Introduction to Interest Rate Models by Antoine SavineAntoine 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.
60 Years Birthday, 30 Years of Ground Breaking Innovation: A Tribute to Bruno...Antoine Savine
- 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.
Notes for Volatility Modeling lectures, Antoine Savine at Copenhagen UniversityAntoine Savine
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.
Stabilise risks of discontinuous payoffs with Fuzzy Logic by Antoine SavineAntoine Savine
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
Notes for Computational Finance lectures, Antoine Savine at Copenhagen Univer...Antoine Savine
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.
Practical Implementation of AAD by Antoine Savine, Brian Huge and Hans-Jorgen...Antoine Savine
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
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.
Value at Risk (VAR) summarizes the worst potential loss over a target period at a given confidence level, accounting for risks across an institution. VAR is calculated using statistical techniques to estimate losses that may occur but are unlikely to be exceeded. It is used to measure market, credit, operational and enterprise-wide risk and determine capital requirements to withstand unexpected losses.
Introduction to Interest Rate Models by Antoine SavineAntoine 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.
60 Years Birthday, 30 Years of Ground Breaking Innovation: A Tribute to Bruno...Antoine Savine
- 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.
Notes for Volatility Modeling lectures, Antoine Savine at Copenhagen UniversityAntoine Savine
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.
Stabilise risks of discontinuous payoffs with Fuzzy Logic by Antoine SavineAntoine Savine
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
Notes for Computational Finance lectures, Antoine Savine at Copenhagen Univer...Antoine Savine
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.
Practical Implementation of AAD by Antoine Savine, Brian Huge and Hans-Jorgen...Antoine Savine
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
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.
Value at Risk (VAR) summarizes the worst potential loss over a target period at a given confidence level, accounting for risks across an institution. VAR is calculated using statistical techniques to estimate losses that may occur but are unlikely to be exceeded. It is used to measure market, credit, operational and enterprise-wide risk and determine capital requirements to withstand unexpected losses.
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.
1. The document discusses switching models, which allow for changes in the behavior of economic and financial variables over time. These switches can be one-time changes or occur frequently.
2. Markov switching models generalize the dummy variable approach to allow for multiple "states of the world" that a variable can occupy. The probability of switching between states is governed by a transition probability matrix.
3. An example application uses a Markov switching model with two states to analyze real exchange rates. This allows for multiple switches between regimes and provides evidence on purchasing power parity theory.
A cancelable swap provides the right but not the obligation to cancel the interest rate swap at predefined dates. Most commonly traded cancelable swaps have multiple exercise dates. Given its Bermudan style optionality, a cancelable swap can be represented as a vanilla swap embedded with a Bermudan swaption. Therefore, it can be decomposed into a swap and a Bermudan swaption. Most Bermudan swaptions in a bank book actually come from cancelable swaps.
Cancelable swaps provide market participants flexibility to exit a swap. This additional feature makes the valuation complex. This presentation provides practical details for pricing cancelable swaps.
Using data from the S&P 500 stocks from 1990 to 2015, we address the uncertainty of distribution of assets’ returns in Conditional Value-at-Risk (CVaR) minimization model by applying multidimensional mixed Archimedean copula function and obtaining its robust counterpart. We implement a dynamic investing viable strategy where the portfolios are optimized using three different length of rolling calibration windows. The out-of-sample performance is evaluated and compared against two benchmarks: a multidimensional Gaussian copula model and a constant mix portfolio. Our empirical analysis shows that the Mixed Copula-CVaR approach generates portfolios with better downside risk statistics for any rebalancing period and it is more profitable than the Gaussian Copula-CVaR and the 1/N portfolios for daily and weekly rebalancing. To cope with the dimensionality problem we select a set of assets that are the most diversified, in some sense, to the S&P 500 index in the constituent set. The accuracy of the VaR forecasts is determined by how well they minimize a capital requirement loss function. In order to mitigate data-snooping problems, we apply a test for superior predictive ability to determine which model significantly minimizes this expected loss function. We find that the minimum average loss of the mixed Copula-CVaR approach is smaller than the average performance of the Gaussian Copula-CVaR.
The talk I gave at WBS 2020 Quant Finance Conference, Spring Edition, on "re-imagining" XVA so as to integrate it naturally into the front office workflow.
The key idea is to represent all FMTMs in spectral form via inline regression (even those FMTMs that are originally generated analytically within forward MC), and to treat this step as completion of "extended model" calibration. The regression coefficients can then be treated as derived market data, somewhat similar to non-parametric local volatility.
Viewed this way, extended model (MC) is generating not only true background factors, but also FMTMs, which can be trivially reconstructed when and where needed, together with the rest of the background factors. Collateral dynamics specification can then be interpreted as XVA "payoff", possibly scripted.
Should You Build Your Own Backtester? by Michael Halls-Moore at QuantCon 2016Quantopian
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.
"Portfolio Optimisation When You Don’t Know the Future (or the Past)" by Rob...Quantopian
We generally assume the past is a good guide to the future, but well do we even know the past? What effect does this uncertainty when estimating inputs have on the notoriously unstable algorithms for portfolio optimization?
I explore this issue, look at some commonly used solutions, and also introduce some alternative methods.
http://profitabletradingtips.com/trading-investing/trading-stock-options
Trading Stock Options
Trading stock options holds two definite advantages over trading stocks directly. A smart trader can certainly make money trading stocks. But by trading stock options the same trader can limit risk and leverage his trading capital.
What Are Options?
Options are contracts that give their buyers the right, not the obligation, to sell or purchase the underlying assets at a future date, within the term of the contract. Whether it is in stock options trading or in trading commodities like corn futures, trading options allows the investor to invest trading capital using a variety of strategies. A call option gives the buyer the right to buy and a put option gives the buyer the right to sell the underlying stock. Sellers are paid a premium for taking on the risk of having to buy or sell at a loss when the buyer chooses to execute his contract. Options are used to both hedge risk and to leverage trading capital.
Hedging Risk and Leveraging Capital
In trading stock options a buyer limits his risk to the premium paid. Let us say that your fundamental and technical analysis of ABC stock indicates that it will soon rise in price. You can buy a hundred share of ABC for $100 a share or $10,000. Or you may see that you can buy a $102 option on ABC for $1. A $102 option means that you can buy to stock for $102 a share at any time up until the end of the contract. Obviously the stock is currently worth $100 a share and you would not want to execute the contract. But, if your analysis is correct the stock price will go up. Let us say that the stock goes up to $110 a share. If you purchased the stock you make $1000 minus fees and commissions. That is a ten percent return on investment. And if the stock price falls to $90 a share you lose $1000, ten percent of your trading capital. But in trading stock options on ABC you pay $100 for a $102 call option on 100 shares. The stock goes up to $110. You execute the contract and purchase the stock for $102 a share and then sell for $110 a share. You make $8 a share or $800 which is a $700 profit or 700% return on invested capital. And, if the stock price falls to $90 you lose your initial $100 and no more.
Short and Long Term
Trading stock options is not just for short term profits. Let us say that you have purchased a hot growth stock. It has multiplied in value
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.
This document provides a summary of credit derivatives and related financial instruments in 3 paragraphs:
Paragraph 1 summarizes the historical development of credit derivatives, including total return swaps pioneered by Bankers Trust to transfer credit risk without transferring loans.
Paragraph 2 explains how credit default swaps (CDS) emerged to meet the need for credit protection, functioning essentially as insurance against bankruptcy where the protection buyer pays premiums to the protection seller.
Paragraph 3 gives an overview of other credit derivative instruments including basket CDS, CDS indexes, synthetic CDOs, credit-linked notes, and credit spread options; it also distinguishes CDS from total return swaps and collateralized debt obligations (CDOs).
