PREPARATORY COURSE

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Module 1: Probability

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Objective: Explain some elementary concepts of the mathematical theory of probability. It exposes which are the probability distributions used in financial and insurance risks and how to estimate parameters. The importance of probability in Solvency II is explained.

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• Introduction to probability

• Combinatorial analysis

• Conditional probability and independence

• Random variables

• Density and distribution functions

• Expectation, variance, moments

• Probability distributions

• Frequency distributions, Poisson, Binomial, Negative Binomial

• Loss distributions, lognormal, EVT, gyh, beta, gamma, weibull, etc.

• Random Vectors

• Distribution fitting and parameter estimation

• Use of probability distributions in Solvency II

• Exercise 1: Probability distribution fits in R

 

Module 2: Statistics

 

Objective: Inferential statistics consists of a set of techniques to obtain, with a certain degree of confidence, information from a population based on information from a sample. Statistics is essential for the construction of models and their validation.

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• Introduction

• Variables and data types

• Descriptive statistics

• Inferential statistics

• Random samples and statistics

• Point estimate

• Estimation by intervals

• Hypothesis tests

• Importance of statistics in Solvency II

• Exercise 2: Descriptive statistics in Python of data from an insurance company

• Exercise 3: Hypothesis testing in R

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Module 3: Finance

 

Objective: To review the concepts of the value of money over time, financial mathematics, valuation of annuities, bonds, and valuation models Capital Asset Pricing Model and Arbitrage pricing theory. The models are essential for the valuation of assets and liabilities of an insurance company.

 

• Value of money over time

• Financial mathematics and annuities

• Bond valuation

• Duration and convexity

• CAPM and APT model

• Stochastic processes

• Monte Carlo simulation

• Exercise 4: Valuation of bonds in Excel

• Exercise 5: Estimation of duration and convexity in Excel

• Exercise 6: Estimation of the CAPM and APT in Excel

• Exercise 7: Monte Carlo Simulation and Stochastic Processes in R

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Module 4: Programming in Python

 

Objective: Explain what the Python programming language is and its functionalities. It explains what Jupyter is and how to install it. Expose basic notions of programming and the libraries that will be used to develop Solvency II models.

 

• Introduction to Python

• Environment and library installation

• Jupyter

• Import and export of data

• basic programming

• statistical tools

• regression libraries

• finance bookstores

• Machine Learning Libraries

• Quantum Libraries

• Exercise 8: Programming in Python

 

Module 5: Programming in R

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Objective: Explain what R and Rstudio are and how to install them. Explain basic notions of R programming and the libraries used to develop Solvency II models.

 

• Introduction to R

• Environment and library installation

• R Studio

• Import and export of data

• Basic programming

• Statistical tools

• Regression libraries

• Finance bookstores

• Actuarial science bookstores

• Exercise 9: programming in R

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Module 6: Machine Learning

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Objective: Automatic machine learning, in English machine learning, essential for systems to be intelligent, allows the development of predictions based on data and improves the projections of traditional models. The use of machine learning and deep learning algorithms is introduced. The benefits of machine learning in risk management for insurance companies are explained.

 

• Introduction Machine Learning

• Differences with statistics

• Supervised and unsupervised models

• decision trees

• Support Vector Machine

• K-means

• Assembly Learning

• Random Forest

• neural networks

• Introduction to ensemble models

• Introduction to Deep Learning

• Exercise 10: Estimation of the Support Vector Machine and Random Forest

• Exercise 11: Deep learning algorithm creation

 

Module 7: Introduction to Financial Risks

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Objective: Lay the theoretical foundations on the financial risks that impact insurance companies, explain the types of risks and the sources of such risks. Understand the probability and impact of events that trigger financial risk in insurance companies.

 

• What is risk?

• Financial risks in insurance companies

• Probability and Impact

• Sources of financial risks

• Differences between financial and non-financial risks

• Market risk

• Interest rate risk

• Liquidity risk

• Credit risk

• Operational risk

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Module 8: Actuarial Sciences

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Objective: Introduction of actuarial sciences for participants without actuarial training. A brief introduction to life and non-life insurance is presented, as well as actuarial mathematics.

 

• What do actuaries do?

