OTM): For call options, if the current price of the underlying asset is lower than the strike price. For put options, if the current price is higher than the strike price.
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Intrinsic Value and Time Value: The premium of an option can be broken down into two components:
- Intrinsic Value: The difference between the current price of the underlying asset and the strike price, if the option were to be exercised immediately. For example, if a call option has an intrinsic value of $5, it means the underlying asset is $5 above the strike price. Intrinsic value only exists for in-the-money options.
- Time Value (Extrinsic Value): The portion of the premium that is not attributable to intrinsic value. It represents the value of the option’s potential to become more profitable before expiration, taking into account factors such as time decay and implied volatility.
The most famous equation in economics
The Black-Scholes equation, developed by Fischer Black and Myron Scholes in 1973, is a mathematical model used to estimate the theoretical price of European-style options. This equation revolutionized the options market by providing a framework for pricing options and understanding their behavior. The model considers several factors, including the current price of the underlying asset, the strike price, the time until expiration, volatility, and risk-free interest rates. The Black-Scholes equation calculates the fair value of an option by balancing the potential gains from exercising the option early with the costs of waiting until expiration. While the model assumes certain idealized conditions, such as constant volatility and no transaction costs, it remains widely used in practice and has paved the way for further developments in option pricing theory.
The Black-Scholes equation is a partial differential equation used to derive the theoretical price of European-style options. The formula for the Black-Scholes option pricing model is as follows:
Where:
and represent the theoretical prices of a call option and a put option, respectively. is the current price of the underlying asset. is the strike price of the option. is the risk-free interest rate. is the time to expiration of the option. is the current time. is the volatility of the underlying asset. and are cumulative distribution functions of the standard normal distribution. and are calculated as follows:
The formula calculates the theoretical price of options by considering the relationship between the current price of the underlying asset, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the asset. It provides a framework for investors and traders to estimate the fair value of options and make informed decisions in the financial markets.
Brownian motion
At first, Brownian motion may seem a little bit out of context, but it is at the heart of what we are studying in this article.
What is Brownian motion?
Brownian motion, named after the botanist Robert Brown who observed the erratic movement of pollen grains in water in 1827, is a stochastic process used to model random motion in various fields, including physics, finance, and biology. In essence, Brownian motion describes the random movement of gas particles due to collisions with surrounding molecules. It’s a cornerstone of stochastic calculus and plays a crucial role in the modeling of diverse phenomena.
How can we define Brownian motion?
Let’s think about the movement of the gas molecule during a small time interval from time
In continuous Brownian motion, the position of a particle at any given time 𝑡 is described by a stochastic process 𝐵(𝑡) that follows a normal distribution with mean 0 and variance
Geometric Brownian motion
Let’s finally understand how Brownian motion relates to the Black-Scholes equation.
The Black-Scholes model relies on the assumption that the price movement of the underlying asset follows a geometric Brownian motion. Geometric Brownian motion is a continuous-time stochastic process where the logarithm of the asset’s price follows a Brownian motion with drift and volatility.
In the context of the Black-Scholes model, geometric Brownian motion is used to model the random fluctuations in the price of the underlying asset over time. This stochastic process helps capture the uncertainty and variability in the asset’s price movement, which is essential for pricing options.
Specifically, the geometric Brownian motion is utilized in the calculation of the option’s price through the derivation of
The geometric Brownian motion assumption allows the Black-Scholes model to incorporate key factors such as drift (captured by the risk-free interest rate 𝑟) and volatility (𝜎) into the pricing of options.
Geometric Brownian Motion (GBM) is a stochastic process used to model the dynamics of asset prices in finance, particularly in options pricing. It’s characterized by the following stochastic differential equation (SDE):
Where:
is the price of the asset at time . is the drift coefficient, representing the expected return of the asset per unit of time. is the volatility coefficient, representing the standard deviation of the asset’s returns per unit of time. is a Wiener process or Brownian motion.
