Github available with all the Pyhton code and applications: __https://github.com/GioPess/Riot-Quantitative-research/blob/main/Levy%20Processes/levyprocesses.ipynb__

1. Theoretical Foundations

Definition of terminologies by Borsa Italiana:

Definition 1.1 Derivatives: ”Derivative financial instruments are contracts whose value depends on the performance of an underlying asset, also known as an underlying asset. The underlying assets can be financial (such as stocks, interest and exchange rates, indices) or real (such as coffee, cocoa, gold, oil, 1 etc.). Derivative financial instruments can be symmetric or asymmetric. In the first case, both parties (buyer and seller) undertake to perform a service on the maturity date, conversely, in asymmetric derivatives, only the seller is obliged to satisfy the buyer’s will. In asymmetric derivatives, in fact, the buyer, by paying a price (called a premium), acquires the right to decide at a future date whether or not to buy and sell the underlying asset. A further distinction concerns derivatives traded on regulated markets and over-the-counter (OTC) derivatives. The former are contracts whose characteristics are standardised and defined by the authority of the market on which they are traded; These characteristics relate to the underlying asset, the duration, the minimum trading size, the liquidation modalities, etc. In Italy, the regulated derivatives market is called IDEM and is managed by Borsa Italiana SpA (there is also the SeDeX market on which securitized derivatives are traded). Instruments such as futures, options, warrants, covered warrants and ETFs circulate on the regulated market. OTC derivatives, on the other hand, are traded bilaterally (directly between the two parties) outside regulated markets; In this case, the contracting parties are free to determine all the characteristics of the instrument; Generally these are swaps and forwards. The main purposes associated with the trading of derivative financial instruments are the following: - Hedging: This is intended to protect the value of a position from undesirable changes in market prices. The use of the derivative instrument makes it possible to neutralize the adverse market trend, balancing the losses/gains on the position to be hedged with the gains/losses on the derivatives market; - speculation: strategies aimed at making a profit based on the expected evolution of the price of the underlying asset; - arbitrage: when a momentary mismatch between the price of the derivative and that of the underlying asset (intended to coincide at the time of expiry of the contract) is exploited, selling the overvalued instrument and buying the undervalued one and thus obtaining a risk-free profit.”

Definition 1.2 Arbitrage: ”Arbitrage activity concerns the simultaneous execution of transactions for the purchase/sale of financial instruments on different markets, at prices that are temporarily misaligned. Arbitrage transactions typically involve (i.) opening long or short positions on certain securities, such as stocks or bonds, (ii.) opening a deposit or taking out a loan, and (iii.) opening long or short positions in derivative instruments. Arbitrage transactions can also be carried out when the same security is listed on several markets and the prices of the different markets are not aligned (also taking into account the transaction costs necessary to carry out the transaction). Arbitrage trades are risk-free as they offer the possibility of making a certain profit (also considering the opportunity cost of the money used in the trade). Of course, the price mismatch conditions that give rise to arbitrage opportunities are only likely to last for brief moments, as arbitrageur activity tends to quickly bring prices back to equilibrium levels.”

Definition 1.3 Hedging: ”A hedging transaction is carried out through the purchase or sale of one or more derivative contracts (forward contracts, futures, swaps or options) whose value depends on the same source of risk that influences the value of the position to be hedged, or on a related source of risk. The logic of hedging is as follows: any increase/decrease in the value of the asset to be hedged may be fully or partially offset by a decrease/increase in the value of the derivative contract. For example, if a trader wants to hedge against falling share prices, he can sign a derivative contract (futures or option) in which the final payoff increases proportionally to the decrease in stock market prices. Depending on whether the trader buys or sells the underlying, the hedge is defined as ”long” or ”short”. The amount of derivatives that need to be subscribed to carry out the hedging transaction depends on the ”hedging ratio”, i.e. an indicator that relates the sensitivity of the asset to be hedged and the sensitivity of the derivative to the source of risk. Covers can be static or dynamic. In the first case, once the hedging operation has been set up, the operator will not have to modify it until it expires; In the second case, on the other hand, the hedging operation required a series of periodic portfolio adjustments. To set up hedging operations, operators refer to the so-called Greeks and in particular to delta, gamma and vega.”

