Stochastic Calculus for Finance: A Practical Guide

Stochastic Calculus for Finance

A Practical Guide

William Johnson

© 2024 by HiTeX Press. All rights reserved.

No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law.

Published by HiTeX Press

For permissions and other inquiries, write to:

P.O. Box 3132, Framingham, MA 01701, USA

Contents

1 Introduction to Stochastic Calculus

1.1 Historical Development of Stochastic Calculus

1.2 Key Concepts and Definitions

1.3 Stochastic Integration Basics

1.4 Properties of Stochastic Integrals

1.5 Financial Motivation and Applications

1.6 Challenges and Misconceptions

2 Probability Theory and Random Variables

2.1 Foundations of Probability Theory

2.2 Discrete and Continuous Random Variables

2.3 Distribution Functions

2.4 Expectation and Variance

2.5 Common Probability Distributions

2.6 Conditional Probability and Independence

2.7 Law of Large Numbers and Central Limit Theorem

3 Brownian Motion and Its Properties

3.1 Concept and Origins of Brownian Motion

3.2 Mathematical Definition and Properties

3.3 Path Characteristics

3.4 Mean and Variance Analysis

3.5 Role in Stochastic Processes

3.6 Multi-Dimensional Brownian Motion

3.7 Applications in Financial Modeling

4 Stochastic Processes in Finance

4.1 Overview of Stochastic Processes

4.2 Markov Processes and Applications

4.3 Poisson Processes and Jump Models

4.4 Geometric Brownian Motion in Stock Prices

4.5 Mean Reversion Models

4.6 Levy Processes and Financial Applications

4.7 Comparison of Stochastic Models

5 Ito Calculus and Stochastic Differential Equations

5.1 Fundamentals of Ito Calculus

5.2 Ito’s Lemma and Applications

5.3 Constructing Stochastic Differential Equations (SDEs)

5.4 Solving Stochastic Differential Equations

5.5 Ito Integrals and Their Properties

5.6 Applications of SDEs in Financial Models

5.7 Comparison with Other Stochastic Calculus

6 Girsanov’s Theorem and Change of Measure

6.1 Concept of Measure and Probability Spaces

6.2 Radon-Nikodym Derivative

6.3 Girsanov’s Theorem in Stochastic Calculus

6.4 Applications of Girsanov’s Theorem

6.5 Martingale Measures and Financial Markets

6.6 Practical Examples of Change of Measure

6.7 Limitations and Challenges

7 Martingales in Continuous Time

7.1 Definition and Basic Properties

7.2 Martingales vs. Submartingales and Supermartingales

7.3 Role of Martingales in Finance

7.4 Stopping Times and the Optional Sampling Theorem

7.5 Continuous-Time Martingales and Brownian Motion

7.6 Doob’s Martingale Convergence Theorems

7.7 Applications in Derivative Pricing

8 Applications of Stochastic Calculus in Option Pricing

8.1 Black-Scholes Model and Option Pricing

8.2 The Role of Brownian Motion in Option Pricing

8.3 Delta Hedging and Risk Management

8.4 Extensions to the Black-Scholes Model

8.5 American Options and Stochastic Differential Equations

8.6 Numerical Methods for Option Pricing

8.7 Real-World Applications and Case Studies

9 Numerical Methods for Stochastic Differential Equations

9.1 Overview of Numerical Solutions in SDEs

9.2 Euler-Maruyama Method

9.3 Milstein Method and Higher-Order Schemes

9.4 Monte Carlo Simulations

9.5 Stability and Convergence of Numerical Methods

9.6 Applications in Financial Modeling

9.7 Challenges and Considerations

10 Risk Management and Stochastic Models

10.1 Foundations of Risk Management

10.2 Value at Risk (VaR) and Conditional Value at Risk (CVaR)

10.3 Stochastic Models for Market Risk

10.4 Credit Risk Modeling and Default Probability

10.5 Liquidity Risk and Model-Based Approaches

10.6 Operational Risk and Stochastic Methods

10.7 Integrating Stochastic Models in Risk Management Frameworks

11 Stochastic Control and Optimal Portfolios

11.1 Principles of Stochastic Control

11.2 Dynamic Programming and Hamilton-Jacobi-Bellman Equation

11.3 Merton’s Portfolio Optimization Problem

11.4 Risk-Return Tradeoff in Stochastic Portfolio Models

11.5 Stochastic Control in Asset Allocation

11.6 Robust Portfolio Optimization

11.7 Applications in Real-World Portfolio Management

11.7.1 Dynamic Asset Allocation

11.7.2 Risk Management and Hedging

11.7.3 Pension Fund Management

11.7.4 Sovereign Wealth Funds and Endowments

11.7.5 Behavioral Portfolio Management

