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Antoine Savine's Global Derivatives 2016 Talk We demonstrate that smoothing, a technique derivatives traders use to stabilise the risk management of discontinuous exotics, is a particular use of Fuzzy Logic. This realisation leads to a general, automated smoothing algorithm. The algorithm is extensively explained and its implementation in C++ is provided as a part of our book with Wiley (2018): Modern Computational Finance: Scripting for Derivatives and xVA

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60 Years Birthday, 30 Years of Ground Breaking Innovation: A Tribute to Bruno...

- Dupire's work from 1992-1996 defined modern finance by establishing conditions for absence of arbitrage, respect of initial yield curves, and respect of initial call prices.
- Dupire showed that models must respect market prices of options through calibration, demonstrating a necessary and sufficient condition for a wide class of diffusion models.
- Dupire's implied volatility formula expresses the implied variance as an average of local variances weighted by probability and gamma, linking market prices to underlying volatility.

A Brief History of Discounting by Antoine Savine

Antoine Savine's lectures and seminars in Bloomberg, NYC and ESILV, Paris
We explain the history and modern construction of interest rate surfaces.

Practical Implementation of AAD by Antoine Savine, Brian Huge and Hans-Jorgen...

Antoine Savine, Brian Huge and Hans-Jorgen Flyger
Global Derivatives 2015 Talk
We identify some of the most delicate challenges of a practical implementation of AAD for exotics, xVA and capital requirement calculations, and explain the solutions.
Danske Bank earned the In-House System of the Year 2015 Risk award after implementing AAD in its regulatory systems.
The practical implementation of AAD is explained in words, mathematics and code in our book with Wiley (2018):
Modern Computational Finance: AAD and Parallel Simulations

Differential Machine Learning Masterclass

Slides for the Differential Machine Learning masterclass given by Brian Huge and Antoine Savine in Barcelona at the Quant Minds International event of December 2021.
This presentation summarises two years of research and development at Danske Bank on the pricing and risk of financial derivatives by machine learning and artificial intelligence.
The presentation develops the themes introduced in two articles in Risk Magazine, October 2020 and October 2021. Those themes are being further developed in the book Modern Computational Finance: Differential Machine Learning, with Chapman and Hall, autumn 2022.

Notes for Volatility Modeling lectures, Antoine Savine at Copenhagen University

1) The document discusses volatility modeling and trading, focusing on modeling volatility.
2) It introduces Black-Scholes option pricing theory, describing how Black-Scholes derives the partial differential equation for option prices and provides a unique solution through arbitrage arguments.
3) The document emphasizes that Black-Scholes is robust to assumptions like stochastic processes and distributions, with the key assumption being continuity of price processes.

Introduction to Interest Rate Models by Antoine Savine

This document provides an introduction to interest rate models. It begins with a simple model involving parallel shifts of the yield curve. However, this model allows for arbitrage opportunities due to convexity. More advanced models incorporate a risk premium and remove arbitrage by including a drift term representing an average steepening of the curve. Under the historical probability measure, changes in forward rates must satisfy certain conditions to avoid arbitrage. Specifically, the drift must include both a volatility term and a risk premium term, with the risk premium being the same for all rates. Markov models further reduce the dynamics to depend only on a single forward rate.

Notes for Computational Finance lectures, Antoine Savine at Copenhagen Univer...

The document discusses computational finance and machine learning in finance. It begins by noting the need for speed in pricing and hedging derivatives, as institutions must compute values and sensitivities rapidly to hedge risk before markets move. Traditional methods become impractical for complex transactions. The document then discusses various techniques to achieve faster computation, including Monte Carlo simulation, adjoint differentiation, leveraging hardware, and machine learning. Regulatory requirements like counterparty valuation adjustment (CVA) further increase computational demands. Overall, the document emphasizes that speed is critical in financial computation and an active area of research.

Financial Cash-Flow Scripting: Beyond Valuation by Antoine Savine

Antoine Savine's QuantMinds and WBS 2018 Presentation
We explore the implementation of scripting in finance and its application to its full potential, in particular for xVA and related regulatory calculation.
The talk is a short summary of some material from our Modern Computational Finance books with Wiley (2018-2019).

60 Years Birthday, 30 Years of Ground Breaking Innovation: A Tribute to Bruno...

- Dupire's work from 1992-1996 defined modern finance by establishing conditions for absence of arbitrage, respect of initial yield curves, and respect of initial call prices.
- Dupire showed that models must respect market prices of options through calibration, demonstrating a necessary and sufficient condition for a wide class of diffusion models.
- Dupire's implied volatility formula expresses the implied variance as an average of local variances weighted by probability and gamma, linking market prices to underlying volatility.

A Brief History of Discounting by Antoine Savine

Antoine Savine's lectures and seminars in Bloomberg, NYC and ESILV, Paris
We explain the history and modern construction of interest rate surfaces.

Practical Implementation of AAD by Antoine Savine, Brian Huge and Hans-Jorgen...

Antoine Savine, Brian Huge and Hans-Jorgen Flyger
Global Derivatives 2015 Talk
We identify some of the most delicate challenges of a practical implementation of AAD for exotics, xVA and capital requirement calculations, and explain the solutions.
Danske Bank earned the In-House System of the Year 2015 Risk award after implementing AAD in its regulatory systems.
The practical implementation of AAD is explained in words, mathematics and code in our book with Wiley (2018):
Modern Computational Finance: AAD and Parallel Simulations

Differential Machine Learning Masterclass

Slides for the Differential Machine Learning masterclass given by Brian Huge and Antoine Savine in Barcelona at the Quant Minds International event of December 2021.
This presentation summarises two years of research and development at Danske Bank on the pricing and risk of financial derivatives by machine learning and artificial intelligence.
The presentation develops the themes introduced in two articles in Risk Magazine, October 2020 and October 2021. Those themes are being further developed in the book Modern Computational Finance: Differential Machine Learning, with Chapman and Hall, autumn 2022.

Notes for Volatility Modeling lectures, Antoine Savine at Copenhagen University

1) The document discusses volatility modeling and trading, focusing on modeling volatility.
2) It introduces Black-Scholes option pricing theory, describing how Black-Scholes derives the partial differential equation for option prices and provides a unique solution through arbitrage arguments.
3) The document emphasizes that Black-Scholes is robust to assumptions like stochastic processes and distributions, with the key assumption being continuity of price processes.