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.
This document discusses duration, a model for measuring interest rate risk. Duration is a measure of how the price of a bond or loan will change in response to changes in interest rates. It represents the weighted average time until cash flows are received. Longer maturity bonds and loans have higher durations, meaning their prices are more sensitive to interest rate changes. Bonds with higher coupons have lower durations. Duration can be used to immunize a financial institution's balance sheet against interest rate risk by matching asset and liability durations.
A step by step guide on how to configure an algorithm for backtesting options strategies using QuantConnect.
GitHub project: https://github.com/rccannizzaro/QC-StrategyBacktest
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/
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.
1) The document introduces the classical linear regression model, which describes the relationship between a dependent variable (y) and one or more independent variables (x). Regression analysis aims to evaluate this relationship.
2) Ordinary least squares (OLS) regression finds the linear combination of variables that best predicts the dependent variable. It minimizes the sum of the squared residuals, or vertical distances between the actual and predicted dependent variable values.
3) The OLS estimator provides formulas for calculating the estimated intercept (α) and slope (β) coefficients based on the sample data. These describe the estimated linear regression line relating y and x.
Counterparty Credit Risk and CVA under Basel IIIHäner Consulting
Financial institutions which apply for an IMM waiver under Basel III need to fullfill a broad set of requirements. We present the quantitative, organizational and operational implications and provide some hand-on guidance how to fulfill the regulatory requirements.
Value at Risk (VaR) is a risk measurement technique used to estimate potential losses that could occur from market risk over a specified time period. The document discusses the need for VaR, how it is defined and calculated using historical simulation, its uses, strengths and weaknesses. It emphasizes that VaR should not be used alone and other risk measures like tail measures and stress testing are also important.
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.
35 page the term structure and interest rate dynamicsShahid Jnu
The document discusses theories and models of the term structure of interest rates. It covers spot and forward rates, the swap rate curve, traditional theories like expectations theory and liquidity preference theory, modern term structure models like CIR and Vasicek, yield curve factor models, and measures of sensitivity to shifts in the yield curve like duration.
Swaps explained. Very useful for CFA and FRM level 1 preparation candidates. For a more detailed understanding, you can watch the webinar video on this topic. The link for the webinar video on this topic is https://www.youtube.com/watch?v=JKBKnxM2Nj4
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.
1. The document discusses switching models, which allow for changes in the behavior of economic and financial variables over time. These switches can be one-time changes or occur frequently.
2. Markov switching models generalize the dummy variable approach to allow for multiple "states of the world" that a variable can occupy. The probability of switching between states is governed by a transition probability matrix.
3. An example application uses a Markov switching model with two states to analyze real exchange rates. This allows for multiple switches between regimes and provides evidence on purchasing power parity theory.
A cancelable swap provides the right but not the obligation to cancel the interest rate swap at predefined dates. Most commonly traded cancelable swaps have multiple exercise dates. Given its Bermudan style optionality, a cancelable swap can be represented as a vanilla swap embedded with a Bermudan swaption. Therefore, it can be decomposed into a swap and a Bermudan swaption. Most Bermudan swaptions in a bank book actually come from cancelable swaps.
Cancelable swaps provide market participants flexibility to exit a swap. This additional feature makes the valuation complex. This presentation provides practical details for pricing cancelable swaps.
Using data from the S&P 500 stocks from 1990 to 2015, we address the uncertainty of distribution of assets’ returns in Conditional Value-at-Risk (CVaR) minimization model by applying multidimensional mixed Archimedean copula function and obtaining its robust counterpart. We implement a dynamic investing viable strategy where the portfolios are optimized using three different length of rolling calibration windows. The out-of-sample performance is evaluated and compared against two benchmarks: a multidimensional Gaussian copula model and a constant mix portfolio. Our empirical analysis shows that the Mixed Copula-CVaR approach generates portfolios with better downside risk statistics for any rebalancing period and it is more profitable than the Gaussian Copula-CVaR and the 1/N portfolios for daily and weekly rebalancing. To cope with the dimensionality problem we select a set of assets that are the most diversified, in some sense, to the S&P 500 index in the constituent set. The accuracy of the VaR forecasts is determined by how well they minimize a capital requirement loss function. In order to mitigate data-snooping problems, we apply a test for superior predictive ability to determine which model significantly minimizes this expected loss function. We find that the minimum average loss of the mixed Copula-CVaR approach is smaller than the average performance of the Gaussian Copula-CVaR.
The talk I gave at WBS 2020 Quant Finance Conference, Spring Edition, on "re-imagining" XVA so as to integrate it naturally into the front office workflow.
The key idea is to represent all FMTMs in spectral form via inline regression (even those FMTMs that are originally generated analytically within forward MC), and to treat this step as completion of "extended model" calibration. The regression coefficients can then be treated as derived market data, somewhat similar to non-parametric local volatility.
Viewed this way, extended model (MC) is generating not only true background factors, but also FMTMs, which can be trivially reconstructed when and where needed, together with the rest of the background factors. Collateral dynamics specification can then be interpreted as XVA "payoff", possibly scripted.
Should You Build Your Own Backtester? by Michael Halls-Moore at QuantCon 2016Quantopian
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.
"Portfolio Optimisation When You Don’t Know the Future (or the Past)" by Rob...Quantopian
We generally assume the past is a good guide to the future, but well do we even know the past? What effect does this uncertainty when estimating inputs have on the notoriously unstable algorithms for portfolio optimization?
I explore this issue, look at some commonly used solutions, and also introduce some alternative methods.
http://profitabletradingtips.com/trading-investing/trading-stock-options
Trading Stock Options
Trading stock options holds two definite advantages over trading stocks directly. A smart trader can certainly make money trading stocks. But by trading stock options the same trader can limit risk and leverage his trading capital.
What Are Options?
Options are contracts that give their buyers the right, not the obligation, to sell or purchase the underlying assets at a future date, within the term of the contract. Whether it is in stock options trading or in trading commodities like corn futures, trading options allows the investor to invest trading capital using a variety of strategies. A call option gives the buyer the right to buy and a put option gives the buyer the right to sell the underlying stock. Sellers are paid a premium for taking on the risk of having to buy or sell at a loss when the buyer chooses to execute his contract. Options are used to both hedge risk and to leverage trading capital.
Hedging Risk and Leveraging Capital
In trading stock options a buyer limits his risk to the premium paid. Let us say that your fundamental and technical analysis of ABC stock indicates that it will soon rise in price. You can buy a hundred share of ABC for $100 a share or $10,000. Or you may see that you can buy a $102 option on ABC for $1. A $102 option means that you can buy to stock for $102 a share at any time up until the end of the contract. Obviously the stock is currently worth $100 a share and you would not want to execute the contract. But, if your analysis is correct the stock price will go up. Let us say that the stock goes up to $110 a share. If you purchased the stock you make $1000 minus fees and commissions. That is a ten percent return on investment. And if the stock price falls to $90 a share you lose $1000, ten percent of your trading capital. But in trading stock options on ABC you pay $100 for a $102 call option on 100 shares. The stock goes up to $110. You execute the contract and purchase the stock for $102 a share and then sell for $110 a share. You make $8 a share or $800 which is a $700 profit or 700% return on invested capital. And, if the stock price falls to $90 you lose your initial $100 and no more.
Short and Long Term
Trading stock options is not just for short term profits. Let us say that you have purchased a hot growth stock. It has multiplied in value
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.
This document provides a summary of credit derivatives and related financial instruments in 3 paragraphs:
Paragraph 1 summarizes the historical development of credit derivatives, including total return swaps pioneered by Bankers Trust to transfer credit risk without transferring loans.