2. Introduction to Stochastic Processes

Let (Ω, F, P) be a probability space and I a real interval of the form [O, T] or R≥0.

Definition 2.1 A measurable stochastic process on Rn is a collection (Xt)t∈I of random variables with values in Rn such that the map

X : I × Ω → Rn , X(t, ω) = Xt(ω)

is measurable with respect to the product σ-algebra B(I) ⊗ F.

A stochastic process can be used to describe a random phenomenon that evolves in time: we can interpret a positive random Xt as the price of a risky asset at time t.

Definition 2.2 Let (Ω, F, P,(Ft)) be a filtered probability space. A real Brownian motion is a stochastic process W = (Wt)t≥0 in R such that

i) W0 = 0 a.s.;

ii) W is Ft-adapted and continuous;

iii) for t < s ≤ 0, the random variable Wt − Ws has normal distribution N0,t−s and is independent of Fs.

Definition 2.3 A geometric Brownian motion is a solution of the stochastic differential equation

where µ, σ ∈ R, i.e. it is a stochastic process S ∈ L 2 such that

2.1 Black-Scholes model (BSM)

In the Black-Scholes model the market consists of a non-risky asset, a bond B and of a risky asset, a stock S. The bond price verifies the equation

where r is the short-term (or locally risk-free) interest rate, assumed to be a constant. Therefore the bond follows a deterministic dynamics: if we set B0 = 1, then Bt = e^rt. The price of the risky asset is a geometric Brownian motion, verifying the equation

where µ ∈ R is the average rate of return and σ ∈ R > 0 is the volatility.

Mathematical finance has been a very active area of research since the derivation of the Black, Merton, and Scholes (BMS) equation and the related model of options valuation in 1973. However, the assumption of constant volatility predicted in BMS was obviously violated by observation on the market of nonconstant volatility over different time horizons and strike prices. To explain this phenomenon, called ’volatility smile/skew’, the researchers have proposed numerous alternatives to the BMS theory. Such attempts can be grouped into three classes:

- Local volatility model: these models assume that the diffusion coefficient of the underlying asset is no longer a constant value but instead a deterministic function of time and of the underlying asset itself: σ = σLV (s, t).

- Stochastic volatility model: In this class of models the volatility itself is considered to be a stochastic process with its own dynamics. Thus, this is a two-factor model, driven by two correlated Wiener processes Wt and Zt.

- Jump model: Introduced by Merton these models considers the underlying asset to follow a Levy process with a drift, a diffusion and a jump term;

3 Introduction to Levy processes

In the field of mathematical finance, you cannot think of analyzing and understanding the models for the valuation of stocks, bonds, derivatives, interest rates, etc. without having basic knowledge of stochastic processes, stochastic integration and probability theory. Financial circles have therefore become a professional outlet, not only for economists but also for graduates in technical-scientific disciplines (e.g. mathematics/physics). The need to solve problems in the economic field, in particular for the valuation and hedging of derivatives, has led, and leads, to the search for models that are able to represent, always with better approximation, the real data. Classical models, based on Brownian motion, show discrepancies. First of all, the sigma parameter (σ) cannot be observed directly and subsequently, empirical studies have shown that the distribution of stock prices shows heavy tails, incompatible with the Gaussian model, as predicted by the Black-Scholes-Merton equation. Such models are replaced by models based on Levy processes, precisely because the latter give a more realistic representation of price movement: abrupt price changes, caused by unexpected economic collapses, political changes or natural disasters, can be better represented by processes with jumps. In fact, the novelty brought by Levy, to overcome the imperfections of Brownian geometric motion, lies in the fact that the log-returns of a stock can make jumps at a given instant (which those of Brownian motion cannot do because they are linked to the Wiener process which is continuous) and that perhaps volatility in turn follows a stochastic process. For these processes, in particular for Levy processes, stochastic integration according to Itˆo will not suffice, but we will also have to consider integration with respect to random measures. In fact, we will consider integer random measures associated with the leaps of a L´evy process. Taking advantage of the fact that the jumps of a Levy process are concentrated on totally inaccessible stopping times, it will suffice to define random measures concentrated on totally inaccessible stopping times. This article aims to provide a comprehensive introduction to L´evy processes, clarifying their fundamental properties and mathematical structures. In addition to the theoretical exposition, we will move on to the practical demonstration, also providing the python codes (soon they will also be uploaded on github)