Introduction to Interest Rate Models by Antoine Savine

This document provides an introduction to interest rate models. It begins with a simple model involving parallel shifts of the yield curve. However, this model allows for arbitrage opportunities due to convexity. More advanced models incorporate a risk premium and remove arbitrage by including a drift term representing an average steepening of the curve. Under the historical probability measure, changes in forward rates must satisfy certain conditions to avoid arbitrage. Specifically, the drift must include both a volatility term and a risk premium term, with the risk premium being the same for all rates. Markov models further reduce the dynamics to depend only on a single forward rate.

Notes for Computational Finance lectures, Antoine Savine at Copenhagen Univer...

The document discusses computational finance and machine learning in finance. It begins by noting the need for speed in pricing and hedging derivatives, as institutions must compute values and sensitivities rapidly to hedge risk before markets move. Traditional methods become impractical for complex transactions. The document then discusses various techniques to achieve faster computation, including Monte Carlo simulation, adjoint differentiation, leveraging hardware, and machine learning. Regulatory requirements like counterparty valuation adjustment (CVA) further increase computational demands. Overall, the document emphasizes that speed is critical in financial computation and an active area of research.

Financial Cash-Flow Scripting: Beyond Valuation by Antoine Savine

Antoine Savine's QuantMinds and WBS 2018 Presentation
We explore the implementation of scripting in finance and its application to its full potential, in particular for xVA and related regulatory calculation.
The talk is a short summary of some material from our Modern Computational Finance books with Wiley (2018-2019).

American Options with Monte Carlo

Pricing American and Bermuda Options by means of the Least Squared Monte Carlo technique developed by Longstaff & Schwartz.

Ch6 slides

This document provides an overview of univariate time series modeling and forecasting. It defines concepts such as stationary and non-stationary processes. It describes autoregressive (AR) and moving average (MA) models, including their properties and estimation. It also discusses testing for autocorrelation and stationarity. The key models covered are AR(p) where the current value depends on p past lags, and MA(q) where the error term depends on q past error terms. Wold's decomposition theorem states that any stationary time series can be represented as the sum of deterministic and stochastic components.

Chapter 07 - Autocorrelation.pptx

The document discusses autocorrelation in econometrics models. It defines autocorrelation as a violation of the assumption that errors are independently distributed over time. The key causes of autocorrelation are omitted variables, model misspecification, and systematic measurement errors. Autocorrelation can be first-order, where the current error depends on the previous error, or higher-order. Autocorrelation biases standard errors and test statistics. Methods to detect autocorrelation include graphical analysis of residuals and formal tests like Durbin-Watson and Breusch-Godfrey. Autocorrelation can be resolved by transforming the model when the autocorrelation coefficient is known.

Pairs Trading: Optimizing via Mixed Copula versus Distance Method for S&P 5...

Pairs Trading: Optimizing via Mixed Copula versus Distance Method for S&P 5...Fernando A. B. Sabino da Silva

We carry out a study to evaluate and compare the relative performance of the distance and mixed copula pairs trading strategies. Using data from the S&P 500 stocks from 1990 to 2015, we find that the mixed copula strategy is able to generate a higher mean excess return than the traditional distance method under different weighting structures when the number of tradeable signals is equiparable. Particularly, the mixed copula and distance methods show a mean annualized value-weighted excess returns after costs on committed and fully invested capital as high as 3.98% and 3.14% and 12.73% and 6.07%, respectively, with annual Sharpe ratios up to 0.88. The mixed copula strategy shows positive and significant alphas during the sample period after accounting for various risk-factors. We also provide some evidence on the performance of the strategies over different market states.Volatility - CH 22 - Hedging with VIX Derivatives | CMT Level 3 | Chartered M...

Volatility - CH 22 - Hedging with VIX Derivatives | CMT Level 3 | Chartered M...Professional Training Academy

Join CMT Level 1, 2 & 3 Program Courses & become a professional Technical Analyst, CMT USA Best COACHING CLASSES. CMT Institute Live Classes by Expert Faculty. Exams are available in India. Best Career in Financial Market.
https://www.ptaindia.com/chartered-market-technician/
Simulation methods finance_1

This document provides an introduction to Monte Carlo simulations in finance. It discusses how Monte Carlo methods can be used to value financial derivatives by simulating asset price paths over time based on stochastic processes, and taking the average of the resulting payoffs. It also describes how Monte Carlo integration can be applied to problems involving the numerical evaluation of multi-dimensional integrals. The document outlines the basic concepts and provides examples of applying Monte Carlo techniques to price European options and estimate the value of pi.

Local and Stochastic volatility

We review local and stochastic volatility models and show transformation from Risk Neutral to Real World measure for Heston Stochastic Volatility.

Financial derivatives (2)

This document provides an overview of financial derivatives, including definitions, common types of derivatives such as options, futures, and swaps. It discusses why derivatives are used to reduce risk, but also notes the risks involved, especially with leveraging. Several case studies are presented where the use of complex derivatives led to major financial catastrophes for the organizations involved.

Pairs Trading

The presentation I gave in my investment class about paris trading. I implemented a experiment using R language to identify good pairs from S&P 100 universe. The algorithm is to perform ADF test on the spread of two random stocks and find out the pairs with stationary spread (co-integrated pairs). Pairs identification period is from 2010/11 to 2012/10, test period is from 2012/11 to 2013/12. Finally I got 33 pairs out of 4950 candidates, and I conduct a summary on the experiment result.

Pairs Trading

This document outlines pairs trading, a market-neutral trading strategy that profits from temporary mispricings between two related assets. It has a history on Wall Street dating back to the 1980s. The strategy involves finding two assets that typically move together, going long one and short the other when their prices diverge significantly, with the expectation that they will converge back to their historical relationship. An example using a drunk man wandering with his dog is provided to analogize how the two assets, though individually random, should remain closely correlated.

Black Scholes Model.pdf

The document provides an overview of the Black-Scholes option pricing model (BSOPM). It describes how BSOPM calculates option prices based on stock price, strike price, time to expiration, risk-free rate, and volatility. It outlines the model's key assumptions and how each variable impacts option value. The document also discusses calculating implied volatility, historical vs implied volatility, and limitations of the BSOPM.

"Quantitative Trading as a Mathematical Science" by Dr. Haksun Li, Founder an...

