A C++ based program which prices the fair value of a participating life insurance whereby the underlying follows a Kou process and the insurer's default occurs only at contract's maturity.
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.
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.
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.
The document provides an overview of the Black-Scholes option pricing model (BSOPM). It describes the key assumptions of the BSOPM, including that the underlying stock pays no dividends, markets are efficient, and prices are lognormally distributed. It also outlines how the BSOPM can be used to calculate theoretical option prices from historical data on the stock price, strike price, time to expiration, interest rate, and volatility. The document discusses implied volatility and how it differs from historical volatility, as well as limitations of the BSOPM.
Black-Scholes Model
Introduction
Key terms
Black Scholes Formula
Black Scholes Calculators
Wiener Process
Stock Pricing Model
Ito’s Lemma
Derivation of Black-Sholes Equation
Solution of Black-Scholes Equation
Maple solution of Black Scholes Equation
Figures
Option Pricing with Transaction costs and Stochastic Volatility
Introduction
Key terms
Stochastic Volatility Model
Quanto Option Pricing Model
Key Terms
Pricing Quantos in Excel
Black-Scholes Equation of Quanto options
Solution of Quanto options Black-Scholes Equation
This chapter discusses basic concepts of option management including European and American call and put options. It covers pricing options, arbitrage opportunities, valuing forward contracts, and put-call parity. Several exercises are provided to practice these concepts, such as calculating option prices in a one-period binomial model, identifying arbitrage opportunities between stock and option prices, and determining upper and lower bounds for option prices.
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.
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.
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.
The document provides an overview of the Black-Scholes option pricing model (BSOPM). It describes the key assumptions of the BSOPM, including that the underlying stock pays no dividends, markets are efficient, and prices are lognormally distributed. It also outlines how the BSOPM can be used to calculate theoretical option prices from historical data on the stock price, strike price, time to expiration, interest rate, and volatility. The document discusses implied volatility and how it differs from historical volatility, as well as limitations of the BSOPM.
Black-Scholes Model
Introduction
Key terms
Black Scholes Formula
Black Scholes Calculators
Wiener Process
Stock Pricing Model
Ito’s Lemma
Derivation of Black-Sholes Equation
Solution of Black-Scholes Equation
Maple solution of Black Scholes Equation
Figures
Option Pricing with Transaction costs and Stochastic Volatility
Introduction
Key terms
Stochastic Volatility Model
Quanto Option Pricing Model
Key Terms
Pricing Quantos in Excel
Black-Scholes Equation of Quanto options
Solution of Quanto options Black-Scholes Equation
This chapter discusses basic concepts of option management including European and American call and put options. It covers pricing options, arbitrage opportunities, valuing forward contracts, and put-call parity. Several exercises are provided to practice these concepts, such as calculating option prices in a one-period binomial model, identifying arbitrage opportunities between stock and option prices, and determining upper and lower bounds for option prices.
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.
The document discusses key concepts related to option pricing models. It provides an overview of the binomial option pricing model (BOPM) and Black-Scholes option pricing model (BSOPM). The BOPM values options using a discrete time approach where the underlying asset price can move up or down over time. The BSOPM uses a continuous time approach to value options based on the stochastic behavior of the underlying asset price over time. Both models are based on the principle of risk neutral valuation and creating a riskless hedge to determine the appropriate discount rate.
The document summarizes key concepts related to the Black-Scholes partial differential equation. It introduces Black-Scholes, which revolutionized finance by finding the fair price of derivatives. The formula was derived from the heat equation and allowed investors to earn maximum profits without risk. It discusses the variables in the Black-Scholes equation like stock price, exercise price, volatility and risk-free rate. An example valuation of a call and put option is shown. The document also covers fundamental concepts like interest rates, probability, expected value, and continuous random variables.
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
This document discusses portfolio optimization and different algorithms used to solve portfolio optimization problems. It begins by formulating the unconstrained and constrained portfolio optimization problems. For the unconstrained problem, it uses quadratic programming to generate the efficient frontier. For the constrained problem, it uses mixed integer quadratic programming and heuristic algorithms like genetic algorithm, tabu search and simulated annealing. It compares the results of these different algorithms and concludes some perform better than others in terms of accuracy and time complexity for portfolio optimization problems with constraints.
The document discusses the Black-Scholes model for pricing options. It provides background on the development of option pricing models over time, leading up to the seminal 1973 Black-Scholes model. It discusses key concepts underlying Black-Scholes such as forming a riskless portfolio to derive the Black-Scholes differential equation. The document also covers properties of the Black-Scholes formula and how it can be applied to calculate option prices and analyze investment strategies.
Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
The document provides an overview of option valuation and pricing models. It discusses intrinsic value, put-call parity, and binomial and Black-Scholes option pricing models. The binomial model uses a tree approach to allow stock prices to move up or down over multiple periods to expiration. The Black-Scholes model provides a closed-form solution and values options based on stock price, strike price, volatility, time to expiration, and risk-free rate. An example applies the Black-Scholes formula to compute prices for a call and put option.
This document provides an introduction to option management concepts including European and American call and put options. It discusses pricing models, arbitrage opportunities, and put-call parity relations. Several exercises are provided to practice these concepts, such as calculating option prices in a binomial model, describing arbitrage strategies, and proving properties of option prices.
This document provides instructions and questions for a final exam on financial markets. It includes 10 multiple choice questions testing concepts such as closed-end funds, net asset value calculations, agency problems, risk-free assets, portfolio theory graphs, risk preferences, capital allocation lines, market portfolios, and rates of return calculations. It also provides definitions and an exercise to solve related to portfolio theory, the efficient frontier, beta in the capital asset pricing model, and determining if a stock is overpriced or underpriced based on its expected return.
This document discusses pricing CDOs using the intensity gamma approach. Some key points:
1) The intensity gamma approach models default correlation through a business time process, where defaults become conditionally independent given the business time path. This addresses issues with the Gaussian copula model.
2) The approach involves simulating business time paths, then calculating default intensities and times to price CDO tranches.
3) The business time process is modeled as a combination of gamma processes and drift. Efficient simulation techniques are discussed to generate the business time paths.
The document discusses pricing the Margrabe option using Monte Carlo simulation and an explicit closed-form solution. It begins by defining the Margrabe option and explaining its use. It then presents Margrabe's closed-form solution, which prices the option as a European call using a change of numeraire approach. Next, it analyzes the option's sensitivity to various parameters. Finally, it outlines different option pricing methods and focuses on Monte Carlo simulation and the change of numeraire approach.
Application of Monte Carlo Methods in FinanceSSA KPI
This document provides an overview of applying Monte Carlo methods in finance. It discusses:
1) Modeling stock prices as stochastic processes using tools like the Wiener process and Ito's lemma.
2) Valuing financial derivatives like options using Monte Carlo techniques to simulate stock price paths and calculate expected values. This allows valuing European and path-dependent options.
3) Using Monte Carlo simulations to calculate value at risk (VaR), a risk measure of the amount of potential losses from market moves.
Financial Cash-Flow Scripting: Beyond Valuation by Antoine SavineAntoine 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).
