In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market, a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free. The easiest way to remember what the risk-neutral measure is, or to explain it to a probability generalist who might not know much about finance, is to realize that it is: 1. * The probability measure of a transformed random variable. Typically this transformation is the utility function of the payoff. The risk-neutral measure would be the measure corresponding to an expectation of the payoff with a linear utility. 2. * An implied probability measure, that is one implied from the current observable/posted/traded prices of the relevant instruments. Relevant means those instruments that are causally linked to the events in the probability space under consideration (i.e. underlying prices plus derivatives), and 3. * It is the implied probability measure (solves a kind of inverse problem) that is defined using a linear (risk-neutral) utility in the payoff, assuming some known model for the payoff. This means that you try to find the risk-neutral measure by solving the equation where current prices are the expected present value of the future pay-offs under the risk-neutral measure. The concept of a unique risk-neutral measure is most useful when one imagines making prices across a number of derivatives that would make a unique risk-neutral measure since it implies a kind of consistency in ones hypothetical untraded prices and, theoretically points to arbitrage opportunities in markets where bid/ask prices are visible. It is also worth noting that in most introductory applications in finance, the pay-offs under consideration are deterministic given knowledge of prices at some terminal or future point in time. This is not strictly necessary to make use of these techniques. (Wikipedia).
QRM L1-1: The Definition of Risk
Welcome to Quantitative Risk Management (QRM). In this first class, we define what risk if for us. We will discuss the basic characteristics of risk, underlining some important facts, like its subjectivity, and the impossibility of separating payoffs and probabilities. Understanding the d
From playlist Quantitative Risk Management
Welcome to Quantitative Risk Management (QRM). In this lesson we introduce the axiomatic approach to risk measures. We give the definition of risk measure and we discuss what its uses for us are in terms of reserve capital quantification. We then define coherent and convex measures. The p
From playlist Quantitative Risk Management
QRM L1-2: The dimensions of risk and friends
Welcome to Quantitative Risk Management (QRM). In this second video, we analyse the dimensions of risk. Risk is in fact an object that we need to consider from different points of view, and that sometimes we cannot even quantify. We will also discuss the importance of statistical thinking
From playlist Quantitative Risk Management
What is Value at Risk? VaR and Risk Management
In todays video we learn about Value at Risk (VaR) and how is it calculated? Buy The Book Here: https://amzn.to/37HIdEB Follow Patrick on Twitter Here: https://twitter.com/PatrickEBoyle What Is Value at Risk (VaR)? Value at risk (VaR) is a calculation that aims to quantify the level of
From playlist Risk Management
Risk Management Lesson 5A: Value at Risk
In this first part of Lesson 5, we discuss Value-at-Risk (VaR). Topics: - Definition of VaR - Loss distribution and confidence level - The normal VaR
From playlist Risk Management
Risk Assessment: Likelihood Determination
http://trustedci.org/ Determining Likelihood of a threat as part of a cyber risk assessment.
From playlist Center for Applied Cybersecurity Research (CACR)
Risk Management Lesson 4A: Volatility
First part of Lesson 4. Topics: - Definitions of volatility - Basic assumptions (do they hold?) - Arch and G-arch models (brief overview)
From playlist Risk Management
FRM: Parametric value at risk (VaR): Pros & Cons
Here is a quick explanation of parametric value at risk (VaR) as a means to illustrating its strengths/weaknesses. Please note: The essence of parametric VaR is "no data:" while historical data is surely used to select a distribution and calibrate its parameters, a parametric VaR leans on
From playlist Value at Risk (VaR): Introduction
FinMath L3-2: Risk-neutral measures and self-financing portfolios
Welcome to Lesson 3 of Financial Mathematics (Part 2). In this second half of the lesson, we discuss important topics like self-financing portfolio, risk neutral measures and their basic properties, and the concept of arbitrage. All these tools are essential in financial mathematics, and t
From playlist Financial Mathematics
Fin Math L6-1: The Black-Scholes-Merton theorem
Welcome to Lesson 6 of Financial Mathematics. This is the lesson of the Black-Scholes-Merton (BSM) theorem. Finally, you might say. But it will also be the lesson of volatility and distortions. A lot of interesting things. In this first video, we focus on the BSM theorem. Topics: 00:00 I
From playlist Financial Mathematics
Fin Math L5-2: A simple exchange rate model
In this second part of Lesson 5, we consider a simple exchange rate model, which allows us to see the Cameron-Martin theorem in action. The model also introduces a particular version of the exponential martingale that will be essential for us later. I ask you to spend some time reasoning a
From playlist Financial Mathematics
Fin Math L4-2: The two fundamental theorems of asset pricing and the exponential martingale
Welcome to the second part of Lesson 4 of Financial Mathematics. In this video we discuss the two fundamental theorems of asset pricing and we introduce the exponential martingale, an essential tool that we will use as the Radon-Nikodym derivative to move from P to Q in the Cameron-Martin
From playlist Financial Mathematics
What's the main mistake that risk managers are making with conduct risk and compliance?
Peter Tyson, Head Of Conduct & Compliance, Standard Life, explains the key mistakes that risk managers are making with conduct and compliance at RiskMinds Insurance 2016.
From playlist Insurance risk: Predict risk in an unpredictable world
4 5 Fundamental theorems of asset pricing Part 1
BEM1105x Course Playlist - https://www.youtube.com/playlist?list=PL8_xPU5epJdfCxbRzxuchTfgOH1I2Ibht Produced in association with Caltech Academic Media Technologies. ©2020 California Institute of Technology
From playlist BEM1105x Course - Prof. Jakša Cvitanić
Fin Math L4-1: Change of measure and the Radon-Nikodym derivative
Welcome to Lesson 4 of Financial Mathematics. In this first part of our lesson we deal with the change of measure, a fundamental operation to guarantee the possibility of finding a proper risk-neutral measure. We therefore introduce Radon-Nikodym derivatives and other related concepts. To
From playlist Financial Mathematics
QRM 10-1: The Greeks for Market Risk
Lesson 10 is devoted to the model building approach to market risk. To use such an approach, we need some basic tools from financial mathematics and basic risk management, an example being the Greeks and duration (which nevertheless is linked to the Greeks). For those of you who are not fa
From playlist Quantitative Risk Management
19. Black-Scholes Formula, Risk-neutral Valuation
MIT 18.S096 Topics in Mathematics with Applications in Finance, Fall 2013 View the complete course: http://ocw.mit.edu/18-S096F13 Instructor: Vasily Strela This is a lecture on risk-neutral pricing, featuring the Black-Scholes formula and risk-neutral valuation. License: Creative Commons
From playlist MIT 18.S096 Topics in Mathematics w Applications in Finance
How do you calculate value at risk? Two ways of calculating VaR
In todays video we learn how to calculate VaR or Value at Risk. Buy The Book Here: https://amzn.to/37HIdEB Follow Patrick on Twitter Here: https://twitter.com/PatrickEBoyle What is VAR? The most popular and traditional measure of risk is volatility. The main problem with volatility, how
From playlist Risk Management
Fin Math L7: The Wang transform and the Lorenz curve in Black-Scholes-Merton
Welcome to Financial Mathematics. In this lesson we continue our discussion about the Wang transform and we also introduce an interesting connection with the Lorenz curve, a very useful instrument originally developed in the inequality studies' literature. As we shall see, the use of Wang
From playlist Financial Mathematics