Presented at QuantCon Singapore 2016, Quantopian's quantitative finance and algorithmic trading conference, November 11th.
Quantitative trading is distinguishable from other trading methodologies like technical analysis and analysts’ opinions because it uniquely provides justifications to trading strategies using mathematical reasoning. Put differently, quantitative trading is a science that trading strategies are proven statistically profitable or even optimal under certain assumptions. There are properties about strategies that we can deduce before betting the first $1, such as P&L distribution and risks. There are exact explanations to the success and failure of strategies, such as choice of parameters. There are ways to iteratively improve strategies based on experiences of live trading, such as making more realistic assumptions. These are all made possible only in quantitative trading because we have assumptions, models and rigorous mathematical analysis.
Quantitative trading has proved itself to be a significant driver of mathematical innovations, especially in the areas of stochastic analysis and PDE-theory. For instances, we can compute the optimal timings to follow the market by solving a pair of coupled Hamilton–Jacobi–Bellman equations; we can construct sparse mean reverting baskets by solving semi-definite optimization problems with cardinality constraints and can optimally trade these baskets by solving stochastic control problems; we can identify statistical arbitrage opportunities by analyzing the volatility process of a stochastic asset at different frequencies; we can compute the optimal placements of market and limit orders by solving combined singular and impulse control problems which leads to novel and difficult to solve quasi-variational inequalities.

"Quantum Hierarchical Risk Parity - A Quantum-Inspired Approach to Portfolio ...

Maxwell will present the methodologies and results behind the algorithm that has been developed by 1QBit, named Quantum Hierarchical Risk Parity, or QHRP.
This is an extension of the work done by Marcos Lopez de Prado on
Hierarchical Risk Parity in his paper "Building Diversified Portfolios that Outperform Out-of-Sample."
QHRP tackles the problem of minimizing the risk of a portfolio of assets using a quantum-inspired approach. Although the ideas surrounding this go back to Markowitz’s mean-variance portfolio optimization of 1952’s Portfolio Selection, we have applied recent quantum-ready machine learning tools to the problem to demonstrate strong performance in terms of a variety of risk measures and lower susceptibility to inaccuracies in the input data.
The quantum-ready approach to portfolio optimization is based on
an optimization problem that can be solved using a quantum annealer. The algorithm utilizes a hierarchical clustering tree that is based on the covariance matrix of the asset returns. The results of real market data used to benchmark this approach against other common portfolio optimization methods will be shared in this presentation.
View the White Paper: https://bit.ly/2k5xTxW.

Ch03

This document discusses the concept of utility and rational choice in economics. It outlines three axioms that rational individuals are assumed to follow: completeness, transitivity, and continuity. Utility represents an individual's ranking of situations based on their preferences. Indifference curves illustrate combinations of goods that provide equal utility. The slope of an indifference curve is the marginal rate of substitution, which decreases as more of one good is substituted for another. Several common utility functions are presented, including Cobb-Douglas, perfect substitutes, and perfect complements.

Should You Build Your Own Backtester? by Michael Halls-Moore at QuantCon 2016

The huge uptake of Python and R as first-class programming languages within quantitative trading has lead to an abundance of backtesting libraries becoming widely available. It can take months, if not years, to develop a robust backtesting and trading infrastructure from scratch and many of the vendors (both commercial and open source) have a huge head start. Given such prevalence and maturity of the available software, as well as the time investment needed for development, is there any benefit to building your own?
In this talk, Mike will argue the advantages and disadvantages of building your own infrastructure, how to develop and improve your first backtesting system and how to make it robust to internal and external risk events. The talk will be of interest whether you are a retail quant trader managing your own capital or are forming a start-up quant fund with initial seed funding.

Risk aversion

This document discusses risk aversion and expected utility theory given uncertainty. It introduces concepts like probability, utility functions, and risk preferences. It explains that individuals do not simply maximize expected wealth but rather expected utility. This is because utility functions capture decreasing marginal utility of wealth as well as risk aversion. Several examples are provided to illustrate concepts like the St. Petersburg paradox and how expected utility differs from certainty equivalent. Five axioms of rational choice under uncertainty are outlined which form the basis of expected utility theory.

Volatility Smiles and Stylised Facts in the Heston Model

Option pricing(call option), simulations, Implied Volatility Smile and some stylized facts are presented in this project. In the Black-Scholes model, it is assumed that the volatility is constant, while the Heston model allows the stochastic volatility. The Black- Scholes model is compared to the Heston model by theoretical and numerical point of views.

Lf 2020 bachelier

A large amount of trivial ramblings on Gaussian distributions caused by reading the book by David and Etheridge on the 1900 Bachelier thesis

Autocorrelation (1)

Autocorrelation measures the correlation of a time series with its past and lagged values. It exists when observations in a time series are correlated with each other. The Durbin-Watson test can detect the presence of autocorrelation by examining the residuals of a regression model. If autocorrelation is present, it violates the assumption that errors are independent and leads to inaccurate test statistics and predictions. Common structures for autocorrelation include autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) processes.

Risk Management - CH 5 -Risk Control | CMT Level 3 | Chartered Market Technic...

Risk Management - CH 5 -Risk Control | CMT Level 3 | Chartered Market Technic...Professional Training Academy

Join CMT Level 1, 2 & 3 Program Courses & become a professional Technical Analyst, CMT USA Best COACHING CLASSES. CMT Institute Live Classes by Expert Faculty. Exams are available in India. Best Career in Financial Market.
https://www.ptaindia.com/chartered-market-technician/
Garbled Circuits for Secure Credential Management Services

Garbled Circuits for Secure Credential Management ServicesOnBoard Security, Inc. - a Qualcomm Company

This presentation discusses the use of Garbled Circuits for improving security and simplifying implementation of Secure Credential Management Systems (SCMS) in the Automotive industryConcurrency in Distributed Systems : Leslie Lamport papers

In computer science, concurrency is the ability of different parts or units of a program, algorithm, or problem to be executed out-of-order or in partial order, without affecting the final outcome. This allows for parallel execution of the concurrent units, which can significantly improve overall speed of the execution in multi-processor and multi-core systems. In more technical terms, concurrency refers to the decomposability property of a program, algorithm, or problem into order-independent or partially-ordered components or units.[1]
A number of mathematical models have been developed for general concurrent computation including Petri nets, process calculi, the parallel random-access machine model, the actor model and the Reo Coordination Language.

American Options with Monte Carlo

Pricing American and Bermuda Options by means of the Least Squared Monte Carlo technique developed by Longstaff & Schwartz.

Ch6 slides

This document provides an overview of univariate time series modeling and forecasting. It defines concepts such as stationary and non-stationary processes. It describes autoregressive (AR) and moving average (MA) models, including their properties and estimation. It also discusses testing for autocorrelation and stationarity. The key models covered are AR(p) where the current value depends on p past lags, and MA(q) where the error term depends on q past error terms. Wold's decomposition theorem states that any stationary time series can be represented as the sum of deterministic and stochastic components.