Paragraph 2 explains how credit default swaps (CDS) emerged to meet the need for credit protection, functioning essentially as insurance against bankruptcy where the protection buyer pays premiums to the protection seller.
Paragraph 3 gives an overview of other credit derivative instruments including basket CDS, CDS indexes, synthetic CDOs, credit-linked notes, and credit spread options; it also distinguishes CDS from total return swaps and collateralized debt obligations (CDOs).
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.
This document discusses duration, a model for measuring interest rate risk. Duration is a measure of how the price of a bond or loan will change in response to changes in interest rates. It represents the weighted average time until cash flows are received. Longer maturity bonds and loans have higher durations, meaning their prices are more sensitive to interest rate changes. Bonds with higher coupons have lower durations. Duration can be used to immunize a financial institution's balance sheet against interest rate risk by matching asset and liability durations.
A step by step guide on how to configure an algorithm for backtesting options strategies using QuantConnect.
GitHub project: https://github.com/rccannizzaro/QC-StrategyBacktest
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/
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.
1) The document introduces the classical linear regression model, which describes the relationship between a dependent variable (y) and one or more independent variables (x). Regression analysis aims to evaluate this relationship.
2) Ordinary least squares (OLS) regression finds the linear combination of variables that best predicts the dependent variable. It minimizes the sum of the squared residuals, or vertical distances between the actual and predicted dependent variable values.
3) The OLS estimator provides formulas for calculating the estimated intercept (α) and slope (β) coefficients based on the sample data. These describe the estimated linear regression line relating y and x.
Counterparty Credit Risk and CVA under Basel IIIHäner Consulting
Financial institutions which apply for an IMM waiver under Basel III need to fullfill a broad set of requirements. We present the quantitative, organizational and operational implications and provide some hand-on guidance how to fulfill the regulatory requirements.
Value at Risk (VaR) is a risk measurement technique used to estimate potential losses that could occur from market risk over a specified time period. The document discusses the need for VaR, how it is defined and calculated using historical simulation, its uses, strengths and weaknesses. It emphasizes that VaR should not be used alone and other risk measures like tail measures and stress testing are also important.
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.
35 page the term structure and interest rate dynamicsShahid Jnu
The document discusses theories and models of the term structure of interest rates. It covers spot and forward rates, the swap rate curve, traditional theories like expectations theory and liquidity preference theory, modern term structure models like CIR and Vasicek, yield curve factor models, and measures of sensitivity to shifts in the yield curve like duration.
Swaps explained. Very useful for CFA and FRM level 1 preparation candidates. For a more detailed understanding, you can watch the webinar video on this topic. The link for the webinar video on this topic is https://www.youtube.com/watch?v=JKBKnxM2Nj4
This document provides an overview of interest rate swaps, including:
- Swaps involve exchanging fixed rate interest payments for floating rate payments between two parties over time.
- Cash flows are calculated based on the notional amount and interest rates, and are exchanged on settlement dates.
- Swaps are valued by calculating the present value of the fixed and floating rate cash flows.
- Swaps allow parties to hedge or take on interest rate risk, as the fixed leg value depends on long-term rates while the floating leg follows short-term rates. Credit risk also exists if counterparties default.
- Common uses of swaps include hedging future debt issuance, creating synthetic floating rate debt, and
Slides on Introduction to Treasury Division in Banking, its Roles, Structure, Products, Equipment, Dealer and Brokers, Dealing Terminology,Dealer Code of Conduct, Money Market, Forex, Swap, Option, IRS, CCS, Derivatives, Types of Financial Markets, Forex Deal example
This document provides an overview of asset swaps and Z-spreads. It begins with an introduction to interest rate swaps, including how they work and their cash flows. It then discusses zero-coupon swaps and how swap curves can be used to derive forward rates and discount factors. The document explains how asset swaps combine a bond investment with an interest rate swap to exchange fixed cash flows for floating rates. Finally, it introduces the concept of Z-spreads which allow fair comparisons of bond yields to swap rates.
7_Analysing and Interpreting the Yield Curve.pptMurat Öztürkmen
The document discusses analysing and interpreting the yield curve. It covers the importance of the yield curve, constructing the curve from discount functions, theories like the expectations hypothesis and liquidity preference theory, the formal relationship between spot and forward rates, interpreting the shape of the curve, and fitting the curve from market data. Specifically, it notes the challenges of fitting the curve given a lack of liquid market data inputs and impact of bid-offer spreads, and recommends using a non-parametric interpolation method like the Svensson model to produce a smoother forward curve.
Interest Rates overview and knowledge insightjustmeyash17
1) Interest rates are determined by factors like expected inflation, default risk, liquidity, and maturity. The relationship between short and long-term interest rates is known as the term structure.
2) The term structure can be upward-sloping, flat, or downward-sloping (inverted). Upward slopes typically occur when short-term rates are low.
3) Three main theories explain the term structure: expectations theory, market segmentation, and liquidity premium theory. The liquidity premium theory best explains the empirical regularities by incorporating both expectations of future rates and investors' preference for liquidity.
A swap is an agreement between two parties to exchange interest payments on a set notional principal over a period of time. In a plain vanilla interest rate swap, one party pays a fixed interest rate while the other pays a floating rate, typically linked to LIBOR. The value of the swap to each party is determined by discounting the expected cash flows using the current LIBOR curve. Midway through the contract, the value depends on how interest rate expectations have changed since inception.
- Interest rate swaps allow parties to exchange interest rate cash flows, with one party paying fixed rates and the other floating rates tied to a reference rate like LIBOR.
- Corporations, money managers, mortgage holders, and dealers use swaps to hedge interest rate risks and manage portfolio exposures.
- The swap spread, the difference between Treasury yields and swap fixed rates, is influenced by factors like funding costs and supply/demand dynamics.
- Forward-starting swaps allow market participants to express views on future interest rate movements over a specific time horizon.
1) The document discusses constructing multiple swap curves to price financial products consistently with different swap markets, including those with and without collateral.
2) It explains how to construct swap curves for a single currency market based on interest rate swaps alone. It then expands this to incorporate cross-currency swaps by deriving separate discounting and index curves.
3) The method is further expanded to consistently incorporate basis spreads observed in tenor swaps between different tenors (e.g. 3-month and 6-month rates) into the construction of discounting and multiple index curves.
This document provides an overview of interest rate swaps, caps, and floors. It defines an interest rate swap as an agreement where two parties exchange periodic interest payments, based on a notional principal amount, with one party paying a fixed rate and the other paying a floating rate. The document outlines key terms used in swaps, how swap rates are calculated, and how swaps can be used for asset/liability management. It also discusses related derivatives like swaptions, rate caps and floors, and how these tools can be used by institutional investors to manage interest rate risk.
This document provides an overview of interest rate swaps, caps, and floors. It defines an interest rate swap as an agreement where two parties exchange periodic interest payments, based on a notional principal amount, with one party paying a fixed rate and the other paying a floating rate. The document outlines key terms used in swaps like notional amount, counterparties, and reset frequency. It also explains how swap rates are quoted in the market and calculated, and how swaps can be used by institutional investors to manage assets and liabilities. Finally, it briefly introduces related financial agreements like swaptions, caps, and floors.
The document discusses factors that determine the money supply and the money multiplier. It defines the monetary base (MB) as currency in circulation plus reserves, and M1 as currency plus checkable deposits. The money multiplier relates these, with M1 equal to the multiplier times MB. The multiplier depends on the currency ratio, reserve ratio, and excess reserves ratio. Changes in these ratios, such as due to bank panics, can impact the money supply by altering the multiplier. The Fed has more control over MB than M1 due to additional influencing factors.