The document discusses the Black-Scholes option pricing model (BSOPM) and its assumptions. It outlines the derivation of the BSOPM, specifying the process the stock price follows and constructing a riskless portfolio to hedge the option. It then presents the BSOPM formula and defines the terms. The document also discusses the standard normal distribution curve used in the formula and provides an example calculation. Finally, it covers extensions such as dividends, the relationship between the BSOPM and binomial option pricing model, and estimating volatility.
Several ways to calculate option probability are outlined, including the derivation that relies on terms from the Black-Scholes (Merton) formula. Programming formulas are provided for Excel. Delta is discussed, as a proxy for option probability and the differences in various volatility measures are described.
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.
This document discusses no arbitrage pricing theory and market risk. It begins by defining no arbitrage pricing as having a zero initial and expiration value. However, it notes that this definition does not guarantee a zero expiration value when holding coupon payments. It then introduces the concepts of present value and forward value, and defines no arbitrage prices that set the present and forward values equal to zero. However, it notes that this introduces market risk, as forward rates are random variables. It concludes by providing examples of interest rate swap valuation and defining market risk probabilities.
The document contains 51 multiple choice questions about financial derivatives. The questions cover topics such as the definition of financial derivatives and what they include, hedging risk with long and short positions, forward and futures contracts, and how they work. Key points addressed are that financial derivatives payoffs are linked to previously issued securities, they help reduce exposure to risk, and perfect hedging in futures markets eliminates both gains and losses.
This document discusses key concepts in portfolio theory, including how to calculate investment returns over single and multiple periods. It defines holding period return (HPR) to measure single period returns and arithmetic average, geometric average, and dollar-weighted return to measure returns over many periods. It also explains how to calculate the expected return, variance, and standard deviation of investments to quantify the expected reward and risk.
Dynamical smart liquidity on decentralized exchanges for lucrative market makingStefan Duprey
This document discusses liquidity provision on Uniswap V3 and proposes algorithms for actively managing liquidity positions. It introduces the concept of concentrated liquidity on Uniswap V3 which allows liquidity providers to specify a price range. It discusses different configurations for liquidity provision including long, short, and market neutral approaches. It then presents algorithms for dynamically exiting liquidity positions based on trend signals, modeling historical fees, and determining optimal price range bounds while minimizing costs like impermanent loss and transaction fees.
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.
The document discusses key concepts related to option pricing models. It provides an overview of the binomial option pricing model (BOPM) and Black-Scholes option pricing model (BSOPM). The BOPM values options using a discrete time approach where the underlying asset price can move up or down over time. The BSOPM uses a continuous time approach to value options based on the stochastic behavior of the underlying asset price over time. Both models are based on the principle of risk neutral valuation and creating a riskless hedge to determine the appropriate discount rate.
The document summarizes key concepts related to the Black-Scholes partial differential equation. It introduces Black-Scholes, which revolutionized finance by finding the fair price of derivatives. The formula was derived from the heat equation and allowed investors to earn maximum profits without risk. It discusses the variables in the Black-Scholes equation like stock price, exercise price, volatility and risk-free rate. An example valuation of a call and put option is shown. The document also covers fundamental concepts like interest rates, probability, expected value, and continuous random variables.
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
This document discusses portfolio optimization and different algorithms used to solve portfolio optimization problems. It begins by formulating the unconstrained and constrained portfolio optimization problems. For the unconstrained problem, it uses quadratic programming to generate the efficient frontier. For the constrained problem, it uses mixed integer quadratic programming and heuristic algorithms like genetic algorithm, tabu search and simulated annealing. It compares the results of these different algorithms and concludes some perform better than others in terms of accuracy and time complexity for portfolio optimization problems with constraints.
The document discusses the Black-Scholes model for pricing options. It provides background on the development of option pricing models over time, leading up to the seminal 1973 Black-Scholes model. It discusses key concepts underlying Black-Scholes such as forming a riskless portfolio to derive the Black-Scholes differential equation. The document also covers properties of the Black-Scholes formula and how it can be applied to calculate option prices and analyze investment strategies.
Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
The document provides an overview of option valuation and pricing models. It discusses intrinsic value, put-call parity, and binomial and Black-Scholes option pricing models. The binomial model uses a tree approach to allow stock prices to move up or down over multiple periods to expiration. The Black-Scholes model provides a closed-form solution and values options based on stock price, strike price, volatility, time to expiration, and risk-free rate. An example applies the Black-Scholes formula to compute prices for a call and put option.
This document provides an introduction to option management concepts including European and American call and put options. It discusses pricing models, arbitrage opportunities, and put-call parity relations. Several exercises are provided to practice these concepts, such as calculating option prices in a binomial model, describing arbitrage strategies, and proving properties of option prices.
This document provides instructions and questions for a final exam on financial markets. It includes 10 multiple choice questions testing concepts such as closed-end funds, net asset value calculations, agency problems, risk-free assets, portfolio theory graphs, risk preferences, capital allocation lines, market portfolios, and rates of return calculations. It also provides definitions and an exercise to solve related to portfolio theory, the efficient frontier, beta in the capital asset pricing model, and determining if a stock is overpriced or underpriced based on its expected return.
This document discusses pricing CDOs using the intensity gamma approach. Some key points:
1) The intensity gamma approach models default correlation through a business time process, where defaults become conditionally independent given the business time path. This addresses issues with the Gaussian copula model.
2) The approach involves simulating business time paths, then calculating default intensities and times to price CDO tranches.
3) The business time process is modeled as a combination of gamma processes and drift. Efficient simulation techniques are discussed to generate the business time paths.
The document discusses pricing the Margrabe option using Monte Carlo simulation and an explicit closed-form solution. It begins by defining the Margrabe option and explaining its use. It then presents Margrabe's closed-form solution, which prices the option as a European call using a change of numeraire approach. Next, it analyzes the option's sensitivity to various parameters. Finally, it outlines different option pricing methods and focuses on Monte Carlo simulation and the change of numeraire approach.
Application of Monte Carlo Methods in FinanceSSA KPI
This document provides an overview of applying Monte Carlo methods in finance. It discusses:
1) Modeling stock prices as stochastic processes using tools like the Wiener process and Ito's lemma.
2) Valuing financial derivatives like options using Monte Carlo techniques to simulate stock price paths and calculate expected values. This allows valuing European and path-dependent options.
3) Using Monte Carlo simulations to calculate value at risk (VaR), a risk measure of the amount of potential losses from market moves.
Financial Cash-Flow Scripting: Beyond Valuation by Antoine SavineAntoine 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).
The document discusses the Black-Scholes option pricing model (BSOPM) and its assumptions. It outlines the derivation of the BSOPM, specifying the process the stock price follows and constructing a riskless portfolio to hedge the option. It then presents the BSOPM formula and defines the terms. The document also discusses the standard normal distribution curve used in the formula and provides an example calculation. Finally, it covers extensions such as dividends, the relationship between the BSOPM and binomial option pricing model, and estimating volatility.
Several ways to calculate option probability are outlined, including the derivation that relies on terms from the Black-Scholes (Merton) formula. Programming formulas are provided for Excel. Delta is discussed, as a proxy for option probability and the differences in various volatility measures are described.
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.