GBM assumes that the asset’s returns are normally distributed and that the percentage change in the asset price is independent of its current value. Ideally, this makes it a suitable model for stock prices over short intervals of time. Nevertheless, these assumptions are easily challenged in the real world.
The Black-Scholes formula is a mathematical model used for pricing options contracts. It’s based on several assumptions, one of which is that the underlying asset follows a geometric Brownian motion. The formula for pricing a European call option using the Black-Scholes model is:
Where:
is the price of the call option at time . is the current price of the underlying asset. is the strike price of the option. is the risk-free interest rate. is the time to expiration of the option. is the cumulative distribution function of the standard normal distribution. and are defined as:
The connection between GBM and the Black-Scholes formula lies in the assumption that the underlying asset follows GBM, which allows for the derivation of the formula through the application of stochastic calculus and the principles of no-arbitrage pricing. By assuming GBM, Black and Scholes were able to derive a closed-form solution for the price of a European call option, which revolutionized the field of quantitative finance and options trading.
Well, so that’s it? No, not at all.
Black-Scholes GBM often doesn’t capture fundamental features of the underlying asset and in addition it also assumes constant volatility that is almost never true in the real world.
What has a hydrologist designing a dam in Egypt to do with option pricing?
In Egypt, Britain’s former colony, hydrologist Harold Edwin Hurst (1880-1978) was tasked to come up with an answer to the following critical question: “What are the ideal dimensions of a dam in the Nile?” Hurst studied numerous time series on water levels (Hurst, 1952) in order to strike a good balance for the following trade-off. If you build it too high, you waste huge amounts of resources; whereas if you build it too low, flooding can lead to huge human and economic disasters. For an anedoctodal account of the story, see Mandelbrot and Hudson (2010). 1
Whilst doing extensive quantitative analysis, Hurst recognised a lot of variability in the levels, where years of large changes of either sign - extreme droughts or flooding - tended to follow each other. The problem boiled down to how one could model this variability and come up with measures to describe the unpredictability. What is the level of persistence or anti-persistence in the water level data? This was very relevant for the dam issue since the level of anti-persistence or unpredictability in the water levels directly relates to the need for ‘overdimensioning’ or safety buffers in terms of the dam’s dimensionality (Hurst, 1956). The reader will probably recognize that the previous problem is a similar trade-off to market risk and capital requirements at financial institutions, with the water levels as PnL and the dam’s dimensions as a capital buffer. Borrowing from Hurst’s work, how can we model the persistence versus antipersistence of financial time series? The answer is rescaled-range analysis (or R/S analysis). It was initially developed by Hurst and later rediscovered by Mandelbrot (Mandelbrot, 2002). It tries to come up with a measure for the persistence of a time series in another way than just regressing the water level on lagged values. R/S analysis uses a power law instead: how does the rescaled range behave for different scales?
Hurst Exponent
The Hurst exponent, named after British hydrologist Harold Edwin Hurst, serves as a crucial measure in quantifying the long-term memory of a time series. In this article, we will delve into its mathematical definition, explore different algorithms to calculate it and elucidate its significance in understanding time series data.
Mathematical Definition:
The Hurst exponent, denoted as 𝐻H, characterizes the persistence or tendency of a time series to exhibit long-range dependence. For a given time series 𝑋X, the Hurst exponent is calculated using the rescaled range (R/S) analysis method. The formula for calculating the Hurst exponent is as follows:
Where:
represents the range of the first 𝑛 data points, denotes the standard deviation of the time series over the same span, denotes the expected value.
How to calculate it?
Rescaled-Range Analysis, often abbreviated as R/S Analysis, is a fundamental technique in time series analysis used to estimate the Hurst exponent, a measure of long-term memory or persistence in a time series. This method, pioneered by Harold Edwin Hurst in the early 20th century, offers insights into the underlying dynamics of complex time series data.