Chapter 07 - Autocorrelation.pptx

The document discusses autocorrelation in econometrics models. It defines autocorrelation as a violation of the assumption that errors are independently distributed over time. The key causes of autocorrelation are omitted variables, model misspecification, and systematic measurement errors. Autocorrelation can be first-order, where the current error depends on the previous error, or higher-order. Autocorrelation biases standard errors and test statistics. Methods to detect autocorrelation include graphical analysis of residuals and formal tests like Durbin-Watson and Breusch-Godfrey. Autocorrelation can be resolved by transforming the model when the autocorrelation coefficient is known.

Pairs Trading: Optimizing via Mixed Copula versus Distance Method for S&P 5...

Pairs Trading: Optimizing via Mixed Copula versus Distance Method for S&P 5...Fernando A. B. Sabino da Silva

We carry out a study to evaluate and compare the relative performance of the distance and mixed copula pairs trading strategies. Using data from the S&P 500 stocks from 1990 to 2015, we find that the mixed copula strategy is able to generate a higher mean excess return than the traditional distance method under different weighting structures when the number of tradeable signals is equiparable. Particularly, the mixed copula and distance methods show a mean annualized value-weighted excess returns after costs on committed and fully invested capital as high as 3.98% and 3.14% and 12.73% and 6.07%, respectively, with annual Sharpe ratios up to 0.88. The mixed copula strategy shows positive and significant alphas during the sample period after accounting for various risk-factors. We also provide some evidence on the performance of the strategies over different market states.Volatility - CH 22 - Hedging with VIX Derivatives | CMT Level 3 | Chartered M...

Volatility - CH 22 - Hedging with VIX Derivatives | CMT Level 3 | Chartered M...Professional Training Academy

Join CMT Level 1, 2 & 3 Program Courses & become a professional Technical Analyst, CMT USA Best COACHING CLASSES. CMT Institute Live Classes by Expert Faculty. Exams are available in India. Best Career in Financial Market.
https://www.ptaindia.com/chartered-market-technician/
Simulation methods finance_1

This document provides an introduction to Monte Carlo simulations in finance. It discusses how Monte Carlo methods can be used to value financial derivatives by simulating asset price paths over time based on stochastic processes, and taking the average of the resulting payoffs. It also describes how Monte Carlo integration can be applied to problems involving the numerical evaluation of multi-dimensional integrals. The document outlines the basic concepts and provides examples of applying Monte Carlo techniques to price European options and estimate the value of pi.

Local and Stochastic volatility

We review local and stochastic volatility models and show transformation from Risk Neutral to Real World measure for Heston Stochastic Volatility.

Financial derivatives (2)

This document provides an overview of financial derivatives, including definitions, common types of derivatives such as options, futures, and swaps. It discusses why derivatives are used to reduce risk, but also notes the risks involved, especially with leveraging. Several case studies are presented where the use of complex derivatives led to major financial catastrophes for the organizations involved.

Pairs Trading

The presentation I gave in my investment class about paris trading. I implemented a experiment using R language to identify good pairs from S&P 100 universe. The algorithm is to perform ADF test on the spread of two random stocks and find out the pairs with stationary spread (co-integrated pairs). Pairs identification period is from 2010/11 to 2012/10, test period is from 2012/11 to 2013/12. Finally I got 33 pairs out of 4950 candidates, and I conduct a summary on the experiment result.

Pairs Trading

This document outlines pairs trading, a market-neutral trading strategy that profits from temporary mispricings between two related assets. It has a history on Wall Street dating back to the 1980s. The strategy involves finding two assets that typically move together, going long one and short the other when their prices diverge significantly, with the expectation that they will converge back to their historical relationship. An example using a drunk man wandering with his dog is provided to analogize how the two assets, though individually random, should remain closely correlated.

Black Scholes Model.pdf

The document provides an overview of the Black-Scholes option pricing model (BSOPM). It describes how BSOPM calculates option prices based on stock price, strike price, time to expiration, risk-free rate, and volatility. It outlines the model's key assumptions and how each variable impacts option value. The document also discusses calculating implied volatility, historical vs implied volatility, and limitations of the BSOPM.

"Quantitative Trading as a Mathematical Science" by Dr. Haksun Li, Founder an...

Presented at QuantCon Singapore 2016, Quantopian's quantitative finance and algorithmic trading conference, November 11th.
Quantitative trading is distinguishable from other trading methodologies like technical analysis and analysts’ opinions because it uniquely provides justifications to trading strategies using mathematical reasoning. Put differently, quantitative trading is a science that trading strategies are proven statistically profitable or even optimal under certain assumptions. There are properties about strategies that we can deduce before betting the first $1, such as P&L distribution and risks. There are exact explanations to the success and failure of strategies, such as choice of parameters. There are ways to iteratively improve strategies based on experiences of live trading, such as making more realistic assumptions. These are all made possible only in quantitative trading because we have assumptions, models and rigorous mathematical analysis.
Quantitative trading has proved itself to be a significant driver of mathematical innovations, especially in the areas of stochastic analysis and PDE-theory. For instances, we can compute the optimal timings to follow the market by solving a pair of coupled Hamilton–Jacobi–Bellman equations; we can construct sparse mean reverting baskets by solving semi-definite optimization problems with cardinality constraints and can optimally trade these baskets by solving stochastic control problems; we can identify statistical arbitrage opportunities by analyzing the volatility process of a stochastic asset at different frequencies; we can compute the optimal placements of market and limit orders by solving combined singular and impulse control problems which leads to novel and difficult to solve quasi-variational inequalities.

"Quantum Hierarchical Risk Parity - A Quantum-Inspired Approach to Portfolio ...

Maxwell will present the methodologies and results behind the algorithm that has been developed by 1QBit, named Quantum Hierarchical Risk Parity, or QHRP.
This is an extension of the work done by Marcos Lopez de Prado on
Hierarchical Risk Parity in his paper "Building Diversified Portfolios that Outperform Out-of-Sample."
QHRP tackles the problem of minimizing the risk of a portfolio of assets using a quantum-inspired approach. Although the ideas surrounding this go back to Markowitz’s mean-variance portfolio optimization of 1952’s Portfolio Selection, we have applied recent quantum-ready machine learning tools to the problem to demonstrate strong performance in terms of a variety of risk measures and lower susceptibility to inaccuracies in the input data.
The quantum-ready approach to portfolio optimization is based on
an optimization problem that can be solved using a quantum annealer. The algorithm utilizes a hierarchical clustering tree that is based on the covariance matrix of the asset returns. The results of real market data used to benchmark this approach against other common portfolio optimization methods will be shared in this presentation.
View the White Paper: https://bit.ly/2k5xTxW.