This document discusses no arbitrage pricing theory and market risk. It begins by defining no arbitrage pricing as having a zero initial and expiration value. However, it notes that this definition does not guarantee a zero expiration value when holding coupon payments. It then introduces the concepts of present value and forward value, and defines no arbitrage prices that set the present and forward values equal to zero. However, it notes that this introduces market risk, as forward rates are random variables. It concludes by providing examples of interest rate swap valuation and defining market risk probabilities.
The document contains 51 multiple choice questions about financial derivatives. The questions cover topics such as the definition of financial derivatives and what they include, hedging risk with long and short positions, forward and futures contracts, and how they work. Key points addressed are that financial derivatives payoffs are linked to previously issued securities, they help reduce exposure to risk, and perfect hedging in futures markets eliminates both gains and losses.
This document discusses key concepts in portfolio theory, including how to calculate investment returns over single and multiple periods. It defines holding period return (HPR) to measure single period returns and arithmetic average, geometric average, and dollar-weighted return to measure returns over many periods. It also explains how to calculate the expected return, variance, and standard deviation of investments to quantify the expected reward and risk.
Dynamical smart liquidity on decentralized exchanges for lucrative market makingStefan Duprey
This document discusses liquidity provision on Uniswap V3 and proposes algorithms for actively managing liquidity positions. It introduces the concept of concentrated liquidity on Uniswap V3 which allows liquidity providers to specify a price range. It discusses different configurations for liquidity provision including long, short, and market neutral approaches. It then presents algorithms for dynamically exiting liquidity positions based on trend signals, modeling historical fees, and determining optimal price range bounds while minimizing costs like impermanent loss and transaction fees.
In this paper, we consider an AAI with two types of insurance business with p-thinning dependent
claims risk, diversify claims risk by purchasing proportional reinsurance, and invest in a stock with Heston
model price process, a risk-free bond, and a credit bond in the financial market with the objective of maximizing
the expectation of the terminal wealth index effect, and construct the wealth process of AAI as well as the the
model of robust optimal reinsurance-investment problem is obtained, using dynamic programming, the HJB
equation to obtain the pre-default and post-default reinsurance-investment strategies and the display expression
of the value function, respectively, and the sensitivity of the model parameters is analyzed through numerical
experiments to obtain a realistic economic interpretation. The model as well as the results in this paper are a
generalization and extension of the results of existing studies.
PROBLEMS IN SELECTION OF SECURITY PORTFOLIOS THE PERFORMANDaliaCulbertson719
PROBLEMS IN SELECTION OF SECURITY PORTFOLIOS
THE PERFORMANCE OF MUTUAL FUNDS IN THE PERIOD 1945-1964
MICHAEL C. JENSEN*
I. INTRODUCTION
A CENTRAL PROBLEM IN FINANCE (and especially portfolio management) has
been that of evaluating the "performance" of portfolios of risky investments.
The concept of portfolio "performance" has at least two distinct dimensions:
1) The ability of the portfolio manager or security analyst to increase re-
turns on the portfolio through successful prediction of future security
prices, and
2) The ability of the portfolio manager to minimize (through "efficient"
diversification) the amount of "insurable risk" born by the holders of
the portfolio.
The major difficulty encountered in attempting to evaluate the performance
of a portfolio in these two dimensions has been the lack of a thorough under-
standing of the nature and measurement of "risk." Evidence seems to indicate
a predominance of risk aversion in the capital markets, and as long as in-
vestors correctly perceive the "riskiness" of various assets this implies that
"risky" assets must on average yield higher returns than less "risky" assets.'
Hence in evaluating the "performance" of portfolios the effects of differential
degrees of risk on the returns of those portfolios must be taken into account.
Recent developments in the theory of the pricing of capital assets by
Sharpe [20], Lintner [15] and Treynor [25] allow us to formulate explicit
measures of a portfolio's performance in each of the dimensions outlined
above. These measures are derived and discussed in detail in Jensen [11].
However, we shall confine our attention here only to the problem of evaluating
a portfolio manager's predictive ability-that is his ability to earn returns
through successful prediction of security prices which are higher than those
which we could expect given the level of riskiness of his portfolio. The founda-
tions of the model and the properties of the performance measure suggested
here (which is somewhat different than that proposed in [11]) are discussed
in Section II. The model is illustrated in Section III by an application of it
to the evaluation of the performance of 115 open end mutual funds in the
period 1945-1964.
A number of people in the past have attempted to evaluate the performance
of portfolios2 (primarily mutual funds), but almost all of these authors have
* University of Rochester College of Business. This paper has benefited from comments and
criticisms by G. Benston, E. Fama, J. Keilson, H. Weingartner, and especially M. Scholes.
1. Assuming, of course, that investors' expectations are on average correct.
2. See for example [2, 3, 7, 8, 9, 10, 21, 24].
389
390 The Journal of Finance
relied heavily on relative measures of performance when what we really need
is an absolute measure of performance. That is, they have relied mainly on
procedures for ranking portfolios. For example, if there are two por ...
Introducing R package ESG at Rmetrics Paris 2014 conferenceThierry Moudiki
The document describes the ESG package in R, which provides tools for generating economic scenarios for use in insurance valuation and capital requirements calculations. It discusses the available risk factors in ESG, including nominal interest rates, equity returns, property returns, and corporate bond returns. It also outlines the package's object-oriented structure and provides examples of how to use ESG to simulate scenarios and calculate insurance liabilities and capital requirements.
The document summarizes Ahmed Ashmawy's M.Sc defense presentation on mixed integer conditional value-at-risk portfolio optimization. It outlines the problem of optimally allocating an investment across multiple stocks while accounting for risk, describes approaches that combine constraint programming and linear programming to solve the mixed integer optimization problem more efficiently, and presents a greedy algorithm that provides near-optimal solutions with improved time performance by iteratively solving relaxed linear programming subproblems. Experimental results demonstrate the greedy approach achieves solutions of similar quality to traditional mixed integer programming solvers but with significantly better scalability to large problem instances.
Derivatives pricing and Malliavin CalculusHugo Delatte
1. The document discusses using Malliavin calculus to compute Greeks, which measure the sensitivity of derivative prices to changes in variables like the underlying asset price. Malliavin calculus can improve the convergence speed for computing Greeks compared to finite difference methods, especially for options with discontinuous payoffs.
2. Traditional approaches to computing Greeks include using binomial trees or the Black-Scholes PDE. Finite difference methods introduce errors from both the Monte Carlo simulation and the derivative approximation. Malliavin calculus avoids differentiating the discontinuous payoff function.
3. The document provides an overview of Malliavin calculus, including the Malliavin derivative operator and integration by parts formula. It then compares
This document discusses estimating covariance matrices for portfolio selection. It introduces a shrinkage estimator that is an optimally weighted average of the sample covariance matrix and single-index covariance matrix. The empirical part compares these estimators to determine which produces the most efficient portfolio with smallest return variability. The sample covariance matrix has problems when the number of assets is large, as it has high variance and its inverse is a poor estimator. Shrinkage aims to improve upon the sample covariance matrix by combining it with a factor model-based estimator.
CVA In Presence Of Wrong Way Risk and Early Exercise - Chiara Annicchiarico, ...Michele Beretta
We will show how to calibrate the main parameter of the model and how we have used it in order to evaluate the CVA and the CVAW of a one derivative portfolio with the possibility of early exercise.