• Introduction to Life insurance

• Introduction to Non-Life insurance

• Type of contracts

• Introduction to Life Insurance Reserves

• Introduction to Non-Life Insurance Reserves

• Margin Based Pricing

• Introduction to actuarial mathematics

• Introduction to Life Insurance

• Introduction to Non-Life Insurance

• Exercise 12: Modeling the distribution of the severity and frequency of claims in Excel and R

• Exercise 14: Simulation of the current values of an Annuity of a life annuity.

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Module 9: Quantum computing and algorithms

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Objective: Quantum computing applies quantum mechanical phenomena. On a small scale, physical matter exhibits properties of both particles and waves, and quantum computing takes advantage of this behavior using specialized hardware. The basic unit of information in quantum computing is the qubit, similar to the bit in traditional digital electronics. Unlike a classical bit, a qubit can exist in a superposition of its two "basic" states, meaning that it is in both states simultaneously.

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• Future of quantum computing in insurance

• Is it necessary to know quantum mechanics?

• QIS Hardware and Apps

• quantum operations

• Qubit representation

• Measurement

• Overlap

• matrix multiplication

• Qubit operations

• Multiple Quantum Circuits

• Entanglement

• Deutsch Algorithm

• Quantum Fourier transform and search algorithms

• Hybrid quantum-classical algorithms

• Quantum annealing, simulation and optimization of algorithms

• Quantum machine learning algorithms

• Exercise 15: Quantum operations

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Module 10: Introduction to quantum mechanics

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• Quantum mechanical theory

• wave function

• Schrodinger's equation

• statistical interpretation

• Probability

• Standardization

• Impulse

• The uncertainty principle

• Mathematical Tools of Quantum Mechanics

• Hilbert space and wave functions

• The linear vector space

• Hilbert's space

• Dimension and bases of a Vector Space

• Integrable square functions: wave functions

• Dirac notation

• operators

• General definitions

• hermitian adjunct

• projection operators

• commutator algebra

• Uncertainty relationship between two operators

• Operator Functions

• Inverse and Unitary Operators

• Eigenvalues and Eigenvectors of an operator

• Infinitesimal and finite unit transformations

• Matrices and Wave Mechanics

• matrix mechanics

• Wave Mechanics

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Module 11: Introduction to quantum error correction

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• Error correction

• From reversible classical error correction to simple quantum error correction

• The quantum error correction criterion

• The distance of a quantum error correction code

• Content of the quantum error correction criterion and the quantum Hamming bound criterion

• Digitization of quantum noise

• Classic linear codes

• Calderbank, Shor and Steane codes

• Stabilizer Quantum Error Correction Codes​

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​Module 12: Quantum Computing II

 

• quantum programming

• Solution Providers

• IBM Quantum Qiskit

• Amazon Braket

• PennyLane

• cirq

• Quantum Development Kit (QDK)

• Quantum clouds

• Microsoft Quantum

• Qiskit

• Main Algorithms

• Grover's algorithm

• Deutsch–Jozsa algorithm

• Fourier transform algorithm

• Shor's algorithm

• Quantum annealers

• D-Wave implementation

• Qiskit Implementation

• Exercise 16: Grover, Fourier Transform and Shor algorithm simulation

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Module 14: Quantum Machine Learning

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• Quantum Machine Learning

• hybrid models

• Quantum Principal Component Analysis

• Q means vs. K means

• Variational Quantum Classifiers

• Variational quantum classifiers

• Quantum Neural Network

  • Quantum Convolutional Neural Network

  • Quantum Long Short Memory LSTM

• Quantum Support Vector Machine (QSVC)

• Exercise 17: Quantum Support Vector Machine

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Module 15: Quantum computing in insurance companies

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• Building Blocks of Payoff Valuation

  • Distribution Loading

  • Payoff Implementation

  • Calculation of the Expected Value

• Amplitude Estimation

  • Amplitude Estimation based on Phase Estimation

  • Amplitude Estimation without Phase Estimation

• Grover's Quantum Search Algorithm

• Insurance-related Payoffs

  • Overall Payoff

• Insurance-related Quantum Circuits

  • Whole life insurance

  • Dynamic Lapse

• Quantum Hardware

  • simulator

  • royal hardware

• Exercise 18: Insurance-related Quantum Circuits

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Module 16: Tensor Networks for Machine Learning

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• What are tensor networks?