11.7.6 Algorithmic and High-Frequency Trading

11.7.7 Conclusion

12 Machine Learning and Stochastic Calculus

12.1 Intersection of Machine Learning and Stochastic Calculus

12.2 Data-Driven Approaches to Stochastic Modeling

12.3 Reinforcement Learning in Stochastic Control

12.4 Neural Networks for Financial Forecasting

12.5 Volatility Forecasting using Machine Learning

12.6 Algorithmic Trading Strategies

12.7 Challenges and Future Directions

13 Advanced Topics in Stochastic Calculus for Finance

13.1 Fractional Brownian Motion and Applications

13.2 Stochastic Volatility Models

13.3 Jump Processes and Lévy Models

13.4 Backward Stochastic Differential Equations (BSDEs)

13.5 Filtering Theory in Finance

13.6 Stochastic Calculus for Interest Rate Models

13.7 Numerical Solutions and Computational Challenges

Preface

In the realm of finance, where complexity intertwines with opportunity, the subtle nuances of market behavior are woven into the fabric of risk and return. Within this intricate tapestry lies the art and science of trading and investment. For those equipped with the right tools, understanding the dynamic movements of financial markets becomes not only feasible but profoundly rewarding. This is where our exploration begins.

Stochastic Calculus for Finance: A Practical Guide stands at the confluence of theory and application, promising to unravel the enigmatic nature of stochastic processes in financial modeling. This book is more than a compilation of mathematical constructs; it is an invitation to discover how these sophisticated models serve as a foundation for informed decision-making in the worlds of trading and investments.

As you turn the pages, you will be introduced to the core principles that underlie stochastic calculus, offering clarity on its practical applications, from option pricing to portfolio optimization. With an eye toward both academic precision and real-world applicability, this volume embarks on elucidating concepts such as Brownian motion, stochastic differential equations, and the revolutionary insights they provide into market mechanics.

Our endeavor is to illuminate the powerful interplay between stochastic calculus and financial instruments. By dissecting the mathematical intricacies and aligning them with strategic financial insights, you will gain a comprehensive understanding poised to enhance your analytical acumen and tactical prowess.

Value awaits those who engage with this text—both in the refined strategies that emerge and the deeper comprehension of market phenomena that is fostered. In bridging the chasm between abstract theory and practical utility, this book endows the reader with more than knowledge; it cultivates a mindset capable of navigating the complexities of modern finance with confidence and foresight.

As you delve into the chapters ahead, anticipate uncovering the mechanisms that drive financial innovation, harnessing state-of-the-art methodologies that are reshaping investment outcomes. This guide aims to equip you not only to observe the waves of financial markets but to understand their undercurrents, enabling you to make calculated decisions with sound reasoning.

In crafting this work, our aspiration is to provide a definitive resource that speaks to both the novice and the seasoned professional—a timeless reference in the fast-evolving landscape of financial analysis and strategy. May it serve as both compass and tool as you engage with the world of financial trading and investment, enlightening your journey through the ever-changing seas of economic transformation.

Chapter 1

Introduction to Stochastic Calculus

Stochastic calculus is a mathematical framework used to model the random behavior of financial markets. It builds upon classical calculus by incorporating probability theory, allowing for the analysis of systems influenced by random forces. This chapter explores the historical development of stochastic calculus, key concepts and definitions, the basics of stochastic integration, and the unique properties of stochastic integrals. It highlights the financial motivations and real-world applications that make stochastic calculus an indispensable tool in modern finance while addressing common challenges and misconceptions faced by learners.