This document discusses risk and return in finance. It defines risk as variability in returns and explains how to measure risk statistically using measures like variance and standard deviation. It also discusses different types of risk like diversifiable and non-diversifiable risk. The document then introduces the Capital Asset Pricing Model (CAPM), which uses an asset's beta to determine its required return based on the security market line. The CAPM model provides a way to quantify risk premiums and determine adequate returns for assets based on their non-diversifiable risk.
Financial Regulation Lecture 9Based on notes by Marlena El.docxericn8
Financial Regulation Lecture 9
Based on notes by Marlena Eley
March 22, 2019
1
1 Concept Review
We have been studying Cooper and Ross’s paper, which demonstrates how
deposit insurance causes moral hazard. Their work uses the Diamond and
Dybvig model and adds:
1. Risky Technology
2. Monitoring by Households (HH)
The introduction of deposit insurance reduces HH’s incentives to monitor.
HHs would have to pay a fixed cost Γ, which is measured in utils (an effort
cost rather than a resource cost), to monitor the bank. When HHs do monitor,
they are able to FORCE the bank to invest in the safe technology. We know
that HHs always prefer the safe long term technology investment to the risky
technology investment because they are risk averse (u′′ < 0).
1.1 LEMMA
Let q denote the probability of a run - this q is exogenously given. A Bank
that maximizes HH’s utility solves the following problem:
if q ≤ q∗ the deposit contract offered allows runs
if q > q∗ the deposit contract offered is a run preventing contract
When q ≤ q∗, banks are providing liquidity insurance, cE > 1, which means
that a run is a potential equilibria. When q > q∗, banks provide contracts
similar to autarky allocations, cE = 1,cL = R.
Intuitively, if runs are not very likely, then the bank prefers to offer a contract
that allows for runs but offers some liquidity insurance, which is valuable to
households.
We’re studying when q ≤ q∗ because we want deposit insurance to be offered
so that moral hazard is an issue.
1.2 Risky Technology
Risky technology is defined as:
(−1, 0,
{
λR with probability ν
0 with probability 1 −ν
Where:
1. λ > 1
2. νλ ≤ 1
2
Sidenote: when νλ = 1, we say this technology has a mean preserving spread.
Moral hazard is introduced here and this implies that rather than strictly
maximizing HH’s utility, Banks are now maximizing profit. The “managers” of
the Banks get whatever is leftover after feeding the consumers.
Banks then maximize profit as follows:
max
i∈[0,1]
ν[iλR + (1 − i−πcE)R− (1 −π)cL]
+ (1 −ν)max(i∗ 0 + (1 − i−πcE)R− (1 −π)cL, 0)
We see that the objective function is linear in i which tells us that there may
be corner solutions where i∗ = 0 (investment is only made in the safe
technology) or i∗ = 1 (investment is only made in the risky technology).
1.3 The Threshold ī
We study this decision by establishing a threshold of indifference, ī - when
they are indifferent between investing everything in the risky technology and
investing in the safe technology.
We can establish ī by looking at the investment decision in the low state,
(1 −ν) when the risky technology fails.
Conditional on being in the low state (1 −ν):
ī = {i ∈ [0, 1] : (1 − i−πcE)R− (1 −π)cL = 0}
∀i > ī, (1 − i−πcE)R− (1 −π)cL < 0, max = 0 therefore invests in risky.
∀i < ī, (1 − i−πcE)R− (1 −π)cL > 0, max = (1 − i−πcE)R− (1 −π)cL therefore invests in safe.
Our next step is to plug these values back into the objective function for each
case.
If i ≥ .
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The document discusses modeling volatility for European carbon markets using stochastic volatility (SV) models. It outlines estimating SV model parameters from market data, simulating conditional volatility distributions, and using these to price options and evaluate market pricing errors. The modeling approach involves projecting returns from an SV model, estimating parameters, and then re-projecting to obtain conditional volatility forecasts for option pricing. Estimated model parameters and implied volatilities from major European carbon exchanges are presented and compared.
The Capital Asset Pricing Model (CAPM) formalizes the relationship between risk and expected return. It states that the expected return of an asset is determined by its sensitivity to non-diversifiable or systematic risk as measured by its beta. Beta measures how an asset's returns co-vary with the market portfolio. According to CAPM, an asset's expected return is equal to the risk-free rate plus its beta multiplied by the market risk premium. Diversification reduces risk by eliminating asset-specific or diversifiable risk, but not market risk.
This document provides an overview of portfolio theory and the Capital Asset Pricing Model (CAPM). It defines key concepts like the efficient frontier, market portfolio, capital market line (CML), beta, and the security market line (SML). The CAPM holds that an asset's expected return is determined by its non-diversifiable risk as measured by its beta. Beta measures how an asset's returns co-vary with the market portfolio. The document provides examples of estimating betas and calculating expected returns using the CAPM framework. It concludes by noting the CAPM is a useful but not perfect model of the risk-return relationship.
This document summarizes a research paper that examines the optimal investment, consumption, and life insurance selection problem for a wage earner. The problem is modeled using a financial market with one risk-free asset and one risky jump-diffusion asset, along with an insurance market composed of multiple life insurance companies. The goal is to maximize the wage earner's expected utility from consumption during life, wealth at retirement or death, by choosing an optimal investment, consumption, and insurance strategy. The authors use dynamic programming to characterize the optimal solution and prove existence and uniqueness of a solution to the associated nonlinear Hamilton-Jacobi-Bellman equation.
Asset Pricing and Portfolio Theory
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Statistical Arbitrage
Pairs Trading, Long-Short Strategy
Cyrille BEN LEMRID

1 Pairs Trading Model 5
1.1 Generaldiscussion ................................ 5 1.2 Cointegration ................................... 6 1.3 Spreaddynamics ................................. 7
2 State of the art and model overview 9
2.1 StochasticDependenciesinFinancialTimeSeries . . . . . . . . . . . . . . . 9 2.2 Cointegration-basedtradingstrategies ..................... 10 2.3 FormulationasaStochasticControlProblem. . . . . . . . . . . . . . . . . . 13 2.4 Fundamentalanalysis............................... 16
3 Strategies Analysis 19
3.1 Roadmapforstrategydesign .......................... 19 3.2 Identificationofpotentialpairs ......................... 19 3.3 Testingcointegration ............................... 20 3.4 Riskcontrolandfeasibility............................ 20
4 Results
22
2
Contents

Introduction
This report presents my research work carried out at Credit Suisse from May to September 2012. This study has been pursued in collaboration with the Global Arbitrage Strategies team.
Quantitative analysis strategy developers use sophisticated statistical and optimization techniques to discover and construct new algorithms. These algorithms take advantage of the short term deviation from the ”fair” securities’ prices. Pairs trading is one such quantitative strategy - it is a process of identifying securities that generally move together but are currently ”drifting away”.
Pairs trading is a common strategy among many hedge funds and banks. However, there is not a significant amount of academic literature devoted to it due to its proprietary nature. For a review of some of the existing academic models, see [6], [8], [11] .
Our focus for this analysis is the study of two quantitative approaches to the problem of pairs trading, the first one uses the properties of co-integrated financial time series as a basis for trading strategy, in the second one we model the log-relationship between a pair of stock prices as an Ornstein-Uhlenbeck process and use this to formulate a portfolio optimization based stochastic control problem.