• Quantum Entanglement

• Tensor networks in machine learning

• Tensor networks in unsupervised models

• Tensor networks in SVM

• Tensor networks in NN

• NN tensioning

• Application of tensor networks in credit scoring models

• Exercise 19: Neural Network using Tensor Networks​

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Module 17: Probabilistic Machine Learning

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• Probability

• Gaussian models

• Bayesian Statistics

• Bayesian logistic regression

• Kernel Family

• Gaussian processes

  • Gaussian processes for regression

• Hidden Markov Model

• Markov chain Monte Carlo (MCMC)

  • Metropolis Hastings algorithm

• Machine Learning Probabilistic Model

• Bayesian Boosting

• Bayesian Neural Networks

• Exercise 20: Gaussian process for regression

• Exercise 21: Bayesian neural networks

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Module 18: Generative AI

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Generative artificial intelligence  is artificial intelligence capable of generating text, images, or other media, using generative models. Generative AI models learn the patterns and structure of their input training data and generate new data that has similar characteristics. Generative AI differs from other types of AI as it is about creating something new that is not modified or copied from its training data. Generative AI is a general-purpose technology used for multiple purposes across many industries. There are many types of multimodal generative AI tasks such as text summarization that produce a shorter version of a piece of text while retaining the main ideas, creating source code from natural language code comments, reasoning through a problem to discover potential new solutions or latent details and assigning a category to a given piece of content such as a document, image, video, or audio clip among other applications.

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• LLMs

• Embeddings

• RAG

• Structured / unstructured insurance data

• Document intelligence

• Prompt Engineering

• AI Agents

• Hallucination Risk

• Governance

• Insurance use cases

• Exercise 22: Embeddings for words, sentences, question answers

• Exercise 23: Embedding Visualization

• Exercise 24: First let's prepare the data for visualization

• Exercise 25: PCA (Principal Component Analysis)

• Exercise 26: Embeddings on Large Dataset

• Exercise 27: Prompt engineering

• Exercise 28: Advanced Prompting Techniques

• Exercise 29: Large Language Models (LLMs)

• Exercise 30: Retrieval Augmented Generation

• Exercise 31: Traditional KMeans to LLM powered KMeans

• Exercise 32: Cluster Visualization

• Exercise 33: Semantic Search

• Exercise 34: Tokens and Words

• Exercise 35: Tokenization in Programming Languages

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SOLVENCY II COURSE, IFRS 17 and STRESS TESTING

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Module 1: Solvency II

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Objective: Explain how Solvency II reflects the new risk management practices to define the necessary capital and manage financial and insurance risks. Solvency Capital Requirement SCR, Minimum Capital Requirement MCR and the three pillars of Solvency II are explained in detail.

 

• The Solvency II Directive and EIOPA

• General structure

• Basel II and III experience

• Implementation Schedule

• Asset Valuation

• Technical Provisions

  • Segmentation

  • Products

• Liability Analysis: Best Estimate and Margin Risk

• Own Resources: Tier 1, Tier 2 and Tier 3

• MCR-Estimate and Calculation

• SCR-Standard Formula

• Internal Model Directives

• Pillar 1: Own resources for solvency

  • Solvency Capital Requirement (SCR)

  • Standard Approach

  • Internal Model

  • Technological aspects and implementation

  • Minimum Capital Requirement (MCR)

• Pillar 2 Supervision Process and Own Risk and Solvency Assessment (ORSA)

  • ORSA definition and scope

  • The ORSA role

• Pillar 3 Transparency Requirements

  • Financial Condition Report

  • Solvency II Directive

• IFRS

  • Residual Margin

  • Risk Margin

  • Best Estimate

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Module 2: Standard Formula Methodology

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Objective: Explain the formulas of the standard approach for the estimation of the SCR and MCR. Understand the risks of insurance companies. Explain in detail the mathematical formulas and possible expected values.

 

• Technical specifications for the preparatory phase part 1

• Dependency Structure

• Risk mitigation techniques

• Market risk

  • Interest rate risk

  • Equity risk

  • real estate risk

  • Spread Risk

  • Concentration Risk

  • liquidity risk

• Credit risk

  • Counterparty risk

  • LGD and PD calculation

• Operational risk

  • Standard Formula

• Underwriting risk: Non-life

  • Reserve Risk

  • Premium Risk

  • catastrophic risk

• Underwriting Risk: Life

  • Mortality Risk

  • Longevity Risk

  • Morbidity Risk

  • Disability Risk

  • Portfolio Fall Risk

  • Expense Risk

  • Revision risk

• Technical Health Risk

• Technical specifications for the preparatory phase part 2

• Determination of the risk-free interest rate

• Exercise 1: Estimation of SCR Mkt of interest rate and Mkt Spread in a bond portfolio and SCR Mkt of equities in a stock portfolio.