Ch03

This document discusses the concept of utility and rational choice in economics. It outlines three axioms that rational individuals are assumed to follow: completeness, transitivity, and continuity. Utility represents an individual's ranking of situations based on their preferences. Indifference curves illustrate combinations of goods that provide equal utility. The slope of an indifference curve is the marginal rate of substitution, which decreases as more of one good is substituted for another. Several common utility functions are presented, including Cobb-Douglas, perfect substitutes, and perfect complements.

Should You Build Your Own Backtester? by Michael Halls-Moore at QuantCon 2016

The huge uptake of Python and R as first-class programming languages within quantitative trading has lead to an abundance of backtesting libraries becoming widely available. It can take months, if not years, to develop a robust backtesting and trading infrastructure from scratch and many of the vendors (both commercial and open source) have a huge head start. Given such prevalence and maturity of the available software, as well as the time investment needed for development, is there any benefit to building your own?
In this talk, Mike will argue the advantages and disadvantages of building your own infrastructure, how to develop and improve your first backtesting system and how to make it robust to internal and external risk events. The talk will be of interest whether you are a retail quant trader managing your own capital or are forming a start-up quant fund with initial seed funding.

Risk aversion

This document discusses risk aversion and expected utility theory given uncertainty. It introduces concepts like probability, utility functions, and risk preferences. It explains that individuals do not simply maximize expected wealth but rather expected utility. This is because utility functions capture decreasing marginal utility of wealth as well as risk aversion. Several examples are provided to illustrate concepts like the St. Petersburg paradox and how expected utility differs from certainty equivalent. Five axioms of rational choice under uncertainty are outlined which form the basis of expected utility theory.

Volatility Smiles and Stylised Facts in the Heston Model

Option pricing(call option), simulations, Implied Volatility Smile and some stylized facts are presented in this project. In the Black-Scholes model, it is assumed that the volatility is constant, while the Heston model allows the stochastic volatility. The Black- Scholes model is compared to the Heston model by theoretical and numerical point of views.

Lf 2020 bachelier

A large amount of trivial ramblings on Gaussian distributions caused by reading the book by David and Etheridge on the 1900 Bachelier thesis

Autocorrelation (1)

Autocorrelation measures the correlation of a time series with its past and lagged values. It exists when observations in a time series are correlated with each other. The Durbin-Watson test can detect the presence of autocorrelation by examining the residuals of a regression model. If autocorrelation is present, it violates the assumption that errors are independent and leads to inaccurate test statistics and predictions. Common structures for autocorrelation include autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) processes.

Risk Management - CH 5 -Risk Control | CMT Level 3 | Chartered Market Technic...

Risk Management - CH 5 -Risk Control | CMT Level 3 | Chartered Market Technic...Professional Training Academy

Join CMT Level 1, 2 & 3 Program Courses & become a professional Technical Analyst, CMT USA Best COACHING CLASSES. CMT Institute Live Classes by Expert Faculty. Exams are available in India. Best Career in Financial Market.
https://www.ptaindia.com/chartered-market-technician/
American Options with Monte Carlo

American Options with Monte Carlo

Ch6 slides

Ch6 slides

Chapter 07 - Autocorrelation.pptx

Chapter 07 - Autocorrelation.pptx

Pairs Trading: Optimizing via Mixed Copula versus Distance Method for S&P 5...

Pairs Trading: Optimizing via Mixed Copula versus Distance Method for S&P 5...

Volatility - CH 22 - Hedging with VIX Derivatives | CMT Level 3 | Chartered M...

Volatility - CH 22 - Hedging with VIX Derivatives | CMT Level 3 | Chartered M...

Simulation methods finance_1

Simulation methods finance_1

Local and Stochastic volatility

Local and Stochastic volatility

Financial derivatives (2)

Financial derivatives (2)

Pairs Trading

Pairs Trading

Pairs Trading

Pairs Trading

Black Scholes Model.pdf

Black Scholes Model.pdf

"Quantitative Trading as a Mathematical Science" by Dr. Haksun Li, Founder an...

"Quantitative Trading as a Mathematical Science" by Dr. Haksun Li, Founder an...

"Quantum Hierarchical Risk Parity - A Quantum-Inspired Approach to Portfolio ...

"Quantum Hierarchical Risk Parity - A Quantum-Inspired Approach to Portfolio ...

Ch03

Ch03

Should You Build Your Own Backtester? by Michael Halls-Moore at QuantCon 2016

Should You Build Your Own Backtester? by Michael Halls-Moore at QuantCon 2016

Risk aversion

Risk aversion

Volatility Smiles and Stylised Facts in the Heston Model

Volatility Smiles and Stylised Facts in the Heston Model

Lf 2020 bachelier

Lf 2020 bachelier

Autocorrelation (1)

Autocorrelation (1)

Risk Management - CH 5 -Risk Control | CMT Level 3 | Chartered Market Technic...

Risk Management - CH 5 -Risk Control | CMT Level 3 | Chartered Market Technic...

Garbled Circuits for Secure Credential Management Services

Garbled Circuits for Secure Credential Management ServicesOnBoard Security, Inc. - a Qualcomm Company

This presentation discusses the use of Garbled Circuits for improving security and simplifying implementation of Secure Credential Management Systems (SCMS) in the Automotive industryConcurrency in Distributed Systems : Leslie Lamport papers

In computer science, concurrency is the ability of different parts or units of a program, algorithm, or problem to be executed out-of-order or in partial order, without affecting the final outcome. This allows for parallel execution of the concurrent units, which can significantly improve overall speed of the execution in multi-processor and multi-core systems. In more technical terms, concurrency refers to the decomposability property of a program, algorithm, or problem into order-independent or partially-ordered components or units.[1]
A number of mathematical models have been developed for general concurrent computation including Petri nets, process calculi, the parallel random-access machine model, the actor model and the Reo Coordination Language.

Fpga creating counter with internal clock

This document describes a design for a counter seven segment display using an FPGA. The author implemented a counter component and a decoder component, then wired them together in a top-level systemSeg entity using port mapping. The counter counts from 0 to 9 and outputs a 4-bit binary coded decimal value. The decoder converts the 4-bit input to a 7-segment display output based on a case statement. The author notes that port mapping is useful for wiring components together in HDL and that designs need to be synchronous with a clock.

Matlab Script - Loop Control

In this lecture we will discuss about another flow control method – Loop control.
A loop control is used to execute a set of commands repeatedly
The set of commands is called the body of the loop
MATLAB has two loop control techniques
Counted loops - executes commands a specified number of times
Conditional loops - executes commands as long as a specified expression is true

IEEE-754 standard format to handle Floating-Point calculations in RISC-V CPUs...