This study was performed to show that under certain assumptions the two approaches are equivalent.
Practitioners most often use a fundamentally driven approach, analyzing the performance of stocks around a market event and implement strategies using back-tested trading levels.
We also study an example of a fundamentally driven strategy, using market reaction to a stock being dropped or added to the MSCI World Standard, as a signal for a pair trading strategy on those stocks once their inclusion/exclusion has been made effective.
This report is organized as follows. Section 1 provides some background on pairs trading strategy. The theoretical results are described in Section 2. Section 3
I2
I1
σp
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Fair valuation of participating life insurance contracts with jump risk
1. EMLYON Business School
Specialized Master Quantitative Finance
(2013 / 2014)
Fair Valuation of Participating Contracts with
Jump Risk
KOUAM KAMGUIA AURELIEN ALEX
Reg.No: 20142165
2. Summary
This report aims to detail the implementation of a Participating Life Insurance con-
tract pricer in accordance with the fair value accounting principle of International
Accounting Standard Board (IASB).
• An inclusion of market and credit risk goes in line with the risk–sensitive
approach promoted by Solvency II and enhances risk management.
• The use of a double exponential random variable creates a rare case of in-
cluding default risk whilst obtaining a closed–form formula.
• A Taylor series approach is sufficiently accurate to approximate the confluent
hypergeometric function within this application.
5. LIST OF TABLES
List of Tables
1 Balance Sheet as at 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 1
6. 1 The Contract
The participating life insurance contract with guarantee (LIC) spans on the interval
[0, T] and at time 0, the balance sheet of the insurer is as follows:
Assets Liabilities
A0 E0
L0
Table 1: Balance Sheet as at 0
where the assets portfolio, A0, is invested partially in equity such that E0 =
(1 − α) A0 and bonds as L0 = αA0 in which α ∈ [0, 1]. The variable α is directly
linked to the financial leverage β = L0
E0
.
As the contract’s name states, the policyholder is guaranteed a minimum rate
of return rg during the life of the contract, in other words she will receive at least
at maturity LT
g = L0ergT
. In practice, the guaranteed rate rg is always less than
the return of a risk–free asset of the same maturity.
Under the conditions that the insurer is fully solvent during the period [0, T],
the policyholders will receive additionally a fraction δ of financial earnings up to
their contribution α to the funding of the portfolio. Formally, this relationship
could be written as: δ AT − LT
g .
Courtois and Quittard-Pinon (2008) priced the LIC under two approaches:
when default occurs solely at maturity and when default could occur anytime
during the contract’s lifetime. For the time being, we will exclusively cover the
first case in the rest of this document.
1
7. 1.1 The Pricing framework
The authors based their LIC pricing under the following assumptions about the
insurance and financial market:
1. there are no transactions costs, taxes, or problems with indivisibilities of
assets
2. there are a sufficient number of investors with comparable wealth levels so
that each investor believes that he can buy and sell as much of an asset as
he wants at the market price.
3. there exists an exchange market for borrowing and lending at the same rate
of interest.
4. short–sales of all assets, with full use of the proceeds, is allowed.
5. The interest rate r is constant.
6. Under the risk–neutral world the firm’s assets portfolio, A , follows a geo-
metric L´evy process notably:
dA
A−
= rdt + σdz + d
Nt
k=1
(Zk − 1) − λζt (1)
where r is the risk–free interest rate, σ is the asset’s volatility, z is a stan-
dard Brownian motion, N is a Poisson process with constant intensity λ, Zk
are strictly positive i.i.d random variables and ζ is here for compensation
purposes.
The random variable Z in the above equation is the novelty of this document
over previous works on LIC pricing because not only it embeds insurer’s
2
8. bankruptcy risk but also, provides quasi–closed–form formula. The random
variables Yk = log(Zk) are i.i.d. and possess a double exponential density:
fY (y) = pη1e−η1y
1{y≥0} + qη2e−η2y
1{y<0} (2)
where η1 is the intensity of positive jumps, η2 is the intensity of negative
jumps, p is the probability of upward jumps, q is the probability of downward
jumps and p + q = 1. All sources of randomness z, N and Yk are independent.
Using Ito’s lemma for jump–diffusions, it can be easily shown that the unique
solution to the firm’s assets portfolio SDE is:
At = A0exp Xt = A0exp r − λζ −
1
2
σ2
t + σzt +
Nt
k=1
Yk (3)
1.1.1 Valuation for default at maturity: generic formulas
At maturity, there are 3 possible outcomes: Within the first case, the insurer is
unable to meet its commitments and the firm is declared bankrupt. Policyholders
receive the residual value of the insurer’s assets, that is AT and the equityholders
nothing. In the second case, the end value of invested assets is greater than
the promised assets value to policyholders but, less than the threshold to deliver
bonuses to policyholders. Finally, the insurer is totally solvent at maturity and
pays the bonus to the policyholders. The mathematical translation of the above
explanation is:
Θ (T) =
AT ifAT < Lg
T
Lg
T ifLg
T ≤ AT ≤
Lg
T
α
Lg
T + δ (αAT − Lg
T ) ifAT >
Lg
T
α
3
9. Therefore, the arbitrage free price of the LIC can be rewritten as:
VL (0) = EQ e−rT
Lg
T + δ (αAT − Lg
T )+
− (Lg
T − AT )+
or more precisely as:
VL (0) = A0 + δαC AT ,
Lg
T
α
− C (AT , Lg
T ) (4)
where C(XT , K) is the price at time 0 of a European call with maturity T, un-
derlying process X and strike K. It’s plain to see that the LIC pricing in this
framework boils down to work out the price of a European call option in which
the underlying process is normally and double exponentially distributed. Luckily,
Kou (2002) developed such a formula.
1.2 European Option Pricing under a Double exponential
jump-diffusion model
Under a double exponential jump-diffusion model, Kou (2002) showed that an
European Option can be computed as follows:
ψc = S(0)Υ(r +
1
2
σ2
− λζ, ˜λ, ˜p, ˜η1, ˜η2, log(K/S(0)), T)
− Ke−rT
Υ(r −
1
2
σ2
− λζ, λ, p, η1, η2, log(K/S(0)), T) (5)
where S(0) is the initial stock price, σ is the underlying instrument volatility,
r, and λ are defined as in 1. η1, η2, p are defined as in 2. K is the strike price.
Furthermore, parameters ˜p, ˜η1, ˜η2, ˜λ and ζ are used to price under the risk-neutral
world. The relationships between ˜p, ˜η1, ˜η2 and ζ and the other variables are as
follows:
˜λ = λ (1 + ζ) , ˜η1 = η1 − 1, ˜η2 = η2 + 1
4
10. and:
˜p =
p
1 + ζ
η1
η1 − 1
and:
ζ = p
η1
η1 − 1
+ q
η2
η2 + 1
− 1
Υ(...) := P{Z(T) ≥ log(K/S(0))} stands for the sum of the double exponential
and normal random vaviables and, equals to the cumulative distribution function
of a standard normal random variable under the European Black-Scholes pricing.
With πn := P(N(T) = n) = e−λT
(λT)n
/n!