Methodology:
The R/S analysis method involves several key steps:
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Segmentation: The time series data is divided into segments of varying lengths. These segments can range from short intervals to the entire length of the time series, allowing for the exploration of different temporal scales.
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Calculating the Rescaled Range (R/S): For each segment length, the rescaled range 𝑅/𝑆 is computed. The rescaled range is defined as the difference between the maximum and minimum values of the cumulative sum of the data, divided by the standard deviation of the data within the segment. Mathematically, it can be expressed as:
Where:
- 𝑅 represents the range (i.e., the difference between the maximum and minimum values) of the cumulative sum of the data within the segment.
- 𝑆 denotes the standard deviation of the data within the same segment.
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Plotting the Rescaled Range: The rescaled range 𝑅/𝑆 is plotted against the segment length on a log-log scale. Each data point on the plot corresponds to a different segment length.
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Estimating the Hurst Exponent: The slope of the log-log plot represents the scaling behavior of the rescaled range with respect to segment length. According to the R/S analysis method, this slope provides an estimate of the Hurst exponent (𝐻).
Meaning of the Exponent:
The value of the Hurst exponent holds significant implications for the underlying dynamics of a time series:
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: Indicates persistent long-term memory or positive autocorrelation. In financial markets, a high Hurst exponent suggests trends may continue for an extended period, potentially indicating momentum or trending behavior. -
: Implies a random walk or no long-term memory. This value is often associated with efficient markets where past price movements do not predict future movements. -
: Suggests anti-persistent behavior or negative autocorrelation. In financial contexts, a low Hurst exponent may indicate mean-reverting behavior, where prices tend to revert to their mean.
Fractals and roughness
At first glance, it may not seem so obvious how hurst exponent, fractals and option pricing relate to each other. I hope that by the end of this chapter, it’ll be a lot more clear.
However, before going on this road let’s review what fractals are.
To introduce fractal geometry we have to talk about Benoît B. Mandelbrot. He is the father of fractal mathematics and not only an incredible mathematician but also an incredible observer. To get you an understanding of what fractals are I’ll use his own words:
“Bottomless wonders spring from simple rules… which are repeated without end”
In the realm of mathematics and natural phenomena, fractals represent a fascinating concept that transcends traditional geometric shapes. These intricate structures exhibit self-similarity across different scales, meaning that as you zoom into a fractal, you encounter similar patterns repeating at increasingly fine levels of detail. Understanding fractals requires grasping the notion of fractal dimensions, a measure of their complexity. Interestingly, the Hurst exponent, a key metric in time series analysis, shares an intimate relationship with fractals and their dimensions.
Fractals:
Fractals, coined by mathematician Benoit Mandelbrot in the 1970s, are geometric shapes or structures that display self-similarity at all scales. Unlike conventional Euclidean geometry, where shapes have integer dimensions (e.g., lines are 1-dimensional, planes are 2-dimensional, and solids are 3-dimensional), fractals possess fractional or non-integer dimensions. This property allows fractals to represent complex natural phenomena such as coastlines, clouds, and even financial markets.
Fractal Dimensions:
Fractal dimensions quantify the complexity or “roughness” of fractals, capturing their self-similar patterns across different scales. Unlike Euclidean dimensions, which are integers, fractal dimensions can take on fractional values. For example:
- A smooth line in traditional geometry has a dimension of 1.
- A plane, like a piece of paper, has a dimension of 2.
- A fractal, such as the famous Koch snowflake or the Mandelbrot set, can have dimensions between integers, indicating their intricate, self-similar structures.
Fractal dimensions provide a more nuanced understanding of complex geometric shapes, allowing us to quantify their irregularity and self-similarity.
Mathematically the fractal dimension
There are different algorithms to calculate this dimension for a time series but the most famous are:
- Higuchi’s algorithm
- Katz’s algorithm
It is very interesting to note that Mandelbrot has proved that, the local properties are reflected in the global ones, resulting in a link between the fractal dimension
Therefore you can estimate the Hurst exponent with good approximation as
Fractional Brownian Motion
Let’s get to the core of this dissertation.