The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portable. RISC-V also uses the IEEE 754 standard.

Verilog-Behavioral Modeling .pdf

Verilog is a HARDWARE DESCRIPTION LANGUAGE (HDL). It is a language used for describing a digital system like a network switch or a microprocessor or a memory or a flip−flop. It means, by using a HDL we can describe any digital hardware at any level. Designs, which are described in HDL are independent of technology, very easy for designing and debugging, and are normally more useful than schematics, particularly for large circuits.
Verilog supports a design at many levels of abstraction. The major three are −
Behavioral level
Register-transfer level
Gate level
Behavioral level
This level describes a system by concurrent algorithms (Behavioural). Every algorithm is sequential, which means it consists of a set of instructions that are executed one by one. Functions, tasks and blocks are the main elements. There is no regard to the structural realization of the design. Verilog Keywords
Words that have special meaning in Verilog are called the Verilog keywords. For example, assign, case, while, wire, reg, and, or, nand, and module. They should not be used as identifiers. Verilog keywords also include compiler directives, and system tasks and functions.
Gate Level Modelling
Verilog has built-in primitives like logic gates, transmission gates and switches. These are rarely used for design work but they are used in post synthesis world for modelling of ASIC/FPGA cells.
Gate level modelling exhibits two properties −
Drive strength − The strength of the output gates is defined by drive strength. The output is strongest if there is a direct connection to the source. The strength decreases if the connection is via a conducting transistor and least when connected via a pull-up/down resistive. The drive strength is usually not specified, in which case the strengths defaults to strong1 and strong0.
Delays − If delays are not specified, then the gates do not have propagation delays; if two delays are specified, then first one represents the rise delay and the second one, fall delay; if only one delay is specified, then both, rise and fall are equal. Delays can be ignored in synthesis.
Gate Primitives
The basic logic gates using one output and many inputs are used in Verilog. GATE uses one of the keywords - and, nand, or, nor, xor, xnor for use in Verilog for N number of inputs and 1 output.

Fekra c++ Course #2

The document provides information about various C++ operators and control flow statements:
- It explains assignment operators like +=, -=, *=, /=, %= and the increment/decrement operators ++ and --.
- Conditional statements like if-else are discussed along with logical operators like &&, || and !. Nested if/else, else-if ladder and dangling else issues are covered.
- Flow charts are introduced as a way to visualize algorithms and conditional logic.
- The switch statement provides an alternative way to write multiple conditional checks compared to nested if/else.
- Examples are provided to illustrate concepts like precedence of logical operators, relational operators, and handling division by zero.

13.ppt

The document discusses modeling synchronous logic circuits in Verilog. It describes basic sequential circuits like latches and registers, and how they can be modeled using if statements in Verilog to infer latches. It also covers D flip-flops, counters, and linear feedback shift registers (LFSRs). LFSRs can be used for applications like pattern generation, encryption, and pseudo-random number generation. Verilog code examples are provided for modeling latches, D flip-flops, LFSRs, and synchronous counters.

Writing more complex models (continued)

Modeling more complicated logic using sequential statements
Skills gained:
1- Model simple sequential logic using loops
2- Control the process execution using wait statements
This is part of VHDL 360 course

Day4 Lab

The document discusses several topics related to digital logic design including:
1. Designing a synchronous counter circuit using a 7-segment display.
2. Using a frequency divider to generate clock signals for updating the counter and display.
3. Implementing a timer delay circuit using a counter and DIP switches to set the maximum value.
4. Designing a PLC counter circuit with an LED display to count input pulses from a switch.

Design considerations

Design considerationsSchool of Design Engineering Fashion & Technology (DEFT), University of Wales, Newport

The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
Electronz_Chapter_12.pptx

This document provides instructions for building a secret locking mechanism using an Arduino, piezo sensor, servo motor, LEDs, and other components. The circuit can detect knock patterns on a surface and unlock if the right sequence is received. A function is defined to check if each knock is valid based on its intensity. If the correct number of valid knocks is received, the servo will unlock and the LEDs will change color to indicate the unlocked status.

Fundamentals of Programming Lecture #1.pptx

This document provides an overview of programming and problem solving concepts. It discusses that a problem is a situation that needs resolution, while a solution removes the problematic situation. Problem solving is the process of deriving a solution. Programming involves creating computer solutions to problems through algorithms expressed as programs. Key steps for problem solving are understanding the problem, planning an algorithm/solution, implementing it as a program, testing the program, documenting it, and maintaining it for changes. Pseudocode and flowcharts are common ways to describe algorithms before implementing them as programs.

Introduction to Selection control structures in C++

The document discusses various control structures in C++ that alter the sequential flow of program execution. It describes selection statements like if, if-else which execute code conditionally based on boolean expressions. Iterative statements like for, while, do-while loops repeat code execution. The switch statement provides a selection from multiple options. Control structures allow solving problems by choosing the appropriate combination of decision making and repetition.

Flip & flop by Zaheer Abbas Aghani

Flip-flops are fundamental building blocks of digital electronics that can store state information. There are several types of flip-flops including D, T, JK, and SR flip-flops. Flip-flops are used as data storage elements, for counting pulses, and synchronizing signals. Counters are digital circuits that store and sometimes display the number of times an event occurs, often in relation to a clock signal. Digital logic design involves the analysis and design of combinational and sequential circuits using techniques like minimization and optimization.

Rate Limiting at Scale, from SANS AppSec Las Vegas 2012

Rate Limits at Scale SANS AppSec Las Vegas.
Rate Limit Everything All the time using a quantized time system with Memcache or Redis. Use this protect resources or discover anomalies.

Writing more complex models

Modeling more complicated logic using sequential statements
Skills gained:
1- Identify sequential environment in VHDL
2- Model simple sequential logic
This is part of VHDL 360 course

1 factor vs.2 factor gaussian model for zero coupon bond pricing final

Financial Algorithms describes the comparison between and relevance of Gaussian one and two factor models in today's interest rate environments across US, European and Asian markets. Negative short rates seem to be the new norm of interest rate markets, especially in Euro-zone & somewhat in US , where poor demand and very low inflation dragging down interest rates in a negative zone. One and Two Factor Gaussian Models under Hull-White Setup can accommodate such scenarios and address the cases of curve steeping of longer end of the zero curve wherein short rates hover in negative zones.

DEF CON 27 - ANDREAS BAUMHOF - are quantum computers really a threat to crypt...