This document provides an overview of swaps, including interest rate swaps, currency swaps, and commodity swaps. It defines a swap as an agreement to exchange cash flows according to specified conditions. Interest rate swaps involve exchanging fixed and floating rate payments. Currency swaps also typically involve an exchange of principal amounts. Commodity swaps define a notional principal in terms of a commodity rather than exchanging principal. Examples of swap structures and cash flows are provided to illustrate how the different types of swaps work.
This document provides an overview of swaps, including interest rate swaps, currency swaps, and commodity swaps. It defines a swap as an agreement to exchange cash flows according to specified conditions. Interest rate swaps involve exchanging fixed and floating rate payments. Currency swaps also typically involve an exchange of principal amounts. Commodity swaps are similar to forward contracts that fix the price of a commodity over time. Examples of swap cash flows are provided to illustrate how the various types of swaps work.
This document discusses discount factors and mark-to-market valuation of cross currency swaps. It begins by explaining how discount factors are derived from risk-free bonds and how rates like Libor and OIS are used as proxies. However, it notes that swap rates cannot be directly used as discount factors since they do not guarantee a fixed payment amount at maturity. The document then discusses how to model the cash flows of interest rate swaps and cross currency swaps, and how to calculate stochastic and implied swap rates to value them using mark-to-market approaches.
This document provides an overview of key finance concepts for managers in a course on finance. It defines cash flows, rates of return, interest rates, time value of money, and timelines. It also explains future value and present value calculations for ordinary annuities and annuities due using relevant formulas. Compounding and discounting are shown to be related concepts for dealing with time value of money.
The foreign exchange market involves the daily global purchase and sale of national currencies totaling $4 trillion per day. It has experienced huge growth recently and vastly exceeds other financial markets in size. While the forex market includes spot transactions, it is largely composed of derivatives contracts used to hedge currency risk. For more information on trading in this massive market, the document recommends visiting the website FX-Arabia.
Similar to A Brief History of Discounting by Antoine Savine (20)
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1. A brief history of discounting
1. Historical platforms for rates and discounting
2. Current platforms for rates and discounting
1. Collateralized transactions
2. Uncollateralized transactions
Antoine Savine
antoine@asavine.com
3. Classical discounting framework
up to mid 1990s
• Yield curve: for a given currency, the term structure of interest rates for all
maturities
– Discount Factor: PV of 1 currency unit delivered at time T
– The curve R(T) gives all relevant rate information
R T T
DF T e
log DF T
R T
T
continuous rate
T R T
yield curve
Feed models
rates for forwards/futures
discount factors for option payoffs
short forward rates for rate exotics
log DF T
f T
T
Value rate transactions
“risk free” loans
swaps, spot or forward starting
T0
lend notional
Tn
receive redemption + interest
N
0 01 , n nN F T T T T
0 0 01 , n n nNDF T N F T T T T DF T
0
0
0
, n
n
n n
DF T DF T
F T T
T T DF T
For the transaction to be fair
“coverage”
rate
4. Interest rate swaps
• Swap: exchange fixed for floating coupons
• Floating leg, assuming Libor is the interest rate, is same as rolling deposits
• Hence and for the fixed leg
• Hence for a par swap
0T nT
1. . ,i iN K cvg T T
1 1. , . ,i i i iN Libor T T cvg T T
fixed leg
floating leg
= = 0
0 nfloat N DF T DF T 1
1
,
n
i i i
i
DF T cvgfix N T TK
annuity A
cvg = day count
0
0 ,
n
n
DF T DF T
K
A T T
5. Construction of the curve
• For a given set of n liquid standard swaps, assume some interpolation scheme on R
– Set R1 to match the market value of swap 1
– Then set R2 to match the market value of swap 2 with R1 constant
– Etc.
• Then use the curve to value a portfolio of custom swaps and other products
• And compute rate risk by finite differences
– Bump the value of the first swap
– Recalibrate the curve
– Reprice the book
– Loop over swaps used for calibration
0.00%
0.50%
1.00%
1.50%
2.00%
0 1 2 3 4 5 6 7 8 9 10
continuous
priced
market
6. Instruments for curve construction
• Short term: 1d to 3m-6m
– Deposits
– Work like single period swaps
– Valued with swap formula with one period
• Long term: 2y to up to 50y
– Swaps
• Medium term: 3m-6m to 2y
– Libor futures
– If futures same as Libor Forwards or FRA (Forward Rate Agreements)
• Work like single period forward starting swaps
• Valued with the same formula
– Problem: futures not the same as FRAs
– We need to “turn” futures into FRAs first
1 2
2
DF T DF T
K
cvg DF T
7. Future and Forward rates
• The only exciting aspect for quants in the world of yiled curves in those days
– “Convexity” bias links higher future rates to covariance between forward and discount rates
– Active field of research in those days with may academic and professional publications
• Intuition for the convexity bias
– I buy the future rate (= sell the future in market language)
– Increase in rates gives positive margin flow that I invest at a higher rate
– Decrease in rates gives negative margin that I borrow at a lower rate
– So buying future rates is better than buying forward rates: no margin flows in that case
– Hence, in rate terms, Fut > FRA
• Impact on the construction of curves
– Market quotes Futures, curve construction takes FRAs
– Need to “remove” convexity bias before construction
8. Future/Forward convexity
• FRA
– Single cash flow on :
– On par when
– That is
– Where is the martingale measure associated with the numeraire
– Also called Tpay-forward measure
• Future
– Continuous margin flows at time t
– We can buy/sell the future on par at any time t, so expected discounted future flows must be 0
– so and since finally
payT ,fixT start endL T T FRA cvg
0 0
exp exp
pay pay
fix
T T
RN RN
s s TE r ds FRA E r ds L
0 0
exp exp ,
pay pay
pay
fix f
pay
i fx ix
T T TRN RN
s s
T
TT TE r ds FRA E r ds L FRAE EL L
payT
Q , exp
payT
RN
pay sDF T E r ds
tdFut
exp 0
t t u
RN v
t v ut t
E r dv dFut
0RN
t tE dFut fix fixT TFut L fix
RN
TFut E L
9. Convexity: change of numeraire formulas
• Radon-Nykodym derivative from forward-neutral to risk-neutral:
• Future:
• We assume a normal diffusion on the FRA
• Then by Girsanov’s theorem:
• If we assume constant volatility for the FRA
And constant volatility for all discount rates
And constant correlation
We finally get:
• Or lift simplifying assumptions and use of a multi-factor interest rate model
0
0,
exp
pay pay
RN
pay
T T
X
s
DF TdQ
dQ r ds
0
0,
exp
pay pay
fix fix fixpay pay
RN
payT TRN
T T TT T
X
s
DF TdQ
Fut E L E L E L
dQ r ds
0
log , ,
fix
pay
fix fix
TTRN
T T pay sE L E L d DF s T dFRA
,
2
fixF r
N N fix pay
T
Fut FRA FRA r T T
10. That’s how we did it in 1995
• A simple methodology
– Lightweight: doable on Excel without code
– Simple maths
– Simple algorithmics: one-dimensional bootstrap (assuming “local” interpolation like linear)
– Little quant involvement
• Based on one assumption: Libor is the interest rate
– We can invest and borrow risk free at Libor
– Libor represents interest rates and interest rates only
• That turns out to be wrong
– Libor is an index of the Interbank Offered rate
– Incorporates information other than interest rates
– After crisis, Libor disconnected from interest rates, which are frozen near zero
11. Basis swaps
• Swap floating for floating on different frequencies, say 3m for 6m
– Both legs are supposed to have same value : floating leg PV does not depend on frequency
– Basis swap theoretically on par
• Yet, 6m systematically trades at significant premium: up to several tens of bp!