1.1

Historical Development of Stochastic Calculus

The journey of stochastic calculus is a fascinating narrative, one that intertwines mathematics and finance, revealing the evolving understanding of randomness and its implications in modeling the seemingly chaotic behaviors observed in financial markets. Stochastic calculus, as a mathematical framework, offers us the tools to incorporate randomness into calculus, thereby transforming how we analyze dynamic systems influenced by uncertainty. Its historical development is marked by significant contributions from eminent mathematicians and scientists, each bringing new insights and techniques that have collectively shaped the discipline into its current form. This section delves into the historical evolution of stochastic calculus, highlighting the key figures and milestones that have been instrumental in its development.

The origins of stochastic calculus can be traced back to the early 20th century, with the pioneering work of the French mathematician Louis Bachelier, often regarded as the father of mathematical finance. In his doctoral thesis, Théorie de la spéculation (1900), Bachelier introduced the concept of Brownian motion to model stock prices, thus laying the groundwork for stochastic calculus. Though his work was initially met with limited attention, it foreshadowed the more comprehensive models that would follow. Bachelier’s use of Brownian motion, which was inspired by earlier physical observations by botanist Robert Brown, offered a mathematical description of the random movement of particles suspended in a fluid, setting a precedent for the stochastic models applied in finance today.

It wasn’t until the late 1920s and 1930s that Bachelier’s ideas gained further attention with the advent of modern probability theory, largely thanks to the pioneering efforts of Andrey Kolmogorov. Kolmogorov’s work provided a rigorous mathematical foundation for probability, publishing Grundbegriffe der Wahrscheinlichkeitsrechnung in 1933, which formalized probability theory using set theory and firmly established probability as a branch of mathematics in its own right. His contributions were crucial in transforming stochastic processes into a robust field of study, offering the necessary tools to describe time-evolving random phenomena.

A parallel wave of development occurred with the contributions of Norbert Wiener in the 1920s, who formalized the concept of Brownian motion, now also known as the Wiener process. Wiener’s work in establishing the mathematical theory of stochastic processes had significant implications for stochastic calculus. The Wiener process became a cornerstone of modern stochastic calculus, particularly in its application as a basic model of randomness, representing continuous-time stochastic processes with stationary independent increments.

The intersection of these mathematical advancements with economic and financial applications would see a significant leap with the contributions of Kiyoshi Itô in the 1940s and 1950s. Itô fundamentally altered the landscape of stochastic calculus with the introduction of Itô calculus, a mathematical framework that extended classical calculus to encompass stochastic processes. His revolutionary concept of the stochastic integral, coupled with Itô’s lemma, provided a mechanism for handling the integration and differentiation of functions of stochastic processes. Itô’s calculus is foundational to the modeling of stochastic differential equations (SDEs), enabling the precise mathematical treatment of continuous-time processes governed by random influences.

Among the most renowned applications of Itô calculus in finance is the Black-Scholes-Merton model, developed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s. This model utilizes stochastic calculus to derive a partial differential equation that estimates the price of financial derivatives. Its introduction marked a paradigm shift in financial markets, furnishing a tangible application of stochastic calculus and demonstrating its utility in pricing options—a breakthrough that earned Merton and Scholes the Nobel Prize in Economic Sciences in 1997.

The powerful influence of stochastic calculus has extended beyond the boundaries of finance, finding applications in fields as diverse as physics, biology, and engineering. Nonetheless, its role in the financial industry remains particularly pronounced, where it has fundamentally transformed risk management and derivative pricing.

Throughout its evolution, the development of stochastic calculus has been characterized by the refinement of conceptual understanding and the practical deployment of mathematical tools to solve complex problems in a world where uncertainty and randomness are intrinsic. Its trajectory is punctuated by the convergence of abstract mathematical theory with tangible financial applications, illustrating an evolving discipline that continues to adapt and grow with the demands of modern financial markets and beyond.

As we reflect on the historical journey of stochastic calculus, it becomes evident that the significant contributions of Bachelier, Kolmogorov, Wiener, and Itô, among others, have crafted a robust framework for understanding and modeling randomness. This sophisticated calculus has not only advanced mathematical comprehension but has also laid the foundation for contemporary practices in financial modeling and analytical techniques, exemplifying the enduring synergy between mathematics and finance.