This document provides an overview of quantum computing and its implications for cryptography. It discusses how quantum computers could break popular asymmetric cryptographic algorithms like RSA by efficiently solving problems like integer factorization that are intractable on classical computers. The document explains Shor's algorithm, which uses quantum Fourier transforms to find the period of exponential functions and derive prime factors in polynomial time, posing a threat to RSA. It also discusses quantum computing concepts like superposition and entanglement that enable this speedup. Overall, the document serves as an introduction to how quantum computers may impact cryptography by breaking algorithms like RSA.

lecture8_Cuong.ppt

This document discusses various topics related to writing Verilog code that will efficiently synthesize to digital logic circuits, including:
- For loops can synthesize if the number of iterations is fixed. Loops will be "unrolled" during synthesis.
- Generate statements allow conditional or parameterized instantiation of modules before simulation.
- Coding styles like using multi-cycle designs or sharing logic units can reduce area compared to naively written code.
- Gated clocks require careful handling to avoid timing issues between clock domains.
- Conditional assignments and always blocks require complete cases or default assignments to avoid unintended latches.
- Examples compare the area and delay of different implementations of multipliers,

Garbled Circuits for Secure Credential Management Services

Garbled Circuits for Secure Credential Management Services

Concurrency in Distributed Systems : Leslie Lamport papers

Concurrency in Distributed Systems : Leslie Lamport papers

Fpga creating counter with internal clock

Fpga creating counter with internal clock

Matlab Script - Loop Control

Matlab Script - Loop Control

IEEE-754 standard format to handle Floating-Point calculations in RISC-V CPUs...

IEEE-754 standard format to handle Floating-Point calculations in RISC-V CPUs...

Verilog-Behavioral Modeling .pdf

Verilog-Behavioral Modeling .pdf

Fekra c++ Course #2

Fekra c++ Course #2

13.ppt

13.ppt

Writing more complex models (continued)

Writing more complex models (continued)

Day4 Lab

Day4 Lab

Design considerations

Design considerations

Electronz_Chapter_12.pptx

Electronz_Chapter_12.pptx

Fundamentals of Programming Lecture #1.pptx

Fundamentals of Programming Lecture #1.pptx

Introduction to Selection control structures in C++

Introduction to Selection control structures in C++

Flip & flop by Zaheer Abbas Aghani

Flip & flop by Zaheer Abbas Aghani

Rate Limiting at Scale, from SANS AppSec Las Vegas 2012

Rate Limiting at Scale, from SANS AppSec Las Vegas 2012

Writing more complex models

Writing more complex models

1 factor vs.2 factor gaussian model for zero coupon bond pricing final

1 factor vs.2 factor gaussian model for zero coupon bond pricing final

DEF CON 27 - ANDREAS BAUMHOF - are quantum computers really a threat to crypt...

DEF CON 27 - ANDREAS BAUMHOF - are quantum computers really a threat to crypt...

lecture8_Cuong.ppt

lecture8_Cuong.ppt

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Governor Olli Rehn: Inflation down and recovery supported by interest rate cu...

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Press conference on the outlook for the Finnish economy
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三、进国企，银行，事业单位，考公务员等等，这些单位是必需要提供真实教育部认证的，办理教育部认证所需资料众多且烦琐，所有材料您都必须提供原件，我们凭借丰富的经验，快捷的绿色通道帮您快速整合材料，让您少走弯路。
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
→ 【关于价格问题（保证一手价格）
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
选择实体注册公司办理，更放心，更安全！我们的承诺：可来公司面谈，可签订合同，会陪同客户一起到教育部认证窗口递交认证材料，客户在教育部官方认证查询网站查询到认证通过结果后付款，不成功不收费！

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Well-crafted financial reports serve as vital tools for decision-making and transparency within an organization. By following the undermentioned tips, you can create standardized financial reports that effectively communicate your company's financial health and performance to stakeholders.

在线办理(TAMU毕业证书)美国德州农工大学毕业证PDF成绩单一模一样

学校原件一模一样【微信：741003700 】《(TAMU毕业证书)美国德州农工大学毕业证PDF成绩单》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才