P{Z(T) ≥ a} =
eση1 2 T
2
σ
√
2πT
∞
n=1
πn
n
k=1
Pn,k(σ
√
Tη1)k
× Ik−1(a − µT; −η1, −
1
σ
√
T
, −ση1
√
T)
+
eση1 2 T
2
σ
√
2πT
∞
n=1
πn
n
k=1
Qn,k(σ
√
Tη2)k
× Ik−1(a − µT; η2,
1
σ
√
T
, −ση2
√
T) + π0Φ(−
a − µT
σ
√
T
)
(6)
where:
Pn,k =
n−1
i=k
P(goingfrom(i, n − i)to(k, 0)) · P(startingfrom(i, n − i))
5
11. =
n−1
i=k
n − k − 1
i − k
·
η1
η1 + η2
i−k
η2
η1 + η2
n−i
pi
qn−i
,
And
Qn,k =
n−1
i=k
n − k − 1
i − k
·
η1
η1 + η2
n−i
η2
η1 + η2
i−k
qi
pn−i
And
In (c; α, β, δ) = −
eαc
α
n
i=0
β
α
n−i
× Hhi (βc − δ) +
β
α
n+1
(2π)
β
× e
αδ
β
+ α2
2β2
Φ −βc + δ +
α
β
when β > 0 , α = 0 and for all n ≥ −1.
In (c; α, β, δ) = −
eαc
α
n
i=0
β
α
n−i
× Hhi (βc − δ) −
β
α
n+1
(2π)
β
∗
× e
αδ
β
+ α2
2β2
Φ βc − δ −
α
β
when β < 0 , α < 0 and for all n ≥ −1. Φ (...) is the standard normal cumulative
distribution function. Hhn (x) is the Hodgkin - Huxley function defined as:
Hhn (x) = 2−n
2 (π)e−x2
2 ×
1F1
1
2
n + 1
2
, 1
2
, 1
2
x2
(2)Γ 1 + 1
2
n
− x
1F1
1
2
n + 1, 3
2
, 1
2
x2
Γ 1
2
+ 1
2
n
where 1F1 (...) is the confluent hypergeometric function and Γ (...) is the gamma
function.
6
12. 2 Computational Implementation
The development of a participating life insurance contract pricer was made in
C++. In order to price a life insurance contract, the end–user will make use of
the “KouFormulas.cpp” file. This file requires the integration of the “LicMath.h”
file which embeds all the necessary mathematical functions for the application.
Moreover, we employed some libraries from Boost. Once the “KouFormulas.cpp”
file opens, she will click on the “Local Windows Debugger” button. The interface
will subsequently request the following information:
1. The initial asset price.
2. The prevailing risk-free interest rate.
3. The asset’s volatility.
4. The average number of jumps per unit of time of the exponential random
variable.
5. The probability value of positive jumps of the double exponential random
variable.
6. The average number of positive jumps per unit of time of the double expo-
nential random variable.
7. The average number of negative jumps per unit of time of the double expo-
nential random variable.
8. The number of simulations required (Enter an integer value between 1 and
50).
9. The leverage coefficient or the proportion of assets invested in risk-free bonds.
7
13. 10. The interest rate guaranteed to the policyholder at the contract’s maturity.
11. The participating coefficient.
12. The maturity of the contract.
2.1 Source code of the file KouFormulas.cpp
// KouFormulas . cpp : Defines the entry point for the console
// a p p l i c a t i o n .
//
// requires LicMath . h
//
#include<cmath>
#include”LicMath . h”
#include<boost /math/ d i s t r i b u t i o n s /normal . hpp>
#include<iostream>
//#i f ! defined ( MSC VER)
using namespace std ;
//#endif
using boost : : math : : normal ;
using namespace std ;
double KouFormulas (double S0 , double K, double int r ,
double sigma , double lambda , double p ,
double eta1 , double eta2 , double n ,
double Expiry )
8
14. {
//Compute the d i f f e r e n t parameters under the risk −neutral measure
double a = log10 (K/S0 ) ;
double variance = pow( sigma , 2 ) ;
double zeta = (( p∗ eta1 )/( eta1 − 1)) + (((1 −p)∗ eta2 )/( eta2 +1)) − 1;
double p est = (p/(1+ zeta )) ∗ ( eta1 /( eta1 −1));
double e t a 1 e s t = eta1 −1;
double e t a 2 e s t = eta2 + 1;
double i n t e n s i t y e s t = lambda ∗ (1+ zeta ) ;
double new drift = (( i n t r + 0.5∗( variance ) − ( lambda ∗ zeta ))∗
Expiry ) ;
double new drift1 = (( i n t r − 0.5∗( variance ) − ( lambda ∗ zeta ))∗
Expiry ) ;
//Compute sum of Pnk and sum of Qnk
double Sum Pnk rn = 0 . 0 ;
double Sum Qnk rn = 0 . 0 ;
double Sum Pnk = 0 . 0 ;
double Sum Qnk = 0 . 0 ;
double k d = 0 . 0 ;
for (unsigned long long int k = 1; k<=n ; k++)
{
k d = static cast<double>(k ) ;
Sum Pnk rn += Pnk(n , k d , eta1 est , eta2 est , p est ) ∗
pow( sigma∗ sqrt ( Expiry )∗
eta1 est , k d ) ∗
In ( k d −1,a − new drift ,− eta1 est ,
−(1/(sigma ∗ sqrt ( Expiry ) ) ) ,
−(sigma∗ e t a 1 e s t ∗ sqrt ( Expiry ) ) ) ;
9
15. Sum Qnk rn += Qnk(n , k d , eta1 est , eta2 est , p est ) ∗
pow( sigma∗ sqrt ( Expiry )∗
eta2 est , k d ) ∗
In ( k d −1,a − new drift , eta2 est ,
(1/( sigma ∗ sqrt ( Expiry ) ) ) ,
−(sigma∗ e t a 2 e s t ∗ sqrt ( Expiry ) ) ) ;
Sum Pnk += Pnk(n , k d , eta1 , eta2 , p) ∗
pow( sigma∗ sqrt ( Expiry )∗ eta1 , k d ) ∗
In ( k d −1,a − new drift1 ,−eta1 ,
−(1/(sigma ∗ sqrt ( Expiry ) ) ) ,
−(sigma∗ eta1 ∗ sqrt ( Expiry ) ) ) ;
Sum Qnk += Qnk(n , k d , eta1 , eta2 , p) ∗
pow( sigma∗ sqrt ( Expiry )∗ eta2 , k d ) ∗
In ( k d −1,a − new drift1 , eta2 ,
(1/( sigma ∗ sqrt ( Expiry ) ) ) ,
−(sigma∗ eta2 ∗ sqrt ( Expiry ) ) ) ;
}
// Standard normal cdf are needed for the remaining of
// t h i s a p p l i c a t i o n
normal distribution <double> nv ( 0 , 1 ) ;
// p o i s s o n d i s t r i b u t i o n <> pv ( lambda ) ;
// define d1 and d2 to measure the r e s p e c t i v e
// cumulative d i s t r i b u t i o n s
double d1 , d2 = 0;