1.2

Key Concepts and Definitions

Understanding stochastic calculus necessitates a grasp of several fundamental concepts that underpin the framework used to model complex financial and probabilistic systems. These concepts provide the building blocks for analyzing stochastic processes and are crucial for anyone aiming to apply stochastic calculus to real-world problems, particularly in finance. In this section, we explore some of the key concepts and provide clear definitions to offer a solid foundation in stochastic calculus. Concepts such as filtration, adapted processes, and stopping times are central to understanding how randomness and temporal structure are accommodated within mathematical models.

Filtration and Measurability

In probability theory, a filtration is an increasing sequence of σ-algebras, {ℱt}t≥0, typically defined on a probability space (Ω,ℱ,ℙ). Here, Ω represents the sample space, ℱ is the σ-algebra, and ℙ is the probability measure. Filtration formalizes the notion of information accumulation over time, capturing the idea that as time progresses, more information becomes available. Within this structure, ℱt represents all the events that can be known at time t.

A process X = {Xt,t ≥ 0} is said to be adapted to the filtration {ℱt}t≥0 if, for every time t ≥ 0, the random variable Xt is ℱt-measurable. This means that Xt is determined by the information available up until time t, signifying that future values do not affect its current value, adhering strictly to causality.

Filtration serves as the backbone of many financial models, such as those involving stock prices, where the availability of historical pricing up to the current time influences the decision-making process of traders and algorithms. For instance, a trading strategy may be designed to be ℱt-adapted, only utilizing past and present information in its calculations, aligning with realistic market practice where future prices remain unknowable.

Adapted Process

An adapted process, as previously mentioned, is a stochastic process that respects the information constraints imposed by a given filtration. Formally, a process {Xt}t≥0 is adapted to {ℱt}t≥0 if Xt ∈ℱt for all times t. Adapted processes are fundamental in financial modeling, where such constraints are essential to ensure models are realistic and enforce that decisions cannot be made using future, unknown information.

Consider a stock price process S = {St,t ≥ 0}, where St represents the price of a stock at time t. This price process must be adapted to the filtration {ℱt}t≥0, which incorporates all available market information up to time t. Consequently, trading strategies, derivative pricing, and risk management tools must be designed by only utilizing such adapted processes to simulate actual market conditions.

Stopping Time

A stopping time is a random variable τ : Ω → [0,∞] associated with a filtration {ℱt}t≥0, satisfying the criterion that for every t ≥ 0, the occurrence or non-occurrence of the event {τ ≤ t} is determined by the information up to time t, i.e., {τ ≤ t}∈ℱt. Stopping times are instrumental in defining the moments at which certain actions, such as decisions or portfolio adjustments, might take place.

Stopping times play a pivotal role in the modeling and execution of trading strategies where decisions must be made based on available information until a certain condition is met. For example, in the realm of American-style options, the decision to exercise the option can be made at any stopping time in a defined interval, allowing flexibility based on varying market conditions.

To better illustrate, consider a trader monitoring a stock and deciding to sell only if it hits a specific high value or drops below a certain low value. The time at which the stock price first reaches one of these bounds can be modeled as a stopping time. This encapsulates the trader’s strategy and aligns with the information structure inherent in the filtration up to that time.

Martingales

A martingale is a stochastic process {Xt,t ≥ 0} that, with respect to a filtration {ℱt}t≥0, models a fair game. Formally, X is a martingale if it satisfies the condition 𝔼[|Xt|] < ∞ for all t ≥ 0, and for all s < t,

almost surely. This property implies that the best prediction for the future value of the process, given all present information, is equal to its present value.

Martingales emerge naturally in the context of fair asset pricing under the no-arbitrage condition. If prices are modeled as martingales, it suggests that on average, there is no gain in expected value when conditioning on past information, reinforcing the efficient market hypothesis within financial theory.

Suppose a trader assumes that a stock’s price process S = {St,t ≥ 0} is a martingale with respect to the reference filtration. This implies no expected profit can be made based merely on past price observations, highlighting the challenge in designing profitable strategies without new, external information or market inefficiencies.

Applications and Implications

These core concepts of filtration, adapted processes, stopping times, and martingales form the basis for more sophisticated models and methods used throughout stochastic calculus. They offer the vocabulary and framework required to develop further theoretical constructs like Itô calculus, stochastic differential equations, and even the valuation of complex financial derivatives.