• Explanation: credit and liquidity for underlying Libor transactions
– Transactions underlying to Libor are “real” transactions with exchange of capital
– They are unsecured (although the basis swap is typically collateralized)
– Hence they carry credit risk on capital and interest
(swaps have no risk on capital and generally none on interest either)
– Credit and liquidity premia on underlying Libor transactions explain the basis swap
• Reflect fact that Libor is not the risk-free rate
0T nT . 6 . 6N L m cvg m
. 3 . 3N L m X cvg m
floating leg
floating leg
0 nfloat N DF T DF T
12. Intuition behind basis swaps
• Example: bank B borrowing at Libor, funding for 6m – 2 possibilities
1. Borrow for 6m at Libor 6m
2. Or borrow for 3m at Libor 3m, then borrow capital and interests for another 3m
– And lock the rate of the rollover loan with a 3m Libor 3m forward
6m
N
1 0.5 0,6N L m
today
3m
N
1 0.25 0,3N L m
today 1 0.25 0,3N L m
1 0.25 0,3 1 0.25 3 ,3N L m L m m
6m
1 0.25 0,3 1 0.25 3 ,3FL mN L m m
1 0.25 0,3 1 0.25 3 ,3N L m L m m
13. Intuition behind basis swaps (2)
• Finally the cash flows in the “3m+3m3m” case resolve to
• Its effective 6m rate is
– According to school books, it must be same as straight 6m rate, therefore
– Or
– Which is same as basis swaps being on par
today
1 0.25 0,3 1 0.25 3 ,3 1 0.5 0,6L m FL m m L m
1 0.5 0,6
3 ,3 4 1
1 0.25 0,3
L m
FL m m
L m
N
1 0.25 0,3 1 0.25 3 ,3FL mN L m m
6m
2 1 0.25 0,3 1 0.25 3 ,3 1L m FL m m
14. Intuition behind basis swaps (3)
• Now suppose B is in financial difficulty in 3m
1. In case B borrowed for 6m, it keeps funding for another 3m,
lender can only hope that B survives another 3m to repay capital and interests
2. But in case it borrowed for 3m with a forward 3m3m:
• It may find it difficult to roll the loan for another 3m liquidity risk
• And/or pay large premium, say Libor3m+200, not Libor credit risk
• Forward Libor only protects against Libor fluctuations, not against own borrowing capacity or spread
– In this case, B is in a better situation in case 1
– Whereas the lender is very clearly in a better situation in case 2
• Hence 6m rate must be higher than “3m+3m3m”
– It is the contrary that would offer arbitrage opportunities
– And this explains the basis swap as a credit/liquidity premium on underlying Libor transactions
0,6 2 1 0.25 0,3 1 0.25 3 ,3 1L m L m FL m m
16. Currency basis swaps
• Swap floating for floating on different currencies, like USD vs EUR
• In theory the swap is on par
• In practice EUR libor trades at significant premium over USD libor
– European banks are seen as less creditworthy and pay credit premium over US banks
– European banks also pay a premium to access USD liquidity
0T nT
. . .EURN L EUR cvg EUR
. . .USDN L USD cvgUSD
USD notional
EUR notional USD notional
EUR notional
= = 0
17. Synthetic cross currency transactions
• If a US bank borrows USD at USD Libor…
• …and adds-on a USDJPY basis swap…
• …then it ends up borrowing JPY at JPY Libor – BS
. . .USDN L USD cvgUSD
USD notional
USD notional
. . .JPYN L JPY BS cvg JPY
. . .USDN L USD cvgUSD
USD notional
JPY notional USD notional
JPY notional
. . .JPYN L JPY BS cvg JPY JPY notional
JPY notional
+
=
18. Intuition behind currency basis swaps
• European bank B with EUR Libor funding, needs USD funding – 2 possibilities
1. Raise USD on markets
• B has lower credit than US banks that borrow at USD Libor
• B pays a premium over USD Libor
2. Borrow EUR at EUR Libor and basis swap into USD
• Must end up paying same rate or there is an arbitrage
• Hence the xCCY basis swap
• Similarly US bank C with USD Libor funding , needs EUR funding, can
1. Either raise EUR, paying less than EUR Libor for better credit
2. Or borrow USD at USD Libor and basis swap into EUR
– Must end up paying same rate, hence basis spread necessary to account for difference in credit
– Note basis spread always reflects a relative difference between different pools of banks
• If some US bank funds at Libor + 100 in USD
• Then it funds at Libor + 100 – BS in EUR
• I borrow in the rate of the currency, but on the basis of my credit
20. The 1997 Asian crisis
• 1997 collapse of Thailand rose a credit crisis that propagated in Asia 1997-1999
• Consequently Asian banks’ credit deteriorated while liquidity premia raised
• Asian basis swaps raised from negligible to ~50bp
• We can no longer discount USD flows at USD Libor and JPY flows at USD JPY
– This is inconsistent and actually arbitrageable
– And note that with low JPY rates and high USDJPY basis US banks fund in JPY at JPY Libor – BS < 0 !!
• The banks who first understood this arbed those slower to react
• Eventually new standards emerged for rate transactions
21. The post 1997 rate machinery
• The industry agreed to use USD Libor 3m as reference for discounting
– Dominated by US and European banks with EURUSD BS insignificant at the time
– USD Libor seen as risk-free reference because “US banks can’t fail”
– Economically, Japanese houses should really discount at JPY Libor but we cut some corners
to avoid operational nightmare of counterparties to a transaction having different valuations
– A somewhat fishy machinery, did the job until the global financial crisis
• Means we do “as if” everybody’s borrows and invests at Libor 3m in USD
• And with xCCY BS add-on we all fund in other ccies at “CCY Libor – xUSD BS”
• Refers to a theoretical discount curve for the currency called “cash curve”
= CCY Rate curve in the USD Libor 3m basis
22. The post 1997 rate machinery (2)
• Libor swaps no longer reference for discount (except vs 3m in the US),
instead we used a “cash curve” per currency = “Libor – BS”
• Floating leg pv no longer but
– Where the forward libors FL are read on a “rate surface” by maturity and Libor frequency
– And DFs are computed on the cash curve
• Construction (and risk management) of curves (now surfaces) much more involved
– Manipulates large amounts of market data
– Construct curves by fitting swaps, BS and xCCY BS together by setting discounts and forward Libors
– Risk less evident to compute and comprehend: what do we freeze when we bump swap rates?
• Discount basis?
• Or final discount rates?