Fractional Brownian Motion (fBm) is a stochastic process that has gained significant attention in various fields, including finance. It is an extension of the classical Brownian Motion. However, fBm differs in that it incorporates a fractal dimension, allowing it to model long-range dependence and exhibit self-similarity over different time scales.
Formally, a fractional Brownian motion
(it starts at 0). - It has stationary increments: for any
, the increment follows a Gaussian distribution with mean and variance . - The Hurst parameter
governs the level of roughness or smoothness of the path, with .
The key feature of fBm is the Hurst parameter
Traditional option pricing models, such as the Black-Scholes model, assume that asset prices follow a geometric Brownian motion, which implies constant volatility and independent increments. However, empirical evidence suggests that asset returns often exhibit volatility clustering and long memory effects, which cannot be captured by the classical Brownian motion.
By incorporating fBm into option pricing models, such as the Fractional Black-Scholes model or the Heston model with fractional volatility, practitioners can better capture the dynamics of asset prices and volatility. This allows for more accurate pricing of options, especially in markets where volatility is non-constant and exhibits long-range dependence.
Additionally, fBm provides insights into the behaviour of financial markets over different time horizons. By estimating the Hurst parameter from historical data, analysts can assess the level of persistence in asset prices and adjust their pricing models accordingly. This flexibility is crucial for risk management and portfolio optimization, as it allows investors to better understand and hedge against potential market movements.
Beyond the classic Black-Scholes equation
As we have seen the Black-Scholes equation has lots of limitations. However, the fractal view of the world may give us some very powerful tools to improve our models. In fact, at first, we may just replace the GBM with the fBm, by doing this we should better capture the dynamics of the asset but the pre-estimated constant volatility term in the equation remains a problem.
The fundamental SDE of this model is:
where:
is the value of the process at time 𝑡t. is the drift coefficient. is the volatility coefficient. is a fractional Brownian motion process governed by the Hurst exponent
In order to solve the constant volatility problem, we may slightly modify the Black-Scholes equation using as variance the variance of the generated fBm. However, it may not be a good idea given that in the Black-Scholes equation, the variance refers to instantaneous volatility while the variance of the fBm would refer to a more comprehensive volatility of the entire stock. To more rigorously assess if this simplified model is really effective requires a lot of work.
Anyway, either the first as well as the second model, would not take into account the variance of the variance, or as we call it a second layer of stochasticity. We therefore propose a third model, probably the best one. This third model includes a stochastic volatility model based on roughness that doing so takes into consideration the second layer of stochasticity.
The fundamental SDE of this final model is:
Let’s review them step by step:
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Stochastic Differential Equation for Asset Price (
): - Here,
represents the asset price at time 𝑡. represents the stochastic volatility at time 𝑡. is a fractional Brownian motion with Hurst parameter 𝐻, which characterises the long-term memory and roughness of the process.
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Volatility Dynamics:
- This equation defines the logarithm of the square root of the volatility process as a function of another process
. This is often done to ensure the positivity of volatility.
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Stochastic Differential Equation for
: represents the logarithm of the square root of the volatility process. determines how the volatility is influenced by the Brownian motion is the differential of a fractional Brownian motion process. is a long-term mean value towards which reverts.
This particular model is called “Rough Fractional Stochastic Volatility” (RFSV) model 2. We’ll not delve into the mathematical details behind this model because it would require mathematics of extraordinary complexity. Indeed the main focus of this article is the intuition behind it and how it works.
The RFSV model better catches the dynamics of the underlying asset taking into account a second layer of stochasticity through a fractal view of the world. The SDE describing this model can be merged into the Black-Scholes equation despite this time we may not be able to solve it analytically but through numerical methods due to its complexity.
This model, and all the enhancements in this field, allow for better pricing of and option or any other kind of asset. They have tons of applications that range from improved risk engineering to optimised trading of energy commodity derivatives.