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INTERNAL AND ORSA MODELS

 

Module 3: Approval of Internal Models

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Objective: Define what internal models are and explain the guidelines on the use of internal models that insurance companies must take into account so that the supervisory authorities approve and continue to allow the use of internal models to calculate solvency capital.

 

• Pre-application

• Application

• Evaluation and right to withdraw the application

• Decision on the application: Terms and conditions

• Monitoring

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Module 4: Own Risk and Solvency Assessment (ORSA)

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Objective: ORSA is the acronym for Own Risk and Solvency Assessment and explains the set of processes used to assess risks according to capital needs. The management framework, the ORSA process and the ORSA reports are explained in detail.

 

• ORSA Scope

• Regulatory context:

• management framework

• ORSA process

• ORSA report

• Government system

• Entity risk

• Stress test and scenario analysis

• Capital requirements and solvency assessment

• Business plan and capital planning

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Term Structure of Interest Rates

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Module 5: Modeling of interest rate term structure (ETTI)

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Objective: The relevance of adequately modeling the term structure of the interest rate or yield curve is crucial for the adequate valuation of the liabilities of the insurance company. It explains how to build the curve, the role of stochastic models and extrapolation methodologies among many other topics.

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• Yield Curve Concept

• nelson siegel

• Yield curve smoothing and term structure models

• Interpolation Methods: Cubic Splines

• Extrapolation Methods: Wilson-Smith

• stochastic modeling

  • Cox–Ingersoll–Ross model

  • Heath–Jarrow–Morton model

• Selecting objective variables

• Principal component analysis

• Selection of scenarios

• Vacicek's model

• Vacicek interest rate model

• Libor Market Model

• Interest rate curve in EIOPA

  • basic curve

  • Last Liquid point

  • volatility adjustment

  • Flow matching adjustment

  • Implementation of extrapolation

Exercise 2: Principal Components Exercise in python

Exercise 3: Estimating Nelson Siegel parameters in python

Exercise 4: Interpolation in Excel

Exercise 5: CIR simulation calculator and Vasicek python

Exercise 6: Caplet and Swaption using Libor Market Model in Excel and VBA

Exercise 7: Wilson-Smith extrapolation method in Excel and R

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IFRS 17

 

Module 6: International Financial Reporting Standard IFRS 17 and IFRS 4

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Objective: The IFRS 17 standard profoundly changes the approach to accounting for insurance, moving from a traditional scheme, based on historical values, to an approach closer to the "economic value" of the contracts. The methodologies and measurements of the insurance contract are explained.​​

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• IFRS 17: Insurance Contract

• Objective and scope

• Typology of insurance contracts

• Disaggregation and classification of IFRS 9 contracts

• Differences with IFRS 4

• Implementation dates

• IFRS 17 enhancements to current accounting practices

• Implementation costs

• Information on profitability

  • Estimation of the present value of future cash flows

  • risk adjustment

  • Contractual service margin

  • Difference in income statement with IFRS 17

• Methodologies and measurement of insurance contracts

  • Building Block Approach (BBA)

  • Variable Fee Approach (VFA)

  • Premium Allocation Approach (PAA)

• IFRS 17 and Solvency II

• IFRS 17 after implementation: lessons learned

• CSM evolution and profitability analytics

• Actual vs expected experience

• Experience variances

• Assumption changes

• IFRS 17 earnings forecasting

• CSM roll-forward analytics

• AI for actuarial cash-flow forecasting

• AI-assisted IFRS 17 anomaly detection

• Automated reconciliation

• IFRS 17 vs Solvency II capital and value metrics

• Exercise 8: Impact of economic scenarios on the balance sheet, income statement and future cash flows by valuation of life insurance contracts, under the IFRS 17 approach, including risk adjustment and contractual service margin in Excel and R.

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VALUATION OF PROVISIONS

 

Module 7: Valuation of Life Insurance provisions

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Objective: The Guidelines on the valuation of Life insurance technical provisions are shown to increase the coherence and convergence of the professional practice of all types and sizes of insurance companies. Fermac Risk shows the European experience of this practice.