– No longer takes 30min on a spreadsheet, heavy machinery in C++ with thousands of lines of code
• Not really complicated but very heavy
– Had to be centralized and dedicated quant teams were put in place
– An so we invented the “linear quant”
0 nN DF T DF T 1 1
1
, ,
n
i i i i i
i
N DF T FL T T cvg T T
23. And then we had the global crisis
• And it all went bananas
– Rates stick to 0 while credit and liquidity premia went nuts
– Libor (even USD) absolutely disconnected from anything to do with interest rates
• So we needed to rethink the whole discounting machinery from the ground
– Acknowledge disconnection between Libor and Rate
– Methodology should work under assumption that Libor is a random number
• Although a published one, with swaps trading on it
• Some IBs were quicker to react and arbed the street for billions
• Subsequently, IBs moved quants from shrinking exotic businesses into linear rates
• And the quant community came out with new standards for discounting
• Best practice today is theoretically clean and incorporates lessons from the crisis
• It is also somehow complex and keeps thousands of quants busy on a daily basis
25. Current practice for discounting
• 2 types of transactions: collateralized and uncollateralized
• Collateralized transactions
– Discount flows at collateral rates
– For floating legs, read forward rates on rate surfaces
– For cash-flows in currencies other than collateral’s (domestic) currency
“Convert” flows in domestic currency using a forward Fx curve built from Fx and xCCY swaps
– Advanced:
• Collateral convexity adjustment when a transaction has different collateral to collateral for curve construction
• Collateral option and “cheapest to deliver” when CSA allows to change collateral
• Uncollateralized transactions
– Makes no sense to discount flows at any kind of “risk free” rate since we know counterparties fail
– Account for potential loss subsequent to counterparty default CVA
– Possible even account for own credit and funding DVA, FVA, … xVA
27. Collateral discounting: example 1
• Transaction = purchase option
• Without collateral
– Must pay premium
– Cost of funding premium: some funding rate r
– Discount payoff at rate r
• With collateral
– Premium reverts immediately as collateral
– No need to fund the premium
– But must compensate collateral at collateral rate c
– Hence, discount payoff at rate c
2 21
2
S SS trV SV S V V
2 21
2
S SS tcV SV S V V
net funding cost
28. Collateral discounting: example 2
• Transaction = swap with negative PV
• Without collateral
– Negative PV = borrowing money
– I expect to pay negative coupons in the future
– In the meantime, I invest the money at some funding rate r
– Discount expected cash flows at rate r
• With collateral
– Pay the negative PV straight away as collateral
– My collateral is compensated at collateral rate c
– Hence, discount expected cash flows at rate c
29. Collateral discounting: maths
• Cash flow CF delivered at future time T, discounted at some funding rate r
• Add flows linked to collateral
– Collateral posting of C=-V is not a cash flow
– The cash-flow is the cost C.r.dt of funding collateral net of its compensation C.c.dt
– Differentiate
– So
• The funding rate r disappears from the equation, discount at collateral rate c
• Note that collateral changes discount rate but not probability measure
, exp ...
T
T t t s T
t
V CF V E r ds V
0 00exp exp , , exp exp
T T s T T s
t t s T u s s s t t t s T u s s s
t t t t t t
V E r ds V r du V c r ds t t E V E r ds V r du V c r ds
0 0 0
exp exp
T T s
t t t t s T t u s s s t t t t t t
t t t
E dV E r r ds V r r du V c r ds V c r ds E cV
0 0 0exp , , exp exp
T T T
t t t s T t t s T s
t t t
E V E c ds V t t V E c ds V c ds CF
30. Valuation of collateralized trades: IRS
• OIS
– OIS = Overnight Indexed Swap
– Trades compounded 1d rate Vs Libor
– EONIA in EUR, Fed Funds in the US, …
– Close to a risk free rate due to very short maturity
• Example / norm for IB swaps: cash collateral in the currency of the swap, earns OIS
– Project floating cash flows from the forward curve surface
– Discount cash flows on the OIS curve
– Discount curve: “Disc = OIS = Libor – LIBOR/OIS BS”
– Build discount and forward curves together to fit market quotes for Libor swaps and OIS basis swaps
31. Valuation of collateralized trades: IRS (2)
• Example 2: cash collateral (earning OIS) in a different currency
– Say: BNP and SocGen trade a USD swap with EUR cash collateral
– Project floating cash flows from the USD forward curve surface
– Discount cash-flows with USD rates but on the EUR OIS basis
– “Disc = USD Libor + EURUSD Libor BS - EUR Libor/OIS BS”
– Intuition:
• EUR collateral creates a context for the transaction where EUR funding = EUR OIS
• So USD funding = EUROIS basis swapped into USD
• Hence, USD CF are discounted on the EUR OIS curve basis swapped into USD
• No direct BS from EUROIS into USD we combine EUR Libor/OIS and EURUSD Libor BS
32. Valuation of collateralized trades
equity and forex forwards
• European equity option, OTC, fully collateralized
– Depends on rates through discounting and forward
– Textbook forward (without dividends):
– Depends on discount factor forward changes with collateral?
– No: equity is compounded at repo rate
Repo = funding rate when posting equity as collateral
Therefore depends on the equity but not on derivative transaction
• What about Fx?
– Fx forward:
– Changes when collateral switched from domestic cash to foreign cash?
– No: change of basis shifts both rates by the same amount, hence rate difference remains constant
In other terms, each DF changes but not the ratio
,
,
tS
F t T
DF t T
,
, exp , ,
,
for
t t d f
dom
DF t T
F t T S S R t T R t T T t
DF t T
33. Valuation of collateralized trades
equity and forex forwards
• FRA and Libor futures
– With discount rate r, FRA solves in X:
– In the context where the curve was built, r is OIS of the currency of the Libor
– If we change collateral to r2=r+b, then FRA now solves
– So if the basis b between r and r2 is independent from the FRA (as a particular case, deterministic),
the FRA remains unchanged when collateral changes,
otherwise it is subject to a convexity bias
• Swaps and forward swaps
– Forward swap = average of FRAs weighted by DFs:
– Depends on discount factors, hence changes with collateral
– Lower collateral rate pushes weights to the back of the curve, generally increase par swap rate
– USD swaps are lower with EUR collateral
1 1
1
, ,
n
i i i i i
i
DF T FL T T cvg T T
F
Annuity
2
1 20
exp , 0
T
RN
sE r ds Lib T T X
2
1 20
exp , 0
T
s s bE r b ds Lib T T X
34. Multi-Curve Linear markets: domestic case
• Multi-Curve Linear market = collection of curves of forward rates,
one for each floating rate reference: OIS, Libor 3m, 6m, 12m, etc.