d1 = ( exp (pow( sigma∗ eta1 est , 2 ) ∗ ( Expiry /2)))/( sigma ∗
10
16. sqrt (2∗ boost : : math : : constants : : pi<double>() ∗ Expiry )) ∗
(gamma q(n+1, i n t e n s i t y e s t )) ∗ Sum Pnk rn +
( exp (pow( sigma∗ eta2 est , 2 ) ∗ ( Expiry /2))/( sigma ∗
sqrt (2∗ boost : : math : : constants : : pi<double>() ∗ Expiry ) ) ) ∗
(gamma q(n+1, i n t e n s i t y e s t )) ∗ Sum Qnk rn +
( Ncdf (0 ,1 , −(a − new drift )/( sigma ∗ sqrt ( Expiry ) ) ) ) ;
d2 = ( exp (pow( sigma∗eta1 , 2 ) ∗ ( Expiry /2)))/( sigma ∗
sqrt (2∗ boost : : math : : constants : : pi<double>() ∗ Expiry )) ∗
(gamma q(n+1,lambda )) ∗ Sum Pnk +
( exp (pow( sigma∗eta2 , 2 ) ∗ ( Expiry /2))/( sigma ∗
sqrt (2∗ boost : : math : : constants : : pi<double>() ∗ Expiry ) ) ) ∗
(gamma q(n+1,lambda )) ∗ Sum Qnk +
( Ncdf (0 ,1 , −(a − new drift1 )/( sigma ∗ sqrt ( Expiry ) ) ) ) ;
return ( S0 ∗ d1 ) − (K∗exp(− i n t r ∗Expiry )∗ d2 ) ;
}
// Define the Payoff of a l i f e insurance contract with p r o f i t s and
// guarantee as in the O. Le Courtois and FQP (2008) paper
double DMLIC(double A0, double int r ,
double alpha , double rg , double delta ,
double sigma , double lambda , double p ,
double eta1 , double eta2 , double n ,
double Expiry )
{
// I n i t i a l i z e the contract parameters
double L0 = alpha ∗ A0;
double K1 LIC = (L0 ∗ exp ( rg ∗Expiry ))/ alpha ;
double K2 LIC = (L0∗exp ( rg ∗Expiry ) ) ;
11
17. return (A0 + ( delta ∗ alpha ∗( KouFormulas (A0, K1 LIC , int r , sigma ,
lambda , p , eta1 , eta2 , n , Expiry ) ) ) −
KouFormulas (A0, K2 LIC , int r , sigma , lambda ,
p , eta1 , eta2 , n , Expiry ) ) ;
}
int main ()
{
cout<<” Please enter the spot Asset price : ” <<endl ;
double A0 = 0 . 0 ;
cin>>A0;
cout<<” Please enter the risk −f r e e i n t e r e s t rate : ” <<endl ;
double r = 0 . 0 ;
cin>>r ;
cout<<” Please enter the V o l a t i l i y l e v e l : ” <<endl ;
double Vol = 0 . 0 ;
cin>>Vol ;
cout<<” Please enter the i n t e n s i t y of the Poisson d i s t r i b u t i o n : ” ;
cout<<endl ;
double lambda = 0 . 0 ;
cin>>lambda ;
cout<<” Please enter the probability value of p o s i t i v e jumps : ” ;
cout<<endl ;
double p = 0 . 0 ;
cin>>p ;
cout<<” Please enter the i n t e n s i t y of p o s i t i v e jumps : ” <<endl ;
12
18. double eta1 = 0 . 0 ;
cin>>eta1 ;
cout<<” Please enter the i n t e n s i t y of negative jumps : ” <<endl ;
double eta2 = 0 . 0 ;
cin>>eta2 ;
cout<<” Please enter the n value ” <<endl ;
int n = 0;
cin>>n ;
cout<<” Please enter the leverage c o e f f i c i e n t value ( alpha ) ” <<endl ;
double alpha = 0;
cin>>alpha ;
cout<<” Please enter the guaranteed rate ” <<endl ;
double rg = 0;
cin>>rg ;
cout<<” Please enter the p a r t i c i p a t i o n c o e f f i c i e n t value ( delta ) ” ;
cout <<endl ;
double delta = 0;
cin>>delta ;
cout<<” Please enter the maturity in years : ” <<endl ;
double T = 0 . 0 ;
cin>>T;
for (n ; n<=70;n++)
{
cout<<”The value of the Life Insurance Contract with default ” ;
13
19. cout<<at Maturity for n : ”<<n<<” i s : ”<<’ ’ ;
cout<<DMLIC(A0, r , alpha , rg , delta , Vol , lambda , p , eta1 , eta2 , n ,T)<<;
cout<<endl ;
}
system ( ”PAUSE” ) ;
return 0;
}
2.2 Source code of the file LicMath.h
// LicMath . h
#ifndef LicMath
#define LicMath
#include<cmath>
#include<iomanip>
#include<iostream>
#include<boost /math/ d i s t r i b u t i o n s . hpp>
#include<boost /math/ d i s t r i b u t i o n s / poisson . hpp>
#include<boost /math/ d i s t r i b u t i o n s /normal . hpp>
#include<boost /numeric/ ublas /matrix . hpp>
#include<boost /numeric/ ublas / vector . hpp>
#include<boost /numeric/ ublas / io . hpp>
#include<boost /math/ s p e c i a l f u n c t i o n s /gamma. hpp>
using namespace std ;
using namespace boost : : math ;
14
20. using namespace boost : : math : : constants ;
static const int g = 7;
static const double Pi =
3.1415926535897932384626433832795028841972 ;
static const double p [ g+2] = {0.99999999999980993 , 676.5203681218851 ,
−1259.1392167224028 , 771.32342877765313 , −176.61502916214059 ,
12.507343278686905 , −0.13857109526572012 , 9.9843695780195716 e −6,
1.5056327351493116 e −7};
typedef unsigned long long int VLint ;
typedef boost : : numeric : : ublas : : vector<int> VectorInt ;
typedef boost : : numeric : : ublas : : vector<double> VectorDbl ;
// Definition the f a c t o r i a l function
double f a c t o r i a l ( VLint n)
{
i f (n ==0 | | n == 1)
return 1;
else
return n ∗ f a c t o r i a l (n−1);
}
// Definition of the Combination function
double nCk( VLint n , VLint k)
{
return f a c t o r i a l (n )/( f a c t o r i a l (k) ∗ f a c t o r i a l (n−k ) ) ;
}
// Definition of the minimum function
15
21. template<class T> T mmin(T& a , T& b)
{
return (a<b) ? a : b ;
}
// Definition of the max function
template<class T>T max(T& a , T& b)
{
return (a<b) ?b : a ;
}
// Definition of the Pnk function
double Pnk( VLint n , VLint k , double eta1 , double eta2 , double p)
{
i f (n < 1)
{
cout<<”n i s equal or greater than 1”<<endl ;
}
i f ( k > n)
{
cout<<”k cannot be higher than n”<<endl ;
}