 

• Deterministic Life Insurance Models

• Deterministic portfolio valuation

• Stochastic portfolio valuation

• Technical Life Risk

• Protection against technical risk of life with options

• Contracts with PB

• Contracts without PB

• unit link

• Variable Annuities

• Reinsurance

• Dynamic Fall Model (Lapse rate)

• Rescue Options

• Profit Sharing Option

• Exercise 9: Life Insurance Portfolio Valuation Tool, includes:

  • Vasicek interest rate simulation

  • Stochastic Mortality Risk Simulation

  • Lapse rate modeling

  • Options using black sholes model.

• Exercise 10: Variable Annuities using Black Sholes model

• Exercise 11: Generative AI in Valuation of Life Insurance provisions

 

​Module 8: Valuation of Non-Life Insurance provisions

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Objective: The Guidelines on the valuation of Non-Life insurance technical provisions are shown to increase the coherence and convergence of the professional practice of all types and sizes of insurance companies. Some traditional and modern techniques for calculating the reserve are explained.

 

• The technical provision for benefits. Regulations in Solvency 2

• Aggregate Claims Modeling

• Frequency Distributions

• Distributions of the claim amount

• Analytical methods

• Monte Carlo Simulation

• Triangle-based methods for calculating Loss Reserving Provisions

  • Grossing up

  • link ratio

  • Chain Ladder

  • Bornhuetter Ferguson

• Stochastic methods for calculating the Provision for benefits.

  • Mack's method

  • Bootstapp Method

• Machine Learning in Non-Life Insurance

  • Claims reserving based on Bayesian neural networks

  • Chain Ladder Neural Network

• Exercise 12: Fit frequency using negative binomial and Poisson

• Exercise 14: Claims amount adjustment using lognormal, gamma, weibull, exponential and G-H in python and R

• Exercise 15: Estimation of accident rate distribution with Monte Carlo simulation in R

• Exercise 16: Chain-Ladder Neural Network

• Exercise 17: Estimating provisions using the Run Off Chain Ladder

• Exercise 18: Estimating provisions using Bootstrap in R

• Exercise 19: Generative AI in Valuation of Non-Life Insurance provisions

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​MARKET RISK

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Module 9: Value at Risk (VaR) and Expected Shorfall in life and non-life lines

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Objective: Explain the concept of the Value at Risk VaR and the Expected Shortfall in the life and non-life branches. The treatment of returns and volatility using GARCH models is explained.

• Introduction to VAR

• VAR in life insurance

• VAR in the non-life business

• Volatility Estimation

  • GARCH(1,1)

  • GARCH Multivariate

  • EWMA

• Volatility Forecasting

• Parametric Models

  • Normal VaR

  • t-student distribution

  • lognormal distribution

• Linear Model for Stocks and Bonds

• Quadratic model for options

• VaR extensions

  • Expected Shortfall or Tail VaR

  • Conditional VaR

• Cash flow mapping

• Exercise 20: Simulation and forecasting volatility using GARCH(1,1) and multivariate model in R

• Exercise 21: estimation of the internal model of VaR and Expected Shortfall in life and non-life insurance

 

Module 10: Parametric VaR with Extreme Value Theory

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Objective: Explain the theory of extreme value to apply it to internal models. This distribution allows estimating the probability of truly extreme events. The pros and cons of these distributions in insurance practice are explained.

 

• EVT Extreme Value Distributions

  • gumbel

  • Frenchet

  • Weibull

• Generalized Pareto distributions

  • Exponential

  • Pareto

  • Beta

• Threshold estimate

• Model Selection

  • Hill and Mean Excess Plot

• Generation of random EVT values

• EVT estimation under Bayesian approach

• Disadvantages of EVT

• Exercise 22: Estimation of Graphs: Mean Excess, Q-Q and Hill plot in R

• Exercise 23: Maximum likelihood parameter estimation of GDP in SAS and R

• Exercise 24: VaR estimation by EVT in R​

• Exercise 25: Quantum VaR estimation 

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Module 11: Historical Simulation and Monte Carlo

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Objective: VaR is explained by Monte Carlo simulation applied to insurance companies. Being the best methodology to estimate the VaR in a time horizon of one year.