3.1 Property:

A Levy process is a stochastic process that follows a random walk with an infinitely divisible one-step distribution. The most fundamental Levy processes are the Brownian Motion and the Poisson process. In this section discuss Levy processes, which are the continuous time counterpart of a random walk. Let us assume that Xt follows a discrete time random walk

where we purposely emphasize that the strong white noise ϵt →t +1 pertains to the time interval [t, t + 1] and where we represent the distribution of the white noise ϵt→t+1. Note that the distribution f1 does not depend on the specific time t because of white noise is identically distributed across time. Also, note that the distribution f1 can feature skewness, multi modality, heavy tails, etc. Consider the aggregate white noise over a time step which is a multiple integer of the unit step:

Each term in the sum is i.i.d., so is the aggregate white noise. Then, it follows that non-overlapping instances of the aggregate white noise represent a white noise in it own right

for a suitable ∆t-step distribution f∆t. If the unit-step distribution f1 has finite variance and step ∆t is large, then the central limit theorem states that the ∆t-step distribution f∆t is approximately normal.

3.2 Infinite divisibility What if the time step ∆t in is not an integer multiple of the unit step 1? For instance, suppose that we want to project the one-week distribution f1 to a one-month distribution f∆t, where ∆t = 30/7. As another example, suppose that we want to ”inject” a binned f1 one-minute distribution for high frequency trading to a ten-second ∆t-step distribution f∆t, where ∆t = 1/6.

The assumption that ∆t is an integer does not seem to play any role. Indeed, it is always possible to apply as in sequence the Fourier transform ℑ, an arbitrary non-integer power ∆t, and the inverse Fourier transform ℑ^−1 to any unit-step pdf f1 and obtain function f∆t. The problem is that if ∆t is not a positive integer the resulting function f∆t is not a true pdf, i.e. a positive function that integrates to one, unless f1 is infinitely divisible.

A Levy process is a process Xt that follows a random walk with an infinitely divisible one-step distribution f1:

Levy process are well defined for any time t ∈ R. Hence, Levy processes as such are are the natural continuous-time generalization of the random walk. Indeed, if the one-step distribution is infinitely divisible, we can re-express the one-step random walk as the sum of i.i.d. white noise ϵt→t+dt = Xt+dt − Xt over time steps of infinitesimal lenght dt, whic we denote more conventionally as a stocastich diffrential equation (SDE):

which is the reason for the nomenclature ”infinitely divisible”. The distribution fdt of the dt-step noise is determined by ”injection” of the one-step distribution f1.

Then, we can proceed as in to obtain the ∆t-step noise ϵt→t+∆t = Xt+∆t - Xt as an infinite sum of infinitesimal (or integral of) i.i.d. variables:

4 Applications in Finance

4.1 Asset price modeling

In the realm of financial modeling, traditional approaches, such as the Black-Scholes model, predominantly utilize geometric Brownian motion to simulate asset price movements. While widely adopted, these models often fall short in accurately representing market behaviors, particularly when it comes to capturing extreme events and sudden price jumps. The introduction of Levy processes in asset price modeling addresses these shortcomings, offering a more comprehensive and realistic approach to understanding financial markets.

4.1.1 The shortcomings of geometric Brownian motion

Geometric Brownian motion, a cornerstone of conventional financial models, is predicated on the assumption that asset returns follow a normal distribution and exhibit time-independent characteristics. However, this assumption frequently diverges from actual market dynamics. Key limitations include the inability to account for leptokurtosis, where return distributions exhibit pronounced heavy tails, and volatility clustering, a phenomenon where periods of high volatility tend to cluster together. These features are commonplace in financial markets but are inadequately represented in the traditional Brownian framework.