i f (p<0 | | p>1)
{
cout<< ”p i s a probability density function , t h e r e f o r e ” ;
cout<<” l i e s within the i n t e r v a l 0 and 1”<<endl ;
}
i f ( eta1 <=1)
16
22. {
cout<< ” eta1 cannot be l e s s than 1”<<endl ;
}
i f ( eta2 <=0)
{
cout<< ” eta2 cannot be l e s s than 0”<<endl ;
}
double Y = 0 . 0 ;
double j d = 0 . 0 ;
for ( VLint i = k ; i<=n−1; i++)
{
j d = static cast<double>( i ) ;
Y += nCk(n−k−1, i−k) ∗ pow(( eta1 )/( eta1+eta2 ) , i−k) ∗
pow(( eta2 )/( eta1+eta2 ) ,n−i )∗ (pow(p , i ))∗( pow(1−p , n−i ) ) ;
}
return Y;
}
// Definition of the Qnk function
double Qnk( VLint n , VLint k , double eta1 , double eta2 , double p)
{
i f (n < 1)
{
cout<<”n i s equal or greater than 1”<<endl ;
}
i f ( k > n)
{
cout<<”k cannot be higher than n”<<endl ;
17
23. }
i f (p<0 | | p>1)
{
cout<< ”p i s a probability density function , ” ;
cout<<” t h e r e f o r e l i e s within the i n t e r v a l 0 and 1”<<endl ;
}
i f ( eta1 <=1)
{
cout<< ”lambda1 cannot be l e s s than 1”<<endl ;
}
i f ( eta2 <=0)
{
cout<< ”lambda2 cannot be l e s s than 0”<<endl ;
}
double X = 0 . 0 ;
double j d = 0 . 0 ;
for ( VLint i = k ; i<=n−1; i++)
{
j d = static cast<double>( i ) ;
X += nCk(n−k−1, j d−k) ∗pow(( eta1 )/( eta1+eta2 ) ,n−j d ) ∗
pow(( eta2 )/( eta1+eta2 ) , j d−k)∗ (pow(p , n−j d ))∗
(pow(1−p , j d ) ) ;
}
return X;
}
/∗Code adapted from the Taylor s e r i e s approximation of the confluent
hypergeometric function 1F1(a ; b ; z ) developed by John Pearson as Part of
his MSc d i s s e r t a t i o n ’ Computation of Hypergeometric Functions ’
18
24. ∗/
long double TaylorConfluentHypergeometric (double a , double b ,
double z )
{
double t o l = pow(10 , −15);
// I n i t i a l i s e a1 , vector of i n d i v i d u a l s terms ,
// and b1 which s t o r e s the sum
// of the computed terms up to that point
boost : : numeric : : ublas : : vector<double> a1 (500);
a1 (1) = 1;
double b1 = 1 . 0 ;
for ( int j =1; j <500;++ j )
{
//Compute the current entry of a1 in terms of l a s t
a1 ( j +1) = ( a+j −1)/(b+j −1) ∗ z/ j ∗a1 ( j ) ;
//Update the sum of computed terms up to that point
b1 = b1+a1 ( j +1);
i f ( fabs ( a1 ( j ))/ fabs ( b1 ) < t o l && fabs ( a1 ( j +1))/
fabs ( b1)< t o l )
break ;
}
return b1 ;
}
19
25. // Definition of the Hodgkin − Huxley function
double Hh( const double& n , const int& x)
{
double A = (pow(2 ,( −n /2)))∗
sqrt ( boost : : math : : constants : : pi<double >())∗
( exp (pow(x ,2)∗ −0.5));
double B = TaylorConfluentHypergeometric ((0.5∗ n )+0.5 ,0.5 ,
(0.5∗pow(x , 2 ) ) ) / ( sqrt (2) ∗ tgamma(1 + (0.5∗ n ) ) ) ;
double C = −(x∗( TaylorConfluentHypergeometric ((0.5∗ n)+1 ,
1 . 5 , ( 0 . 5 ∗ pow(x , 2 ) ) ) ) ) / ( tgamma((0.5)+(0.5∗ n ) ) ) ;
return A ∗ (B+C) ;
}
// Definition of the In function
double In ( const double& n , const double& c , const double& alpha ,
const double& beta , const double& delta1 )
{
// Declare v a r i a b l e s which w i l l hold your values
double r e s u l t 1 = 0 . 0 ;
normal snd ( 0 , 1 ) ;
double j d = 0 . 0 ;
i f ( beta > 0 && alpha != 0)
{
for ( int i = 0; i<=n ; i++)
{
j d = static cast<double>( i ) ;
r e s u l t 1 += (pow( beta /alpha , n−j d )) ∗
20
26. Hh( j d , ( beta ∗ c ) − delta1 ) +
(pow(( beta / alpha ) , n+1) ∗
( sqrt (2 ∗ boost : : math : : constants : : pi<double >())/
beta ) ∗ ( exp ( ( ( alpha ∗ delta1 )/( beta )) +
pow( alpha ,2)/(2 ∗pow( beta , 2 ) ) ) ) ∗
( cdf ( snd ,(− beta ∗ c ) + ( delta1 ) +
( alpha / beta ) ) ) ) ;
}
}
i f ( beta < 0 && alpha < 0)
{
for ( int i = 0; i !=n ; i++)
{
j d = static cast<double>( i ) ;
r e s u l t 1 += (pow( beta /alpha , n−j d )) ∗
Hh( j d , ( beta ∗ c ) − delta1 ) − (pow(( beta / alpha ) , n+1) ∗
( sqrt (2 ∗ boost : : math : : constants : : pi<double >())/ beta ) ∗
( exp ( ( ( alpha ∗ delta1 )/( beta )) + pow( alpha ,2)/
(2 ∗pow( beta , 2 ) ) ) ) ∗( cdf ( snd , ( beta ∗ c ) −
( delta1 ) − ( alpha / beta ) ) ) ) ;
}
}
return −(exp ( alpha ∗ c )/ alpha ) ∗ r e s u l t 1 ;
}
// Definition of the normal p r o b a b i l i t y density function
double npdf (double mu, double sigma , double x)
{
return (1.0 / ( sqrt (2.0 ∗
boost : : math : : constants : : pi<double >())) ∗
sigma ) ∗ exp( −0.5 ∗ (pow(x−mu, 2 ) /
21
27. (pow( sigma , 2 ) ) ) ) ;
}
/∗ Approximation of the normal cumulative d i s t r i b u t i o n from
Haug .E (1998) ”The Complete Guide to Option Pricing Formulas”
∗/
double Ncdf (double mu, double sigma , double x)
{
double a1 , a2 , a3 , k ;
double r e s u l t = 0 . 0 ;
a1 = 0.4361836;
a2 = −0.1201676;
a3 = 0.937298;
k = 1.0 / (1.0 + (0.33267 ∗ x ) ) ;
i f (x >= 0)
{
r e s u l t = 1 − npdf (mu, sigma , x) ∗ ( a1 ∗ k + ( a2 ∗ k ∗ k) +
( a3 ∗ k ∗ k ∗ k ) ) ;
}
else
{
r e s u l t = 1 − Ncdf (mu, sigma , −x ) ;
}
return r e s u l t ;
}
22
29. References
Courtois, O. L. and Quittard-Pinon, F. (2008). Fair valuation of participating life
insurance contracts with jump risk. The Geneva Risk and Insurance Review, 33.
Kou, S. (2002). A jump diffusion model for option pricing. Management Science,
48(8):1086–1101.
24