• Each curve implements some interpolation scheme
provides forward rates for all maturities
– Forward OIS rates can be read on the OIS curve for all future dates
– Forward Libor 3m rates can be read on the Libor 3m curve
– In case something like the 4m Libor is needed, interpolated from the Libor 3m and Libor 6m curves
• Forward rates do not depend on collateral if we ignore convexity adjustments
35. Multi-Curve Linear Markets (2)
• For a transaction with cash collateral in the domestic currency,
the curve that corresponds to collateral rate in general OIS
is used for discounting cash flows
• Where
– X is the reference curve: OIS, Libor 3m, 6m, 12m, …
– is the timeline for the floating frequency of X
• For OIS, every business day from today
• For Libor 3m, every 3m
• Etc
–
– F is the forward rate X picked on the curve by maturity
– cvg is the conventional day count for the floating rate X
1 1
0
1 1
1 , ,1 , ,
X n T
X n T n T
X i i i i
i
DF T
F T T cvg T TF T T cvg T T
iT
max , in T i T T
0T n T T
compound
discount
36. Valuation with a MCL market
• Discounting with cash collateral with collateral rate reference X
• Fixed cash flow CF paid at date T
• Floating cash flow with floating rate Y for period
we pick the forward rate on the Y curve (e.g. Libor3m) and discount it on the X curve (in general, OIS)
• Fixed swap leg
• Floating swap leg
Ref Y (e.g. Libor 3m)
XPV DF T CF
2 1 2 1 2, ,X YPV DF T F T T cvg T T 1 2,T T
0T nT
1. . ,i iN K cvg T T
1
1
,
n
X i i i X
i
PV NK DF T cvg T T NK annuity
0T nT
1 1. , . ,i i i iN Y T T cvg T T
1 1
1
, ,
n
X i Y i i i i
i
PV N DF T F T T cvg T T
37. Building a MCL market
• Build MCL market = set forward rates for all references
so as to zero the PV of selected par swaps and basis swaps
• Swaps are cash collateralized with collateral rate = OIS in the currency
• First build OIS and main Libor curve (3m for USD, 6m for EUR, …)
so as to fit Libor swaps and OIS/Libor basis swaps together
– Bi-dimensional bootstrap
– In case of a smooth interpolation, bootstrap cannot be used and global fit is necessary
– Build short term out of spot Libor and OIS, OIS depos and OIS swaps, and Libor futures
– Incorporate Central Bank meeting dates and end of year effects in interpolation
• Other curves Y are fitted to Libor/Y basis swaps
by setting forward Ys
with DFs and forward Libors known at this stage
38. • MCL markets cannot value cash flows in another currency
• xCCY BS CCY1 vs CCY2 (assuming both MCL have been constructed)
• Assume USD Fed Funds collateral
• We know how to PV cash-flows on the USD leg
• But for the EUR leg, we can PV only with EUR collateral
For PV under USD collateral, we need another curve
Discount curve for EUR cash flows under USD collateral
Cross Currency MCL markets
0T nT
. . .EURN L EUR cvg EUR
. . .USDN L USD cvgUSD
USD notional
EUR notional USD notional
EUR notional
EUR
USDDF
39. • Forward Fx (under collateral X):
• We saw that F is independent of X
since X affects identically the numerator and denominator
• So
• Since S and are known from domestic markets,
The curve of forward Fx and the curve of carry the same information
• It is standard practice to store forward Fx curves in xCCY MCL markets
More precisely we usually store forward Fx factors = forward Fx / spot Fx
Forward Fx Curves
for
X
dom
X
DF T
F T S
DF T
for
dom
dom
dom
DF T
F T S
DF T
dom
domDF T
for
domDF
40. Valuation with a xCCY market
• Discount cash flows in the currency of the collateral same as domestic case
• Discount cash flows in a foreign currency:
– “Convert” in the domestic currency with the forward fx for payment date
– Discount as a domestic cash flow on the collateral rate curve
• Fixed cash flow paid at date T in foreign currency
• Floating foreign cash flow with floating rate Y for period
we pick the forward rate on the foreign Y curve (e.g. some Libor)
we convert in domestic currency by applying the forward fx read on the xCCY curve for the payment date
we discount on the domestic X curve (in general, OIS)
dom dom forPV DF T FFX T CF
2 2 1 2 1 2, ,dom dom YPV DF T FFX T F T T cvg T T
1 2,T T
41. Building a xCCY market
• Build xCCY market = set forward Fx curve
• As usual, apply interpolation scheme to provide fFx factors of all maturities
• Set the fFx curve so as to zero the PV of market traded par xCCY BS
• After domestic MCL have been constructed, USD leg has fixed value
• Apply to value EUR leg
• Set so that the EUR leg matches the USD leg
• Repeat for all quoted maturities in xCCY swaps
0T nT
. . .EURN L EUR cvg EUR
. . .USDN L USD cvgUSD
USD notional
EUR notional USD notional
EUR notional
2 2 1 2 1 2, ,dom dom YPV DF T FFX T F T T cvg T T
.FFX
42. Comparing 1997 and current setups
• 1997 setup = particular case of current
– With implicit assumption = all transactions have USD collateral with collateral rate = USD Libor 3m
– Which is never the case
• But prior to the crisis, USD OIS/Libor and EURUSD xCCY BS too low to really matter
– So the assumption was “almost” right
– In any case, spreads were too low to provide arbitrage opportunities
– And banks were reluctant to undertake the massive developments required for the current setup
43. Collateral convexity: intuition
• I buy a forward F under some CSA X
– Full collateral X
– Collateral compensated at some rate x
• Then we change to collateral Y
– Compensated at rate y=x+b where b is the basis spread between X and Y
– Suppose b is positively correlated to F
– When F raises I receive collateral that I must compensate at a higher than expected rate
– When F drops I post collateral that earns a lower than expected rate
– So I am worst off with collateral Y
– Hence, under the new CSA, my forward is lower
44. Collateral convexity: maths
• Consider a floating cash-flow referenced by some index L under collateral X
– We have
– Where
• When we construct a market under collateral X
– We extract forwards
– They are not risk-neutral expectations but “forward-X” neutral expectations
• But when we value this cash-flow under collateral Y:
.,
0
exp 0,
X Tpaypay
fix fix
T
RN X X DF
s T pay TPV E r ds L DF T E L
, exp exp ,
T T
X RN X X
t st t
DF t T E r ds f t u du
.,
0, ,
X Tpay
fix
X DF
L fix pay TF T T E L
.,
0
exp 0,
Y Tpaypay
fix fix
T
RN Y Y DF
s T pay TPV E r ds L DF T E L
still the
risk-neutral measure
different discount different forward
45. Collateral convexity: change of numeraire
• Change of collateral from X to Y: forward changes from to
– Radon-Nykodym
– b=ry-rx: basis spread
• We assume a normal diffusion for F, then by Girsanov’s theorem:
• If we also assume constant (normal) volatility for F and all bases, and correlation:
• For 10Y3m Libor with nvol = 0.40%, basis nvol = 0.20% and correl = 25% we get ~1bp
• The formula is in T^2 so for a 20Y we get ~4bp
.,
., .,
0 0
0
exp exp
0, 0, exp
pay pay
Y Tpay
X T X Tpay pay pay
T T
Y X Y X
DF s s s
Y X TDF DF Y X
pay pay
t s
r r ds b ds
dQ
DF T DF TdQ E b ds
.,X Tpay
fix
DF
TE L
.,Y Tpay
fix
DF
TE L
., .,
0 0
,
log , , , , , , ,
,
Y T X Tpay pay fix fix
fix fix
Y
T TpayDF DF X X Y X X Y X Y X
T T s pay pay sX
pay
DF s T
E L E L d dF F dB s T T s dF B t T R t T R t T
DF s T
,
2
fixY X F B
N N fix pay
T
F F F B T T
46. Collateral option
• Some CSAs allow to post collateral in a choice of currencies and/or assets
– Example: USD swap with choice of USD or EUR collateral
– Giving an option that belongs to the party posting collateral
– That party has negative PV
– It will want to apply maximum discount
– By posting higher yielding (= cheapest to deliver) collateral
– So it is always the highest rate = highest basis that prevails
– Right now it is best posting EURs than USDs
– But according to forward currency basis curves, will reverse in the future
47. Collateral option (2)
• Value of the collateral option: general case
–
– Note
– Where LHS is the base PV discounted with collateral 1 and RHS is the adjustment for collateral option
– Note is the basis from collateral 1 to collateral 2
– And finally the collateral option is worth
• Value of the collateral option: deterministic case
–
– With
1 2
0
exp
T
RN
s sPV E c c ds CF
1 1 1 2 1
0 0 0
exp exp
T T s
RN RN
s u s s s sPV E c ds CF E c du PV c c c ds
1 2 1 2 1 12
s s s s s sc c c c c b
1 12
0 0
exp
T s
RN
u s sE c du PV b ds
PV DF T E CF
1 2
0
exp 0, 0,
T
DF T f u f u du
49. Counterparty Value Adjustments
• Uncollateralized transaction
– Can produce loss if counterparty defaults while PV is positive
– But does not produce a gain if default of negative PV
• Example: payer swap on an increasing curve
– Negative cash-flows followed by positive expected (actually forward) cash flows
– Synthetic loan to counterparty
– With loss arising from default
50. CVA
• On default, PV is netted across transactions with a counterparty or “netting set”
• When we trade without collateral we give away
a put on the netting set contingent to default
• This is a “real option” with value
– DF: here means
– RR: recovery rate
– S: survival probability
– E: risk-neutral expectation
• Option pricing, not estimation
• Model parameters must be calibrated
– For instance, S must be implied from CDS whenever possible
– Because CVA is an option that is meant to be hedged
– For instance, the loss from the payer swap can be avoided by buying CDS on the counterparty
– CVA = cost of the dynamic hedge, computed as a RN expectation as per Feynman-Kac theorem
1
1
1 max 0,
n
RN
i i i i
i
CVA E DF T RR S T S T PV T
CDS implied probability to
default between Ti-1 and Ti
0
exp
T
sDF T r ds
51. Uncollateralized transactions
• Following Lehman, we can no longer discount cash flows at some “risk free” rate
– Must incorporate CVA
– CVA is computed per netting set, not per transaction
• Current practice:
– Value each transaction as risk-free: base value
– Compute CVA on the netting set
– Allocate CVA charge across transactions: incremental or marginal method
52. CVA allocation: incremental method
• Idea: allocate to each new trade the additional CVA due to trade
compared to former CVA without the trade
• Denote the CVA for some netting set with the first n trades in the set
(in chronological order)
• Incremental CVA for trade n: may be positive or negative
• Benefit: the contribution for each trade is charged in accordance to
change in CVA and then remains constant
• Drawback: allocation depends on the order of the trades
The same transaction is charged very differently depending on booking time
nCVA
1n nCVA CVA
53. CVA allocation: marginal method
• Marginal CVA for some trade =
notional times change in CVA for a marginal increase in notional
• We remind that
• And with m trades in the NS, with notionals N and unit values V
• So the marginal CVA for trade j is
• And immediately (also as a consequence of Euler’s theorem)
we see that the CVA is the sum of marginal CVAs
• Benefit: intellectually sound, easy and fast, charge does not depend on order
• Drawback: all charges change retroactively when new trades are introduced!