The concept of a filtration ensures that any model incorporating randomness adheres to a structured and logical accumulation of information, critical for applications in financial engineering and beyond. Adapted processes are designed to respect these information constraints, ensuring realistic simulations and analyses. Stopping times introduce flexibility, necessary for decisions dependent on stochastic thresholds, while martingales provide insight into the expected behavior of financial markets under fair game assumptions.

In practical application, the use of these fundamental tools facilitates the construction and implementation of sophisticated trading algorithms. Such algorithms might rely on historical daily and intraday price data aggregated through a filtration, making decisions only when certain stochastic criteria, modeled by stopping times, are met. Martingale properties are especially valuable in defining strategies that avoid biased predictions based on historical trends alone, thus aligning with efficient market assumptions.

As we delve further into stochastic calculus, these definitions and concepts will recur, reinforcing their centrality in the analysis and interpretation of systems governed by randomness. Together, they form the underpinnings of a mathematical framework that adeptly captures the intricacies of dynamic, uncertain environments, exemplified by the vibrant and often unpredictable landscape of modern financial markets.

1.3

Stochastic Integration Basics

At the heart of stochastic calculus lies the concept of stochastic integration, a powerful mathematical tool that extends the notion of integration to domains influenced by randomness. While classical integrals operate within deterministic settings where paths and functions are known, stochastic integrals cater to processes that incorporate uncertainty and variability, such as those encountered in financial markets. This section aims to distill the fundamental aspects of stochastic integration, exploring its significance, challenges, and applications.

Deterministic vs. Stochastic Integration

In classical calculus, the integral of a function over a domain represents the accumulation of quantities or change in values. This integral relies on well-defined paths or deterministic functions. In contrast, stochastic integration involves functions that evolve in accordance with stochastic processes, characterized by inherent randomness.

The necessity for stochastic integration arises in scenarios where the variable of integration is itself a random process. For example, consider a stock price modeled by a stochastic process, where the need to accumulate changes in its value over time based on a series of random events requires a form of integration that can accommodate uncertainty in both the integrand and the integration path.

The Itô Integral

A key development in stochastic calculus was Kiyoshi Itô’s formulation of the stochastic integral, known as the Itô integral, which handles the integration of stochastic processes such as Brownian motion. Unlike classical integral calculus, where limits and continuity typically imply a smooth path, the Itô integral applies to functions associated with non-smooth, continuous-time paths — a crucial adaptation for processes like Brownian motion, characterized by nowhere differentiable trajectories.

The Itô integral is typically represented as:

where {Xt}t≥0 is an adapted, integrable process, and {Wt}t≥0 is a Wiener process (standard Brownian motion).

The integration in the Itô calculus framework takes place in a mean-square sense, making it fundamentally different from the Riemann-Stieltjes integral. This stems from the stochastic nature of Wt, contributing to an increment variance that greatly influences the properties of the Itô integral.

Properties of the Itô Integral

The Itô integral has several distinctive properties that distinguish it from classical integrals, lending it unique abilities in modeling and analysis:

1. Non-Determinism: The Itô integral respects the inherent randomness of the integrated process. As such, ∫ 0T X t dWt is itself a random variable, whose distribution encompasses the stochastic nature of Wt. 2. Itô Isometry: Itô calculus introduces the Itô isometry, an invaluable result linking the expectation of the integral’s square to the integrand, enabling easier computation and analysis. Formally, for a process Xt,

This property simplifies handling stochastic integrals by providing a pathway to evaluate expectations solely based on the deterministic properties of the integrand.

3. Martingale Property: The Itô integral of an adapted process with respect to a Brownian motion is a martingale. This property plays a pivotal role in financial mathematics, particularly in the absence of arbitrage, confirming the martingale structure of asset prices under a risk-neutral measure.

An Example of Itô Integral Application

To illustrate the practical use of the Itô integral, consider a simplified model where the future value of a stock St can be predicted by incorporating randomness, represented by a Wiener process. Suppose dSt = μSt dt + σSt dWt, where μ is the drift coefficient and σ is the volatility.

This stochastic differential equation (SDE) can be interpreted as the stock price having a deterministic trend component μSt dt and a stochastic component σSt dWt. Solving this SDE involves integrating both parts over time, employing Itô calculus to accommodate the stochastic nature of Wt and provide a realistic simulation of St’s trajectory over time.