4.1.2 Advantages of Levy processes in financial modeling

Levy processes offer significant enhancements over traditional models by incorporating elements that are more reflective of real-world financial market conditions. Key attributes include:

Incorporation of Jumps: Levy processes can model sudden and significant changes in asset prices, aligning more closely with the sporadic nature of financial markets.

Heavy-Tailed Distributions: These processes are better suited to model distributions with fat tails, providing a more accurate representation of the probability of extreme events in asset returns.

Flexibility and Adaptability: Levy processes allow for a more adaptable modeling framework, capable of fitting a wide range of empirical data and capturing the complex dynamics of asset returns.

Enhanced Correlation Structure: They offer a refined approach to understanding the correlation structures between different assets, an aspect crucial for portfolio optimization and risk management.

By integrating Levy processes, financial models become more robust, capable of capturing the intricacies and anomalies prevalent in real-world markets. This advancement aids in more accurate risk assessment, portfolio management, and strategic decision-making in finance.

4.2 Risk management

Risk management in finance is a critical discipline focused on identifying, assessing, and mitigating potential risks that can affect investments and financial activities. Traditional risk management models often rely on assumptions of normal distribution and predictable market behavior, which can lead to significant underestimation of risks, especially in volatile markets. The integration of Levy processes into risk management strategies offers a more nuanced and effective approach to handling financial uncertainties.

4.2.1 Limitations of conventional risk management models

Traditional models in risk management, such as Value at Risk (VaR) and other standard methodologies, typically assume a Gaussian distribution of asset returns. This assumption can be problematic because:

Underestimation of Extreme Events: These models often fail to account for the high likelihood of extreme market movements, leading to inadequate preparation for market crises.

Oversimplification of Market Dynamics: The assumption of linear and normal market movements oversimplifies the complex nature of financial markets.

4.2.2 Incorporating Levy processes for robust risk assessment

Levy processes, with their ability to model non-normal distributions and sudden market jumps, provide a more comprehensive framework for risk management:

Capturing Extreme Market Events: By accommodating heavy tails in the distribution, Levy processes allow for a more accurate estimation of the risks of extreme market events, enhancing stress testing and scenario analysis.

Dynamic Risk Modeling: They enable the modeling of risks in a more dynamic and realistic manner, acknowledging the non-linear and unpredictable nature of markets.

Improved Tail Risk Measurement: The incorporation of Levy processes aids in better tail risk measurement, crucial for understanding and preparing for potential severe losses.

Enhanced Portfolio Optimization: By providing a more accurate depiction of risk, these processes allow for more effective portfolio optimization strategies, balancing returns against the potential for extreme losses.

Incorporating Levy processes into risk management not only provides a more realistic understanding of market dynamics but also equips financial professionals with the tools to better anticipate and mitigate potential risks. This approach leads to more informed decision-making and robust risk management strategies in the unpredictable world of finance.

4.3 Option pricing

Option pricing is a fundamental aspect of financial engineering, involving the determination of fair values for options – financial derivatives that provide the right, but not the obligation, to buy or sell an asset at a predetermined price. Traditional methods of option pricing, such as the Black-Scholes model, often rely on simplistic assumptions about market behavior, which can lead to inaccuracies in real-world scenarios. Integrating Levy processes into option pricing models presents a more sophisticated and accurate approach, particularly for options with complex features or in volatile markets.

4.3.1 Limitations of traditional option pricing models

The Black-Scholes model and similar traditional methods are built on the premise of geometric Brownian motion, which assumes a log-normal distribution of asset prices and constant volatility. However, these assumptions have inherent limitations: Inadequate Handling of Market Anomalies: Traditional models struggle to account for leptokurtosis (heavy-tailed distributions) and volatility clustering, common in financial markets. Oversimplification of Asset Price Movements: By assuming a continuous and smooth price movement, these models overlook the possibility of sudden jumps or drops in prices.