1
1
1
n
RN
i i i i
i
CVA E DF T RR S T S T PV T
1
m
j j
j
PV N V
1 0
1
1 1
i
n
RN
j i i i j j iPV T
ij
CVA
N E DF T RR S T S T N V T
N
1
m
j
j j
CVA
CVA N
N
54. Calculation of CVA
• Challenges…
– The most exotic option ever: credit contingent put on a whole netting set
– Netting sets may have options an even exotics
– Model dimensionality
– Calculation time
– Risk sensitivities
• …such that many IBs need to recourse to crude approximations…
Like represent a exotics as a vanillas
for instance Bermuda as simple swap of same expected duration
or analytically valued European swaption
• …and/or compute on large racks of servers
55. Efficient calculation of CVA
• We have been working on efficient implementation
for CVA for the last couple of years
• And published results in our series of talks “CVA on iPad mini”
– Used and adapted our experience with exotics and hybrids
– Used Longstaff-Schwartz regressions in a particular way
– Incorporated new algorithms such as parallel computing, AAD and branching
• Short version Global Derivatives 2014, Amsterdam
• Long version Aarhus 2014
• Email cvaCentral@aSavine.com for slides
56. What model for CVA?
• Challenge: model all variables impacting the whole netting set
– Rate curves in all currencies in the NS
– Fx for all currencies involved
– All equity, credit, inflation, etc. indices referred to in the transactions
– All at once, consistent and calibrated
• Rate models must be multi-factor
– Otherwise all rates within currency are perfectly correlated
– And netting effect between reverse transaction with non-matching maturities is overestimated
– Choice between Markov (muti-factor Cheyette) and non-Markov (BGM) models
– Models calibrated to todays prices, curves and volatility surfaces
– Best practice requires stochastic volatility, especially for NS with exotics
• Quants reuse models developed for exotics…
– CMS spread steepeners: MF rate models with SV
– Callable Power Duals / chooser TARNs: Fx with LV/SV coupled with rate models
• …But ultra high dimensionality is something new
– Never in exotics did we need to deal with such a number of variables
– Even chooser TARNs (typically up to 5 currencies) implemented single factor IR on each ccy
57. CVA as stochastic discounting
• CVA formula under numeraire N:
• Knowing that:
• We rewrite as:
• At this point it is clear that cash-flow evaluation is quadratic
• But if we use Longstaff-Schwartz proxies for the indicators:
• And reverse the order of the sums:
0 1
1
1
1 max 0,
i
n
N
i i i
i T
CVA N E RR S T S T PV T
N
i i
j
n
jN
i T T
j i T
CF T
PV T N E
N
0 0 1 0
1
1 1
i
j
n n
jN
i i PV T
i j i T
CF T
CVA N E RR S T S T
N
0 0 1 0
1
1 1
i
j
n n
jN
i i V
i j i T
CF T
CVA N E RR S T S T
N
0 0 1 0
1 1
1 1
i
j
jn
jN
i i V
j i T
CF T
CVA N E RR S T S T
N
58. CVA reformulation (2)
• Then we can write:
• Where:
• And so:
• Computing a CVA means discounting cash-flows with an exotic stochastic notional
• The cash-flows are evaluated once, the exotic discount is calculated along the path
• We use proxies for the indicators,
the cash flows themselves are calculated without approximation
0
1
j
j
n
TN
j
j T
CVA E CF T
N
1 0
1
1 1j i
j
T i i V
i
RR S T S T
1 1 0
1 1j j j
T T j j V
RR S T S T
59. Efficient calculations of CVA (2)
• Recipe = “iPad mini” summary
• Prepare the infrastructure
– Use scripting for representation of cash-flows
– Compress trades in a netting set into a “super swap”
– Use LSM pre-simulations to turn early exercises into path-dependency
• Reformulate CVA problem with LSM proxies and “CVA-discounting”
• Turbo-charge
– Multi-thread the simulations and LSM
– Use AAD for risk
60. xVA
• Common practice: compute, in addition to and in the same way as CVA
– The quite controversial DVA = “own CVA” or benefit from own default
– Note DVA = -CVA for counterparty
• And/or funding adjustment
– FBA = benefit from investing excess cash at a my funding rate (buy back my bonds) > risk free rate …
– …when PV is negative, remember negative PV = synthetically borrow excess cash through transaction
– Note funding spread reflects CDS, hence FBA redundant with DVA (and meant to replace it)
– We also have FCA, cost from borrowing at higher rate when PV is positive
– And the total Funding Valuation Adjustment FVA = FBA - FCA
1
1
1 min 0,
n
RN
i i i i
i
DVA E DF T myRR myS T myS T PV T
1 1 1 1
1
, , min 0,
n
RN
i i F i i RiskFree i i i i i
i
FBA E DF T myS T myR T T R T T T T PV T
1 1 1 1
1
, , max 0,
n
RN
i i F i i RiskFree i i i i i
i
FCA E DF T myS T myR T T R T T T T PV T
61. xVA (2)
• Finally we get the adjusted value for uncollateralized transactions:
– PV* = risk free PV + FVA – CVA
– Note that FVA is linear and easy to compute from transaction PVs,
while the split into FBA and FCA is non-linear, heavy to compute and model dependent
– FBA and FCA usually computed together with CVA (same methodology) as FVA split is useful info
– Note applying FVA is same as changing discount basis from risk free (OIS) to my funding
– And then we still need to apply CVA
• For a clear and transparent presentation on FVA, FBA, FCA see Burgard (2014)
• Finally banks are recently most interested in KVA: Capital Valuation Adjustment
– Credit capital charge is sort of like CVA, but more complicated
– Can be calculated with similar methodologies
– KVA = capital cost of making a transaction
– Out of scope in this presentation
– For a clear and complete presentation, see Flyger (2014)