 

• VaR Historical Simulation

  • Adjust for volatility

• VaR Monte Carlo simulation

  • Simulation with a risk factor

  • Simulation with multiple risk factors

  • Variance Reduction Methods

• Normal Multivariate Distribution and T-Student

• VaR Monte Carlo based on Gaussian copula and t-student copula

• Exercise 23: VaR estimation: using Monte Carlo Simulation and Historical Simulation in Excel and R

• Exercise 24: Historical Simulation Backtesting

• Exercise 25: VaR using Gaussian copula and tStudent in SAS and R

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Module 12: Market Risk

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Objective: Good market risk practices for insurance companies are explained. The results of the SCR under the standard formula are compared against the internal models.

 

• Standard Formula on Market Risk

• Sub-modules in market risk

  • Interest rate risk

  • Equity risk

  • real estate risk

  • Spread Risk

  • Concentration Risk

  • liquidity risk

• SCR VaR 99.5%

• Internal and partial models

• Internal Market Risk Model

• VaR for interest rate risk

  • Stochastic Process Selection

  • VaR using stochastic process of an asset

  • Simulation with principal components

  • Scenario Simulation

• VaR for interest rate risk with principal components

• Spread VaR

• exchange rate VaR

• Equity VaR

• Concentration and correlation risk modeling

• VaR of Options

  • Delta Normal VaR

  • Delta Gamma VaR

  • Monte Carlo simulation

• Boundary Structure

• AI-Enhanced Investment & Market Risk for Insurers

  • Expected Shortfall

  • Dynamic correlation

  • Market regime detection

  • AI volatility forecasting

  • Yield-curve regime detection

  • Credit-spread stress

  • Investment portfolio concentration

  • Alternative assets

  • Private credit

  • Infrastructure investments

  • AI portfolio revaluation

  • Climate-sensitive investment risk

• Exercise 26: Estimating the VaR of options with Monte Carlo simulation in Excel and R

• Exercise 27: Cash Flow mapping and VaR estimation of a bond portfolio

• Exercise 28: VaR estimation of non-life and life risk

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​Module 13: Stress Testing and Backtesting

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Objective: Stress testing is one of the best tools for managing market risk. Consider exceptional but plausible events. Insurance companies will also need to validate internal models with backtesting.

 

• Stress Testing Approaches

• Historical Stress Testing

• Reverse Stress Test

• Stress testing in correlation

• Stress testing on volatility

• Multivariate stress testing

• Backtesting

  • Kupiec`s Test

• Frequency Conditional Coverage

• Analysis of losses in the tail of the distribution

• Clean and dirty backtesting

• Exercise 29:  AI Investment Portfolio Stress Engine

  • Rates + Spreads + Equity + FX + Property

  • → Asset Value
    → Own Funds
    → SCR
    → Solvency Ratio.

• Exercise 30: Backtesting of VaR in Excel

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CREDIT RISK

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Module 14: Credit Risk Structural and Reduced Form Models

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Objective: Structural credit risk models require financial information from the company and have proven to be efficient during the pandemic. These models help measure the credit risk of fixed income investments, particularly bonds.

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• Structural Models

  • Merton's model

  • KMV model

• Reduced form models

  • Jarrow-Turnbull Model

  • Duffie and Singleton Model

  • Neutral default probabilities

  • Conversion of default currents into discrete PDs

  • Adjustment of reduced form models to historical databases

  • Construction of default probability curves

  • Validation with Falkenstein and Boral Test

  • Jump to default

  • zero coupon bonds

  • voucher with coupons

  • convertible bonds

  • CDS Valuation

• Insurance investment credit risk

  • Sovereign risk

  • Corporate bond portfolios

  • Reinsurer counterparty risk

  • Private Credit

  • Rating migrations

  • Credit spread deterioration

  • Wrong-way risk

  • AI credit migration models

  • XGBoost for downgrade/default prediction

  • Explainable AI

  • Geopolitical credit overlays

• Exercise 31: Structural model in R and Excel

• Exercise 32:  AI Credit Migration & Reinsurer Stress

• Rating Migration → Spread → PD → Bond Value → Counterparty Loss → SCR

• Exercise 33: Bonus and CDS valuation in Python

• Exercise 34: Internal model of market and credit risk

  • CIR simulation of fixed and variable income interest rates

  • Jarrow-Turnbull-Lando model for credit risk with transition matrices.

  • Comparison against standard formulas.

 

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