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- 1. Danske Bank Stabilize risks of discontinuous payoffs with Fuzzy Logic Antoine Savine
- 2. Fuzzy Logic Danske Bank 2 Discontinuous payoffs with Monte-Carlo • Discontinuous profiles produce unstable risks with Monte-Carlo − Example: 110 1y Digital in Black-Scholes MC, vol = 20% − 10,000 to 250,000 simulations − Seed changed between pricings − Decent convergence on price, risk all over the place
- 3. Fuzzy Logic Danske Bank Malliavin’s calculus 3 • The industry developed 2 families of methods to deal with this problem: • One is the Likelihood Ratio Method (LRM) and more general Malliavin’s calculus − Apply − Compute pathwise − No need to differentiate a discontinuous payoff, differentiate log-likelihood instead − Tractability, efficiency and stability depend on model , , , , , log , , log , i i i i i i X a a X X a a X Ea f X f X a X dX a a a X f X a X dX f X a X dX a X f X a X a X dX E f X a X log , ia a X
- 4. Fuzzy Logic Danske Bank Payoff smoothing 4 • Payoff smoothing is considerably simpler (and surprinsingly effective): Replace discontinuous payoffs by ”close” continuous ones − Digital Call spread − Barrier Smooth barrier • Benefit: work on payoffs only, no work required on models • Problem: to identify (and smooth) all discontinuities we need to know exactly how the payoff is calculated • Today, this is a manual process: Traders rewrite payoffs one by one using a ”call-spread” or ”smooth” function: smooth(x,p,n,e) = 0 e/2-e/2 n p x
- 5. Fuzzy Logic Danske Bank Smoothing a digital 5 • Digital Call Spread • Dig = CS = finite difference continuous approximation • Alt. interpretation: spread the notional evenly across strikes between (K-e/2,K+e/2) C K
- 6. Fuzzy Logic Danske Bank Smoothing a barrier 6 • Spread notional across barriers in (B-e/2,B+e/2) • Example: 1y 110-120 RKO in Black-Scholes (20%) • Weekly monitoring, barrier shifted by 0.60 std • Smooth barrier: knock-out part of the notional depending on how far we land in (B-e/2,B+e/2) total KO partial KO no KO
- 7. Fuzzy Logic Danske Bank Smoothing everything 7 • How can we generalize digital/barrier smoothing to any payoff? • We need to identify (and smooth) all discontinuities • Hence, we need to know exactly how the payoff is calculated • We need access to the ”source code” of the payoff • ”Hard coded” payoffs only allow evaluation against scenarios: ”compiled code” • But scripted payoffs provide full information: ”source code”
- 8. Fuzzy Logic Danske Bank Scripting Languages 8 • Simple ”programming” language optimized for the description of cash-flows • Describe the cash-flows in any transaction: vanillas, exotics, even CVA/KVA • And then evaluate the script with some model • Example: simplified Autocallable − If spot declines 3 years in a row, lose 50% of notional − Otherwise make 10% per annum until spot raises above initial level
- 9. Fuzzy Logic Danske Bank The many benefits of scripting cash-flows 9 • Run-time pricing and risks of transactions − Users create/edit a transaction by scripting its cash-flows − Then produce value and risk by sending the script to a model • Homogeneous representation of all cash flows in all transactions − Software can manipulate, merge and compress cash-flows across transactions − Crucial for ”derivatives on netting sets” like VAR/CVA/XVA/KVA/RWA • Crucially for us, provide ”source code” cash-flow information to the software − So the software can, among (many) others: − Obviously, evaluate cash-flows in different scenarios − Detect contingency, path-dependence or early exit and select the appropriate model − Optimise the calculation of value and risk sensitivities − Detect (and smooth) discontinuities
- 10. Fuzzy Logic Danske Bank Identifying discontinuities 10 • In programming, incl. scripting, discontinuities come from control flow − If this then that pattern − Call to discontinuous functions (meaning themselves implement if this then that statements) • Autocallable: − 3 control flows − Unstable risk
- 11. Fuzzy Logic Danske Bank Manual smoothing 11 • Replace if this then that by smooth • Does the job, but hard to write and read and error prone, even in simple cases
- 12. Fuzzy Logic Danske Bank Automatic smoothing 12 • Automatically turn − Nice, easy, natural scripts − Into smoothed scripts that produce stable risks • We have access to the script we can identify control flows • We only need to smooth them all
- 13. Fuzzy Logic Danske Bank Fuzzy Logic to the rescue 13 • Turns out we don’t need to transform the script at all! • We just need to evaluate differently the conditions and the conditional statements • In classical (sharp) logic, a condition is true or false − Schrodinger’s cat is dead or alive • In fuzzy logic, a condition is true to a degree = degree of truth or DT − Schrodinger’s cat is dead and alive − For instance, it can be 65% alive and 35% dead • Note: DT is not a probability Financial interpretation: condition applies to DT% of the notional − A barrier is 65% alive means 35% of the notional knocked out, the rest is still going − It does not mean that it is hit with proba 35%, which makes no sense on a given scenario
- 14. Fuzzy Logic Danske Bank Fuzzy Logic 14 • Invented in the 1960s • Not to smooth financial risks • But to build the first expert systems • Automate expert opinions − Expert: it is dangerous for the engine to go fast and be hot − Programmer: what do you mean by ”fast” and ”hot”? − Expert: say fast = ”>100mph” and hot = ”>50C” Code: if speed > 100 and heat > 50 then danger = true else danger = false endIf Makes no sense: (99.9,90) safe, (100.1,50.1) dangerous !!
- 15. Fuzzy Logic Danske Bank 1 2 1 2 1 2 1 2 1 2or DT C and C DT DT DT C C DT DT DT DT More Fuzzy Logic 15 • With fuzzy logic: − isFast and isHot are DTs = smooth functions resp. of speed and heat valued in [0,1] − DTs combine like booleans alt. Code: danger = fuzzyAnd( isFast( speed), isHot( heat)) Results make sense: (99.9,90) 50% dangerous, (100.1,50.1) 25% dangerous 1 2 1 2 1 2 1 2 min , or max , DT C and C DT DT DT C C DT DT
- 16. Fuzzy Logic Danske Bank Back to payoff smoothing 16 • Digital smoothing = apply fuzzy logic to if spot>strike then pay 1 • Barrier smoothing = apply fuzzy logic to if spot>barrier then alive = 0 • Classical smoothing exactly corresponds to the replacement of sharp logic with fuzzy logic in the evaluation of conditions • Reversely, we can systematically apply fuzzy logic to evaluate all conditions • Fuzzy logic for control flow effectively removes discontinuities • And stabilises risk sensitivities
- 17. Fuzzy Logic Danske Bank Evaluation of ”if this then that” with fuzzy logic 17 • Evaluation with sharp logic 1. Evaluate condition C 2. If C is true, evaluate statements between then and else or endIf 3. If C is false, evaluate statements between else and endIf if any • Evaluation with fuzzy logic 1. Record state S0 (all variables and products) just before the ”if” statement 2. Evaluate the degree of truth (DT) of the condition 3. Evaluate ”if true” statements 4. Record resulting state S1 5. Restore state to S0 6. Evaluate ”else” statements if any 7. Record resulting state S2 8. Set final state S = DT * S1 + (1-DT) * S2
- 18. Fuzzy Logic Danske Bank Fuzzy Logic implementation 18 • No change in scripts • Change the evaluation of if this then that statements only • Effectively stabilises risks sensitivities with minimal PV impact • (With proper optimization) just as fast as normal • Easily flick between sharp (pricing) and fuzzy (risk) evaluation
- 19. Fuzzy Logic Danske Bank Advanced Fuzzy Logic: condition domains 19 • Consider the script • The variable digPays is valued in {0,1} • To apply fuzzy logic / call spread to digPays > 0 makes no sense • Evidently the DT of digPays > 0 is digPays itself • This is easy to correct but we must first compute the domains of all conditions • Fortunately, we have access to the script • We can write code that traverses the script and calculates condition domains • This is not as hard as it sounds
- 20. Fuzzy Logic Danske Bank Advanced Fuzzy Logic: beware the cat 20 • We are used to conditions being true or false − Schrodinger’s cat is dead or alive − The cat cannot be dead and alive − If the cat is alive then it can’t be dead, etc. • We may write scripts based on such logic, like for instance: • With fuzzy logic, where spot > 100 to a degree and <= 100 to a degree − The logic that underlies the script is not applicable − And the result is a bias in the value by orders of magnitude! − With fuzzy logic, best stick to the decription of cash-flows and refrain from injecting additional ”logic” X has to be overwritten by either 0 or 1 right?
- 21. Fuzzy Logic Danske Bank Advanced Fuzzy Logic: what epsilon? 21 • Fuzzy logic applies a ”call spread” to conditions: − Too wide risk of valuation bias − Too narrow unstable risks − What is a reasonable size? • Typical e for a condition expr > 0 = fraction of the std dev of expr • Alternative = set e such that Proba ( - e/2 < expr < e/2) = target • Solution: use pre-simulations to estimate the first 2 moments of all conditions • And set their epsilon accordingly