4.3.2 Advantages of Levy processes in option pricing

Incorporating Levy processes into option pricing models brings several key benefits:

Accurate Modeling of Price Jumps: Levy processes allow for the modeling of abrupt changes in asset prices, aligning the theoretical pricing of options more closely with observed market behaviors.

Handling Heavy-Tailed Distributions: These processes are adept at modeling the heavy tails of asset return distributions, providing a more realistic assessment of the probabilities of extreme price movements.

Enhanced Volatility Modeling: Levy processes offer a framework for incorporating varying levels of volatility, a critical factor in option pricing. Flexibility for Complex Options: For options with path-dependent features or those in highly volatile markets, L´evy processes provide a more flexible and accurate pricing tool.

By utilizing Levy processes in option pricing models, financial practitioners can achieve a more nuanced and precise valuation of options. This approach is particularly advantageous in environments characterized by rapid market shifts or when dealing with complex derivative products, ensuring more reliable and efficient financial strategies.

4.3.3 Option pricing models based on Levy processes

Merton Jump-Diffusion Model - The Merton Jump-Diffusion model is an extension of the classic Black-Scholes model, incorporating random jumps in asset prices alongside the continuous Brownian motion. This model addresses one of the key limitations of traditional models – their inability to account for sudden, significant changes in asset prices. Key features include:

Modeling of Sudden Price Jumps: It integrates jump processes, allowing for the simulation of sudden and large movements in asset prices, which are often observed during market shocks or significant news events.

Flexibility in Volatility and Price Dynamics: The model provides a more flexible approach to volatility, acknowledging that asset price movements are not always smooth and continuous.

Improved Accuracy for Short-Term Options: The model is particularly effective for pricing short-term options where the likelihood of price jumps is higher.

Variance gamma model - The Variance Gamma model, another application of Levy processes, further refines option pricing by introducing a process that captures both the jumps and the ’fat tails’ characteristic of asset return distributions. Its highlights include:

Handling of Heavy-Tailed Distributions: This model is adept at capturing the leptokurtosis observed in asset return distributions, offering a more accurate reflection of the risk of extreme price movements.

Versatility in Modeling Price Changes: By accommodating both small and large price changes, the Variance Gamma model offers a comprehensive view of potential market movements.

Applicability to a Range of Financial Instruments: Its robustness makes it suitable for pricing a wide variety of options and other derivative instruments, especially in markets known for their volatility and unpredictability.

5. Conclusion

Levy processes, in the field of financial modeling, constitute a class of stochastic processes of particular relevance, by virtue of their peculiar properties and their ability to capture complex phenomena observed in financial markets. A Levy process is characterized by independent steady-state increments and jumps, thus introducing a flexible approach for describing the price dynamics of financial assets. The presence of stationary increases within Levy’s processes implies that continuous price changes over time are modeled in a coherent manner that is representative of the persistence of market trends. This provides a significant advantage over purely Gaussian models, which often underestimate the frequency and magnitude of changes in financial markets. The distinguishing feature of Levy’s processes, however, lies in their ability to model sudden jumps in asset prices. These jumps can be interpreted as reflections of unforeseen events or relevant announcements, which generate significant impacts on the markets. The inclusion of this jumping component helps to capture the discontinuous and non-linear nature of price movements, making Levy processes particularly suitable for modeling extreme market conditions or fat-tail events. The practical application of Levy’s processes extends to a variety of financial areas, including option valuation, risk management, and portfolio simulation. Their flexibility allows financial operators to deal with complex scenarios and adapt models to the specific needs of the market. In addition, proper modeling of jumps in Levy’s processes provides greater accuracy in the evaluation of options and the management of risk associated with sudden events. In conclusion, L´evy’s processes, due to their unique combination of steady-state increments and jumps, represent a sophisticated tool in financial modeling, offering a more accurate and detailed analysis of complex market dynamics. Their adaptability and ability to capture non-linear phenomena make them highly valuable tools for traders and financial analysts who are confronted with the complexity and uncertainty inherent in global markets.

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