The Itô Lemma

A cornerstone of stochastic calculus, the Itô Lemma, extends the chain rule of classical calculus to stochastic processes, accommodating functions of stochastic integrals. For a twice-differentiable function f(t,Xt) and a process {Xt} satisfying dXt = μ(t,Xt) dt + σ(t,Xt) dWt, the Itô Lemma expresses the differential:

This formula is crucial for deriving the dynamics of functionals of stochastic processes, widely used in option pricing where asset prices follow stochastic processes, allowing for derivation of models like the Black-Scholes formula.

Implications for Financial Models

Within financial markets, stochastic integration forms the theoretical core of derivative pricing, risk management, and hedging strategies. The ability to integrate stochastic processes with Itô’s calculus empowers analysts and traders to build models that reflect realistic market behavior, capturing volatility and price fluctuations effectively.

For instance, stochastic models of interest rates, such as the Vasicek or Hull-White models, rely upon stochastic differential equations solved through Itô integration. Financial practitioners utilize these models to estimate and hedge against risks associated with fluctuating interest rates.

Challenges and Considerations

Despite its strengths, working with Itô calculus and stochastic integration presents challenges, particularly for those new to the framework. The non-intuitive nature of the stochastic integral, alongside its reliance on properties such as sigma-algebras and adaptedness, requires a strong foundational understanding of probability theory.

Moreover, when applying stochastic calculus to modeling, practitioners must carefully choose model parameters—such as volatility and drift—in real-world settings to achieve meaningful and accurate forecasts. Inadequate estimation can lead to impaired model performance and poor predictive power, impacting financial decision-making.

Conclusion of Stochastic Integration Exploration

In the exploration of stochastic integration basics, we have seen how this mathematical tool transcends classical integration to address the dynamic qualities of random processes. Itô integrals and Itô’s lemma remain vital tools in financial mathematics, enabling effective quantification and modeling of market uncertainty and assisting practitioners in devising strategies and products aligned with the nature of modern financial markets.

By approaching stochastic integration with a structured understanding of its properties and functions, one can unlock its potential to address real-world problems where uncertainty prevails, making it an indispensable aspect of both theoretical and applied domains.

1.4

Properties of Stochastic Integrals

Stochastic integrals, particularly those involving Brownian motion, are a cornerstone of modern mathematical finance and stochastic calculus. Understanding their properties is essential for effectively modeling and analyzing systems where uncertainty plays a key role. Stochastic integrals differ fundamentally from their deterministic counterparts, and they possess unique characteristics that enable them to capture the intricacies of stochastic processes, such as those observed in financial markets. This section delves into these distinctive properties, illustrating their implications and applications through detailed explanations and examples.

Linearity

One foundational property of stochastic integrals is their linearity. If {Xt}t≥0 and {Y t}t≥0 are integrable processes, adapted to a common filtration {ℱt}t≥0, and a and b are real numbers, the linearity of stochastic integrals can be expressed as:

This property simplifies the analysis of complex stochastic integrals by allowing them to be broken down into simpler components. For instance, when assessing risk in a portfolio influenced by multiple factors, the linearity of the corresponding integrals allows for straightforward computation of their contributions to the overall risk.

Mean and Variance

One of the more striking departures from traditional integration observed in stochastic integrals is the nature of their mean and variance. For an Itô integral of the form ∫ 0T X t dWt, where Xt is a deterministic, square-integrable process, the expectation is zero because of the martingale property:

This reflects the martingale nature of the Itô integral, hinting at the lack of guaranteed trends or drift in unbiased, fair-market models.

The variance of an Itô integral is more complex and reveals the deep stochastic nature of the process:

The variance grows as the integral of the square of the integrand over time, in stark contrast to deterministic integrals where variance does not naturally arise since paths are predefined.

Path-Dependent Nature

Stochastic integrals are inherently path-dependent, meaning they rely fundamentally on the trajectory taken by the underlying stochastic process. Each realization of the process contributes differently to the value of the integral, reinforcing the necessity to treat stochastic integrals within a probabilistic framework.

Consider trading strategies that rely on the integrated performance of an asset price, modeled via a stochastic process. The final value derived from such a strategy will depend heavily on the realized path of the asset price, crucially affecting both the expected outcome of the strategy and the estimations of risk involved.

Quadratic Variation

A particularly unique